Noncontact Atomic Force Microscope Dissipation Reveals a Central Peak of ${\mathrm{SrTiO}}_3$ Structural Phase Transition

The critical fluctuations at second order structural transitions in a bulk crystal may affect the dissipation of mechanical probes even if completely external to the crystal surface. Here, we show that noncontact force microscope dissipation bears clear evidence of the antiferrodistortive phase transition of ${\mathrm{SrTiO}}_3$, known for a long time to exhibit a unique, extremely narrow neutron scattering “central peak.” The noncontact geometry suggests a central peak linear response coupling connected with strain. The detailed temperature dependence reveals for the first time the intrinsic central peak width of order 80 kHz, 2 orders of magnitude below the established neutron upper bound.

Figure 1

(a) Experimental AFM dissipation $W$ as a function of temperature. Raw data, taken at different surface spots and different tip sample distances $z$. The sharp peak corresponds to the critical temperature of ${\mathrm{SrTiO}}_3$ in the bulk region under the tip. (b) Low temperature ($T=5\mathrm{K}$) STM image of ${\mathrm{SrTiO}}_3(100)$ surface. The image is obtained at constant current $I=10\mathrm{pA}$ and bias voltage $U=1\mathrm{V}$. The length of the scale bar is equal to 20 nm. More details are given in the Supplemental Material [12].) (c) The distance dependence of the dissipation $W$, taken as the maximum of the peak shown in (a), at four different spots on the sample. A fit to the experimental data, $W\propto z^{-p}$, is shown in red, with $p\sim 4.2$. This exponent is close to the value $p=4$ expected for phononic dissipation, as appropriate for coupling to acoustical surface fluctuations of an insulating bulk material.

Figure 2

$\mathrm{Im}{\chi }_{\mathrm{CP}}(\omega )$, the low-energy central peak component of $\mathrm{Im}\chi (\omega )$ on a log-log scale for various temperatures, showing a peak at ${\omega }_{\text{peak}}(T)\sim t^{2\gamma }$ that moves towards 0 as $T\to T_c$. Upper inset: sketch of the full $\mathrm{Im}\chi (\omega )$ on a linear $\omega$ scale, showing the broad soft-phonon Lorentzian at ${\omega }_{\infty }$ with the sharp low-energy central peak. For clarity we used here a high value of ${\omega }_{\text{low}}$ to show both peaks on the same scale. Lower inset: cross section of 3D simulation of a tip perturbing a semi-infinite crystal through a vdW interaction. The tip (red dots) is shown as a truncated pyramid where every atom exerts a $-C/r^6$ vdW potential on the crystal atoms (blue dots), which are held together by a harmonic potential. Arrows (magnified for clarity) represent on a log scale the atom displacements from the relaxed positions.

Figure 3

Experimental dissipation $W$ above $T_c$ as a function of temperature (black dots). Inset: the same data on a linear temperature scale, showing data both below and above $T_c$. Red and green lines, fit above and below $T_c$ according to Eq. (5); pink dashed line, fit above $T_c$ excluding the high plateau data. The value of ${\omega }_{\text{low}}$ is 156 kHz (solid curve) and 44 kHz (dashed curve), both within a factor 2 of 83 kHz, the value obtained from a simple fit of the saturation temperature (see text). $\overline{t}$ is roughly the temperature where, upon cooling from above $T_c$, the dissipation levels off to a plateau. Horizontal error bars corresponding to $\delta t\sim {10}^{-4}$ are small and are omitted.