Abstract
Unusual behavior in quantum materials commonly arises from their effective lowdimensional physics, reflecting the underlying anisotropy in the spin and charge degrees of freedom. Here we introduce the magnetotropic coefficient k = ∂^{2}F/∂θ^{2}, the second derivative of the free energy F with respect to the magnetic field orientation θ in the crystal. We show that the magnetotropic coefficient can be quantitatively determined from a shift in the resonant frequency of a commercially available atomic force microscopy cantilever under magnetic field. This detection method enables part per 100 million sensitivity and the ability to measure magnetic anisotropy in nanogramscale samples, as demonstrated on the Weyl semimetal NbP. Measurement of the magnetotropic coefficient in the spinliquid candidate RuCl_{3} highlights its sensitivity to anisotropic phase transitions and allows a quantitative comparison to other thermodynamic coefficients via the Ehrenfest relations.
Introduction
Correlated quantum materials governed by strong electronic interactions commonly host a variety of competing and coexisting electronic phases, such as the copper and ironbased highT_{c} superconductors where charge ordering, hightemperature superconductivity, and magnetism occur in close proximity^{1}. Mapping the associated phase diagram is a critical first step to understanding their physics. These phases are commonly characterized by anisotropic behavior that reflects the microscopic anisotropy in the spin and charge degrees of freedom. Prominent examples include anisotropy in the magnetic susceptibility of the cuprates^{2,3,4}, the identification of hiddenorder phases in URu_{2}Si_{2} and SmB_{6}^{5,6} and the electronic nematicity of the ironbased superconductors^{7,8}. While anisotropy is an essential ingredient for the complex phases that emerge in quantum materials, its experimental signatures can be subtle.
An established and highly sensitive technique to directly probe small anisotropies in correlated metals and exotic magnets is torque magnetometry^{9,10,11}. When a sample with an anisotropic magnetization M is placed in an external magnetic field B, it experiences a torque τ = M × B. This torque can be measured with high accuracy by mounting a crystal onto a cantilever^{12,13,14,15,16,17}. Because the overall susceptibility is small, we assume that the local field B is approximately equal to the applied field H, and use B throughout.
Both the magnetic torque τ = ∂F/∂θ and the magnetization M = −∂F/∂B are first derivatives of the free energy F, and thus these thermodynamic potentials provide sensitive and essential information at phase transitions (Fig. 1a). Second derivatives of the free energy, however, such as the heat capacity C = −T∂^{2}F/∂T^{2}, the magnetic susceptibility χ = −∂^{2}F/∂B^{2}, and the elastic moduli \(c_{ijkl} = \partial ^2F{\mathrm{/}}\partial \epsilon _{ij}\partial \epsilon _{kl}\) often provide more fundamental insights into a material. These quantities can be directly related to physical properties, such as the density of states, and are the essential quantities to formulate microscopic theories. Unlike first derivatives, they exhibit discontinuities at secondorder phase transitions and their magnitudes can be related to one another through the Ehrenfest relations^{18}.
Here, we develop a technique to measure the curvature of a sample’s free energy with respect to magnetic field orientation k = ∂^{2}F/∂θ^{2}—the thermodynamic coefficient directly linked to magnetic anisotropy. We name this the magnetotropic coefficient as it describes rigidity with respect to rotation in a magnetic field. With the sample mounted onto a resonating cantilever, the magnetotropic response acts to reduce or enhance stiffness of the total system, leading to a shift in the resonant frequency (Fig. 1c). This is in contrast with the magnetic torque, which bends the cantilever to a new equilibrium position but does not lead to a frequency shift. The measured frequency is highly sensitive to the magnetic anisotropy of a sample, which is the basis of resonant torsion magnetometry^{19,20}. We demonstrate the sensitivity of resonant torsion magnetometry and highlight the importance of its thermodynamic properties on two materials, one with charge and the other with spindominated anisotropy. We measure quantum oscillations in the Weyl semimetal NbP^{21,22} and the antiferromagnetic phase boundary of the spinliquid candidate RuCl_{3}^{23,24,25,26,27,28,29}.
Results
Kinematics of resonant torsion
To detect the magneotropic coefficient, we use the Akiyama Probe (AProbe)—a selfoscillating and selfsensing cantilever designed for scanning probe microscopy^{30}. The Aprobe is made of two separate resonators: a silicon Ushaped cantilever and a quartz tuning fork (see Fig. 1b and Methods)^{30}. The benefits of repurposing the Aprobe for resonant torsion magnetometry are threefold: the relatively large spring constant of the silicon cantilever (5 N m^{−1}^{30}) allows us to extend ultrasensitive and dynamic cantilever magnetometry^{31,32,33,34,35,36} to macroscopic sample sizes; placement of the sample on the silicon cantilever (rather than one leg of a quartz tuning fork) eliminates complications that arise from the center of mass motion of the tuning fork coupling to the resonance mode^{37,38}; and electrical readout of the Aprobe eliminates the need for optical detection of the resonant frequency, making setup relatively straightforward and more robust compared to previous approaches.
To elucidate the physical distinction between the magnetic torque and the magnetotropic coefficient and to describe the measurement, we briefly review the energetics of the resonating sample. In the harmonic approximation, the energy of a cantilever with effective stiffness K, moment of inertia I (see Methods) and an attached sample can be written as
where Δθ describes the angle of rotation of the sample with respect to a fixed magnetic field. Note that this is opposite in sign to θ discussed elsewhere, which describes rotation of the magnetic field with respect to fixed crystal axes.
The first two terms describe the kinetic and potential energies of the bare cantilever and together determine the base oscillation frequency, \(\omega _0^2 = K{\mathrm{/}}I\). We parameterize the motion of the lever as it vibrates by an angle Δθ at the tip of the lever where the sample is mounted (Fig. 1c). The last two terms in Eq. (1) describe the anisotropic energy of the measured sample in the applied magnetic field. Both the torque and the magnetotropic coefficient appear as coefficients in a Taylor expansion of the free energy F(θ, B), and they manifest themselves in distinct physical responses of the sample. The torque shifts the equilibrium angle about which the lever oscillates to Δθ_{τ} = τ/(K + k) (Fig. 1c). The magnetotropic coefficient encodes the curvature of the free energy with respect to the rotation angle, and appears as a shift in the oscillation frequency (ω_{0} + Δω)^{2} = (K + k)/I. For small frequency shifts, this can be expanded as
Therefore, k can be directly determined by a measurement of the resonant frequency of the cantilever.
Linear magnetic response
In general, the functional form of the magnetization M(B) can include terms other than those linear in magnetic field. These are common in magnetic materials, even at very low fields. Therefore, the torque τ = M(B) × B, and subsequently the magnetotropic coefficient k = ∂τ/∂θ, can also carry a more complex form. We first focus on the simple case of the linear response regime (M_{i} = χ_{ij}B_{j}), however, to illustrate the different behaviors of τ and k. Here, the free energy F(θ, B) = (1/4)(χ_{j} − χ_{i})B^{2} cos2θ requires the angle dependences of the torque τ ∝ sin 2θ and the magnetotropic coefficient
to strongly differ. Here, θ is defined as the angle of rotation of magnetic field with respect to the ith crystal direction in a righthanded spherical coordinate system.
We observe the expected angle dependence for the magnetotropic coefficient in a resonant torsion measurement of RuCl_{3} at low fields within the linear regime (Fig. 2). Importantly, the signal of resonant torsion is maximal for fields along the axes of symmetry, a disadvantageous field orientation for conventional torque measurements because the signal goes to zero. Even in the vicinity of these directions (gray lines in Fig. 2), the torque is subject to an undesirable interaction effect (see methods), which contributes minimally to the magnetotropic coefficient. While the magnetic torque and the magnetotropic coefficient are simply related to each other in the linear magnetic regime, we later capture the nonlinear response in RuCl_{3} at higher magnetic fields and show that it conveys new information about the magnetic anisotropy, different from the magnetic torque.
High sensitivity de Haasvan Alphen
In order to demonstrate the sensitivity of the technique, we measure quantum oscillations in the Weyl semimetal NbP^{21,22} up to 3 T (Fig. 3). This semimetal is nonmagnetic, and its entire magnetic response at low fields is due to the weak Landau diamagnetism of the conduction electrons, as well as the Berry paramagnetism arising from its nontrivial topology^{39,40}. With the magnetic field applied along the crystallographic c axis, where the magnetic torque signal is zero, we can resolve quantum oscillations in fields well below 1 T. The quantum oscillation frequencies for this field orientation agree with those reported in the literature^{21,22}. With a characteristic response bandwidth of 1 Hz, the smallest detectable frequency shift is Δf/f = 6 × 10^{−9} = Δk/K, where K is the effective bending stiffness of the lever (see Methods). With K = 180 nJ rad^{−2}, the smallest detectable magnetotropic coefficient is Δk = 1.1 × 10^{−15} J rad^{−2}, equivalent to 1.2 × 10^{8} μ_{B} at 1 T. This can be used to estimate the required mass of a metallic crystal that can be investigated with resonant torsion magnetometry. Even in only weakly anisotropic metals (1% anisotropy), which would contribute 0.01 μ_{B} per formula unit, only 10^{12} formula units are needed to resolve a signal at the demonstrated sensitivity. For a 3 Å unit cell size, this corresponds to a 3 μm^{3} sample size or a sample weight of 0.1 ng for a sample density of 5 g cm^{−3}. Resonant torsion magnetometry is thus ideally suited to investigate anisotropy when only the smallest samples exist in single crystal form.
Thermodynamics and phase transitions
The magnetotropic coefficient can provide valuable insight into the thermodynamics of phase transitions via the Ehrenfest relations. k can be more formally defined as a member of a matrix of second derviatives (thermodynamic coefficients) of the free energy when temperature T, volume V, magnetic field B, and magnetic field orientation θ are independent variables. The relation of k to other thermodynamic coefficients is directly apparent from the behavior of the thermodynamic potential in the T, V, B, and θ variables
We can derive the Ehrenfest relation that relates a discontinuous jump in the resonant torsion to other thermodynamic coefficients. If we assume that T_{c}(θ)_{V,B} is the boundary of a secondorder phase transition induced by the magnetic field angle measured at a fixed volume V and magnetic field B, then continuity of all first derivatives (S, P, M, τ) across such a boundary, ΔS = 0 and Δτ = 0, requires that discontinuous jumps in the three thermodynamic coefficients C, (∂S/∂θ) = −(∂τ/∂T), and k are all related to each other:
Here, ΔX indicates the jump of X across the phase boundary and dT* and dθ* are short segments along the phase boundary in the T − θ phase plane, such that dT*/dθ* = (∂T_{c}/∂θ)_{B}. The Ehrenfest relation connecting the jump in the magnetotropic coefficient Δk and the jump in the heat capacity ΔC is
where the derivative is to be taken along the phase boundary at fixed magnetic field. Similarly, Ehrenfest relations between the jumps in k, χ, and C give \({\mathrm{\Delta }}k =  {\mathrm{\Delta }}\chi \left( {\partial B_{\mathrm{c}}{\mathrm{/}}\partial \theta } \right)_T^2\) and \({\mathrm{\Delta }}\chi = \left( {{\mathrm{\Delta }}C{\mathrm{/}}T_{\mathrm{c}}} \right)\left( {\partial T_{\mathrm{c}}{\mathrm{/}}\partial B} \right)_\theta ^2\), where the derivatives in the two relations must be taken along the phase boundary at fixed temperature and at a fixed field orientation, respectively.
In order to verify experimentally these thermodynamic relations, we refer again to RuCl_{3}, an effective spin1/2 quantum magnet that orders antiferromagnetically at T_{N} = 7 K^{26}. Below this temperature, longrange order can be suppressed with a magnetic field of ~8 T for fields applied within the honeycomb planes^{28,29}, with recent evidence suggesting a spinliquid state at higher magnetic fields^{23}. We measure RuCl_{3} at T = 1.3 K—well within the antiferromagnetically ordered state^{27,29}—to observe the evolution of the magnetic torque and the resonant torsion as we cross the secondorder phase boundary with increasing magnetic field (Fig. 4). The torque (Fig. 4a) is inferred from the piezoresistively detected bending amplitude of a Seiko Instruments cantilever^{14}. With small fields at an angle ~10° away from the honeycomb planes, both τ and k respond quadratically to the applied magnetic field. For this field orientation, we observe the suppression of longrange order at ~9 T. Across the phase boundary (gray line in Fig. 4), τ shows a break in slope crossing over to linear behavior at higher magnetic fields, whereas k experiences a discontinuous jump. Akin to the advantages of techniques sensing the magnetic susceptibility compared to magnetization, detecting k offers a more appropriate means for identifying magnetic phase transitions.
The experimentally observed jump Δk ≈ 6 J rad^{−2} mol^{−1} in this configuration (Fig. 4b) can be directly compared to heat capacity measurements under magnetic field. (∂T_{c}/∂θ)_{B} can be estimated from the angle dependence of the resonant torsion of RuCl_{3} at fixed temperatures. One such scan at T = 1.3 K and B = 17.5 T shows a pronounced anomaly at the phase boundary of the longrange ordered state (blue vertical line in Fig. 5). Entry into the ordered state is marked by a jump down at the phase boundary, as required by Eq. (6). Similar measurements at various fixed magnetic fields allow to map out the phase boundary of the antiferromagnetically ordered state (Fig. 5b). The derivative (∂T_{c}/∂θ)_{B} = (∂T_{c}/∂B)_{θ} (∂B_{c}/∂θ)_{T} at T = 1.3 K, B = 10 T, and θ = 102° can be estimated as (∂B_{c}/∂θ)_{T} ≈ 2.8 T rad^{−1} and (∂T_{c}/∂B)_{θ} ≈ 25 K T^{−1}. The heat capacity jump at the antiferromagnetic transition at T = 1.3 K has been reported as ΔC/T_{c} ~ 1.7 mJ mol^{−1} K^{−2}^{28}. Thus, the righthand side of Eq. (6) gives ~8 J rad^{−2} mol^{−1}, in agreement with the size of the measured jump Δk of 6 J rad^{−2} mol^{−1} found above. This quantitative agreement is remarkable, especially given the uncertainties of the derivatives due to the complex shape of the phase boundary.
Discussion
The magnetotropic coefficient provides valuable thermodynamic information and complements the magnetic torque. The direct measurement of k highlights secondorder phase transitions by discontinuous jumps that can be related to anomalies in other thermodynamic measurements, such as the heat capacity. Resonant torsion allows direct access to the magnetic anisotropy when the magnetic field is aligned along the principal magnetic axes—a blindspot for conventional torque magnetometry. Finally, the ability to measure shifts in the resonant frequency of lever vibrations much more precisely than the amplitude of lever deflections results in better than part per 100 million sensitivity and the opportunity to measure subnanogram samples.
Methods
Characteristics of the cantilever
The Aprobe, originally designed for atomic force microscopy (AFM), consists of a piezoelectic quartz tuning fork, with a silicon cantilever glued to the ends (Fig. 6). Electrical grounding of the silicon tip can be made via the blob of silver epoxy on one contact (Fig. 6a). Each tuning fork leg is 2400 μm long and the crosssectional area is 124 × 214 μm^{2} (Fig. 6b). The silicon cantilever is l = 310 μm long, w = 30 μm wide (Fig. 6a), and 3.7 μm thick^{30}. We estimate the mass of the lever to be 100 ng, much smaller than the mass of the tuning fork.
In Eq. (1) in the main text, the potential and kinetic energy of the vibrating cantilever is described with a bending stiffness K and a moment of inertia I. The bending stiffness K is the coefficient of elastic energy stored in the silicon cantilever when the tip of the lever is bent by an angle Δθ. The stored elastic energy, therefore, depends not only on the geometry (width, length, etc.) of the lever, but also on the shape of the resonance mode.
In the main text, we calibrate the shift in frequency using known values of the anisotropic susceptibility in the linear response regime, and its connection to the magnetotropic coefficient k = (χ_{c} − χ_{a})B^{2} cos(2θ). Alternatively, we can estimate the magnitude of the magnetotropic coefficient k for RuCl_{3} using Eq. (2) in the main text. First, we need to take into account the shape of the bending mode of the cantilever to determine the bending stiffness K.
The bending shape of the silicon cantilever is described by ζ(z, t), where z is the distance along the lever from the point of attachment and ζ is the vertical displacement of the lever^{41}. At resonance, the motion of the lever is described by ζ(z, t) = ζ^{(n)}(z)sin(2πf^{(n)}t), where ζ^{(n)}(z) is the shape of the nth mode and f^{(n)} is its frequency. The shape of the lever is found from elastic equations derived from an energy functional E = \((1{\mathrm{/}}2)\rho A{\int}_{{\rm{d}}z} ({\rm{d}}\zeta (z,t){\mathrm{/}}{\rm{d}}t)^2\) + \((1{\mathrm{/}}2)YI_c{\int}_{{\rm{d}}z} \left( {{\rm{d}}^2\zeta (z,t){\mathrm{/}}{\rm{d}}z^2} \right)^2\), where the first term is the kinetic energy (ρ is the density and A is the crosssectional area) and the second term is the potential energy (Y is Young’s modulus and I_{c} is the moment of inertia of the lever’s crosssection with respect to its center of mass^{41}). The second derivative d^{2}ζ(z, t)/dz^{2} in the elastic energy represents the inverse radius of local curvature. Evaluating the second term for the fundamental vibration mode ζ^{(0)} (normalized as dζ(z)/dzz = tip of the lever = Δθ(t)) gives K^{(0)} = 1.63(YI_{c}/L), which for the silicon lever evaluates to K^{(0)} = 180 nJ rad^{−2}. Similarly, evaluating the first term results in the moment of inertia for the same mode as I^{(0)} = 0.13ρAL^{3} = 0.19 × 10^{−17} J Hz^{−2}.
For RuCl_{3}, calibration of the magnetotropic coefficient using the linear response regime k = (χ_{c} − χ_{a})B^{2} cos(2θ) yields 1 Hz = 0.321 J rad^{−2} mol^{−1}. From the dimensions, we estimate the sample mass to be ~20 ng, which corresponds to 25 picomol per unit cell of RuCl_{3} (where the unit cell contains 4 formula units). This gives 8 pJ rad^{−2} for the magnetotropic coefficient k of the sample. Using K^{(0)} = 180 nJ rad^{−2} in Eq. (2), the righthand side evaluates to 2.2 × 10^{−5}, which is close to the expectation on the lefthand side for a 1 Hz shift in frequency.
The two gold contacts wrap around each leg to create quadrupolar electric field lines (Fig. 6c) when a voltage is applied. In resonant torsion magnetometry, the gold contacts are effectively used as a driver and a pickup. Applying a voltage induces motion due to the piezoelectric properties of the quartz. The large mechanical motion of the silicon cantilever on resonance drives a piezoelectric current that is detected in our measurement. In addition to this piezoelectric current, a background current is present at all frequencies due to the parasitic capacitance of the wires connecting the cantilever in the cryostat to the room temperature electronics.
The pickup voltage near a resonance has a standard Lorentzian shape, V = V_{BG} + A/[ω − ω_{0} + iΓ/2], where V_{BG} is a background voltage due to the background current discussed above. ω_{0} and Γ are the resonant frequency and linewidth, respectively (Fig. 7b). In the complex plane, plotting the imaginary versus real parts of the Lorentzian traces a circle. Any anharmonic deviation from Eq. (1) of the main text leads to a distortion of this circle. We find that the response of the Akiyama probe deviates from a circle, signaling the nonlinear response regime of the lever, when driven with oscillating voltages in excess of 100 mV (V_{osc} in Fig. 8). We also checked optically that a driving voltage of 1 V leads to a displacement of the lever of about 2 degrees. Our typical driving voltage of 10 mV therefore is accompanied by much smaller angular displacements, ensuring the validity of Eq. (1).
Tracking the resonant frequency
Our measurement requires a method for following the resonant frequency as a function of temperature, magnetic field, and magnetic field orientation. Typically, this can be achieved with a phaselocked loop (PLL) controlled lockin amplifier. We used the readily available (PLL/PID) option of the Zurich Instruments midfrequency lockin (MFLI) amplifier for fast and sensitive response to shifts in the resonant frequency as a function of magnetic field only. As a function of temperature and in some situations, such as measurement across a sharp phase transition, we use our custom softwareimplemented PLL. This allows us to obtain additional information at frequencies near the resonance, but it significantly slows down the measurement speed. Below we explain a critical limitation with the hardwareimplemented PLL due to the background impedance detected in our measurement. We discuss how we overcome this with a capacitance compensation circuit (Fig. 8) when using the hardwareimplemented PLL, and how this is resolved in software when frequency scans through the resonance are necessary.
For robust tracking, the hardware PLL requires a large phase swing across the resonant frequency. The largest phase swing of 360° is obtained when the reference point for the phase is inside of the Lorentzian circle (Z_{1} in Fig. 7). The PLL implementation of the Zurich MFLI calculates the phase with respect to zero voltage only. Therefore, to achieve a large phase swing across the resonance, the circle of the Lorentzian must be in the vicinity of zero voltage in the complex plane. In the actual measurement, this circle is shifted away from zero (blue curve in Fig. 7) due to the finite background voltage arising from parallel capacitance in the cables (outside and inside of the measurement probe). The circle diameter, which is proportional to the amplitude of the drive voltage and inversely proportional to the linewidth of the resonance, is typically 10–100 μV for a resonance in air with a qfactor of ~2000. In vacuum, the qfactor is in excess of 10,000 and the circle diameter increases above 100 μV, leading to a larger phase swing for the same given background voltage.
There are three different situations for the phase swing on resonance. First, the zero voltage is inside the circle (Z_{1} in Fig. 7) and the phase swing is monotonic as a function of frequency. This is most preferred for successful tracking. Second, the zero voltage lies outside of the circle and the phase swing is nonmonotonic (the phase returns to the same value on both sides of the resonance). When the angle range (visible from Z_{2} in Fig. 7) of the Lorentzian circle is not too small (roughly >30°), the PLL implementation of the Zurich MFLI can reliably still lock in to the resonance, but is sometimes an unstable situation. For example, if the system goes through a phase transition and the resonance becomes much broader, it may be lost. Third, the border of the circle crosses the zero voltage at the tail of the resonance (which never happens). This corresponds to no background voltage V = A/[ω − ω_{o} + iΓ/2] and a 180° phase swing. In the special case for the border of the circle near the zero voltage, the phase first swings slowly 180°, followed by an abrupt additional 180° swing either up or down depending upon the position of the resonance (either slightly inside or slightly outside of the circle)—this is usually the situation that arises with use of a capacitance compensation circuit (Fig. 8).
In order to improve the phase swing when the background voltage is large (or the circle diameter is not large enough), we incorporate a modified capacitance compensation circuit (Fig. 8) based on the one recommended by NANOSENSORS™^{30}. The first stage following the input (upper left) acts as a buffer. An effective negative capacitance is then added in parallel with that of the measurement cables and the Aprobe. The piezoelectric current, as well as the background current, is then detected at the current–voltage converter (marked by the red cross) and then amplified. At zero magnetic field, the background capacitance is nulled with the potentiometer until the phase shift on resonance is maximized, allowing successful tracking with the hardware PLL. Because the background capacitance changes as a function of temperature, we use our custom program to adaptively follow the resonance with temperature. Our softwareimplemented PLL uses feedback from previous frequency scans to measure phase from arbitrary points in the complex voltage plane^{42,43}.
Regarding the linewidth
These measurements also allow us to measure the linewidth evolution with temperature and magnetic field, which provides relaxation time information about anisotropic fluctuations. Excluding any experimental artifacts, the linewidth is directly determined by the energy dissipation per oscillation cycle^{44}, which in this measurement will be associated with magnetically anisotropic fluctuations. Here we discuss some experimental factors that are unrelated to the physics in the sample. This is especially important because causality (expressed via Kramers–Kronig relations) requires that changes in the linewidth are accompanied by related changes in the frequency^{44}; when the linewidth decreases by an amount ΔΓ, the frequency increases by a comparable amount. For example, bringing the lever into vacuum at room temperature removes the dissipation associated with air friction around the lever, which typically decreases the linewidth by about 20 Hz. This is accompanied by a 20 Hz increase in frequency that is observed in our measurement. This is of concern only when the frequency shifts are smaller than the linewidth. In our RuCl_{3} measurements, the linewidth in vacuum at cryogenic temperatures is a couple of Hz and the frequency shifts observed in nanogramsized samples under magnetic field/temperature are typically 100–1000s of Hz. Normally, we use 10 mbar of helium4 exchange gas, which allows for a qfactor that can be as much as 30,000 at cryogenic temperatures. We have also observed unexpected frequency shifts with temperature that we believe result from partial covering of the lever with grease (used for sample attachment). The grease freezes below about 200 K, effectively increasing the bending stiffness of the lever by as much as 1%.
Magnetotropic coefficient in the linear magnetic regime
We now refer to the linear response regime (M_{i} = χ_{ij}B_{j}) to compare the magnitude of the frequency shift Δω/ω_{0} and the average deflection angle Δθ_{τ}. The free energy of the sample as it rotates in a magnetic field is
where α = χ_{j} − χ_{i} is the anisotropic susceptibility restricted to the plane of vibration. In the linear regime, τ = 2F(θ, B) tan 2θ and k = 4F(θ, B) so that both the frequency shift and deflection angle can be written as a function of the free energy
This shows that relative frequency shift in the linear regime is of the same order of magnitude as the bending angle:
Although the shift in frequency is accompanied with (and is proportional to) the average bending angle of the lever due to the magnetic torque, the former is not caused by the latter. These are two independent phenomena and they will start to affect each other if anharmonic effects become important—both in the response of the lever due to a large Δθ_{τ} or in the sample due to a nonlinear magnetic response. In particular, for nonlinear magnetization, Δω and Δθ_{τ} are not related in a simple way.
Comparison with conventional torque magnetometry
As stated in the main text, resonant torsion magnetometry allows us to probe magnetic anisotropy in highly anisotropic systems along the crystallographic directions—often the main goal of an experiment. In order to detect the intrinsic magnetic anisotropy with conventional torque magnetometry, one must always apply field off of these directions to avoid interaction effects. This arises because the magnetic torque signal goes to zero near the principal directions (gray line in Fig. 2). Here (just like at all field orientation angles), the lever reorients in the magnetic field due to the torque. The sample then experiences a new torque due to the reorientation, but the change in these two torques is of the order of the total torque signal size. This behavior leads to a nonlinear response near the crystallographic axes^{10}, which can be avoided by measuring the magnetotropic coefficient instead of torque.
We find that use of the Aprobe allows us to overcome several other systematic challenges associated with torque that is inferred from the angular deflection of a piezoresistive cantilever^{12,14}. These include the additional deflection due to the force from magnetic field gradients, a magnetoresistive contribution, and an asymmetry in the response of the amplitudedetection levers. In our resonant torsion measurements, we are always operating within the linear response regime of the lever. The shift in the resonant frequency of the vibrating cantilever is capacitively inferred from an impedance measurement, alleviating the effects of magnetoresistance. Furthermore, shifts in the frequency of lever vibrations produced by spatial inhomogeneities of the magnetic field are much smaller than the corresponding shift in the average deflection angle of the lever, as demonstrated herein. In addition, our experimental setup allows for easy rotation with respect to the applied magnetic field, even within the confines of a pulsed field magnet. The high eigenfrequency of the cantilever and the high qfactor allow for fast response and high sensitivity on the ~10 ms timescale of pulsed magnetic fields. In this environment, where magnetoresistance can dominate in piezoresistive torque magnetometry, resonant torsion can be a marked advancement.
Data availability
All relevant data are available upon request from the authors.
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Acknowledgements
The authors would like to thank the electronics workshop at the Max Planck Institute for Chemical Physics of Solids, particularly Dominic Hibsch, Wolfgang Geyer, and Torsten Breitenborn. We also thank Terunobu Akiyama for helpful discussions. Synthesis and characterization of the NbP single crystals was performed at Los Alamos National Laboratory under the auspices of the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. The portion of this work completed at the National High Magnetic Field Laboratory is supported through the National Science Foundation Cooperative Agreement numbers DMR1157490 and DMR1644779, The United States Department of Energy, and the State of Florida. M.D.B. acknowledges studentship funding from EPSRC under grant no. EP/I007002/1. R.D.M. acknowledges support from LANL LDRDDR 20160085 topology and strong correlations. K.A.M. and P.J.W.M. acknowledge support of the Max Planck Society.
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K.A.M., R.D.M., A.S. and P.J.W.M. conceived of the experiment; A.E., N.J.G., E.D.B., M.S. and M.B. prepared and characterized the samples; K.A.M., M.D.B., K.R.S., J.B.B., E.S., R.D.M. and A.S. performed the experiments. K.A.M., B.J.R., F.A., E.S., R.D.M., A.S. and P.J.W.M. analyzed the data. K.A.M., B.J.R., A.S. and P.J.W.M. wrote the manuscript with input from all coauthors.
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Modic, K.A., Bachmann, M.D., Ramshaw, B.J. et al. Resonant torsion magnetometry in anisotropic quantum materials. Nat Commun 9, 3975 (2018). https://doi.org/10.1038/s4146701806412w
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