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发布日期: Jun 1, 2020
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Magnetic and electronic phase transitions probed by nanomechanical resonators

Results .Antiferromagnetic mechanical resonators . FePS 3 is an Ising-type antiferromagnet with a Néel temperature in bulk in the range of TN ?~?118–123?K5 , 13 , 15, exhibiting a distinct feature in its specific heat near TN 15. Ionic layers in FePS 3 are stacked in van der Waals planes, that can be exfoliated to thin the crystal down with atomic precision5. Using mechanical exfoliation and all-dry viscoelastic stamping16, we transfer thin flakes of FePS 3 over circular cavities etched in an oxidised Si wafer, to form membranes (see the inset in Fig.? 1 a). Suspended FePS 3 devices with thicknesses ranging from 8 to 45?nm are placed in a cryostat and cooled down to a temperature of 4?K. The resonance frequency of the nanodrums is then characterized using a laser interferometry technique17(see Fig.? 1 a and “Methods”).Fig. 1: Characterisation of mechanical resonances in a thin antiferromagnetic FePS 3 membrane.a Laser interferometry setup. Red interferometric detection laser: λred =? 632?nm. Blue actuation laser diode: λblue ?=?405?nm. VNA, vector network analyzer, CM, cold mirror; PBS, polarizing beam splitter; PD, photodiode; LD, laser diode. Inset: optical image of a FePS 3 membrane, including electrodes introducing an option for electrostatic control of strain in the membrane. Flake thickness: 45.2? ±?0.6?nm; membrane diameter: d ?= ?10?μm. Scale bar: 30? μ m. b – d Amplitude ( A ) and phase ( ? ) of the fundamental resonance at three different temperatures for the device shown in ( a ). Filled dots, measured data; solid lines, fit of the mechanical resonance used to determine f0 and Q 17. Full size image The resonance frequency of the fundamental membrane mode, f0 ( T ), is measured in the temperature range from 4 to 200?K. Typical resonances are shown in Fig.? 1 b–d in the antiferromagnetic phase (80?K), near the transition (114?K) and in the paramagnetic phase (132?K), respectively. Figure? 2 a shows f0 ( T ) of the same FePS 3 membrane (solid blue curve). Near the phase transition, significant changes in amplitude, resonance frequency, and quality factor are observed.Fig. 2: Mechanical and thermal properties of a FePS 3 resonator with membrane thickness of 45.2?±?0.6?nm. In all panels, dashed vertical lines indicate the detected transition temperature, TN ?= 114? ±? 3?K as determined from the peak in the temperature derivative of \({f}_{0}^{2}\). a Solid blue line—measured resonance frequency as a function of temperature. Solid magenta line—temperature derivative of \({f}_{0}^{2}\). b Solid blue line—experimentally derived specific heat and corresponding thermal expansion coefficient. Solid magenta line—the theoretical calculation of the magnetic specific heat as reported in Takano et al.15added to the phononic specific heat from Debye model (dashed magenta line) with a Debye temperature of ΘD ?= ?236?K15. c Mechanical quality factor Q ( T ) of the membrane fundamental resonance. d Solid orange line—measured mechanical damping Q?1 ( T ) as a function temperature. Solid blue line—normalized cv ( T )? T term20 , 21(see Supplementary equation ( 14 )), with cv ( T ) taken from ( b ). Full size image Resonance and specific heat . To analyze the data further, we first analyze the relation between f0 and the specific heat. The decrease in resonance frequency with increasing temperature in Fig.? 2 a is indicative of a reduction in strain due to thermal expansion of the membrane. The observed changes can be understood by considering the resonance frequency of a bi-axially tensile strained circular membrane: $${f}_{0}(T)=\frac{2.4048}{\pi d}\sqrt{\frac{E}{\rho }\frac{\epsilon (T)}{(1-\nu )}},$$ (1)where E is the Young’s modulus of the material, ν its Poisson’s ratio, ρ its mass density, ? ( T ) the strain and T the temperature. The linear thermal expansion coefficient of the membrane, αL ( T ), and silicon substrate, αSi ( T ), are related to the strain in the membrane18as \(\frac{{\rm{d}}\epsilon (T)}{{\rm{d}}T}\approx -({\alpha }_{{\rm{L}}}(T)-{\alpha }_{{\rm{Si}}}(T))\), using the approximation \({\alpha }_{{{\rm{SiO}}}_{2}}\ll {\alpha }_{{\rm{Si}}}\)(see Supplementary Note? 1 ). By combining the given expression for \(\frac{{\rm{d}}\epsilon (T)}{{\rm{d}}T}\)with equation ( 1 ) and by using the thermodynamic relation αL ( T )?=? γcv ( T )/(3 KVM )19between αL ( T ) and the specific heat (molar heat capacity) at constant volume, cv ( T ), we obtain: $${c}_{{\rm{v}}}(T)=3{\alpha }_{{\rm{L}}}(T)\frac{K{V}_{{\rm{M}}}}{\gamma }=3\left({\alpha }_{{\rm{Si}}}-\frac{1}{{\mu }^{2}}\frac{{\rm{d}}[{f}_{0}^{2}(T)]}{{\rm{d}}T}\right)\frac{K{V}_{{\rm{M}}}}{\gamma }.$$ (2)Here, K is the bulk modulus, γ the Grüneisen parameter, VM =? M / ρ the molar volume of the membrane and \(\mu =\frac{2.4048}{\pi d}\sqrt{\frac{E}{\rho (1-\nu )}}\), that are assumed to be only weakly temperature dependent. The small effect of non-constant volume ( ν ?≠?0.5) on cv is neglected.We use the equation ( 2 ) to analyze f0 ( T ) and compare it to the calculated specific heat for FePS 3 from literature15. In doing so, we estimate the Grüneisen parameter following the Belomestnykh???Tesleva relation \(\gamma \approx \frac{3}{2}\left(\frac{1+\nu }{2-3\nu }\right)\) 19 , 22. This is an approximation to Leont’ev’s formula23, which is a good estimation of γ for bulk isotropic crystalline solids within ?~10% of uncertainty19. Furthermore, we use literature values for the elastic parameters of FePS 3 as obtained from first-principles theoretical calculations24to derive E ?= ?103?GPa, ν ?= 0.304 and ρ ?= 3375?kg?m ?3 (see Supplementary Note? 2 ).Detecting phase transitions . In Fig.? 2 a, the steepest part of the negative slope of f0 ( T ) (solid blue curve) leads to a large peak in \(\frac{{\rm{d}}({f}_{0}^{2}(T))}{{\rm{d}}T}\)(solid magenta curve) near 114?K, the temperature which we define as TN and indicate by the vertical dashed lines. In Fig.? 2 b the specific heat curve of FePS 3 (blue solid line) as estimated from the data in Fig.? 2 a and equation ( 2 ) is displayed. The results are compared to a theoretical model for the specific heat of FePS 3 (magenta solid line in Fig.? 2 b), which is the sum of a phononic contribution from the Debye model (magenta dashed line) and a magnetic contribution as calculated by Takano et al.15. It is noted that other, e.g. electronic contributions to cv ( T ) are small and can be neglected in this comparison, as is supported by experiments on the specific heat in bulk FePS 3 crystals15. The close correspondence in Fig.? 2 b between the experimental and theoretical data for cv ( T ) supports the applicability of equation ( 2 ). It also indicates that changes in the Young’s modulus near the phase transition, that can be of the order of a couple of percent25 , 26, are insignificant and that it is the anomaly in cv of FePS 3 which produces the observed changes in resonance frequency and the large peak in \(\frac{{\rm{d}}({f}_{0}^{2})}{{\rm{d}}T}\)visible in Fig.? 2 a.Effect of strain . The abrupt change in cv ( T ) of the membrane can be understood from Landau’s theory of phase transitions10. To illustrate this, we consider a simplified model for an antiferromagnetic system, like FePS 3 , with free energy, F , which includes a strain-dependent magnetostriction contribution (see Supplementary Note? 3 ). Near the transition temperature and in the absence of a magnetic field it holds that: $$F={F}_{0}+[a(T-{T}_{{\rm{N}}})+\zeta (\epsilon )]{L}_{z}^{2}+B{L}_{z}^{4}.$$ (3)Here, a and B are phenomenological positive constants, L zis the order parameter in the out-of-plane direction and ζ ( ? ) ?= η ij ? ij, a strain-dependent parameter with η ija material-dependent tensor, that includes the strain and distance-dependent magnetic exchange interactions between neighboring magnetic moments. By minimizing equation ( 3 ) with respect to L z, the equilibrium free energy, \({F}_{\min }\), and order parameter are obtained (see Supplementary Note? 3 ). Two important observations can be made. Firstly, strain shifts the transition temperature according to: $${T}_{{\rm{N}}}^{ }(\epsilon )={T}_{{\rm{N}}}-\frac{\zeta (\epsilon )}{a},$$ (4)where \({T}_{{\rm{N}}}^{ }\)is the Néel temperature, below which free energy minima \({F}_{\min }\)with finite order ( L z?≠?0) appear. Secondly, since close to the transition the specific heat follows \({c}_{{\rm{v}}}(T)=-T\frac{{\partial }^{2}{F}_{\min }}{\partial {T}^{2}}\), this general model predicts a discontinuity in cv of magnitude \({T}_{{\rm{N}}}^{ }\frac{{a}^{2}}{2B}\)at the transition temperature \({T}_{{\rm{N}}}^{ }\), in accordance with the experimental jump in cv ( T ) and \(\frac{{\rm{d}}({f}_{0}^{2}(T))}{{\rm{d}}T}\)observed in Fig.? 2 a and b.Temperature-dependent Q-factor . We now analyze the quality factor data shown in Fig.? 2 c, d. Just above TN , the quality factor of the resonance (Fig.? 2 c) shows a significant increase as the temperature is increased from 114 to 140?K. The observed minimum in the quality factor near the phase transition, suggests that dissipation in the material is linked to the thermodynamics and can be related to thermoelastic damping. We model the thermoelastic damping according to Zener20and Lifshitz-Roukes21that report dissipation of the form Q?1 ?= ? βcv ( T )? T , where β is the thermomechanical term (see Supplementary Note? 4 ). Since we have obtained an estimate of cv ( T ) from the resonance frequency analysis (Fig.? 2 b), we use this relation to compare the experimental dissipation Q?1 ( T ) (orange solid line) to a curve proportional to cv ( T )? T (blue solid line) in Fig.? 2 d. Both the measured dissipation and the thermoelastic term display a peak near TN ~?114?K. The close qualitative correspondence between the two quantities is an indication that the thermoelastic damping related term indeed can account for the temperature dependence of Q ( T ) near the phase transition. We note that the temperature-dependent dissipation in thin membranes is still not well understood, and that more intricate effects might play a role in the observed temperature dependence.Electrostatic strain . Equation ( 4 ) predicts that the transition temperature is strain-dependent due to the distance-dependent interaction coefficient ζ ( ? ) between magnetic moments. To verify this effect, we use an 8? ±? 0.5?nm thin sample of FePS 3 suspended over a cavity of 4?μm in diameter. A gate voltage \({V}_{{\rm{G}}}^{{\rm{DC}}}\)is applied between the flake and the doped bottom Si substrate to introduce an electrostatic force that pulls the membrane down and thus strains it (see Supplementary Figs.? 4 and 5 ). As shown in Fig.? 3 a, the resonance frequency of the membrane follows a W-shaped curve as a function of gate voltage. This is due to two counteracting effects27: at small gate voltages capacitive softening of the membrane occurs, while at higher voltages the membrane tension increases due to the applied electrostatic force, which causes the resonance frequency to increase.Fig. 3: Resonance frequency and transition temperature tuning with a gate voltage.a Resonance frequency as a function of gate voltage at 50?K. Inset: schematics of the electrostatic tuning principle. b Resonance frequency as a function of temperature for six different voltages. Inset: optical image of the sample, t ?=? 8 ±?0.5?nm. Scale bar: 16?μm. c Derivative of \({f}_{0}^{2}\)as a function of gate voltage and temperature. Blue arrow, line colors and legend indicate the values of \({V}_{{\rm{G}}}^{{\rm{DC}}}\). Dashed gray lines indicate the decrease in transition temperature \(\Delta {T}_{{\rm{N}}}={T}_{{\rm{N}}}^{ }({V}_{{\rm{G}}}^{{\rm{DC}}})-{T}_{{\rm{N}}}(0\, {\mathrm{V}})\)with increasing \({V}_{{\rm{G}}}^{{\rm{DC}}}\). d Blue solid dots—shift in TN as a function of \({V}_{{\rm{G}}}^{{\rm{DC}}}\)extracted from the peak position in ( c ). Vertical blue bars—error bar in Δ T Nestimated from determining the peak position in ( c ) within 2% accuracy in the measured maximum. Orange solid line—model of electrostatically induced strain Δ? as a function of \({V}_{{\rm{G}}}^{{\rm{DC}}}\)(see Supplementary Note? 5 ). Full size image Figure? 3 b shows f0 ( T ) for six different gate voltages. The shift of the point of steepest slope of f0 ( T ) with increasing \({V}_{{\rm{G}}}^{{\rm{DC}}}\)is well visible in Fig.? 3 b and even more clear in Fig.? 3 c, where the peak in \(\frac{{\rm{d}}({f}_{0}^{2})}{{\rm{d}}T}\)shifts 6?K downward by electrostatic force induced strain. The observed reduction in \({T}_{{\rm{N}}}^{ }\)as determined by the peak position in \(\frac{{\rm{d}}({f}_{0}^{2})}{{\rm{d}}T}\)qualitatively agrees with the presented model and its strain dependence from equation ( 4 ), as shown in Fig.? 3 d indicative of a reduced coupling of magnetic moments with increasing distance between them due to tensile strain. .