Imaging stress and magnetism at high pressures using a nanoscale quantum sensor
S. Hsieh, P. Bhattacharyya, C. Zu, T. Mittiga, T. J. Smart, F. Machado, B. Kobrin, T. O. Höhn, N. Z. Rui, M. Kamrani, S. Chatterjee, S. Choi, M. Zaletel, V. V. Struzhkin, J. E. Moore, V. I. Levitas, R. Jeanloz, N. Y. Yao
IImaging stress and magnetism at high pressures usinga nanoscale quantum sensor
S. Hsieh, , , ∗ P. Bhattacharyya, , , ∗ C. Zu, , ∗ T. Mittiga, T. J. Smart, F. Machado, B. Kobrin, , T. O. H ¨ohn, , N. Z. Rui, M. Kamrani, S. Chatterjee, S. Choi, M. Zaletel, V. V. Struzhkin, J. E. Moore, , V. I. Levitas, , R. Jeanloz, N. Y. Yao , , † Department of Physics, University of California, Berkeley, CA 94720, USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, 80799 Munich, Germany Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015, USA Departments of Mechanical Engineering and Material Science and Engineering,Iowa State University, Ames, IA 50011, USA † To whom correspondence should be addressed; E-mail: [email protected]
Pressure alters the physical, chemical and electronic properties of matter. Thedevelopment of the diamond anvil cell (DAC) enables tabletop experimentsto investigate a diverse landscape of high-pressure phenomena ranging fromthe properties of planetary interiors to transitions between quantum mechan-ical phases. In this work, we introduce and utilize a novel nanoscale sensingplatform, which integrates nitrogen-vacancy (NV) color centers directly intothe culet (tip) of diamond anvils. We demonstrate the versatility of this plat-form by performing diffraction-limited imaging ( ∼ nm) of both stress fieldsand magnetism, up to pressures ∼ GPa and for temperatures ranging from a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec − K. For the former, we quantify all six (normal and shear) stress compo-nents with accuracy . . GPa, offering unique new capabilities for charac-terizing the strength and effective viscosity of solids and fluids under pressure.For the latter, we demonstrate vector magnetic field imaging with dipole accu-racy . − emu, enabling us to measure the pressure-driven α ↔ (cid:15) phasetransition in iron as well as the complex pressure-temperature phase diagramof gadolinium. In addition to DC vector magnetometry, we highlight a com-plementary NV-sensing modality using T noise spectroscopy; crucially, thisdemonstrates our ability to characterize phase transitions even in the absenceof static magnetic signatures. By integrating an atomic-scale sensor directlyinto DACs, our platform enables the in situ imaging of elastic, electric andmagnetic phenomena at high pressures. A tremendous amount of recent attention has focused on the development of hybrid quan-tum sensing devices, in which sensors are directly integrated into existing toolsets ranging frombiological imaging to materials spectroscopy ( ). In this work, we demonstrate the versa-tility of a novel platform based upon quantum spin defects combined with static high pressuretechnologies (
5, 6 ). 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 objectivesin 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 sensi-tive in situ imaging of signals within the high pressure chamber. In the former case, measuringthe local stress environment permits the direct observation of inhomogeneities in plastic flowand the formation of line defects. In the latter case, the ability to spatially resolve field distri-butions can provide a direct image of complex order parameters and textured phenomena such2s magnetic domains. Unfortunately, the enormous stress gradients generated near the samplelimit the utility of most conventional tabletop spectroscopy techniques; as a result, one is oftenrestricted to measuring bulk properties averaged over the entire DAC geometry.Our approach to these challenges is to utilize an ensemble of NV centers ( ∼ ppm den-sity) implanted ∼ nm from the surface of the diamond anvil culet (Fig. 1A,B). Each NVcenter represents an atomic-scale defect (i.e. a substitutional nitrogen impurity adjacent to avacancy) inside the diamond lattice and exhibits an S = 1 electronic spin ground state ( ).In the absence of external fields, the | m s = ± i spin sublevels are degenerate and separatedby D gs = (2 π ) × .
87 GHz from the | m s = 0 i state. Crucially, both the nature and energyof these spin states are sensitive to local changes in stress, temperature, magnetic and electricfields (Fig. 1C) ( ). These spin states can be optically initialized and read out, as well ascoherently manipulated via microwave fields. Their energy levels can be probed by performingoptically detected magnetic resonance (ODMR) spectroscopy where one measures a change inthe NV’s fluorescence intensity when an applied microwave field is on resonance between twoNV spin sublevels (Fig. 1D), thus enabling sensing of a variety of external signals over a widerange of environmental conditions (
1, 16, 17 ).Here, we focus on the sensing of stress and magnetic fields, wherein the NV is governed bythe Hamiltonian, H = H + H B + H S , with H = D gs S z (zero-field splitting), H B = γ B ~B · ~S (Zeeman splitting), and H S = [ α ( σ xx + σ yy ) + β σ zz ] S z + [ α ( σ yy − σ xx ) + β (2 σ xz )] ( S y − S x ) + [ α (2 σ xy ) + β (2 σ yz )] ( S x S y + S y S x ) capturing the NV’s response to the local diamondstress tensor, ↔ σ (Fig. 1C). Note that in the above, γ B ≈ (2 π ) × . MHz/G is the gyromagneticratio, { α , , β , } are the stress susceptibility coefficients ( ), ˆ z is the NV orientation axis,and ˆ 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 possiblecrystallographic orientations of the NV (Fig. 1D). By extracting the energy shifting and splitting3f the spin sublevels for each NV orientation group, one obtains an overconstrained set ofequations 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 opposinganvils compressing either a beryllium copper or rhenium gasket ( ). The sample chamber de-fined by the gasket and diamond-anvil culets is filled with a pressure-transmitting medium (ei-ther a 16:3:1 methanol/ethanol/water solution or cesium iodide) to provide a quasi-hydrostaticenvironment. Microwave excitation is applied via a 4 µ m thick platinum foil compressed be-tween the gasket and anvil pavilion facets, while scanning confocal microscopy (with a trans-verse diffraction-limited spot size ∼ nm, containing ∼ 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 ofdiamond (i.e. (111)-cut and (110)-cut culet). For a generic stress environment, the intrinsicdegeneracy associated with the four NV orientations is not sufficiently lifted, implying thatindividual resonances cannot be resolved. In order to resolve these resonances while preservingthe stress contribution, we sequentially tune a well-controlled external magnetic field to beperpendicular 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 projectionof the external magnetic field along their symmetry axes. Crucially, this enables one to resolvethe stress information encoded in the remaining NV orientation, while the other three groups ofNVs are spectroscopically split away. Using this method, we obtain sufficient information toextract the full stress tensor, as depicted in Fig. 2. A number of intriguing features are observedat the interface between the culet and the sample chamber, which provide insight into bothelastic (reversible) and plastic (irreversible) deformations.At low pressures ( P = 4 . GPa), the normal stress along the loading axis, σ ZZ , is spatially4niform (Fig. 2A), while all shear stresses, { σ XY , σ XZ , σ Y Z } , are minimal (Fig. 2B) ( ). Theseobservations are in agreement with conventional stress continuity predictions for the interfacebetween a solid and an ideal fluid ( ). Moreover, σ ZZ is consistent with the independentlymeasured pressure inside the sample chamber (via ruby fluorescence), demonstrating the NV’spotential as a built-in pressure scale ( ). In contrast to the uniformity of σ ZZ , the field profilefor the mean lateral stress, σ ⊥ ≡ ( σ XX + σ Y Y ) , exhibits a concentration of forces toward thecenter of the culet (Fig. 2A). Using the measured σ ZZ as a boundary condition, we performfinite element simulations to reproduce this spatial pattern ( ).Upon increasing pressure ( P = 13 . GPa), a pronounced spatial gradient in σ ZZ emerges(Fig. 2B inset). This qualitatively distinct feature is consistent with the solidification of thepressure-transmitting medium into its glassy phase above P g ≈ . GPa ( ). Crucially, thisdemonstrates our ability to characterize the effective viscosity of solids and liquids under pres-sure. To characterize the sensitivity of our system, we perform ODMR spectroscopy with astatic applied magnetic field and pressure under varying integration times and extract the fre-quency uncertainty from a Gaussian fit. We observe a stress sensitivity of { . , . , . } GPa/ √ Hz for hydrostatic, average normal, and average shear stresses, respectively. This is con-sistent with the theoretically derived stress sensitivity, η S ∼ ∆ νξC √ Nt = { . , . , . } GPa/ √ Hz, respectively, where N is the number of NV centers, ∆ ν is the linewidth, ξ is therelevant stress susceptibility, t is the integration time, and C is an overall factor accounting formeasurement infidelity ( ). In combination with diffraction-limited imaging resolution, thissensitivity opens the door to measuring and ultimately controlling the full stress tensor distribu-tion 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 thecase of stress, we observe a magnetic sensitivity of µ T/ √ Hz, in agreement with the theoreti-5ally estimated value, η B ∼ δνCγ B √ Nt = 8 . µ T/ √ Hz. Assuming a point dipole located a distance d ∼ µ m from the NV layer, this corresponds to an experimentally measured magnetic momentsensitivity: . × − emu/ √ Hz (Fig. 1F).Sensitivity in hand, we begin by directly measuring the magnetization of iron as it undergoesthe pressure-driven α ↔ (cid:15) phase transition from body-centered cubic (bcc) to hexagonal close-packed (hcp) crystal structures ( ); crucially, this structural phase transition is accompanied bythe depletion of the magnetic moment, and it is this change in the iron’s magnetic behavior thatwe image. Our sample chamber is loaded with a ∼ µ m polycrystalline iron pellet as well asa ruby microsphere (pressure scale), and we apply an external magnetic field B ext ∼ G. Asbefore, by performing a confocal scan across the culet, we acquire a two-dimensional magneticresonance map (Fig. 3). At low pressures (Fig. 3A), near the iron pellet, we observe significantshifts in the eight NV resonances, owing to the presence of a ferromagnetic field from the ironpellet. As one increases pressure (Fig. 3B), these shifts begin to diminish, signaling a reductionin the magnetic susceptibility. Finally, at the highest pressures ( P ∼ GPa, Fig. 3C), themagnetic field from the pellet has reduced by over two orders of magnitude.To quantify this phase transition, we reconstruct the full vector magnetic field producedby 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 NVlayer inside the culet, assuming the iron pellet generates a dipole field ( ). This enables usto extract an effective dipole moment as a function of applied pressure (Fig. 3G). In order toidentify the critical pressure, we fit the transition using a logistic function ( ). This procedureyields the transition at P = 16 . ± . (Fig. 3J).In addition to changes in the magnetic behavior, another key signature of this first ordertransition is the presence of hysteresis. We investigate this by slowly decompressing the dia-mond anvil cell and monitoring the dipole moment; the decompression transition occurs at P = . ± . GPa (Fig. 3J), suggesting a hysteresis width of approximately ∼ GPa, consistentwith a combination of intrinsic hysteresis and finite shear stresses in the methanol/ethanol/waterpressure-transmitting medium ( ). Taking the average of the forward and backward hystere-sis pressures, we find a critical pressure of P c = 13 . ± . GPa, in excellent agreement withindependent measurements by M¨ossbauer spectroscopy, where P c ≈ GPa (Fig. 3J) ( ).Next, we demonstrate the integration of our platform into a cryogenic system, enabling usto make spatially resolved in situ measurements across the pressure-temperature ( P - T ) phasediagram of materials. Specifically, we investigate the magnetic P - T phase diagram of therare-earth element gadolinium (Gd) up to pressures P ≈ GPa and between temperatures T = 25 − K. Owing to an interplay between localized 4f electrons and mobile conduc-tion electrons, Gd represents an interesting playground for studying metallic magnetism; inparticular, the itinerant electrons mediate RKKY-type interactions between the local moments,which in turn induce spin-polarization of the itinerant electrons ( ). Moreover, much like itsother rare-earth cousins, Gd exhibits a series of pressure-driven structural phase transitions fromhexagonal 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 ofmagnetic ordering and metastable transition dynamics leads to a complex magnetic P - T phasediagram that remains the object of study to this day ( ).In analogy to our measurements of iron, we monitor the magnetic ordering of a Gd flakevia the NV’s ODMR spectra at two different locations inside the culet: close to and far awayfrom 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 paththrough the P - T phase diagram (Fig. 4C, blue curves). In addition to these DC magnetome-try measurements, we also operate the NV sensors in a complementary mode, i.e. as a noisespectrometer. 7e begin by characterizing Gd’s well-known ferromagnetic Curie transition at ambient pres-sure, which induces a sharp jump in the splitting of the NV resonances at T C = 292 . ± . K(Fig. 4D). As depicted in Fig. 4A, upon increasing pressure, this transition shifts to lower tem-peratures, and consonant with its second order nature ( ), we observe no hysteresis; this moti-vates us to fit the data and extract T 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 theCurie temperature at a rate: dT C /dP = − . ± . K/GPa, consistent with prior studies viaboth DC conductivity and AC-magnetic susceptibility ( ). Surprisingly, this linear decreaseextends well into the Sm-type phase. Upon increasing pressure above ∼ GPa (path [b] inFig. 4C), we observe the loss of ferromagnetic (FM) signal (Fig. 4B), indicating a first orderstructural transition into the paramagnetic (PM) dhcp phase ( ). In stark contrast to the previ-ous Curie transition, there is no revival of a ferromagnetic signal even after heating up ( ∼ K) and significantly reducing the pressure ( < . GPa).A few remarks are in order. The linear decrease of T C well beyond the ∼ GPa structuraltransition between hcp and Sm-type is consistent with the “sluggish” equilibration betweenthese two phases at low temperatures ( ). The metastable dynamics of this transition arestrongly pressure and temperature dependent, suggesting that different starting points (in the P - T phase diagram) can exhibit dramatically different behaviors ( ). To highlight this, we probetwo 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 thelocal magnetic field across the structural transition into the PM dhcp phase at ∼ GPa (Fig. 4Cpath [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 ∼ K (Fig. 4C path[d], green triangle). In general, these two transitions are extremely challenging to probe viaDC magnetometry since their signals arise only from small differences in the susceptibilities8etween the various phases ( ).To this end, we demonstrate a complementary NV sensing modality based upon noise spec-troscopy, 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 depolariza-tion time, T , as one crosses the phase transition (Fig. 4D). Normally, the NV’s T time islimited by spin-phonon interactions and increases dramatically as one decreases temperature.Here, we observe a strikingly disparate behavior. In particular, using nanodiamonds drop-caston a Gd foil at ambient pressure, we find that the NV T is nearly temperature independent in theparamagnetic phase, before exhibiting a kink and subsequent decrease as one enters the ferro-magnetic phase (Fig. 4D). We note two intriguing observations: first, one possible microscopicexplanation for this behavior is that T is dominated by Johnson-Nyquist noise from the thermalfluctuations of charge carriers inside Gd (
31, 32 ). Gapless critical spin fluctuations or magnonsin the ordered phase, while expected, are less likely to cause this signal ( ). Second, we ob-serve that the Curie temperature, as identified via T -noise spectroscopy, is ∼ K higher thanthat observed via DC magnetometry (Fig. 4D). Similar behavior has previously been reportedfor the surface of Gd (
26, 33 ), suggesting that our noise spectroscopy could be more sensitive tosurface physics.In summary, we have developed a hybrid platform that integrates quantum sensors into dia-mond anvil cells. For the first time, the full stress tensor can be mapped across the sample andgasket, as a function of pressure. This provides essential information for investigating mechan-ical phenomena, such as pressure-dependent yield strength, viscous flow of fluids and plasticdeformation 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 toobtain via either numerical finite-element simulations or more conventional experimental meth-ods ( ). In the case of magnetometry, the high sensitivity and close proximity of our sensor9nables 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 ( ) andsingle grain remnant magnetism ( ), as well as phenomena that exhibit spatial textures such asmagnetic skyrmions ( ) and superconducting vortices ( ).While our work utilizes NV centers, the techniques developed here can be readily extendedto 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 sen-sitivities ( ). In addition, one can consider defects in other anvil substrates beyond diamond;indeed, recent studies have shown that moissanite (6H silicon carbide) hosts optically activedefects 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 supportlarger 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 environ-ments, opening up an enormous new range of experiments for quantitatively characterizingmaterials 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, andD.-H. Lee. We are especially grateful to M. Doherty and M. Barson for sharing their raw dataon stress susceptibilities. We thank C. Laumann for introducing us to the idea of integrating NVcenters 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.10 unding
This work was supported as part of the Center for Novel Pathways to Quantum Coherencein 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 ac-knowledges support from the National Science Foundation Graduate Research Fellowship un-der 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.
References
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50 nm layer of NV centers is embedded into the diamondanvil directly below the sample chamber. ( C ) Stress (top) both shifts and splits the | m s = ± i sublevelsat first order; in particular, the shifting is characterized by Π z = α ( σ xx + σ yy ) + β σ zz , and the splittingis characterized by Π ⊥ = [ α ( σ yy − σ xx ) + β (2 σ xz )] + [ α (2 σ xy ) + β (2 σ yz )] . An axial magneticfield (bottom) splits the | m s = ± i sublevels at first order, but a transverse magnetic field leads to shiftsonly at second order. ( D ) ODMR spectrum from an NV center ensemble under an applied magnetic field.( E ) Each pair of resonances in (D) corresponds to one of the four NV crystallographic orientations. ( F )Comparison of high pressure magnetometry techniques. The system characterized in this work is shownhere assuming a sample suspended in a pressure medium µ m away from the culet (black open circle).We project that by exfoliating a sample directly onto the culet surface and using nm implanted NVcenters, the distance from the sample can be significantly reduced, thus improving dipole accuracy (openred circles). Inductive methods (pickup coils [green diamonds] and SQUIDs [blue squares]) integrate themagnetization of a sample over their area ( ). In contrast, high energy photon scattering techniques (x-ray magnetic circular dichroism [orange hexagons], and M¨ossbauer spectroscopy [pink triangles]) probeatomic scale magnetism ( ). Note that the length scale for these methods is shown here as the spot sizeof the excitation beam. =13.6 GPaP=4.9 GPa A B
Position (μm) Position (μm) S t r e ss ( G P a ) Position (μm)
GPa GPa
Position (μm)
50 μm . G P a (GPa) Figure 2:
Full tensorial reconstruction of the stresses in a (111)-cut diamond anvil. ( A ) Spatiallyresolved maps of the loading stress (left) and mean lateral stress (right), σ ⊥ = ( σ XX + σ Y Y ) , acrossthe culet surface. In the inner region, where the culet surface contacts the pressure-transmitting medium(16:3:1 methanol/ethanol/water), the loading stress is spatially uniform, while the lateral stress is con-centrated towards the center; this qualitative difference is highlighted by a linecut of the two stresses(below), and reconstructed by finite element analysis (orange and purple dashed lines). The black pixelsindicate where the NV spectrum was obfuscated by the ruby microsphere. ( B ) Comparison of all stresstensor components in the fluid-contact region at P = P = P = σ ZZ (inset). .5 3.0 3.5Frequency (GHz)02040 A B CD E FG H I J Y P o s i t i o n ( m ) F r e q u e n c y S p li tt i n g ( M H z )
10 µm
Pressure (GPa) D i p o l e S t r e n g t h ( e m u ) CompressionDecompression
Figure 3:
Imaging iron’s α ↔ (cid:15) phase transition. Applying an external magnetic field ( B ext ∼ G)induces a dipole moment in the polycrystalline iron pellet which generates a spatially varying magneticfield across the culet of the diamond anvil. By mapping the ODMR spectra across the culet surface, we re-construct the local magnetic field which characterizes the iron pellet’s magnetization. ( A - C ) Comparisonbetween the measured ODMR spectra (dark regions correspond to resonances) and the theoretical reso-nance positions (different colors correspond to different NV crystallographic orientations) across verticalspatial cuts at pressures 9.6 GPa, 17.2 GPa and 20.2 GPa, respectively (16:3:1 methanol/ethanol/watersolution). ( D - F ) Map of the measured energy difference of a particular NV crystallographic orientation(blue lines in (A-C)). Black pixels correspond to ODMR spectra where the splitting could not be ac-curately extracted owing to large magnetic field gradients ( ). ( G - I ) Theoretical reconstruction of theenergy differences shown in (D-F). Data depicted in (A-C) are taken along the thin black dashed lines.( J ) Measured dipole moment of the iron pellet as a function of applied pressure at room temperature, forboth compression (red) and decompression (blue). Based on the hysteresis observed ( ∼ GPa), we findthe critical pressure P c = 13 . ± . GPa, in excellent agreement with previous studies ( ). A B
Temperature (K) ( s ) T LaserMW
280 300 32050100150
200 250 300
Temperature (K) N o r m a li z e d S p li tt i n g [a] P = . G P a P = . G P a P = G P a Temperature (K)
250 30001
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Temperature (K) S p li tt i n g ( M H z ) [b] NV Layer
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Temperature (K) D e p o l a r i z a t i o n t i m e T ( s ) S p li tt i n g ( M H z ) Temperature (K)
Pressure (GPa) T e m p e r a t u r e ( K ) hcp Sm-type dhcp PMFM PMAFM PM [a] [b] [c][d]
Figure 4:
Magnetic P - T phase diagram of gadolinium. A ∼ µ m × µ m × µ m polycrystallineGd foil is loaded into a beryllium copper gasket with a cesium iodide pressure medium. An externalmagnetic field, B ext ∼ G, induces a dipole field, B Gd , detected by the splitting of the NVs (rightinset, (B)). ( A ) The FM Curie temperature T C decreases with increasing pressure up to ∼ GPa. NVsplittings for three P - T paths, labeled by their initial pressure P , are shown. The P - T path for run[a] ( P = 0 . GPa) is shown in (C). (
Inset A ) depicts the cool-down (blue) and heat-up (red) of asingle P - T cycle, which shows negligible hysteresis. ( B ) If a P - T path starting in hcp is taken intothe dhcp phase (at pressures & GPa) ( ), the FM signal is lost and not reversible. Such a P - T path [b], is shown in C . On cool-down (dark blue), we observe the aforementioned Curie transition,followed by the loss of FM signal at . GPa,
K. But upon heat-up (red) and second cool-down (lightblue), the FM signal is not recovered. (
Left Inset ) When the pressure does not go beyond ∼ GPa,the FM signal is recoverable ( ). ( C ) Magnetic P - T phase diagram of Gd. At low pressures, weobserve the linear decrease of T C (black line) with slope − . ± . K/GPa, in agreement with previousmeasurements ( ). This linear regime extends into the Sm-type phase (black dashed line) due to theslow dynamics of the hcp → Sm-type transition ( ). When starting in the Sm-type phase, we no longerobserve a FM signal, but rather a small change in the magnetic field at either the transition from Sm-typeto dhcp (orange diamonds) or from PM to AFM (green triangle), depending on the P - T path. The bottomtwo phase boundaries (black lines) are taken from Ref. ( ). ( D ) At ambient pressure, we observe a Curietemperature, T C = 292 . ± . K, via DC magnetometry (blue data). Using nanodiamonds drop-cast ontoa Gd foil (and no applied external magnetic field), we find that the depolarization time ( T ) of the NVs isqualitatively different in the two phases (red data). T is measured using the pulse sequence shown in thetop right inset. ( Bottom inset ) The T measurement on another nanodiamond exhibits nearly identicalbehavior. upplementary Material for “Imaging stress andmagnetism at high pressures using a nanoscalequantum sensor” S. Hsieh, , , ∗ P. Bhattacharyya, , , ∗ C. Zu, , ∗ T. Mittiga, T. J. Smart, F. Machado, B. Kobrin, , T. O. H ¨ohn, , N. Z. Rui, M. Kamrani, S. Chatterjee, S. Choi, M. Zaletel, V. V. Struzhkin, J. E. Moore, , V. I. Levitas, , R. Jeanloz, N. Y. Yao , , † Department of Physics, University of California, Berkeley, CA 94720, USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, 80799 Munich, Germany Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015, USA Departments of Mechanical Engineering and Material Science and Engineering,Iowa State University, Ames, IA 50011, USA † To whom correspondence should be addressed; E-mail: [email protected]
Contents a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec .1 Theoretical sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Experimental sensitivity and accuracy . . . . . . . . . . . . . . . . . . . . . . 73.3 Comparison to other magnetometry techniques . . . . . . . . . . . . . . . . . 8 P - T phase diagram of Gd . . . . . . . . . . . . . . . . . . . . 366.5 Noise spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.6 Theoretical analysis of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The nitrogen-vacancy (NV) center is an atomic defect in diamond in which two adjacent carbonatoms are replaced by a nitrogen atom and a lattice vacancy. When negatively charged (byaccepting a electron), the ground state of the NV center consists of two unpaired electrons ina spin triplet configuration, resulting in a room temperature zero-field splitting D gs = (2 π ) × .
87 GHz between | m s = 0 i and | m s = ± i sublevels. The NV can be optically initialized intoits | m s = 0 i sublevel using a laser excitation at wavelength λ = 532 nm . After initialization,a resonant microwave field is delivered to coherently address the transitions between | m s = 0 i and | m s = ± i . At the end, the spin state can be optically read-out via the same laser excitationdue to spin-dependent fluorescence spectroscopy ( ).The presence of externals signals affects the energy levels of the NV, and, in general, liftsthe degeneracy of the | m s = ± i states. Using optically detected magnetic resonance (ODMR)to characterize the change in the energy levels one can directly measure such external signals.More specifically, combining the information from the four possible crystallographic orientationof the NV centers, enables the reconstructuction of a signal’s vector (e.g. magnetic field) ortensorial (e.g. stress) information. 3 Experimental details
All diamond anvils used in this work are synthetic type-Ib ([N] . ppm) single crystal di-amonds cut into a 16-sided standard design with dimensions 0.2 mm diameter culet, 2.75 mmdiameter girdle, and 2 mm height (Almax-easyLab and Syntek Co., Ltd.). For stress mea-surement, both anvils with (111)-cut-culet and (110)-cut-culet are used, while for magenticmeasurement on iron and gadolinium, (110)-cut-culet anvil is used. We perform C + ion im-plantation (CuttingEdge Ions, 30 keV energy, × cm − ) to generate a ∼
50 nm layer ofvacancies near the culet surface. After implantation, the diamonds are annealed in vacuum( < − Torr) using a home-built furnace with the following recipe: 12 hours ramp to 400 ◦ C,dwell for 8 hours, 12 hours ramp to 800 ◦ C, dwell for 8 hours, 12 hours ramp to 1200 ◦ C, dwellfor 2 hours. During annealing, the vacancies become mobile, and probabilistically form NVcenters with intrinsic nitrogen defects. After annealing, the NV concentration is estimated tobe around 1 ppm as measured by flourescence intensity. The NV centers are photostable aftermany iterations of compression and decompression up to 27 GPa, with spin-echo coherencetime T ≈ µs , mainly limited by nitrogen spin bath.The miniature diamond anvil cell body is made of nonmagnetic Vascomax with cubic boronnitride backing plates (Technodiamant). Nonmagnetic gaskets (rhenium or beryllium copper)and pressure media (cesium iodide, methanol/ethanol/water) are used for all experiments. We address NV ensembles integrated inside the DAC using a home-built confocal microscope.A 100 mW 532 nm diode-pumped solid-state laser (Coherent Compass), controlled by anacousto-optic modulator (AOM, Gooch & Housego AOMO 3110-120) in a double-pass con-figuration, is used for both NV spin initialization and detection. The laser beam is focused4hrough the light port of the DAC to the NV layer using a long working distance objective lens(Mitutoyo 378-804-3, NA 0.42, for stress and iron measurements; Olympus LCPLFLN-LCD20X, NA 0.45, for gadolinium measurement in cryogenic environment), with a diffraction-limitspot size ≈ nm. The NV fluorescence is collected using the same objective lens, spec-trally separated from the laser using a dichroic mirror, further filtered using a nm long-passfilter, and then detected by a fiber coupled single photon counting module (SPCM, ExcelitasSPCM-AQRH-64FC). A data aquisition card (National Instruments USB-6343) is used for flu-orescence counting and subsequent data processing. The lateral scanning of the laser beam isperformed using a two-dimensional galvanometer (Thorlabs GVS212), while the vertical focalspot position is controlled by a piezo-driven positioner (Edmund Optics at room temperture;attocube at cryogenic temperature). For gadolinium measurements, we put the DAC into aclosed-cycle cryostat (attocube attoDRY 800) for temperature control from − K. TheAOM and the SPCM are gated by a programmable multi-channel pulse generator (SpinCorePulseBlasterESR-PRO 500) with ns temporal resolution.A microwave source (Stanford Research Systems SG384) in combination with a 16W am-plifier (Mini-Circuits ZHL-16W-43+) serves to generate signals for NV spin state manipula-tion. The microwave field is delivered to DAC through a 4 µ m thick platinum foil compressedbetween the gasket and anvil pavilion facets, followed by a 40 dB attenuator and a 50 Ω termi-nation. In this work, we use continous-wave optically detected magnetic resonance (ODMR) spec-troscopy to probe the NV spin resonances. The laser and microwave field are both on forthe entire measurement, while the frequency of the microwave field is swept. When the mi-crowave field is resonant with one of the NV spin transitions, it drives the spin from | m s = 0 i | m s = ± i , resulting in a decrease in NV fluorescence. The magnetic field sensitivity for continuous-wave ODMR ( ) is given by: η B = P G γ B ∆ ν C√R , (1)where γ B is the gyromagnetic ratio, P G ≈ . is a unitless numerical factor for a Gaussianlineshape, ∆ ν = 10 MHz is the resonance linewidth,
C ≈ . is the resonance contrast, and R ≈ . × s − is the photon collection rate. One can relate this to magnetic momentsensitivity by assuming that the field is generated by a point dipole located a distance d fromthe NV center (pointing along the NV axis). Then the dipole moment sensitivity is given by η m = P G γ B ∆ ν C√R πd µ , (2)where µ is the vacuum permeability.Analogous to Eq. 1, the stress sensitivity for continuous-wave ODMR is given by η S = P G ξ ∆ ν C√R , (3)where ξ is the susceptibility for the relevant stress quantity. More specifically, ξ is a tensordefined by: ξ αβ = (cid:12)(cid:12)(cid:12)(cid:12) δf α δσ β (cid:12)(cid:12)(cid:12)(cid:12) σ (0) (4)where f α , α ∈ [1 , are the resonance frequences associated with the 4 NV crytallographicorientations; σ (0) is an initial stress state; and δσ β is a small perturbation to a given stresscomponent, e.g. β ∈ { XX, Y Y, ZZ, XY, XZ, Y Z } . For optimal sensitivity, we consider6erturbations about an unstressed state (i.e. σ (0) = ) . The resulting susceptibilities for stresscomponents in a (111)-cut diamond frame are ξ αβ = (2 π ) × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [MHz / GPa] . In the main text and in Table 1, we compute the sensitivity using the maximum susceptibilityfor each stress component: ξ ( max ) β = max α ξ αβ (5) In order to characterize the sensitivity of our system, we perform ODMR spectroscopy on asingle resonance. We fit a Gaussian lineshape to this resonance and observe the fitting error onthe center frequency as a function of the total integration time, T (Fig. 1). In particular, we fitthe time scaling behavior of the fitting error to AT − / , where A , divided by the susceptibilityof interest, characterizes the experimental sensitivity for a given signal. For T & s, the ex-perimental accuracy saturates due to systematic noise, which we define here as the “systematicaccuracy” for each type of signal.For scalar signals (e.g. axial magnetic fields, temperature, etc.), the accuracy is directlyproportional to the minimum fitting error. For stress components, however, determining theaccuracy is more complicated as the relation between resonance frequencies and the full stresstensor is a multi-dimensional, nonlinear function (Section 4.1). To this end, we quantify the Equivalently, one can begin from any hydrostatic stress, i.e. σ (0) ∼ I . Non-hydrostatic stress, however, willgenerally reduce the stress susceptibilities, as will the presence of electric or magnetic fields. The Z axis is normal to the diamond surface, and the XZ plane contains two of the NV axes (the vertical axisand one of the three non-vertical axes). M agne t i c f i e l d a cc u r a cy ( µ T ) Integration time (s) 0.050.10.20.4 F r equen cy a cc u r a cy ( M H z ) Figure 1: Scaling of magnetic field accuracy as a function of total integration time on a singleresonance. Right axis corresponds to standard deviation of center frequency fitting. Solid linecorresponds to a fit to AT − / where A is the sensitivity reported in the main text and T isthe total integration time. Dashed line corresponds to the scaling predicted by Eq. 1. Theexperimental accuracy saturates for T & s due to systematic noise.accuracy of each stress component using a Monte Carlo procedure. We begin with an unstressedstate, which corresponds to the initial set of frequencies f (0) α = D gs . We then apply noise to eachof the freqencies based on the minimum fitting error determined above—i.e. f (0) α + δf α , where δf α are sampled from a Gaussian distribution with a width of the fitting error—and calculatethe corresponding stress tensor using a least-squared fit (Sec. 4.1). Repeating this procedureover many noise realizations, we compute the standard deviation of each stress component. Theresults of this procedure are shown in Table 1. In this section, we discuss the comparison of magnetometry techniques presented in Fig. 1F ofthe main text. For each sensor, the corresponding dipole accuracy (as defined in Section 3.2)is plotted against its relevant “spatial resolution,” roughly defined as the length scale withinwhich one can localize the source of a magnetic signal. In the following discussion, we specifythe length scale plotted for each method in Fig. 1F of the main text. We consider two broad8ignal (unit) Theo Sensitivity Exp Sensitivity Accuracy(unit/ √ Hz) (unit/ √ Hz) (unit)Hydrostatic stress (GPa) .
017 0 .
023 0 . Average normal stress (GPa) .
022 0 .
03 0 . Average shear stress (GPa) .
020 0 .
027 0 . Magnetic field ( µ T) . . Magnetic dipole (emu), . × − . × − . × − floating sample ( d = 5 µ m)Magnetic dipole (emu), . × − . × − . × − exfoliated sample ( d = 5 nm) ( ∗ ) Magnetic dipole (emu), . × − . × − . × − exfoliated sample,single NV ( d = 5 nm) ( † ) Electric field (kV/cm), . . . single NV ( † ) Temperature (K), . .
55 0 . single NV ( † ) Table 1: NV sensitivity and accuracy for various signals. Sensitivity is calculated using Eqs. 2-3. We also report the typical fitting error of the center frequency for the relevant experimentsin the main text. Gray rows correspond to projected sensitivity given an exfoliated sampleatop ( ∗ ) an ensemble of nm depth NV centers or ( † ) a single nm depth NV center with ∆ ν = 1 MHz , C = 0 . , R = 10 s − . Magnetic dipoles are reported in units of emu, where emu = 10 − A · m . 9ategories of high pressure magnetometers.The first category encompasses inductive methods such as pickup coils ( ) and super-conducting quantum interference devices (SQUIDs) ( ) . Magnetic dipole measurementaccuracies are readily reported in various studies employing inductive methods. We estimatethe relevant length scale of each implementation as the pickup coil or sample bore diameter.The second class of magnetometers comprises high energy methods including M ¨ossbauerspectroscopy ( ) and x-ray magnetic circular dichroism (XMCD) ( ), which probeatomic scale magnetic environments. For the M¨ossbauer studies considered in our analysis, wecalculate magnetic dipole moment accuracies by converting B -field uncertainties into magneticmoments, assuming a distance to the dipole on order of the lattice spacing of the sample. Weassess the length scale as either the size of the absorbing sample or the length scale associatedwith the sample chamber/culet area. For XMCD studies, we accept the moment accuraciesreported in the text. Length scales are reported as the square root of the spot size area. Notably,we emphasize that both methods provide information about atomic scale dipole moments ratherthan a sample-integrated magnetic moment; these methods are thus not directly comparable toinductive methods.We compare these methods alongside the NV center, whose accuracy is defined in Sec-tion 3.2 and shown in Table 1. For the current work, we estimate a length scale ∼ µ m, corre-sponding to the approximate distance between a sample (suspended in a pressure-transmittingmedium) and the anvil culet. By exfoliating a sample onto the diamond surface, the diffraction-limit ∼ nm bounds the transverse imaging resolution for ensemble NV centers; this limitcan be further improved for single NV centers via super-resolution techniques ( ). Under the category of inductive methods, we also include the “designer anvil” which embeds a pickup coildirectly into the diamond anvil. Stress tensor
In this section, we describe our procedure for reconstructing the full stress tensor using NVspectroscopy. This technique relies on the fact that the four NV crystallographic orientationsexperience different projections of the stress tensor within their local reference frames. In par-ticular, the full Hamiltonian describing the stress interaction is given by: H S = X i Π z,i S z,i + Π x,i (cid:0) S y,i − S x,i (cid:1) + Π y,i ( S x,i S y,i + S y,i S x,i ) (6)where Π z,i = α (cid:0) σ ( i ) xx + σ ( i ) yy (cid:1) + β σ ( i ) zz (7) Π x,i = α (cid:0) σ ( i ) yy − σ ( i ) xx (cid:1) + β (cid:0) σ ( i ) xz (cid:1) (8) Π y,i = α (cid:0) σ ( i ) xy (cid:1) + β (cid:0) σ ( i ) yz (cid:1) (9) σ ( i ) is the stress tensor in the local frame of each of NV orientations labeled by { i = 1 , , , } ,and { α , , β , } are stress susceptibility parameters (Section 4.3.3). Diagonalizing this Hamil-tonian, one finds that the energy levels of each NV orientation exhibit two distinct effects: the | m s = ± i states are shifted in energy by Π z,i and split by ⊥ ,i = 2 q Π x,i + Π y,i . Thus, theHamiltonian can be thought of as a function that maps the stress tensor in the lab frame to eightobservables: H S ( σ (lab ) ) = { Π z, , Π ⊥ , , Π z, , Π ⊥ , , ... } . Obtaining these observables throughspectroscopy, one can numerically invert this function and solve for all six components of thecorresponding stress tensor.In practice, resolving the resonances of the four NV orientation groups is not straightforwardbecause the ensemble spectra can exhibit near degeneracies. When performing ensemble NVmagnetometry, a common approach is to spectroscopically separate the resonances using anexternal bias magnetic field. However, unlike magnetic contributions to the Hamiltonian, stress11hat couples via Π ⊥ is suppressed by an axial magnetic field. Therefore, a generic magneticfield provides only stress information via the shifting parameters, Π z,i , which is insufficient forreconstructing the full tensor.To address this issue, we demonstrate a novel technique that consists of applying a well-controlled external magnetic field perpendicular to each of the NV orientations. This techniqueleverages the symmetry of the NV center, which suppresses its sensitivity to transverse magneticfields. In particular, for each perpendicular field choice, three of the four NV orientations exhibita strong Zeeman splitting proportional to the projection of the external magnetic field alongtheir symmetry axes, while the fourth (perpendicular) orientation is essentially unperturbed .This enables one to resolve Π z,i for all four orientations and Π ⊥ ,i for the orientation that isperpendicular to the field. Repeating this procedure for each NV orientation, one can obtain theremaining splitting parameters and thus reconstruct the full stress tensor.In the following sections, we provide additional details regarding our experimental proce-dure and analysis. In Section 4.2, we describe how to use the four NV orientations to calibratethree-dimensional magnetic coils and to determine the crystal frame relative to the lab frame.In Section 4.3, we discuss our fitting procedure, the role of the NV’s local charge environment,and the origin of the stress susceptibility parameters. In Section 4.4, we present the results ofour stress reconstruction procedure for both (111)- and (110)-cut diamond. In Section 4.5, wecompare our experimental results to finite element simulations. A transverse magnetic field leads to shifting and splitting at second order in field strength. We account for theformer through a correction described in Section 4.3, while the latter effect is small enough to be neglected. Morespecifically, the effective splitting caused by magnetic fields is ( γ B B ⊥ ) /D gs ≈ − MHz, which is smallerthan the typical splitting observed at zero field. .2 Experimental details To apply carefully aligned magnetic fields, we utilize a set of three electromagnets that areapproximately spatially orthogonal with one another and can be controlled independently viathe application of current. Each coil is placed >
10 cm away from the sample to reduce themagnetic gradient across the (200 µ m ) culet area .To calibrate the magnetic field at the location of the sample, we assume that the field pro-duced by each coil is linearly porportional to the applied current, I . Our goal is then to find theset of coefficients, a mn such that B m = X m a mn I n , (10)where B m = { B X , B Y , B Z } is the magnetic field in the crystal frame and n = { , , } indexesthe three electromagnets. We note that this construction does not require the electromagnets tobe spatially orthogonal.To determine the nine coefficients, we apply arbitrary currents and measure the Zeemansplitting of the four NV orientations via ODMR spectroscopy. Notably, this requires the abil-ity to accurately assign each pair of resonances to their NV crystallographic orientation. Weachieve this by considering the amplitudes of the four pairs of resonances, which are pro-portional to the relative angles between the polarization of the excitation laser and the fourcrystallagraphic orientations. In particular, the | m s = 0 i ↔ | m s = ± i transition is driven bythe perpendicular component of the laser field polarization with respect to the NV’s symmetryaxis. Therefore, tuning the laser polarization allows us to assign each pair of resonances to aparticular NV orientation.In order to minimize the number of fitting variables, we choose magnetic fields whose pro-jection along each NV orientation is sufficient to suppress their transverse stress-induced energy We note that the pressure cell, pressure medium and gasket are nonmagnetic. γ B B (cid:29) Π ⊥ . As a result, the spectrum measured at each magnetic field is deter-mined by (a) the stress-induced shift Π z,i for each NV orientation, which is constant for allapplied fields, and (b) the applied vector magnetic field { B X , B Y , B Z } . Sequentially applyingdifferent currents to the electromagnet coils and determining the subsequent vector magneticfield at the sample three times, we obtain sufficient information to determine the matrix a mn aswell as the shift Π z for all NV orientations. We find that the calibration technique is precise towithin 2 % . To determine the orientation of the crystal frame (i.e. the [100] diamond axis) with respect tothe lab frame, we apply an arbitrary magnetic field and measure its angle (a) in the lab framevia a handheld magnetometer, and (b) in the crystal frame via the Zeeman splittings (see 4.2).Together with the known diamond cut, this provides a system of equations for the rotationmatrix, R c , that relates the lab frame and the crystal frame: R c ˆ B ( lab ) = ˆ B ( crystal ) , R c ˆ Z = ˆ e ( crystal ) (11)where ˆ Z = (0 , , > is the longitudinal axis in the lab frame, and ˆ e ( crystal ) is the unit vectorperpendicular to the diamond cut surface in crystal frame, e.g. ˆ e ( crystal ) ∝ (1 , , > for the(111)-cut diamond. We solve for R c by numerically minimizing the least-squared residue ofthese two equations.However, we note that the magnetic field determined by the Zeeman splittings contains anoverall sign ambiguity. To account for this, we numerically solve Eq. (11) using both signsfor ˆ B ( crystal ) and select the solution for R c with the smaller residue. Based on this residue, weestimate that our calibration is precise to within a few degrees.14 .3 Analysis Having developed a technique to spectrally resolve the resonances, we fit the resulting spectrato four pairs of Lorentzian lineshapes. Each pair of Lorentzians is defined by a center frequency,a splitting, and a common amplitude and width. To sweep across the two-dimensional layer ofimplanted NV centers, we sequentially fit the spectrum at each point by seeding with the best-fitparameters of nearby points. We ensure the accuracy of the fits by inspecting the frequencies ofeach resonance across linecuts of the 2D data (Fig. 2B).Converting the fitted energies to shifting ( Π z,i ) and splitting parameters ( Π ⊥ ,i ) requires us totake into account two additional effects. First, in the case of the shifting parameter, we subtractoff the second-order shifting induced by transverse magnetic fields. In particular, the effectiveshifting is given by Π z, B ≈ ( γ B B ⊥ ) /D gs , which, under our experimental conditions, corre-sponds to Π z, B ≈ − MHz. To characterize this shift, one can measure each of the NVorientations with a magnetic field aligned parallel to its principal axis, such that the transversemagnetic shift vanishes. In practice, we obtain the zero-field shifting for each of the NV orien-tations without the need for additional measurements, as part of our electromagnet calibrationscheme (Section 4.2). We perform this calibration at a single point in the two-dimensional mapand use this point to characterize and subtract off the magnetic-induced shift in subsequent mea-surements with arbitrary applied field. Second, in the case of the splitting parameter, we correctfor an effect arising from the NV’s charge environment. We discuss this effect in the followingsection. The final results for the shifting ( Π z,i ) and splitting ( Π ⊥ ,i ) parameters for the (111)-cutdiamond at 4.9 GPa are shown in Fig. 2C. 15 BC MHz MHz MHzMHzMHzMHz
M M
50 μm
Figure 2: Stress reconstruction procedure applied to the (111)-cut diamond at 4.9 GPa. ( A ) Atypical ODMR spectrum with the resonances corresponding to each NV orientation fit a pair ofLorentzian lineshapes. ( B ) A linecut indicating the fitted resonance energies (colored points)superimposed on the measured spectra (grey colormap). ( C ) 2D maps of the shifting ( Π z,i ) andsplitting parameters ( Π ⊥ ,i ) for each NV orientation across the entire culet.16 .3.2 Effect of local charge environment It is routinely observed that ensemble spectra of high-density samples (i.e. Type 1b) exhibit alarge ( − MHz) splitting even under ambient conditions. While commonly attributed toinstrinsic stresses in the diamond, it has since been suggested that the splitting is, in fact, due toelectric fields originating from nearby charges ( ). This effect should be subtracted from thetotal splitting to determine the stress-induced splitting.To this end, let us first recall the NV interaction with transverse electric fields: H E = d ⊥ (cid:2) E x ( S y − S x ) + E x ( S x S y + S y S x ) (cid:3) (12)where d ⊥ = 17 Hz cm/V. Observing the similarity with Eq. (6), we can define ˜Π x = Π s ,x + Π E ,x (13) ˜Π y = Π s ,y + Π E ,y (14)where Π S , { x,y } are defined in Eq. (7) and Π E , { x,y } = d ⊥ E { x,y } . The combined splitting forelectric fields and stress is then given by ⊥ = 2 (cid:0) (Π s ,x + Π E ,x ) + (Π s ,y + Π E ,y ) (cid:1) / . (15)We note that the NV center also couples to longitudinal fields, but its susceptibility is ∼ times weaker and is thus negligible in the present context.To model the charge environment, we consider a distribution of transverse electric fields.For simplicity, we assume that the electric field strength is given by a single value E , and itsangle is randomly sampled in the perpendicular plane. Adding the contributions from stress and17lectric fields and averaging over angles, the total splitting becomes ˜Π ⊥ , avg = Z dθ (Π S , ⊥ + Π E , ⊥ + 2Π S , ⊥ Π E , ⊥ cos θ ) / = 1 π q Π s , ⊥ − Π E , ⊥ EllipticE − S , ⊥ Π E , ⊥ q Π S , ⊥ − Π E , ⊥ + q Π S , ⊥ + Π E , ⊥ EllipticE − s , ⊥ Π E , ⊥ q Π S , ⊥ + Π E , ⊥ (16)where EllipticE ( z ) is the elliptic integral of the second kind. This function is plotted in Fig. 3A,and we note its qualitative similarity to a quadrature sum.To characterize the intrinsic charge splitting ( Π E , ⊥ ), we first aquire an ODMR spectrum foreach diamond sample under ambient conditions. For example, for the (111)-cut diamond, wemeasured Π E , ⊥ ≈ . MHz. For subsequent measures under pressure, we then subtract offthe charge contribution from the observed splitting by numerically from inverting Eq. (16) andsolving for Π s , ⊥ . A recent calibration experiment established the four stress susceptibilities relevant to this work( ). In this section, we discuss the conversion of their susceptibilities to our choice of basis(the local NV frame), and we reinterpret their results for the splitting parameters taking intoaccount the effect of charge.In their paper, Barson et. al. define the stress susceptilities with respect diamond crystalframe: Π z = a ( σ X X + σ YY + σ ZZ ) + 2 a ( σ YZ + σ ZX + σ X Y ) (17) Π x = b (2 σ ZZ − σ X X − σ YY ) + c (2 σ X Y − σ YZ − σ ZX ) (18) Π y = √ b ( σ X X − σ YY ) + c ( σ YZ − σ ZX )] (19)18here X YZ are the principal axes of the crystal frame. Their reported results are { a , a , b, c } =(2 π ) × { . , − . , . , . } MHz/GPa.To convert these susceptibilities to our notation (Eq. 6), one must rotate the stress tensorfrom the crystal frame to the NV frame, i.e. σ xyz = Rσ X YZ R > . The rotation matrix thataccomplishes this is: R = − √ − √ q √ − √ √ √ √ . (20)Applying this rotation, one finds that the above equations become (in the NV frame) Π z = ( a − a )( σ xx + σ yy ) + ( a + 2 a ) σ zz (21) Π x = ( − b − c )( σ yy − σ xx ) + ( √ b − √ c )(2 σ xz ) (22) Π x = ( − b − c )(2 σ xy ) + ( √ b − √ c )(2 σ yz ) (23)Thus, the conversion between the two notations is (cid:18) α β (cid:19) = (cid:18) −
11 2 (cid:19) (cid:18) a a (cid:19)(cid:18) α β (cid:19) = (cid:18) − − √ − √ (cid:19) (cid:18) bc (cid:19) (24)In characterizing the splitting parameters ( b and c ), Barson et. al. assumed a linear depen-dence between the observed splitting and Π S , ⊥ . However, our charge model suggests that for Π S , ⊥ . Π E , ⊥ the dependence should be nonlinear. To account for this, we re-analyze their datausing Eq. 16 as our fitting form, rather than a linear function as in the original work. The resultsare shown in Fig. 3 for two NV orientation groups measured in the experiment: (110) and (100) , where ( · · · ) denotes the crystal cut and the subscript is the angle of the NV group withrespect to the crystal surface. From the fits, we extract the linear response, Π s , ⊥ /P , for the twogroups. These are related to the stress parameters by b − c and b , respectively. Using these19 B C
Figure 3: Interplay between stress and random electric fields. ( A ) Theoretical curve (blue) forthe total splitting in the presence of stress and electric fields, Eq. (16). We compare this to aquadratic sum (red). ( B - C ) Measured splitting parameter (blue) for uniaxial pressure applied toa (110)-cut and (100)-cut diamond, reprinted with permission from ( ). We fit the data using(a) a linear function (orange), ˜Π ⊥ = Π E , ⊥ + Π S , ⊥ , and (b) the aforementioned theoretical curve,Eq. (16) (green). Both fits include two free parameters: Π E , ⊥ and a = Π S , ⊥ /P . We report thebest-fit value for the latter parameter in the inset.relations and the results of the fits, one finds { b, c } = (2 π ) × {− . , . } MHz/GPa .Finally, we convert these and the original reported for { a , a } to the NV frame using Eq. 24.This leads to the susceptibilites that we use for our analysis: { α , β , α , β } = (2 π ) × { . , − . , − . , − . } MHz/GPa . (25) In this section, we discuss our stress reconstruction results for (a) the (111)-cut diamond at4.9 GPa and 13.6 GPa (Fig. 4), and (b) the (110)-cut diamond at 4.8 GPa (Fig. 5). The stresstensors were obtained by numerically minimizing the least-squared residue with respect to themeasured shifting and splitting parameters (i.e. Π z,i , Π ⊥ ,i ). While ideally we would measureall eight observables, in this experiment we measured only six: all four shifting parameters and Note that the overall sign of these parameters cannot be determined through these methods, as the energysplitting is related to the quadrature sum of Π x and Π y . To determine the sign, one would need to measure thephase of the perturbed states ( ). σ ZZ and σ ⊥ = ( σ XX + σ Y Y ) , i.e. the two azimuthally symmetric normal components.We can estimate the accuracy of the reconstructed tensors from the spatial variations of σ ZZ at 4.9 GPa. Assuming the medium is an ideal fluid, one would expect that σ ZZ to be flatin the region above the gasket hole. In practice, we observe spatial fluctuations characterizedby a standard deviation ≈ . GPa; this is consistent with the expected accuracy based onfrequency noise (Table 1). The errorbars in the reconstructed stress tensor are estimated usingthe aforementioned experimental accuracy.Interestingly, the measured values for σ ZZ differs from the ruby pressure scale by ∼ .This discrepancy is likely explained by inaccuracies in the susceptibility parameters; in particu-lar, the reported susceptibility to axial strain (i.e. β ) contains an error bound that is also ∼ .Other potential sources of systematic error include inaccuracies in our calibration scheme or thepresence of plastic deformation.Finally, we note that, in many cases, our reconstruction procedure yielded two degeneratesolutions for the non-symmetric stress components; that is, while σ ZZ and σ ⊥ have a uniquesolution, we find two different distributions for σ XX , σ XY , etc. This degeneracy arises from thesquared term in the splitting parameter, Π ⊥ ,i = 2 q Π x,i + Π y,i , and the fact we measure onlysix of the eight observables. In Fig. 4 and Fig. 5 (and Fig. 2B of the main text), we show thesolution for the stress tensor that is more azymuthally symmetric, as physically motivated byour geometry. Using equations from elasticity theory under the finite element approach, a numerical simula-tion was coded in ABAQUS for the stress and strain tensor fields in the diamond anvil cell.The diamond anvil cell is approximately axially symmetric about the diamond loading axis, in21
Pa GPa GPaGPaGPaGPa (cid:25)(cid:28)(cid:3)(cid:612)(cid:84)
GPa GPa GPaGPaGPaGPa (cid:28)(cid:23)(cid:3)(cid:612)(cid:84) A B Figure 4: Stress tensor reconstruction of (111)-cut diamond at ( A ) 4.9 GPa and ( B ) 13.6 GPa.In the former case, we reconstruct both the inner region in contact with the fluid-transmittingmedium, and the outer region in contact with the gasket. In the latter case, we reconstruct onlythe inner region owing to the large stress gradients at the contact with the gasket; note that theblack pixels in the center indicates where the spectra is obscured by the ruby flourescence. Asdescribed in the main text, both pressures exhibit inward concentration of the normal lateralstress ( σ XX and σ Y Y ). In contrast, the normal loading stress is uniform for the lower pressureand spatially varying at the higher pressure, indicating that the pressure medium has solidified.22
Pa GPa GPaGPaGPaGPa
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Figure 5: Stress tensor reconstruction of (110)-cut diamond at 4.8 GPa pressure. Analogous tothe (111)-cut at low pressure, we observe an inward concentration of lateral stress and a uniformloading stress in the fluid-contact region. 23
Figure 6: ( A ) Diamond geometry, ( B ) anvil tip with distribution of the applied normal stress,( C ) distribution of the applied shear stress. Normal stress σ ZZ at the culet and zero shear stress σ RZ along the pressure-transmitting medium/anvil boundary ( r ≤ µm ) are taken from exper-iment. Normal and shear contact stresses along all other contact surfaces are determined fromthe best fit of the mean in-plane stress distribution σ ⊥ = 0 . σ RR + σ ΘΘ ) to experiment (maintext Fig. 2A and Fig. 7)this case the crytallographic (111) axis (i.e. the Z axis). This permits us to improve simulationefficiency by reducing the initially 3D tensor of elastic moduli to the 2D axisymmetric cylin-drical frame of the diamond as follows. Initially, the tensor can be written in 3D with cubicaxes c = 1076 GPa, c = 125 GPa, c = 577 GPa. Next, we rotate cubic axes such that the(111) direction is along the Z axis of the cylindrical coordinate system. Finally, the coordinatesystem is rotated by angle θ around the Z axis and the elastic constants are averaged over ◦ rotation. The resulting elasticity tensor in the cylindrical coordinate system is . . . . . . . . [GPa] . The geometry of the anvil and boundary conditions (Fig. 6) are as follows:1. The top surface of the anvil is assumed to be fixed. The distribution of stresses or dis-24lacements along this surface does not affect our solution close to the diamond culet lineAB.2. The normal stress σ ZZ along the line AB is taken from the experimental measurements(main text Fig. 2A and 7). The pressure-transmitting medium/gasket boundary runsalong the innermost 47 µ m of this radius.3. Along the pressure-transmitting medium/anvil boundary ( r ≤ µm ) and also at thesymmetry axis r = 0 (line AE) shear stress σ RZ is zero. Horizontal displacements at thesymmetry axis are also zero.4. Normal and shear contact stresses along all other contact surfaces are determined fromthe best fit to the mean in-plane stress distribution σ ⊥ = 0 . σ RR + σ ΘΘ ) measured inthe experiment (main text Fig. 2A and Fig. 7 ). We chose to fit to σ ⊥ rather than to othermeasured stresses is because it has the smallest noise in experiment. With this, the normalstress on the line BD with the origin at point B is found to be σ c = 3 . × x − . × x + 4 . × x − x + 4 . , (26)where σ c is in units of GPa, and the position x along the lateral side is in units of mm.The distribution of the normal stresses is shown in Fig. 6B and Fig. 8.5. At the contact surface between the gasket and the anvil, a Coulomb friction model isapplied. The friction coefficient on the culet is found to be 0.02 and along the inclinedsurface of the anvil (line BD) is found to vary from 0.15 at point B to 0.3 at 80 µ m fromthe culet. The distribution of shear stresses is shown in Fig. 6C and Fig. 8.6. Other surfaces not mentioned above are stress-free.The calculated distributions of the stress tensor components near the tip of the anvil areshown in Fig. 9. 25 (A) (B) Figure 7: ( A ) Distribution of applied normal stress σ ZZ and the mean in-plane stress σ ⊥ alongthe culet surface of the diamond from the experiment and FEM simulations. ( B ) Distribution ofthe mean in-plane stress σ ⊥ (experimental and simulated) as well as the simulated radial σ RR and circumferential σ ΘΘ stresses along the culet surface of the diamond. Figure 8: Distribution of applied normal and shear stress along the lateral surface of the diamonddetermined from the best fit of the mean in-plane stress distribution σ ⊥ to experiment (main textFig. 2A and Fig. 7). In this section, we discuss the study of the pressure-induced α ↔ (cid:15) transition in iron. Inparticular, we provide the experimental details, describe the model used for fitting the data, and26 Figure 9: Calculated distributions of the components of stress tensor in the anvil for r < and z < µm .outline the procedure to ascertain the transition pressure.For this experiment, the DAC is prepared with a rhenium gasket preindented to µ mthickness and laser drilled with a µ m diameter hole. We load a ∼ µ m iron pellet,extracted from a powder (Alfa Aesar Stock No. 00737-30), and a ruby microsphere for pres-sure calibration. A solution of methanol, ethanol and water (16:3:1 by volume) is used as thepressure-transmitting medium.The focused laser is sequentially scanned across a 10 ×
10 grid corresponding to a ∼ × µ m area of the NV layer in the vicinity of the iron pellet, taking an ODMR spectrum at eachpoint. As discussed in the main text, the energy levels of the NV are determined by both themagnetic field and the stress in the diamond. Owing to their different crystallographic orienta-tions, the four NV orientations in general respond differently to these two local parameters. Asa result, for each location in the scan, eight resonances are observed.A large bias magnetic field ( ∼ G), not perpendicular to any of the axes, is used tosuppress the effect of the transverse stress in the splitting for each NV orientation. However,the longitudinal stress still induces an orientation-dependent shift of the resonances which is27igure 10: (A)
Example of a typical spectrum with a fit to eight free Gaussians. Resonance pairsare identified as in Fig. 1D of the main text: NV4 has the strongest magnetic field projection andNV1 has the weakest. (B)
Example spectrum for which resonances are broadened and shifted.In this case we cannot correlate any resonances in the spectrum to specific NV orientations.nearly constant across the imaging area, as measured independently (Fig 2C).By analyzing the splittings of the NV resonances across the culet, we can determine thelocal magnetic field and thereby reconstruct the dipole moment of the iron pellet.To estimate the error in pressure, a ruby fluorescence spectrum was measured before andafter the ODMR mapping, from which the pressure could be obtained ( ). The pressure wastaken to be the mean value, while the error was estimated using both the pressure range and theuncertainty associated with each pressure point. The eight resonances in a typical ODMR spectrum are fit to Gaussian lineshapes to extract theresonance frequency (Fig 10A). Resonances are paired as in Fig. 1D of the main text: fromoutermost resonances to innermost, corresponding to NV orientations with the strongest mag-netic field projection to the weakest, respectively. Once identified, we calculate the splitting andmagnetic field projection for each NV orientation.28e note that there are two regimes where our spectra cannot confidently resolve and identifyall the eight resonances. First, at high pressure, the resonance contrast for some NV orienta-tions is diminished, possibly due to a modification of the frequency response of the microwavedelivery system. Second, close to or on top of the iron pellet, the resonances are broadened; weattribute this to the large magnetic field gradients (relative to the imaging resolution) caused bythe sample. The resulting overlap in spectral features obfuscates the identity of each resonance(Fig. 10B). In both cases, we fit and extract splittings only for the orientations we could identifywith certainty.
We model the magnetization of our pellet sample as a point dipole at some location within thesample chamber. The total magnetic field is then characterized by the external applied field, B , the dipole of the sample, d , and the position of the dipole, r . Because of the presence of alarge applied field, we observe that the magnetization of the sample aligns with B , and thus,we require only the strength of the dipole to characterize its moment, d = D ˆ B . We expectthe external magnetic field and the depth of the particle to remain nearly constant at differentpressures. This is indeed borne out by the data, see Sec. 5.4. As a result, we consider the externalmagnetic field B = ( − , − , G and depth of the iron pellet r Z = − µ m to be fixed.Due to the dipole of the iron pellet, the magnetic field across the NV layer at position x isgiven by: B ( x ) = B + µ π | x | (3ˆ x ( d · ˆ x ) − d ) , (27)where hats represent unit vectors. At each point, the local field induces a different splitting, ∆ ( i ) , to the 4 NV crytallographic orientations i ∈ { , , , } , measured by diagonalizing theHamiltonian H = D gs S z + B ( i ) z S z + B ( i ) ⊥ S x , where B z = | B · ˆ z ( i ) | is the projection of B onto29he axis of the NV, and B ( i ) ⊥ = q | B | − ( B ( i ) z ) , its transverse component. D gs is the zerofield splitting of the NV. For each choice of D , r X and r Y , we obtain a two dimensional map of { ∆ ( i ) } . Performing a least squares fit of this map against the experimental splittings determinesthe best parameters for each pressure point. The error in the fitting procedure is taken as theerror in the dipole strength D . Although the α ↔ (cid:15) structural phase transition in iron is a first order phase transition, we donot observe a sharp change in the dipole moment of the sample, observing instead a cross-overbetween the two magnetic behaviors. We attribute this to the non-hydrostatic behavior of thesample chamber at high pressures. As a result, different parts of the iron pellet can experiencedifferent amounts of pressure and, thus, undergo a phase transition at different applied pressures.The measured dipole moment should scale with the proportion of the sample that has undergonethe phase transition. This proportion, p ( P ) , should plateau at either or on different sidesof the phase transition, and vary smoothly across it. To model this behavior we use a logisticfunction: p ( P ) = 1 e B ( P − P c ) + 1 . (28)The dipole strength is then given by: D = p ( P ) D α + [1 − p ( P )] D (cid:15) , (29)where D α ( D (cid:15) ) is the dipole moment of the sample in the α ( (cid:15) ) structural phase and /B corresponds to the width of the transition, thus its uncertainty. During the decompression, around GPa, we observed a significant drift of the pressure dur-ing measurement of the ODMR spectra. Unfortunately, the starting pressure was close to the30 ata Reconstructed 8009001000 M H z
10 µm
Figure 11: Measured map of the splittings of one of the NV orientations (left). Near the top ofthe plot we observe a much stronger splitting compared to the bottom of the plot. Throughoutthe measurement, the shift in the pressure induced a shift in the dipole moment of the sample.We consider 3 different regions (seperated by horizontal lines) corresponding to 3 differentdipole strengths. The reconstructed map of the splittings is shown on the right in agreementwith the data. From the center and the spread of dipole strengths, we extract the dipole momentand its error. Black bar corresponds to µ m .transition pressure, and the drift in pressure led to a very large change in the pellet’s dipole mo-ment throughout the scanning measurement. This is clear in the measured data, Fig. 11, withthe top-half of the map displaying a significantly larger shift with respect to the bottom-half.To extract the drift in the dipole moment, we divide the two-dimensional map into threedifferent regions, each assumed to arise from a constant value of the dipole moment of thepellet. By fitting to three different dipole moments (given a fixed position, r X and r Y ) weobtain an estimate of the drift of the dipole moment that allows us to compute an errorbar of thatmeasurement. The estimated dipole moment at this pressure point is taken as the midpoint of thethree extracted values, D max + D min , while the error is estimated by the range, D max − D min . In this section we present additional data where we have allowed both the external magneticfield and the depth of the iron pellet to vary in the fitting procedure. The result of the fittingprocedure is summarized in Fig. 12. 31n particular, we expect the external magnetic field and the depth of the pellet to remainconstant at different pressures. Indeed, we observe this trend in the extracted parameters,Fig. 12(A,B). Using the mean and standard deviation, we estimate these values and their er-rors, quoted in Sec. 5.2. The final fitting procedure with these values fixed is presented in themain text.
We use a custom-built closed cycle cryostat (Attocube attoDRY800) to study the P - T phasediagram of Gd. The DAC is placed on the sample mount of the cryostat, which is incorporatedwith a heater and a temperature sensor for temperature control and readout.For this experiment, we use beryllium copper gaskets. The Gd sample is cut from a 25 µ mthick Gd foil (Alfa Aesar Stock No. 12397-FF) to a size of ∼ µ m × µ m and loaded withcesium iodide (CsI) as the pressure-transmitting medium. A single ruby microsphere loadedinto the chamber is used as a pressure scale.For each experimental run, we start with an initial pressure (applied at room temperature K) and cool the cell in the cryostat. Due to contraction of the DAC components withdecreasing temperature, each run of the experiment traverses a non-isobaric path in P - T phasespace, Fig. 14A. Using fiducial markers in the confocal scans of the sample chamber, we trackpoints near and far from the Gd sample throughout the measurement. By performing ODMRspectroscopy at these points for each temperature, we monitor the magnetic behavior of thesample. More specifically, comparing the spectra between the close point (probe) against thefar away one (control), Fig. 13, enables us to isolate the induced field from the Gd sample.32 Pressure (GPa) M a g n e t i c F i e l d ( G )
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Pressure (GPa) P o s i t i o n ( m ) AC B
Pressure (GPa) D i p o l e S t r e n g t h ( e m u ) -Iron -Iron CompressionDecompression B X B Y B Z B r X r Y r Z Figure 12: Result of fitting procedure when the external magnetic field and the depth of theiron pellet is allowed to vary at each pressure. ( A )[( B )] External magnetic field [position of thepellet] extracted as a function of pressure (circles correspond to compression while diamondscorrespond to decompression). Across the entire range of pressures, the extracted externalmagnetic field and the depth of the iron pellet is approximately constant. In the final fitting pro-cedure, these values are fixed to their extracted mean (dashed lines). Shaded regions correspondto a standard deviation above and below the mean value. ( C ) Dipole strength of the iron pel-let, extracted when all seven parameters ( B X , B Y , B Z , D, r X , r Y , r Z ) are fitted. The resultingtransitions occur at . GPa and . GPa for compression and decompression, respectively.Comparing with the width of the transition ( . GPa), these values are in excellent agreementwith those presented in the main text. 33 B ProbeControlrubyGd
Figure 13: ( A ) The protocol for obtaining P - T phase map of Gd relies on monitoring theODMR spectrum versus temperature and pressure at a point of interest (probe) near the sample.To verify that the observed signal is from the Gd flake, one can perform the same measurementon a control point further away from the sample. ( B ) The difference in the splitting between theprobe and control points isolates the magnetic field generated by the Gd sample, allowing us tomonitor the magnetic behavior of the sample. There are three different transitions we which to locate in the study of the Gd’s P - T phasediagram: a magnetic transition from PM dhcp to FM dchp; structural phase transitions, eitherhcp → dhcp or Sm-type ↔ dhcp; and a magnetic phase transition from PM Sm-type to AFMSm-type.In order to extract the transition temperature of the paramagnet to ferromagnet transitionfrom our data, we model the magnetization of our sample near the magnetic phase transitionusing a regularized mean field theory.The magnetism of gadolinium is well-described by a three dimensional Heisenberg magnetof core electrons ( ). In the presence of an external magnetic field, the free energy near thecritical point is expanded in even powers of the magnetization with a linear term that couples tothe external magnetic field: f = − Bm + α T − T C ) m + β m , (30)34here m is the magnetization, B is the external magnetic field, α and β the expansion coef-ficients, T the temperature, and T C the transition temperature. In this treatment, we implicitlyassume that α and β do not vary significantly with pressure and thus can be taken to be constantacross paths in P - T phase space. The magnetization m min is then obtained by minimizing thefree energy.Because our observation region extends far away from the transition, we observe a plateau-ing of the splittings that emerges from the microscopics of Gd. Using R as the regularizationscale and ˜ A as the maximum magnetization of the sample we propose the simple regularizationscheme: m ( T, P ) = ˜
A m min m min + R . (31)The splitting of the NV group, up to some offset, is proportional to the magnetization ofthe sample. This proportionality constant, A , captures he relation between magnetization andinduced magnetic field, the geometry of sample relative to the measurement spot, as well as thesusceptibility of the NV to the magnetic field. The splitting of the NV is then given by: ∆ = A m min m min + R + c (32)where we incorporated ˜ A into A as well. Normalizing α and β with respect to B , we obtain sixparameters that describe the magnetization profile, directly extracting T C .In the case of the first order structural phase transitions, similar to that of iron, we take thesusceptibility to follow a logistic distribution. We model the observed splitting as: ∆ = Ae B ( T − T C ) + 1 + c (33)Fitting to the functional form provides the transition temperature T C . Error bar is taken as largestbetween /B and the fitting error.In the case of the paramagnetic to antiferromagnetic transition, we use the mean field sus-ceptibility across the phase transition of the system. The susceptibility across such transition is35eaked at the transition temperature: χ ( T ) ∝ T − θ p T > T c C L ( H/T ) T − θ p L ( H/T ) T < T c (34)where C is chosen to ensure continuity of χ , L ( x ) is the derivative of the Langevin function L ( x ) at, H is a meaasure of the applied field, and θ p is the assymptotic Curie point. Finally, wefit the observed splitting to: ∆ = Aχ ( T ; T c , H, θ p ) + c (35)where, as before, A captures both the geometric effects, as well as the response of the chosenNV group to the magnetic field. In this section we present the data for the different paths taken in P - T phase and the resultingfits. Table 2 summarizes the observations for all experimental runs. Fig. 14 contains the dataused in determining the linear pressure dependence of the hcp phase. Fig. 15 comprises thedata used in determining the transition to the dhcp phase, either via the FM hcp to PM dhcptransition, Fig. 15B, or via the difference in susceptibilities between PM Sm-type and PM dhcpof Gd, Fig. 15C and D. We emphasize that in the blue path, we begin the experiment below GPa and thus in the hcp structure, while for the orange and green, we begin above GPa, sowe expect the system to be in Sm-type. Finally, Fig. 16 contains the data where we observe achange in the susceptibility of Gd that occurs at the purported Sm-type PM to AFM transition. P - T phase diagram of Gd The rich magnetic behavior of Gd is partially dependent on its structural phases, captured inthe sequence: hexagonal closed packed (hcp) to Samarium (Sm) type at ∼ GPa, and then to36
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Figure 14: (A)
Paths in the P - T phase space that inform about the hcp PM phase to thehcp FM phase. (B-O) Measured NV splitting and corresponding fit. The resulting transitiontemperatures are highlighted in (A) with squares. Shaded region corresponds to the part of thespectrum fitted. 37
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Figure 15: (A)
Paths in the P - T phase space that inform about the transition to the PM dhcpphase. (B-D) Measured NV splitting and corresponding fit. The resulting transition tempera-tures are highlighted in (A) with squares. We interpret (B) as a transition from FM hcp to PMdhcp, while (C),(D) as a transition from PM Sm-type to PM dhcp. Shaded region correspondsto the part of the spectrum fitted.
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A B
Pressure (GPa) T e m p e r a t u r e ( K ) Figure 16: (A)
Path in the P - T phase space where a signal consistent with the purported AFMtransition in Sm-type Gd is seen (B) . Shaded region corresponds to the part of the spectrumfitted. 38un Direction Phase transition Remarks, visible in Fig.1 Heat-up hcp (FM) −→ hcp (PM) New sample, Fig. 14B2 Cool-down hcp (PM) −→ hcp (FM) Fig. 14C3 Cool-down hcp (PM) −→ hcp (FM) Fig. 14D4 Cool-down No observation Probably starting in Sm dueto large initial pressure5 Cool-down hcp (PM) −→ hcp (FM) New sample, Fig. 14E6 Heat-up hcp (FM) −→ hcp (PM) Fig. 14F7 Cool-down hcp (PM) −→ hcp (FM) Fig. 14G8 Heat-up hcp (FM) −→ hcp (PM) Fig. 14H9 Cool-down hcp (PM) −→ hcp (FM) Fig. 14I, 15B −→ dhcp (PM)10 Cool-down Weak evidence for Probably starting in Sm dueSm (PM) −→ Sm (AFM) to metastability, Fig. 16B11 Cool-down hcp (PM) −→ hcp (FM) New sample, Fig. 14J12 Heat-up hcp (FM) −→ hcp (PM) Fig. 14K13 Cool-down hcp (PM) −→ hcp (FM) Fig. 14L14 Cool-down Weak evidence for Probably starting in Sm dueSm (PM) −→ dhcp (PM) to large initial pressure15 Cool-down Weak evidence for Probably starting in Sm dueSm (PM) −→ dhcp (PM) to metastability, Fig. 15C16 Heat-up Weak evidence for Fig. 15Ddhcp (PM) −→ Sm (PM)17 Cool-down hcp (PM) −→ hcp (FM) New sample, Fig. 14M18 Heat-up hcp (FM) −→ hcp (PM) Fig. 14N19 Cool-down hcp (PM) −→ hcp (FM) Fig. 14Oand start of transition to dhcp (PM)Table 2: Summary of all experimental runs in the P - T phase diagram, indexing either a decreaseor increase in temperature during this path, and the observed phase transitions. Each group ofruns, between double lines in the table, corresponds to a different sample.39ouble hexagonal closed packed (dhcp) at ∼ GPa. In particular, while the paramagnetic (PM)phase of hcp orders to a ferromagnet (FM), the PM phase of Sm-type orders to an antiferromag-net (AFM) ( ). Similarly, dhcp undergoes a PM to magnetically ordered phase transition.For experimental runs with initial pressures < GPa (runs 1-3, 5-9, 11-13, 17-19), weobserve a PM ↔ FM phase transition in hcp Gd. In agreement with previous studies, wesee a linear decrease of the Curie temperature with increasing pressure up to ∼ GPa ( ). Notably, prior studies have shown a structural transition from hcp to Sm-type at GPa(
25, 27, 28 ), which is believed to be “sluggish” (
23, 25 ). This is indeed consistent with ourobservation that the linear dependence of the Curie temperature persists well into the Sm-typeregion, suggesting the existence of both structural phases over our experimental timescales.Furthermore, in run 9 (Table 2 and Fig. 16A,B), we observe a complete loss of FM signalwhen pressures exceed ∼ GPa at ∼ K, in good agreement with the previously reportedphase transition from hcp (FM) to dhcp (PM) structure ( ). Upon performing a similar pathin P - T space (run 19), we observe the same behavior. In contrast to the previous slow hcp toSm-type transition, we believe that the equilibrium timescale for the hcp (FM) to dhcp (PM)transition is much faster at this temperature.After entering the dhcp structure (run 9), we no longer observe a clear FM signal from thesample even after heating to K and depressurizing < . GPa. This can be explained bythe retention of dhcp or Sm-type structure in the sample. Previous studies, suggesting that theSm-type phase in Gd is metastable up to ambient pressure and temperature ( ), corroboratethat our sample is likely still in the Sm-type structural phase. It is not too surprising, that bycontinuing to cool down and walking along a slightly different P - T path, we observe only asmall change in the NV splitting at ∼ K and ∼ GPa as we cross the purported Sm-typePM to AFM phase boundary (run 10 in Table 2) (
23, 25, 27 ).Moreover, the metastable dynamics of hcp to Sm-type transitions are strongly pressure and40emperature dependent, suggesting that different starting points (in the P - T phase diagram) canlead to dramatically different behaviors. Indeed, by preparing the sample above 2 GPa at roomtemperature (run 4), we no longer detect evidence for a ferromagnetic Curie transition, hintingthe transition to the Sm-type structure. Instead, we only observe a small change in the NV split-ting at ∼ GPa and ∼ K, which could be related to the presence of different paramagneticsusceptibilities of the Sm-type and dhcp structural phases. Interestingly, by cycling temperatureacross the transition (run 14-16 in Table 2), we observe negligible hysteresis, suggesting fastequilibration of this structural transition.
In order to perform magnetic noise spectroscopy of Gd at temperatures ranging from 273 K to340 K, we attach a small chunk of Gd foil (100 µ m × µ m × µ m) close to a microwavewire on a Peltier element with which we tune the temperature. Instead of mm-scale diamondsas before, we use nano-diamonds ( Adamas , ∼
140 nm average diameter) drop-cast onto the Gdfoil to minimize the distance to the surface of our sample.With no external field applied, all eight resonances of the NVs inside the nano-diamonds arefound within our resolution to be at the zero-field splitting D gs for either para- and ferromag-netic phase of Gd, leading to a larger resonance contrast since we can drive all NVs with thesame microwave frequency. Measuring the NV’s spin relaxation time T under these circum-stances is equivalent to ascertaining the AC magnetic noise at ∼ T . First, we applya 10 µ s laser pulse to intialize the spin into the | m s = 0 i state. After laser pumping, we letthe spin state relax for a variable time τ , before turning on a second laser pulse to detect thespin state (signal bright). We repeat the exact same sequence once more, but right before spindetection, an additional NV π -pulse is applied to swap the | m s = 0 i and | m s = ± i populations41 Contrast
T i m e ( m s ) Figure 17: Plots of T measurements below and above the magnetic phase transition in Gd.The green (orange) curve was measured at 320 K (276 K) and yields T = 91 ± µ s (66 ± µ s),indicating a clear reduction of the spin polarization lifetime in the ferromagnetic phase. Astretched exponential function with exponent α = 0.6 (0.65) was used for fitting.(signal dark). The difference between signal bright and dark gives us a reliable measurement ofthe NV polarization (Fig. 4D top inset in main text) after time τ . The resulting T curve exhibitsa stretched exponential decay ∝ e − ( τ/T ) α , with α ∼ . (Fig. 17).By sweeping the Peltier current over a range of ∼ K to
K, therefore determining the temperature dependence of T .This procedure is performed on two different nano-diamonds on top of the Gd flake toconfirm that the signal is not an artifact. Furthermore, this is contrasted with an additional mea-surement at a nano-diamond far away from the Gd foil, exhibiting no temperature dependenceof T . 42 Contrast
T i m e ( m s ) Figure 18: Plots of T measurements away from the Gd flake at 315 K (grey curve) and 286 K(red curve). The resulting spin polarization lifetimes T = 243 ± µ s (315 K) and 247 ± µ s(286 K) are the identical within the errorbar. T The depolarization time T of NV centers shows a distinct drop when we decrease the temper-ature T to across the ferromagnetic phase transition of Gd, Fig. 4D of the main text. Assumingthat Johnson noise is the main contribution, because we are working at a fixed small transitionfrequency ( ω ∼ . GHz) and in the thermal limit ( ~ ω (cid:28) k B T ), we can consider the DClimit. In this case, we have T ∝ ρ ( T ) /T , where ρ ( T ) = 1 /σ ( T ) is the DC resistivity ( ).Importantly, previously measurements of the resistivity curve for Gd show a kink at T C , witha sharper temperature dependence below T C (
30, 31 ). However, this sudden change in slope isinsufficient to explain our observations of T ; in particular, given the magnitude of the resistiv-ity, the change in temperature dominates the T behavior. This implies that T should increasein the ferromagnetic phase if the sole contribution is bulk Johnson noise, whereas observations43ndicate otherwise.A hint to the resolution of this puzzle comes from two observations. First, NV centers drop-cast onto Gd samples are very close to the sample, and hence far more sensitive to the surfacethan the bulk. Second, the surface of Gd is well known to show a higher ferromagnetic transitiontemperature than the bulk; the drop in T starts at a larger temperature ( ≈ K) compared tothe bulk T C ≈ K. These observations strongly suggest that the NV is detecting a large dropof surface resisitivity as we lower T across the surface critical temperature, and this dominatesover the small drop of bulk resistivity in the observed behavior.In order to quantitatively estimate the relative contribution of the surface to the bulk, wewrite down, following Ref. ( ), the contribution to the noise for a single two-dimensionallayer at a distance z from the probe for a sample with conductivity σ ( T )1 T ∝ N ( ω ) = k B T µ σ ( T )16 πz . (36)Here we have assumed that the optical conductivity has a smooth dc limit (true for typicalmetals) and taken the extreme thermal limit to neglect the small frequency dependence of σ . Gdhas a hcp structure with c ≈ a , so we approximate the sample as being composed of decoupledtwo-dimensional layers and add their individual contributions to the noise. If the distance fromthe surface to the probe is d , the surface thickness is D (infinite bulk thickness), and the surfaceand bulk conductivity are denoted by σ s and σ b respectively, then we have: T ∝ T (cid:20)Z d + Dd dz σ s ( T ) z + Z ∞ d + D dz σ b ( T ) z (cid:21) = T σ s ( T ) (cid:18) d − d + D (cid:19) + T σ b ( T ) d + D . (37)Eq. (37) makes it explicit that when
D/d is an O (1) number (i.e. the surface thickness is ofthe order of sample-probe distance) the surface and bulk contributions are comparable. On theother hand, if D/d (cid:28) , the bulk noise dominates. For our drop-cast nano-diamonds on thesurface of Gd, we can estimate D ≈ nm, given the distinct surface signatures in the densityof states even 6 layers deep ( ). We also estimate the average distance as approximately half44he radius of a nano-diamond, d ≈ nm. Therefore, we see that, for our samples, a largerise in surface conductivity can cause a significant increase in magnetic noise, even if the bulkconductivity remains roughly constant across the transition to the ferromagnetic phase. Hence,we conjecture that an enhanced surface conductivity below the surface critical temperature T c,s is responsible for the observed drop in T .The sharp drop of surface resisitivity below the surface ordering temperature can be due toseveral reasons. It can be caused by the critical behavior of surface magnetism, or a differentelectron-magnon coupling on the surface because the surface electrons have more localizedwave-functions. Here, we provide one consistent picture for the drop in surface resisitivity interms of a distinct surface criticality relative to the bulk.From Ref. ( ) we know that both the bulk residual resistivity and the phonon con-tribution to the resistivity is quite small, and electron scattering below the bulk T C is domi-nated by magnetic excitations. Since T C = 292 K is much larger than the Debye temperature Θ D ≈ K (
31, 33 ), the phonon contribution to scattering is expected to be linear in T near T C . Above T C , the slope dρ/dT for Gd is very small. Hence the majority of scatteringbelow T C takes place due to magnetic correlations, which, below T C , changes resistivity by dρ/dT ∝ t β − where t = | T C − T | /T C ( ). β can be significantly different from 1, leading toa cusp in ρ ( T ) at T C . For the bulk, we can write: ρ b ( T ) = ρ b ( T C ) − α ph (cid:18) T C − TT C (cid:19) − α mag (cid:18) T C − TT C (cid:19) β Θ( T C − T ) (38)Above T C , the singularity in dρ/dT is of the form t − α . However, for both Heisenberg andIsing universality classes of ferromagnetic transitions, α is close to zero ( α ≈ − . ), and thesurface enhancement of the surface density of states is negligible. Therefore, for T > T C we assume that the surface conductivity is identical to the bulk conductivity. Moreover, thescattering from uncorrelated core-spins should be constant at high temperatures away from T C ,45o the slope dρ/dT is entirely from phonons for T (cid:29) T C . Using this relation, we can estimate α ph ≈ µ Ωcm using the data for T between and
K ( ). Using the data for ρ at T = 280 K in Ref. ( ) to extract α mag and β ≈ . for the three dimensional Heisenbergmodel, which is believed to describe quite well the ordering of local moments in Gd ( ), weobtain α mag : ρ b ( T ) − ρ b ( T C ) = − µ Ωcm = − α ph (cid:18) (cid:19) − α mag (cid:18) (cid:19) . = ⇒ α mag ≈ µ Ωcm (39)This gives the bulk resistivity as a function of temperature, but it does not replicate the exper-imental observations, purple line in Fig. 19. We now postulate a similar critical behavior atthe surface but with surface critical exponent β s for the magnetization. On a two-dimensionalsurface, the Mermin-Wagner theorem forbids the spontaneous breaking of a continuous spin-rotation symmetry at a non-zero temperature ( ). For a surface ferromagnetic phase transition,we must have theory with reduced symmetry. Given the easy axis anisotropy in Gd (
22, 30 ), thesurface magnetic phase transition is plausibly in the Ising universality class, with β s = 0 . ( ). Therefore, on the surface, we have: ρ s ( T ) = ρ s ( T c,s ) − α ph,s (cid:18) T c,s − TT c,s (cid:19) − α mag,s (cid:18) T c,s − TT c,s (cid:19) . Θ( T c,s − T ) (40)In absence of evidence otherwise, we take α ph,s = α ph (same value as in the bulk). However, α mag,s can be significantly enhanced relative to the bulk value. This can be due to severalreasons. The surface electrons can be more localized than the bulk, therefore increasing theelectron core-spin coupling. Further, the surface local moments can have a larger net spin S relative to the bulk which orders more slowly. Since the electron-spin scattering cross-sectionis proportional to S ( S + 1) ( ), a fully polarized core 4f state with S = 7 / will have a largerscattering rate with an itinerant electron compared to a partially polarized state with S < / .The exact value of α mag,s thus depends on delicate surface physics; here we treat it as a freeparameter. Fig. 19 shows a good fit to our data with the estimates α mag,s = 7 α mag ≈ µ Ωcm ,46 �� ��� ��� ��� ��� ������������������� � ( � ) � � ( μ � ) Figure 19: The purple curve shows T taking only the bulk contribution to Johnson noise intoaccount. The red curve shows T taking both surface and bulks contribution into account, with T C = 292 K and T c,s = 302 K. The blue dots are experimental data.surface thickness D = 10 nm ≈ c , and sample-probe distance d = 50 nm (we have used anoverall proportionality factor for the fit).We note that spin-fluctuations in Gd can also cause cause the NV polarization to relax. Al-though such fluctuations are negligible in the paramagnetic phase as our sample-probe distanceis much larger than the lattice spacing ( ), gapless critical fluctuations and spin-wave modescan indeed have a larger contribution to magnetic noise. However, the magnon contribution isrelated to magnon occupancies and decreases with decreasing temperature ( ), implying that T should increase as one lowers temperature in the ferromagnetic phase. This is inconsistentwith the behavior we observe. Bulk critical spin-fluctuations should make the largest contribu-tion at T C , which is also not observed. An even more involved theoretical analysis is requiredto rule out critical surface spin-fluctuations. This analysis is left for future work. References
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