Imaging stress and magnetism at high pressures using a nanoscale quantum sensor

TL;DR

This paper presents a nanoscale quantum sensor integrated into diamond anvils for high-resolution imaging of stress and magnetism at high pressures, enabling new insights into phase transitions and material properties.

Contribution

It introduces a novel NV-center-based sensing platform embedded in diamond anvils for in situ high-pressure imaging of stress and magnetic fields.

Findings

Achieved diffraction-limited imaging (~600 nm) of stress and magnetism up to 30 GPa.

Quantified all six stress components with accuracy <0.01 GPa.

Demonstrated vector magnetic field imaging and phase transition detection in iron and gadolinium.

Abstract

Pressure alters the physical, chemical and electronic properties of matter. The development of the diamond anvil cell (DAC) enables tabletop experiments to investigate a diverse landscape of high-pressure phenomena ranging from the properties of planetary interiors to transitions between quantum mechanical phases. In this work, we introduce and utilize a novel nanoscale sensing platform, which integrates nitrogen-vacancy (NV) color centers directly into the culet (tip) of diamond anvils. We demonstrate the versatility of this platform by performing diffraction-limited imaging (~600 nm) of both stress fields and magnetism, up to pressures ~30 GPa and for temperatures ranging from 25-340 K. For the former, we quantify all six (normal and shear) stress components with accuracy $<0.01$ GPa, offering unique new capabilities for characterizing the strength and effective viscosity of solids and fluids under pressure. For the latter, we demonstrate vector magnetic field imaging with dipole accuracy $<10^{-11}$ emu, enabling us to measure the pressure-driven $\alpha\leftrightarrow\epsilon$ phase transition in iron as well as the complex pressure-temperature phase diagram of gadolinium. In addition to DC vector magnetometry, we highlight a complementary NV-sensing modality using T1 noise spectroscopy; crucially, this demonstrates our ability to characterize phase transitions even in the absence of static magnetic signatures. By integrating an atomic-scale sensor directly into DACs, our platform enables the in situ imaging of elastic, electric and magnetic phenomena at high pressures.

Full text

Imaging stress and magnetism at high pressures using a nanoscale quantum sensor

S. Hsieh,1,2,∗ P. Bhattacharyya,1,2,∗ C. Zu,1,∗ T. Mittiga,1 T. J. Smart,3 F. Machado,1

B. Kobrin,1,2 T. O. Höhn,1,4 N. Z. Rui,1 M. Kamrani,5 S. Chatterjee,1 S. Choi,1

M. Zaletel,1 V. V. Struzhkin,6 J. E. Moore,1,2 V. I. Levitas,5,7 R. Jeanloz,3 N. Y. Yao1,2,†

1Department of Physics, University of California, Berkeley, CA 94720, USA

2Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

3Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA

4Fakultät für Physik, Ludwig-Maximilians-Universität München, 80799 Munich, Germany

5Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA

6Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015, USA

7Departments of Mechanical Engineering and Material Science and Engineering,

Iowa State University, Ames, IA 50011, USA

*†*To whom correspondence should be addressed; E-mail: norman.yao@berkeley.edu

Pressure alters the physical, chemical and electronic properties of matter. The development of the diamond anvil cell (DAC) enables tabletop experiments to investigate a diverse landscape of high-pressure phenomena ranging from the properties of planetary interiors to transitions between quantum mechanical phases. In this work, we introduce and utilize a novel nanoscale sensing platform, which integrates nitrogen-vacancy (NV) color centers directly into the culet (tip) of diamond anvils. We demonstrate the versatility of this platform by performing diffraction-limited imaging (\sim$$600 nm) of both stress fields and magnetism, up to pressures \sim$$30 GPa and for temperatures ranging from $25-340$ K. For the former, we quantify all six (normal and shear) stress components with accuracy $\lesssim 0.01$ GPa, offering unique new capabilities for characterizing the strength and effective viscosity of solids and fluids under pressure. For the latter, we demonstrate vector magnetic field imaging with dipole accuracy $\lesssim 10^{-11}$ emu, enabling us to measure the pressure-driven $\alpha\leftrightarrow\epsilon$ phase transition in iron as well as the complex pressure-temperature phase diagram of gadolinium. In addition to DC vector magnetometry, we highlight a complementary NV-sensing modality using $T_{1}$ noise spectroscopy; crucially, this demonstrates our ability to characterize phase transitions even in the absence of static magnetic signatures. By integrating an atomic-scale sensor directly into DACs, our platform enables the in situ imaging of elastic, electric and magnetic phenomena at high pressures.

A tremendous amount of recent attention has focused on the development of hybrid quantum sensing devices, in which sensors are directly integrated into existing toolsets ranging from biological imaging to materials spectroscopy  (?, ?, ?, ?). In this work, we demonstrate the versatility of a novel platform based upon quantum spin defects combined with static high pressure technologies  (?, ?). In particular, we instrument diamond anvil cells with a layer of nitrogen-vacancy (NV) centers directly at the culet, enabling the pursuit of two complementary objectives in high pressure science: first, to understand the strength and failure of materials under pressure (e.g. the brittle-ductile transition) and second, to discover and characterize new phases of matter (e.g. high temperature superconductors)  (?, ?, ?, ?). Achieving these goals hinges upon the sensitive in situ imaging of signals within the high pressure chamber. In the former case, measuring the local stress environment permits the direct observation of inhomogeneities in plastic flow and the formation of line defects. In the latter case, the ability to spatially resolve field distributions can provide a direct image of complex order parameters and textured phenomena such as magnetic domains. Unfortunately, the enormous stress gradients generated near the sample limit the utility of most conventional tabletop spectroscopy techniques; as a result, one is often restricted to measuring bulk properties averaged over the entire DAC geometry.

Our approach to these challenges is to utilize an ensemble of NV centers (\sim$$1 ppm density) implanted \sim$$50 nm from the surface of the diamond anvil culet (Fig. 1A,B). Each NV center represents an atomic-scale defect (i.e. a substitutional nitrogen impurity adjacent to a vacancy) inside the diamond lattice and exhibits an $S=1$ electronic spin ground state  (?). In the absence of external fields, the $|m_{\textrm{s}}=\pm 1\rangle$ spin sublevels are degenerate and separated by $D_{\textrm{gs}}=(2\pi)\times 2.87~{}\mathrm{GHz}$ from the $|m_{\textrm{s}}=0\rangle$ state. Crucially, both the nature and energy of these spin states are sensitive to local changes in stress, temperature, magnetic and electric fields (Fig. 1C)  (?, ?, ?, ?). These spin states can be optically initialized and read out, as well as coherently manipulated via microwave fields. Their energy levels can be probed by performing optically detected magnetic resonance (ODMR) spectroscopy where one measures a change in the NV’s fluorescence intensity when an applied microwave field is on resonance between two NV spin sublevels (Fig. 1D), thus enabling sensing of a variety of external signals over a wide range of environmental conditions  (?, ?, ?).

Here, we focus on the sensing of stress and magnetic fields, wherein the NV is governed by the Hamiltonian, $H=H_{0}+H_{\textrm{B}}+H_{\textrm{S}}$, with $H_{0}=D_{\textrm{gs}}S_{z}^{2}$ (zero-field splitting), $H_{\textrm{B}}=\gamma_{\textrm{B}}\vec{B}\cdot\vec{S}$ (Zeeman splitting), and $H_{\textrm{S}}=\left[\alpha_{1}(\sigma_{xx}+\sigma_{yy})+\beta_{1}\sigma_{zz}\right]S_{z}^{2}+\left[\alpha_{2}(\sigma_{yy}-\sigma_{xx})+\beta_{2}(2\sigma_{xz})\right](S_{y}^{2}-S_{x}^{2})+\left[\alpha_{2}(2\sigma_{xy})+\beta_{2}(2\sigma_{yz})\right](S_{x}S_{y}+S_{y}S_{x})$ capturing the NV’s response to the local diamond stress tensor, $\overset{\scriptscriptstyle\smallleftrightarrow}{\sigma}$ (Fig. 1C). Note that in the above, $\gamma_{\textrm{B}}\approx(2\pi)\times 2.8$ MHz/G is the gyromagnetic ratio, $\{\alpha_{1,2},\beta_{1,2}\}$ are the stress susceptibility coefficients  (?), $\hat{z}$ is the NV orientation axis, and $\hat{x}$ is defined such that the $xz$-plane contains one of the carbon-vacancy bonds (Fig. 1E). In general, the resulting ODMR spectra exhibit eight resonances arising from the four possible crystallographic orientations of the NV (Fig. 1D). By extracting the energy shifting and splitting of the spin sublevels for each NV orientation group, one obtains an overconstrained set of equations enabling the reconstruction of either the (six component) local stress tensor or the (three component) vector magnetic field  (?).

In our experiments, we utilize a miniature DAC (Fig. 1A,B) consisting of two opposing anvils compressing either a beryllium copper or rhenium gasket  (?). The sample chamber defined by the gasket and diamond-anvil culets is filled with a pressure-transmitting medium (either a 16:3:1 methanol/ethanol/water solution or cesium iodide) to provide a quasi-hydrostatic environment. Microwave excitation is applied via a 4 $\mu$m thick platinum foil compressed between the gasket and anvil pavilion facets, while scanning confocal microscopy (with a transverse diffraction-limited spot size \sim$$600 nm, containing \sim$$10^{3} NVs) allows us to obtain two-dimensional ODMR maps across the culet.

We begin by probing the stress tensor across the culet surface using two different cuts of diamond (i.e. (111)-cut and (110)-cut culet). For a generic stress environment, the intrinsic degeneracy associated with the four NV orientations is not sufficiently lifted, implying that individual resonances cannot be resolved. In order to resolve these resonances while preserving the stress contribution, we sequentially tune a well-controlled external magnetic field to be perpendicular to each of the different NV orientations  (?). For each perpendicular field choice, three of the four NV orientations exhibit a strong Zeeman splitting proportional to the projection of the external magnetic field along their symmetry axes. Crucially, this enables one to resolve the stress information encoded in the remaining NV orientation, while the other three groups of NVs are spectroscopically split away. Using this method, we obtain sufficient information to extract the full stress tensor, as depicted in Fig. 2. A number of intriguing features are observed at the interface between the culet and the sample chamber, which provide insight into both elastic (reversible) and plastic (irreversible) deformations.

At low pressures ($P=4.9$ GPa), the normal stress along the loading axis, $\sigma_{ZZ}$, is spatially uniform (Fig. 2A), while all shear stresses, $\{\sigma_{XY}$, $\sigma_{XZ}$, $\sigma_{YZ}\}$, are minimal (Fig. 2B) (?). These observations are in agreement with conventional stress continuity predictions for the interface between a solid and an ideal fluid  (?). Moreover, $\sigma_{ZZ}$ is consistent with the independently measured pressure inside the sample chamber (via ruby fluorescence), demonstrating the NV’s potential as a built-in pressure scale  (?). In contrast to the uniformity of $\sigma_{ZZ}$, the field profile for the mean lateral stress, $\sigma_{\perp}\equiv\frac{1}{2}(\sigma_{XX}+\sigma_{YY})$, exhibits a concentration of forces toward the center of the culet (Fig. 2A). Using the measured $\sigma_{ZZ}$ as a boundary condition, we perform finite element simulations to reproduce this spatial pattern (?).

Upon increasing pressure ($P=13.6$ GPa), a pronounced spatial gradient in $\sigma_{ZZ}$ emerges (Fig. 2B inset). This qualitatively distinct feature is consistent with the solidification of the pressure-transmitting medium into its glassy phase above $P_{g}\approx 10.5$ GPa  (?). Crucially, this demonstrates our ability to characterize the effective viscosity of solids and liquids under pressure. To characterize the sensitivity of our system, we perform ODMR spectroscopy with a static applied magnetic field and pressure under varying integration times and extract the frequency uncertainty from a Gaussian fit. We observe a stress sensitivity of $\{0.023,0.030,0.027\}$ GPa/$\sqrt{\textrm{Hz}}$ for hydrostatic, average normal, and average shear stresses, respectively. This is consistent with the theoretically derived stress sensitivity, $\eta_{S}\sim\frac{\Delta\nu}{\xi C\sqrt{Nt}}=\{0.017,0.022,0.020\}$ GPa/$\sqrt{\textrm{Hz}}$, respectively, where $N$ is the number of NV centers, $\Delta\nu$ is the linewidth, $\xi$ is the relevant stress susceptibility, $t$ is the integration time, and $C$ is an overall factor accounting for measurement infidelity  (?). In combination with diffraction-limited imaging resolution, this sensitivity opens the door to measuring and ultimately controlling the full stress tensor distribution across a sample.

Having characterized the stress environment, we now utilize the NV centers as an in situ magnetometer to detect phase transitions inside the high-pressure chamber. Analogous to the case of stress, we observe a magnetic sensitivity of $12~{}\mathrm{\mu}$T/$\sqrt{\textrm{Hz}}$, in agreement with the theoretically estimated value, $\eta_{B}\sim\frac{\delta\nu}{C\gamma_{\textrm{B}}\sqrt{Nt}}=8.8~{}\mathrm{\mu}$T/$\sqrt{\textrm{Hz}}$. Assuming a point dipole located a distance $d\sim 5~{}\mu$m from the NV layer, this corresponds to an experimentally measured magnetic moment sensitivity: $7.5\times 10^{-12}$ emu/$\sqrt{\textrm{Hz}}$ (Fig. 1F).

Sensitivity in hand, we begin by directly measuring the magnetization of iron as it undergoes the pressure-driven $\alpha\leftrightarrow\epsilon$ phase transition from body-centered cubic (bcc) to hexagonal close-packed (hcp) crystal structures  (?); crucially, this structural phase transition is accompanied by the depletion of the magnetic moment, and it is this change in the iron’s magnetic behavior that we image. Our sample chamber is loaded with a \sim$$10~{}\mathrm{\mu m} polycrystalline iron pellet as well as a ruby microsphere (pressure scale), and we apply an external magnetic field {\bf B}_{\textrm{ext}}$$\sim$$180 G. As before, by performing a confocal scan across the culet, we acquire a two-dimensional magnetic resonance map (Fig. 3). At low pressures (Fig. 3A), near the iron pellet, we observe significant shifts in the eight NV resonances, owing to the presence of a ferromagnetic field from the iron pellet. As one increases pressure (Fig. 3B), these shifts begin to diminish, signaling a reduction in the magnetic susceptibility. Finally, at the highest pressures ($P\sim 22$ GPa, Fig. 3C), the magnetic field from the pellet has reduced by over two orders of magnitude.

To quantify this phase transition, we reconstruct the full vector magnetic field produced by the iron sample from the aforementioned two-dimensional NV magnetic resonance maps (Fig. 3D-F). We then compare this information with the expected field distribution at the NV layer inside the culet, assuming the iron pellet generates a dipole field  (?). This enables us to extract an effective dipole moment as a function of applied pressure (Fig. 3G). In order to identify the critical pressure, we fit the transition using a logistic function  (?). This procedure yields the transition at $P=16.7\pm 0.7~{}\mathrm{GPa}$ (Fig. 3J).

In addition to changes in the magnetic behavior, another key signature of this first order transition is the presence of hysteresis. We investigate this by slowly decompressing the diamond anvil cell and monitoring the dipole moment; the decompression transition occurs at $P=10.5\pm 0.7$ GPa (Fig. 3J), suggesting a hysteresis width of approximately \sim$$6 GPa, consistent with a combination of intrinsic hysteresis and finite shear stresses in the methanol/ethanol/water pressure-transmitting medium  (?). Taking the average of the forward and backward hysteresis pressures, we find a critical pressure of $P_{\textrm{c}}=13.6\pm 3.6$ GPa, in excellent agreement with independent measurements by Mössbauer spectroscopy, where $P_{\textrm{c}}\approx 12~{}$GPa (Fig. 3J)  (?).

Next, we demonstrate the integration of our platform into a cryogenic system, enabling us to make spatially resolved in situ measurements across the pressure-temperature ($P$-$T$) phase diagram of materials. Specifically, we investigate the magnetic $P$-$T$ phase diagram of the rare-earth element gadolinium (Gd) up to pressures $P\approx 8~{}$GPa and between temperatures $T=25-340$ K. Owing to an interplay between localized 4f electrons and mobile conduction electrons, Gd represents an interesting playground for studying metallic magnetism; in particular, the itinerant electrons mediate RKKY-type interactions between the local moments, which in turn induce spin-polarization of the itinerant electrons  (?). Moreover, much like its other rare-earth cousins, Gd exhibits a series of pressure-driven structural phase transitions from hexagonal close-packed (hcp) to samarium-type (Sm-type) to double hexagonal close-packed (dhcp) (Fig. 4) (?). The interplay between these different structural phases, various types of magnetic ordering and metastable transition dynamics leads to a complex magnetic $P$-$T$ phase diagram that remains the object of study to this day  (?, ?, ?).

In analogy to our measurements of iron, we monitor the magnetic ordering of a Gd flake via the NV’s ODMR spectra at two different locations inside the culet: close to and far away from the sample (the latter to be used as a control)  (?). Due to thermal contraction of the DAC (which induces a change in pressure), each experimental run traces a distinct non-isobaric path through the $P$-$T$ phase diagram (Fig. 4C, blue curves). In addition to these DC magnetometry measurements, we also operate the NV sensors in a complementary mode, i.e. as a noise spectrometer.

We begin by characterizing Gd’s well-known ferromagnetic Curie transition at ambient pressure, which induces a sharp jump in the splitting of the NV resonances at $T_{\textrm{C}}=292.2\pm 0.1$ K (Fig. 4D). As depicted in Fig. 4A, upon increasing pressure, this transition shifts to lower temperatures, and consonant with its second order nature  (?), we observe no hysteresis; this motivates us to fit the data and extract $T_{\textrm{C}}$ by solving a regularized Landau free-energy equation  (?). Combining all of the low pressure data (Fig. 4C, red squares), we find a linear decrease in the Curie temperature at a rate: $dT_{\textrm{C}}/dP=-18.7\pm 0.2$ K/GPa, consistent with prior studies via both DC conductivity and AC-magnetic susceptibility  (?). Surprisingly, this linear decrease extends well into the Sm-type phase. Upon increasing pressure above $\sim 6$ GPa (path [b] in Fig. 4C), we observe the loss of ferromagnetic (FM) signal (Fig. 4B), indicating a first order structural transition into the paramagnetic (PM) dhcp phase  (?). In stark contrast to the previous Curie transition, there is no revival of a ferromagnetic signal even after heating up (\sim$$315 K) and significantly reducing the pressure ($<0.1$ GPa).

A few remarks are in order. The linear decrease of $T_{\textrm{C}}$ well beyond the \sim$$2 GPa structural transition between hcp and Sm-type is consistent with the “sluggish” equilibration between these two phases at low temperatures  (?). The metastable dynamics of this transition are strongly pressure and temperature dependent, suggesting that different starting points (in the $P$-$T$ phase diagram) can exhibit dramatically different behaviors  (?). To highlight this, we probe two different transitions out of the paramagnetic Sm-type phase by tailoring specific paths in the $P$-$T$ phase diagram. By taking a shallow path in $P$-$T$ space, we observe a small change in the local magnetic field across the structural transition into the PM dhcp phase at \sim$$6 GPa (Fig. 4C path [c], orange diamonds). By taking a steeper path in $P$-$T$ space, one can also investigate the magnetic transition into the antiferromagnetic (AFM) Sm-type phase at \sim$$150 K (Fig. 4C path [d], green triangle). In general, these two transitions are extremely challenging to probe via DC magnetometry since their signals arise only from small differences in the susceptibilities between the various phases  (?).

To this end, we demonstrate a complementary NV sensing modality based upon noise spectroscopy, which can probe phase transitions even in the absence of a direct magnetic signal  (?). Specifically, returning to Gd’s ferromagnetic Curie transition, we monitor the NV’s depolarization time, $T_{1}$, as one crosses the phase transition (Fig. 4D). Normally, the NV’s $T_{1}$ time is limited by spin-phonon interactions and increases dramatically as one decreases temperature. Here, we observe a strikingly disparate behavior. In particular, using nanodiamonds drop-cast on a Gd foil at ambient pressure, we find that the NV $T_{1}$ is nearly temperature independent in the paramagnetic phase, before exhibiting a kink and subsequent decrease as one enters the ferromagnetic phase (Fig. 4D). We note two intriguing observations: first, one possible microscopic explanation for this behavior is that $T_{1}$ is dominated by Johnson-Nyquist noise from the thermal fluctuations of charge carriers inside Gd  (?, ?). Gapless critical spin fluctuations or magnons in the ordered phase, while expected, are less likely to cause this signal  (?). Second, we observe that the Curie temperature, as identified via $T_{1}$-noise spectroscopy, is \sim$$10 K higher than that observed via DC magnetometry (Fig. 4D). Similar behavior has previously been reported for the surface of Gd  (?, ?), suggesting that our noise spectroscopy could be more sensitive to surface physics.

In summary, we have developed a hybrid platform that integrates quantum sensors into diamond anvil cells. For the first time, the full stress tensor can be mapped across the sample and gasket, as a function of pressure. This provides essential information for investigating mechanical phenomena, such as pressure-dependent yield strength, viscous flow of fluids and plastic deformation of solids, and may ultimately allow control of the deviatoric- as well as normal-stress conditions in high pressure experiments. Crucially, such information is challenging to obtain via either numerical finite-element simulations or more conventional experimental methods  (?). In the case of magnetometry, the high sensitivity and close proximity of our sensor enables one to probe signals that are beyond the capabilities of existing techniques (Fig. 1F); these include for example, nuclear magnetic resonance (NMR) at picoliter volumes  (?) and single grain remnant magnetism  (?), as well as phenomena that exhibit spatial textures such as magnetic skyrmions  (?) and superconducting vortices  (?).

While our work utilizes NV centers, the techniques developed here can be readily extended to other atomic defects. For instance, recent developments on all-optical control of silicon-vacancy centers in diamond may allow for microwave-free stress imaging with improved sensitivities  (?). In addition, one can consider defects in other anvil substrates beyond diamond; indeed, recent studies have shown that moissanite (6H silicon carbide) hosts optically active defects that show promise as local sensors  (?). In contrast to millimeter-scale diamond anvils, moissanite anvils can be manufactured at the centimeter-scale or larger, and therefore support larger sample volumes that ameliorate the technical requirements of many experiments. Finally, the suite of sensing capabilities previously demonstrated for NV centers (i.e. electric, thermal, gryroscopic precession etc.) can now straightforwardly be extended to high pressure environments, opening up an enormous new range of experiments for quantitatively characterizing materials at such extreme conditions which can test, extend and validate first-principles theory.

Acknowledgements

We gratefully acknowledge fruitful discussions with Z. Geballe, G. Samudrala, R. Zieve, J. Jeffries, E. Zepeda-Alarcon, M. Kunz, I. Kim, J. Choi, K. de Greve, P. Maurer, S. Lewin, and D.-H. Lee. We are especially grateful to M. Doherty and M. Barson for sharing their raw data on stress susceptibilities. We thank C. Laumann for introducing us to the idea of integrating NV centers into diamond anvil cells. We thank D. Budker, J. Analytis, A. Jarmola, M. Eremets, R. Birgeneau, F. Hellman, R. Ramesh for careful readings of the manuscript.

Funding

This work was supported as part of the Center for Novel Pathways to Quantum Coherence in Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-AC02-05CH11231. SH acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1752814. VIL and MK acknowledge support from Army Research Office (Grant W911NF-17-1-0225).

Author Contributions

All authors contributed extensively to all aspects of this work.

Competing interests

The authors declare no competing financial interests.

Data and materials availability

The data presented in this study are available from the corresponding author on request.