Temperature dependence of the $^{13}$C hyperfine structure of the negatively-charged nitrogen-vacancy center in diamond

TL;DR

This study investigates how the hyperfine interactions of the NV center in diamond vary with temperature from 10 K to 700 K, revealing minimal changes and insights into the defect's electronic structure and stability.

Contribution

It provides detailed experimental and theoretical analysis of temperature effects on the $^{13}$C hyperfine structure in NV centers, enhancing understanding for quantum sensing applications.

Findings

Hyperfine interaction change is small across temperature range

Changes are mainly due to orbital hybridization variations

Defect structure tends toward undistorted at higher temperatures

Abstract

The nitrogen-vacancy (NV) center is a well utilized system for quantum technology, in particular quantum sensing and microscopy. Fully employing the NV center's capabilities for metrology requires a strong understanding of the behavior of the NV center with respect to changing temperature. Here, we probe the NV electronic spin density as the surrounding crystal temperature changes from 10 K to 700 K by examining its $^{13}$C hyperfine interactions. These results are corroborated with \textit{ab initio} calculations and demonstrate that the change in hyperfine interaction is small and dominated by a change in the hybridization of the orbitals constituting the spin density. Thus indicating that the defect and local crystal geometry is returning towards an undistorted structure at higher temperature.

Figures

Equations

$$H_{hf}=\mathbf{S}\cdot \mathbf{D}\cdot \mathbf{S}+{\gamma }_e\mathbf{S}\cdot \mathbf{B}+\mathbf{S}\cdot \mathbf{A}\cdot \mathbf{I},$$

$$\displaystyle\mathbf{A}_{\text{diag}}=\left(\begin{array}[]{ccc}A_{\perp}&0&0\\ 0&A_{\perp}&0\\ 0&0&A_{\parallel}\end{array}\right).$$

$$\begin{array}{rl}{A}_{\parallel }& =f+2d\\ A_{\perp }& =f-d.\end{array}$$

$$\psi$$

$$1$$

$$f$$

$$=3777\times (1-{\left|c_p\right|}^2)\eta \text{MHz},$$

$$d$$

$$=107.4\times {\left|c_p\right|}^2\eta \text{MHz},$$

$$\displaystyle R=\left(\begin{array}[]{ccc}-\frac{1}{3}&0&\frac{2\sqrt{2}}{3}\\ 0&1&0\\ -\frac{2\sqrt{2}}{3}&0&-\frac{1}{3}\\ \end{array}\right).$$

$${\delta }_{hf}$$

$${\Delta }_{hf}$$

$${+324A_{\parallel }{A}_{\perp }D+{\left(4A_{\parallel }^2+A_{\perp }^2\right)}^2]}^{\frac{1}{2}}$$

$${\Delta }_{hf}$$

$$\Delta D$$

$$\begin{array}{rl}\Delta D& \approx -a_2{T}^2-a_3{T}^3-\\ & (b_4+a_4)T^4-(b_5+a_5)T^5+\cdots \end{array}$$

Full text

I Introduction

The negatively charged nitrogen-vacancy (NV) center in diamond is a leading system in quantum technology. Due to its atom-like size, bright fluorescence, ability to optical initialize and read-out its electron spin, and long coherence time in ambient conditions, the NV center has been rapidly adopted for nanoscale quantum sensing/microscopy and quantum information processing (QIP). The quantum sensing applications of the NV center take advantage of the susceptibilities of the ground-state spin-resonances to magnetic fields loubser1977optical ; balasubramanian2008nanoscale ; maze2008nanoscale ; taylor2008high , electric fields dolde2011electric ; dolde2014charge , temperature Acosta2010 ; chen2011temperature ; neumann2013high ; toyli2012measurement ; toyli2013fluorescence and strain nanomechanical_sensing_barson ; teissier2014strain ; ovartchaiyapong2014dynamic . Precision quantum sensing and high-fidelity QIP operations require these susceptibilities to be well characterized and understood. The magnetic, electric and strain responses have been well characterized and understood. However, there remains contention regarding the origin of the temperature susceptibility. Initial models considering only strain from thermal expansion failed to describe the observed behaviour chen2011temperature ; Acosta2010 ; toyli2012measurement . To rectify the unexplained temperature dependence, Doherty et al PhysRevB.90.041201 added a quadratic spin-phonon interaction to phenomenologically explain the thermal dependence. To fully accept this model, further first principles modelling and experimental validation is required. Here, we use the hyperfine interaction to probe the electronic behaviour during thermal expansion via experiment and first principles modelling.

During thermal expansion/contraction or mechanical stress, the nuclei surrounding the NV change position, causing the NV center’s electron orbitals to move and change doherty2014electronic ; nanomechanical_sensing_barson . The effect of this change on the hyperfine interaction is included in two ways: (1) the orbital hybridization (or bond-angle) of the atomic orbitals constituting the unpaired spin density changes, and (2) the spin-density of the electron at the location of the nuclear spin changes. These processes are depicted in figure 1. Precise information of these processes will improve our fundamental understanding of the NV  electronic orbitals and how they change with temperature.

The hyperfine interaction is due to two contributions: (1) a dipolar interaction between the nuclear spin and the local distribution of electron spin density and (2) the Fermi contact interaction of the electron spin density at the nuclear spin loubser1978electron . Both of these contributions are strongly dependent on the relative positions of the nuclear and electron orbitals. As such, the hyperfine interaction can be used as a atomscopic probe of the effect of thermal expansion on the NV  electron orbitals.

This investigation first outlines the hyperfine interaction theoretically and its relation to the NV  electronic wavefunction. Experimental tests are then performed and first principles modelling is used to provide further insight into the results. The key outcomes are that the change in hyperfine interaction is negligible and the dominant change in the electron spin density is a variation in atomic orbital hybridization.

II Theoretical Details

The NV  center is an axially symmetric defect with three-fold $C_{3v}$ symmetry (figure 1). The electronic ground-state (${}^{3}A_{2}$) consists of an spin-triplet with a zero-field splitting of $D\approx 2.87$ GHz (at room temperature) between the $m_{s}=$ 0 and $m_{s}=$ $\pm 1$ spin levels (figure 1). The hyperfine interaction of the NV  ground-state with a nearby nucleus can be described by the following spin-Hamiltonian doherty2012theory ,

[TABLE]

where $\mathbf{S}$ are the spin-operators; $\mathbf{D}$ is the zero-field (or spin-spin) tensor; $\gamma_{e}$ is the electron gyromagnetic ratio; $\mathbf{B}$ is the applied magnetic field; $\mathbf{A}$ is the hyperfine tensor; and $\mathbf{I}$ are the nuclear spin operators. The temperature dependence of the zero-field splitting has been measured previously and can be described by a power series $D(T)\approx\sum d_{i}T^{i}$ Acosta2010 ; PhysRevB.90.041201 ; chen2011temperature ; toyli2012measurement . Note that the nuclear Zeeman interaction is negligible for the magnetic fields considered here ($<$ 100 G). In the diagonalized form there are only two unique components to $\mathbf{A}$, an axial ($A_{\parallel}$) and a non-axial ($A_{\perp}$) component.

[TABLE]

These components are given with respect to a reference frame that is not the NV coordinate system felton2009hyperfine ; He1993paramagnetic-ii ; loubser1977optical ; loubser1978electron . As such, a rotation is required to transform the hyperfine tensor to the NV coordinate system $\mathbf{A}=R\cdot\mathbf{A}_{\text{diag}}\cdot R^{T}$, see appendix A for details. The hyperfine parameters for a first shell 13C  are $A_{\parallel}=199.7(2)$ MHz and $A_{\perp}=120.3(2)$ MHz felton2009hyperfine . Solving the Hamiltonian either numerically or by approximation (see appendix A) gives the 13C  zero-field hyperfine splitting $\Delta_{hf}\approx 127$ MHz and average resonance $\delta_{hf}=2876$ MHz. Due to the non-diagonal terms in $A$ and the large size of the hyperfine perturbation relative to the zero-field splitting, there is a small amount of $D$ dependence in $\Delta_{hf}$ and a small amount of hyperfine dependence in $\delta_{hf}$. The large splitting $\Delta_{hf}$ enables the easy identification of NV  centers with a first shell 13C  upon examination of the optically detected magnetic resonance (ODMR) spectra.

The hyperfine components of $\mathbf{A}$ can be re-described by the Fermi contact term $f$ and the dipolar term $d$ loubser1978electron ,

[TABLE]

The contact term is spatially isotropic and satisfied by a spherically distributed $s-$orbital. There is no dipolar contribution from an $s-$ orbital, as such, some $p-$ (or higher) orbital character is required. Using these arguments, the contribution to the total molecular orbitals (MOs) from a atomic orbital at the nuclear spin is a hybrid orbital $\psi$ which is a linear combination of $s-$ ($\phi_{s}$) and $p-$orbitals ($\phi_{p}$) loubser1978electron ,

[TABLE]

where the Fermi contact $f$ and dipolar terms $d$ can be described as He1993paramagnetic-ii ,

[TABLE]

where $\left|\phi_{s}(0)\right|^{2}$ is the probability per unit volume of $s-$orbital at the location of the 13C  nucleus and $\langle\phi_{p}|\frac{1}{r^{3}}|\phi_{p}\rangle$ is the average $1/r^{3}$ value of the $\phi_{p}$ orbital. $\eta$ is the electron spin-density at the nucleus, $g_{n}$ is the 13C  nuclear $g-$factor and $\mu_{N}$ is the nuclear magneton. The atomic parameters $\left|\phi_{s}(0)\right|^{2}$ and $\langle\phi_{p}|\frac{1}{r^{3}}|\phi_{p}\rangle$ are numerically determined and available in published tables ayscoughEPR1967 ; MORTON1978577 . These relationships show that measuring $A_{\parallel}$ and $A_{\perp}$ to determine $f$ and $d$ allows for the determination of the temperature dependent changes to the NV  orbital hybridization $\left|c_{s}\right|^{2}$ / $\left|c_{p}\right|^{2}$ and the spin-density $\eta$.

III Experimental details and results

The measurements included low temperature ODMR of measurements of single NV centers and high temperature measurements of an ensemble of NV centers. For high temperature ODMR detection a lock-in amplifier (Stanford SR830) was used with a time-constant of 1 ms. The microwave signal was amplitude modulated at 100% depth at a modulation frequency of 1.7 kHz (R&S SMIQ03B). The increased sensitivity from the lock-in amplifier allowed for 13C  ODMR resonances to be easily measured above background noise, (figure 3(a)), even for the relatively low ${\sim}1\%$ natural abundance of 13C  expected in the ensemble. The optical transition of NV- was excited using a 500 mW 532 nm CW laser into a Nikon LU Plan $100\times$/0.8 WD 3.5 mm objective lens. The microwave ground state transition was excited using a simple coaxial circular loop short ($R\sim 3$ mm) and an amplifier providing 16 mW of microwaves into the short. The sample was mounted onto a home-made iron hotplate with a vacuum chuck (500 Torr) ensuring good thermal contact on the sample and a feedback loop controlling the temperature.

For low temperature single site 13C  measurements, a liquid helium cold finger continuous flow cryostat was used with a scanning confocal microscope. A 0.9 NA Nikon LU Plan Fluor air objective was mounted on a 3 axis piezo scanning stage within the vacuum space of the cryostat. Microwaves were provided to the sample via a 25 $\mu$m wire soldered across the sample. Since single NV centers with an adjacent first shell 13C  were required, an alpha-numerical grid and marker system was scribed into the diamond using focussed ion beam (FIB). Due to the low probability of finding a single NV center with a 13C  in the first shell, a search algorithm was employed that recorded the position of bright spots and performed CW ODMR at each bright spot to establish if that bright spot was an NV center with a first shell 13C. For each temperature the desired NV center was re-found using the marking system due to thermal expansion of the cold-finger.

Measurements were obtained using CW ODMR spectra for both single and ensembles of NV  centers. In these measurements, fluorescence is continually measured as a frequency sweep of microwaves is applied directly to the sample. An example of the 13C  spectra can be seen in figure 2(a). Repeating for a range of temperatures, comparisons of the peak frequencies of the two hyperfine resonances ($\omega_{1},\omega_{2}$) with the mean frequency $\delta_{hf}=(\omega_{1}+\omega_{2})/2$ and difference in frequency $\Delta_{hf}=(\omega_{1}-\omega_{2})$ can be made.

As described in the last section and equations (14-15) in appendix A, there is a component of $D(T)$ present in the separation of the spin-resonances, as such a small temperature dependence in $\Delta_{hf}$ inherited from $D(T)$ is expected. Despite this, the difference of the two resonances ($\Delta_{hf}$) shown in figure 2(b) show no obvious change within experimental error, this indicates there is no measurable change in the hyperfine interaction. As expected, the mean frequency of the two hyperfine resonances follows the well known $D$ temperature shift, shown in figure 2(a). There is no observable change in $\Delta_{hf}$ that can be attributed to a change in the hyperfine parameters $A_{\parallel}$ or $A_{\perp}$. The high-temperature data shows a large amount of scatter in the data points. The error bars are only indicative of the error in the fit and measurable experimental parameters, as such, there must be an larger unmeasured and unknown random error in the measurement. This is probably due to some thermal dependent fluctuations in magnetization of the iron hotplate or stray magnetism from the heater element circuitry, see appendix C. Note that the error analysis on the fitting was performed by a Monte-Carlo method, in which the raw spectra was repeatedly modulated by normal distributed random noise and re-fitted. The amplitude of this normally distributed random noise is equal to the standard deviation of the residuals from the initial fit.

IV ab initio calculations

Since the temperature dependence of the hyperfine parameters was not able to be measured, further ab initio investigation were pursued. We performed calculations using the Vienna Ab initio Simulation Package (VASP) vasp1996Kresse . The calculations used a supercell containing 512 atoms, a plane-wave cut-off energy of 600 eV, Gamma point sampling and the Perdew-Burke-Ernzhof (PBE) functional PBEBurke1996 . The hyperfine tensor was evaluated in the NV  ground state for the nearest-neighbour carbon atoms to the vacancy using the inbuilt hyperfine routine of VASP.

Temperature was simulated via the thermal expansion of the diamond lattice. This was implemented in the calculations by varying the lattice constant of the supercell. For each lattice constant, the atomic geometry was allowed to relax prior to the evaluation of the hyperfine tensor. A linear fit to each hyperfine tensor component was performed to obtain the components as continuous functions of lattice constant (figure 3(a)). These were then converted to functions of temperature by employing the X-ray crystallography data PhysRevB.65.092102 ; reeber1996thermal for the lattice constant as a function of temperature. Strong agreement with the experimental determined hyperfine parameters was obtained with the parameters with $A_{\parallel}\approx 199.6$ MHz and $A_{\perp}\approx 119.5$ MHz at zero temperature. These results are within 1$\sigma$ and 4$\sigma$ respectively of the experimental values published by Felton et al felton2009hyperfine .

Figures 3(c-d) show the behaviour of $\left|c_{p}\right|^{2}$ and $\eta$ due to changes in temperature. These results show that the spin-density reduces slightly and the orbitals are becoming more $s-$ type and less $p-$ type. Previous ab initio studies larsson2008electronic have shown that the NV center self-distorts away from tetrahedral coordination at low temperature. The nearest-neighbour carbon atoms move towards being in-plane with the next-to-nearest carbon atoms. As a result, the bonds between the nearest-neighbour and next-to-nearest carbon atoms become more like $sp^{2}$ bonds. The dangling bond that constitutes the spin density at the nearest-neighbour carbon atoms thus becomes more like a pure $p-$orbital. These ab initio results show that, for increasing temperature, the dangling orbital is returning towards $sp^{3}$, which in turn implies that the local crystal is returning towards tetrahedral structure. This is shown in figure 3(e) by the large displacements of the nearest-neighbour carbons towards the vacancy.

The results also show that the spin density associated with the nearest-neighbour carbon atoms is decreasing with expansion. This must be the consequence of the spin density being distributed over more carbon atoms with expansion. This can be understood as the consequence of the confining electrostatic potential of the defect becoming shallower as the lattice expands, thus allowing the electrons to spread over more distant sites.

The overall increase in magnitude of the hyperfine parameters with expansion is consistent with this picture because the small decrease in spin density at the nearest-neighbour carbon atoms is more than compensated by the increase in $s-$orbital contribution to the spin density at those atoms. This increase in $s-$orbital increases the Fermi contact contribution, which is the dominant contribution to the parameters.

V Conclusion

High and low temperature spectroscopy of the NV–13C  hyperfine interaction was performed to obtain the changes in the hyperfine parameters due to changing temperature. It was experimentally found that the change in the hyperfine parameters due to changes in temperature was not resolved by our measurement. It would be possible to perform ENDOR Raman-heterodyne manson1990raman or pulsed NMR measurements in the future to improve the accuracy of the hyperfine terms. Ab initio simulations corroborated this evidence and found that the spin-density doesn’t change significantly and there is a slightly larger change in orbital hybridization such that the bonds change hybridization towards to perfect $sp^{3}$ bonding at higher temperature. This seems to fit with the temperature model proposed by Doherty et al PhysRevB.90.041201 . That is, the change in spin-density is small so that the change of spin-spin interaction due to thermal expansion predominately comes from the fact that atoms are further apart only. Ultimately, this means that the change in spin-resonances from thermal expansion is small, supporting the addition of the extra quadratic spin-phonon interaction.

This is the first ever attempt to atomscopically measure the spin-density and hybridization ratio of the NV center’s orbitals due to deformations of the diamond lattice. The small magnitude of these changes have positive implications for quantum information applications using a 13C  spin cluster Neumann1326 ; dutt2007quantum . As the resonances will not shift due to temperature fluctuations reducing the experimental requirements on thermal control or heat management. These results also rule out any benefit of using the 13C  spin temperature susceptibility directly for thermometry applications, despite the increased coherence time of nuclear spins.

VI Acknowledgements

This work was supported by funding from the Australian Research Council (DP140103862, DE170100169). MSJB acknowledges support from the DAAD-GO8 joint research cooperative scheme and the Robert and Helen Crompton award. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. JW acknowledges funding from the DFG and the EU via ERC grant SMeL and ASTRIQS as well as QIA.

Appendix A Hyperfine tensor rotation

The most widely accepted hyperfine parameters are given by Felton et al felton2009hyperfine where, $A_{\parallel}$ and $A_{\perp}$ are 199.7(2) MHz and 120.3(2) MHz respectively. However, these values are given in reference to the nuclear coordinate system at an angle ($\theta$, $\varphi$) = ($125.26^{\circ}$, $45^{\circ}$) from the crystallographic ([100],[010],[001]) coordinate system. This coordinate system is aligned along $[11\bar{1}]$, which is at an angle $\theta=\arccos\left(\frac{-1}{3}\right)$ from $[\bar{1}\bar{1}\bar{1}]$. To convert to the NV spin coordinate system (where $z$ is parallel to [111],[$\bar{1}\bar{1}\bar{1}$]) a rotation operation needs to be applied to the hyperfine tensor. Ignoring the angle $\varphi$ as the system is axially symmetric about [111], the rotation matrix is,

[TABLE]

Using this rotation gives the non-diagonal hyperfine tensor in the NV coordinate frame $\mathbf{A}=R\cdot\mathbf{A}_{\text{diag}}\cdot R^{T}$

Apply second-order degenerate perturbation theory the average spin-resonance frequency is

[TABLE]

and the difference in spin-resonance frequency is

[TABLE]

For all of the analysis in the this report, the numerical solutions to the Hamiltonian were used, this approximation is used only to demonstrate the behaviour of the spin-resonances.

Appendix B Full temperature range fit for $D(T)$

The multiple published expressions of the polynomial describing the temperature shift $\Delta D(T)$Acosta2010 ; PhysRevB.90.041201 ; chen2011temperature ; toyli2012measurement have been for narrower ranges in temperature presented here. Whilst these descriptions agree with each other within their prescribed temperature region they give wildly different results for predictions outside of their prescribed temperature range, as shown in figure 4(b). This divergence is a problem general to using a truncated polynomial or power series to describe a function. To circumvent this problem, we combined multiple expressions to obtain new polynomial parameters for the full useful temperature range of the NV  spin from liquid He temperature to 700 K.

Doherty et alPhysRevB.90.041201 described the temperature shift with a combination of the pressure shift due to the thermal expansion of the diamond ($a_{i}$ terms) and extra quadratic electron-phonon process ($b_{i}$ terms). The pressure shift is simply calculated from the hydrostatic stress parameter $A=14.6$ (MHz/GPa) doherty2014electronic , the bulk modulus of diamond $B=442$ (GPa/strain) and the volumetric strain due to thermal expansion $\Delta V/V$. Doherty et al’s semi-empirical treatment resulted in suitable fit (for $T<300$ K) using just two free parameters for the electron-phonon process $b_{4}$ and $b_{5}$.

[TABLE]

We adopt this method by refitting the $b_{i}$ terms to artificial $D$ data constructed from Toyli et al toyli2012measurement and Doherty et al. This data was shifted to ensure a smooth curve, since we only fit the change in $D$ any change in offset is not important. However, Doherty et al’s model utilized the thermal expansion definition provided by Sato et al PhysRevB.65.092102 which, for similar reasons, is only valid for temperatures up to 300 K, as shown in figure 4(a). As such, to further the temperature range we use the definition by Reeber et al reeber1996thermal , which is not provided in a convenient power series. To accurately describe the data we find it is necessary to fit up to the sixth power of $T$ and find the parameters $b_{4}=-1.44(8)\times 10^{-9}$, $b_{5}=3.1(3)\times 10^{-12}$ and $b_{6}=-1.8(3)\times 10^{-15}$, this fit is shown in figure 4(c).

Appendix C Ensemble ODMR data analysis

The change in separation of the central ODMR spin-resonances (NV centers without a first shell 13C) demonstrated some unexpected behaviour with changing temperature. As shown in figure 5. This behaviour was observed to not be monotonic with increasing temperature and was repeatable across separate experimental runs. This was also present in the separation of the first shell 13C ODMR spin-resonances. It was decided that this must be some remnant magnetization of the iron hotplate that has some thermal dependence, or some other non-linear process that effects the ODMR spectra. Another possibility is a change in strain or 14N hyperfine resonances, though how this occurs in a non-monotonic fashion is not clear. Since this behaviour was present in the both the 13C  and non-hyperfine centers a like it was decided to simply subtract this effect from the separation of the 13C  spin-resonances to separate the values of $\Delta_{hf}$.