We present a method for measuring thermal expansion under tunable uniaxial stresses and show measurements of the thermal expansion of Mn3Sn, a room temperature antiferromagnet that exhibits a spontaneous Hall effect, under uniaxial stresses of up to 1.51 GPa compression. The measurement of thermal expansion provides thermodynamic data about the nature of phase transitions, and uniaxial stress provides a powerful tuning method that does not introduce disorder. Mn3Sn exhibits an anomaly in its thermal expansion near ∼270 K, associated with a first-order change in its magnetic structure. We show that this transition temperature is suppressed by 54.6 K by 1.51 GPa compression along [0001]. We find the associated entropy change at the transition to be ∼ 0.1 J mol−1 K−1 and to vary only weakly with applied stress.

The application of uniaxial stress modifies lattice constants and can provide powerful insight into the electronic structure of materials. Its effects often differ qualitatively from those of hydrostatic stress. For example, the superconducting critical temperature of YBa2Cu3O6.67 is enhanced by hydrostatic stress but suppressed by in-plane uniaxial stress.1 To date, several techniques have been combined with uniaxial stress, including magnetic susceptibility,2 transport,3 nuclear magnetic resonance,4 and muon spin rotation.5 Dilatometry, the measurement of the thermal expansion of a material, is also an attractive technique to apply to uniaxially stressed samples because it provides extremely high-precision bulk thermodynamic information. It is orders of magnitude more sensitive to changes in lattice parameter than x-ray or neutron diffraction.6 It has, for example, been utilized in measurements of electron-lattice coupling in electronically and magnetically ordered phases6,7 and to probe thermodynamic effects of quantum criticality.8–10 

In the majority of modern dilatometers, samples are compressed between two anvils.9 Therefore, a straightforward method to measure thermal expansion under uniaxial stress is to configure the dilatometer to also apply substantial force through these anvils.11 It has been shown, however, that very high uniaxial stresses and high stress homogeneity can be achieved by preparing samples as narrow beams, embedding the ends in epoxy, and applying force through the epoxy and along the sample length.12 In this Letter, we explore methods for dilatometry measurements built around this sample configuration. Our goal is to maintain high sensitivity while applying GPa-level uniaxial stresses. Although this is a common measurement for engineering materials, our goal is to explore methods appropriate for samples that are more challenging to handle, due to the small size of available samples, and/or unfavorable mechanical properties such as brittleness and tendency to cleave.

We apply these methods to Mn3Sn, in which the Mn moments form a triangular spin structure below TN420 K.13,14 Recently, this chiral antiferromagnet has attracted considerable attention due to its anomalous transport properties15,16 associated with the presence of magnetic Weyl fermions16,17 and the potential for applications in spintronics devices.18,19 However, stoichiometric Mn3Sn undergoes a first-order transition below TH 270 K from the triangular spin structure to a spin spiral,20,21 in which the topological transport properties are lost.22,23 While room-temperature application of topological transport properties is, therefore, possible, this transition prevents their study at low temperature, where greater precision is available through the suppression of thermal fluctuations. By growing off-stoichiometric Mn3+xSn1−x, this transition can be suppressed, but measurement precision then becomes limited by defects rather than thermal fluctuations.

The transition at TH is accompanied by an expansion along [0001], and our goal here is to suppress TH as far as possible through compressive uniaxial stress, σ, which introduces no disorder when the sample deformation is elastic, applied along [0001]. From the Clausius–Clapeyron relationship, we estimate that σ4 GPa, where σ<0 denotes compression, would be required to obtain TH0. Here, we achieve σ=1.51 GPa and show through thermal expansion measurements that, at this stress, TH is suppressed by 54.6 K.

A piezoelectric-driven uniaxial stress apparatus, which incorporates both force and displacement sensors, has been presented in Ref. 12. Such a device could, in principle, be used for dilatometry measurements. In that device, the sample is coupled directly (i.e., through a high-stiffness mechanical link) to piezoelectric actuators. Force is thus delivered at a high spring constant because piezoelectric actuators are, in general, high-stiffness devices that generate only small displacements. Consequently, the force applied to the sample is not automatically independent of length changes that the sample might undergo and must instead be held constant through feedback.

The approach we explore here is to place a spring of low spring constant between the piezoelectric actuators and the sample; we term this the conversion spring, as it converts displacement from the actuators into a force on the sample. The advantage is that force is delivered to the sample at a low spring constant. For materials such as Mn3Sn that undergo first-order transitions, this means that the force on the sample does not change drastically even if the sample length changes abruptly. The disadvantage is, to achieve high stresses, the actuators must be long and/or the sample small. Here, the actuators can generate a displacement of ∼100 μm. For the force on the sample to be well-controlled, the majority of this displacement must go into the conversion spring. If we specify that ∼90% should go into the conversion spring, there is only ∼10 μm left to compress the sample, as well as the epoxy that holds it and any other coupling elements to the sample. Therefore, to reach high strains, we limit the sample length to ∼300 μm.

Our design is presented in Fig. 1. Parts, unless otherwise noted, are made out of Grade 2 titanium. The device is composed of a main body that contains an integrated moving block, guided by flexures. Movement of this block is driven by three piezoelectric actuators (PI PICMA® P-885.91) that sit below the main body, are connected to the body by L-shaped brackets, and are joined at their opposite ends by a rectangular prism, a “bridge.” Macor® caps are fixed on both ends of the actuators for electrical isolation, important for high voltage operation in exchange gas. With the actuators fixed together in this manner, expansion of the outer actuators translates the moving block toward the sample carrier and compresses the sample. Expansion of the inner actuator causes tension to the sample.

The sample is mounted onto a detachable carrier composed of two parts, one moving and the other fixed, joined by flexures; the sample spans the moving and fixed parts. The carrier incorporates the conversion spring. It also incorporates a second capacitive sensor, composed of two blocks separated by 10μm, one attached to the moving part and the other to the fixed part. This sensor measures changes in the length of the sample and so is termed the displacement sensor. The rest of the carrier serves as part of the shielding for this capacitor. The carrier is coupled to the cell through attachment wires, here 200 μm-diameter beryllium copper, which transmit longitudinal applied force to the sample while attenuating large inadvertent torques that may be applied when the carrier is clamped to the rest of the apparatus. A schematic of the essential mechanical connections of the apparatus is shown in Fig. 1(b).

Measured capacitances are converted to displacement, d, through C=ε0A/(d+d0)+Coffset, where A is the capacitor area and Coffset is the stray capacitance. A is 7.7 and 7.4 mm2 for the force and displacement capacitors, respectively. d0 is the initial capacitor plate separation, 150 μm for the force and 10 μm for the displacement capacitors. The stray capacitance was found to be negligible for both capacitors. The spring constants of the conversion spring and carrier flexures were measured at room temperature (see the supplementary material) and are 5.6×104 and 3.5×104 N m−1, respectively. To obtain force, the length change measured by the displacement sensor was subtracted from that measured by the force sensor, to obtain the displacement applied to the conversion spring.

Single crystals of Mn3Sn were grown from Sn flux using the Bridgman method, following protocols reported in a previous study.16 After alignment using a Laue diffractometer, single crystals were spark eroded and polished into bar-like shapes and then screened through a combination of magnetometry and heat capacity measurements. The samples selected for the study here have 280>TH>260 K, corresponding to Mn3+xSn1−x compositions with 0.009<x<0.02, as measured by inductively coupled plasma spectroscopy.24 To provide a control measurement, the thermal expansion of a sample with TH = 267 K was measured using a conventional capacitive dilatometer.9 

Measurements were performed in an Oxford Instruments TeslatronTM PT with a Cernox thermometer mounted to the outer frame of the stress dilatometer. The temperature was swept at a rate of 0.25 K min−1. An Andeen-Hagerling capacitance bridge, AH2550A, and a Keysight LCR meter, E4980AL, were used to measure the carrier and force capacitors, respectively. To maintain a constant force sensor reading, the applied voltage to the actuators was driven with a proportional-integral-derivative controller. We find, with this experimental configuration, a noise level of 79 pm /HZ on the displacement sensor, shown in Fig. 2. The noise was determined by measuring the carrier capacitance every second with a sample under a small, constant applied force ∼0.38 N for two hours. The temperature during that time period varied by 1.03 K, which accounts for the 5 nm background length change. The noise is determined by subtracting a smoothed background from the data. Figure 2(b) demonstrates that the dominant contribution to this noise is fluctuations in the applied force from the feedback on the force capacitor. It could be improved by increasing the sensitivity of the force capacitor, e.g., by reducing its d0, or lengthening the feedback time constant.

Samples are epoxied across a gap between two 50 μm-thick titanium foils. Sample1, shown in Fig. 3, was prepared with a uniform cross section of 30 μm × 36 μm. TH is, as expected, suppressed by applied pressure, though the transition broadens dramatically as |σ| is increased. We attribute this broadening to extended stress-gradient regions in the sample ends, which arise because force is transferred to the sample through the mounting epoxy over a substantial length scale, and the displacement sensor measures the total displacement across the carrier, including deformation of the epoxy and sample ends embedded in the epoxy.

Therefore, to concentrate stress efficiently, we found it essential to mill sample 2 into a bow-tie shape, using a Thermo Fisher Scientific Helios G4 PFIB UXe Xe2+ plasma focused ion beam instrument, as shown in Fig. 4(c). The two wide end portions of the sample are epoxied to the carrier and serve to couple applied force from the carrier into the central, necked portion of the sample, with a cross section of 32 μm × 36 μm. By shaping the sample in this way, the shear stress within the epoxy layers is reduced, reducing the contribution to the measurement from the end portions. The sample is then epoxied across the gap between mounting foils, and additional foils are epoxied on top of the tabs to make the mounting symmetric, which improves stress homogeneity by reducing bending.25 

The results are shown in Figs. 4 and 5. Panels (a) and (b) of Fig. 4 show the length change of sample 2 and its derivative for forces from −0.02 to 1.78 N, equivalent to stresses of 0.01 to −1.51 GPa in the neck, applied along [0001]. Two transitions are apparent: one that remains at 260 K independent of the applied stress and the other that shifts to lower temperature as the sample is compressed. We attribute the stress-independent transition to the transition in the tabs of the sample, where the applied stress is low, and the transition that shifts to the transition in the necked portion of the sample. This is in contrast to the broadening seen in sample 1 and shows that the shaping of the sample gives a sharp dichotomy between high- and low-stress sample regions. Thus, the contributions from the two sections can be cleanly separated. At −1.51 GPa, TH is suppressed by 54.6 K–210.9 K. The width of the transition is nearly constant up to -1.33 GPa. Above |σ| 1.4 GPa, the transition is broadened, which may be due to bending-induced strain inhomogeneity or plastic deformation prior to the sample fracture above |σ| 1.6 GPa. Figure 4(e) shows the hysteretic behavior vs temperature. The first-order nature of the transition is shown by the step-like change in sample length and hysteresis in TH between increasing- and decreasing-temperature ramps. At high force, the difference in TH on cooling and warming remains the same as for low applied forces. However, the possible onset of plastic deformation in the sample and/or its epoxy mounting may explain the change in slope between the cooling and warming curves.

Figure 5(a) shows heat capacity data from a Mn3Sn sample (black) with TH280 K and a sample of reported composition Mn3.03Sn0.97 (red) with TH270 K reproduced from Ref. 26. While the transition temperatures of the samples differ, the change in entropy at the transition appears to be similar. Figure 5(b) shows the dependence of TH on uniaxial pressure for sample 2. The hysteresis between increasing-T and decreasing-T ramps is nearly stress-independent. Figure 5(c) shows the fractional length change of ΔL/L at TH for sample 2 and that of an unstressed sample measured with a conventional dilatometer. Data points from sample 2 are not included for pressures below |σ| 0.5 GPa, as the overlap between transitions in the neck and tab portions of the sample creates too much uncertainty. Our data in Figs. 5(b) and 5(c) can be related to the change in entropy at TH, ΔS, using the Clausius–Clapeyron relation:

|dTH/dσ|=(ΔL/L)Vmol/ΔS,
(1)

which is valid for first-order transitions. The negative sign on the left hand side of the equation is due to our convention that compression is negative. With this and our zero-pressure dilatometry and heat capacity data, we estimate the zero pressure dTH/dσc slope to be 40.1 K GPa−1. dTH/dσ of sample 2 is found to be 33.3 K GPa1 at low |σ|, in reasonable agreement with this estimate, and this shows that the calibration of our pressure axis is correct to within 20%. There may be some sample-to-sample variation, and we note in addition that our goal at this stage of development is not a high-precision calibration but to explore the methodology needed for high sensitivity at high stresses. As |σ| is increased to large values, TH deviates below this slope, see Fig. 5(b). Applying the Clausius–Clapeyron relationship to the data of panel (c), we find that the entropy change associated with the transition is mildly suppressed from its zero pressure value to 0.1 J mol−1 K−1 above |σ| 0.7 GPa, as shown in Fig. 5(d).

From the estimated Young's modulus of Mn3Sn along the [0001] axis (between 118 and 161 GPa, supplementary material), the sample strain at σ=1.51 GPa is 1%. As a comparison, it was found that 8% substitution of the Sn site with Mn can suppress the spin spiral transition completely while changing the c and a lattice constants by only ∼ −0.3%,20 which indicates that the changes in TH due to Sn substitution are not primarily due to the effect of the substitution on the lattice constants. A similar case of composition sensitivity in a relatively high-temperature magnetic transition was observed in FeRh.28 In Mn3Sn, it is striking that the spiral transition at TH is acutely sensitive to changes in composition, and yet, in comparison, mildly responds to lattice strains of 1%. This unexpected response to strain had to be directly measured; it could not be extrapolated from measurements at zero stress using the Clausius–Clapeyron relationship. Our results suggest that, with further technical refinement, suppression of TH to zero will be achievable, and the methodology presented here will be applicable to a range of other materials.

See the supplementary material for an explanation of the characterization of the device flexures and how force is determined on the sample. In addition, an explanation of our Young's modulus estimate and finite element analysis simulations are shown.

M.I. and K.R.S. contributed equally to this work.

The authors thank Alexander Steppke and Hilary Noad for their technical expertise and fruitful scientific discussions. We also thank Thomas Lühmann for his help with LabVIEW programming. This work was partially supported by CREST (Nos. JPMJCR18T3 and JPMJCR15Q5), Japan Science and Technology Agency, by New Energy and Industrial Technology Development Organization (NEDO), by Grants-in-Aids for Scientific Research on Innovative Areas (Nos. 15H05882 and 15H05883) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, and by Grants-in-Aid for Scientific Research (No. 19H00650) from the Japanese Society for the Promotion of Science (JSPS). M.I. is supported by a JSPS Research Fellowships for Young Scientists (DC1). C.W.H. and A.P.M. acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) through SFB 1143 (Project-Id 247310070). C.W.H. has 31% ownership of Razorbill Instruments, a United Kingdom company that markets piezoelectric-based uniaxial stress cells.

The data that support the findings of this study are available within the article and its supplementary material. Raw data of this study are available from the corresponding author upon reasonable request.

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Supplementary Material