Spatial Correlation between Fluctuating and Static Fields over Metal and Dielectric Substrates

We report spatially resolved measurements of static and fluctuating electric fields over conductive (Au) and nonconductive (${\mathrm{SiO}}_2$) surfaces. Using an ultrasensitive “nanoladder” cantilever probe to scan over these surfaces at distances of a few tens of nanometers, we record changes in the probe resonance frequency and damping that we associate with static and fluctuating fields, respectively. We find static and fluctuating fields to be spatially correlated. Furthermore, the fields are of similar magnitude for the two materials. We quantitatively describe the observed effects on the basis of trapped surface charges and dielectric fluctuations in an adsorbate layer. Our results are consistent with organic adsorbates significantly contributing to surface dissipation that affects nanomechanical sensors, trapped ions, superconducting resonators, and color centers in diamond.

Figure 1

Experimental setup. (a) A nanoladder cantilever oscillates parallel to the surface in the $x$ direction. It is scanned over a sample surface consisting of ${\mathrm{SiO}}_2$ and a pattern of Au that is 250 nm thick and electrically grounded. (b) Schematic representation of the diamond tip interacting with electrical (magnetic) fields generated by charge (spin) defects close to the sample surface. A bright orange area indicates an adsorbant layer, and $q_{\mathrm{tip}}$ represents a tip charge with a distance $\mathrm{\Delta }$ to the tip apex.

Figure 2

Scanning results. (a) Maps of the cantilever frequency $f$ recorded at a distance $d=30\mathrm{nm}$ over Au and (b) $d=80\mathrm{nm}$ over ${\mathrm{SiO}}_2$. The colors range from red (4500) to white (5000 Hz) in (a) and from blue (4200) to white (5210 Hz) in (b). Both scale bars are 100 nm long. Insets show the tip over the substrate, ${\mathrm{SiO}}_2$ (blue) and Au (red). (c)–(f) Line scans of the resonance frequency $f$ and noncontact friction ${\mathrm{\Gamma }}_{\mathrm{NCF}}$ over Au (c),(e) and ${\mathrm{SiO}}_2$ (d),(f). The lines corresponds to $d=30$, 45, 100, and 150 nm for Au and $d=45$, 60, 75, and 100 nm for ${\mathrm{SiO}}_2$ (bottom to top). Lines are offset for better visibility by 1.5 kHz each in (c), 2 kHz each in (d), $4\times {10}^{-15}\mathrm{kg}{\mathrm{s}}^{-1}$ each in (e) and $6\times {10}^{-15}\mathrm{kg}{\mathrm{s}}^{-1}$ each in (f). Shaded areas denote errors estimated from repeated measurements. Lateral drift may have occurred between the different line scans in (d) and (f). With our compact nanopositioners, drift can be caused by piezo creep after $z$ approaches performed to ensure correct $d$ calibration between the line scans.

Figure 3

Quantitative data analysis. (a) Maximum and minimum $f$ as a function of $d$ over Au and (b) over ${\mathrm{SiO}}_2$. Filled and open squares correspond to our measurements and to model calculations, respectively. Solid lines are phenomenological fits using $f_0\pm {\beta }_{\pm }/(d+\mathrm{\Delta }{)}^{{\nu }_{\pm }}$. For Au, ${\beta }_{-}=3\times {10}^{-23}\mathrm{Hz}{\mathrm{m}}^{3.5}$ and ${\nu }_{-}=3.5$, ${\beta }_{+}=2.3\times {10}^{-12}\mathrm{Hz}{\mathrm{m}}^2$ and ${\nu }_{+}=2$ (${\beta }_{+}=1.25\times {10}^{-12}\mathrm{Hz}{\mathrm{m}}^2$ for the dashed line). For ${\mathrm{SiO}}_2$, ${\beta }_{-}=7.5\times {10}^{-23}\mathrm{Hz}{\mathrm{m}}^{3.5}$ and ${\nu }_{-}=3.5$, ${\beta }_{+}=1.8\times {10}^{-19}\mathrm{Hz}{\mathrm{m}}^3$ and ${\nu }_{+}=3$. See main text and Supplemental Material for details on the model [51]. (c) Measured ${\mathrm{\Gamma }}_{\mathrm{NCF}}$ and corresponding model for Au and (d) for ${\mathrm{SiO}}_2$. Squares indicate the maxima and minima of a linescan at a distance $d$. The shaded areas corresponds to the model predictions for varying $f$ [cf. solid lines in (a)–(b)] for $h=1.0\mathrm{nm}$ (dark shade) and for $h$ between 0.4 and 2.0 nm (bright shade). We use $q_{\mathrm{tip}}=q_e$ for both models, $\epsilon =2$ and $\tan \theta =0.01$ for Au, and $\epsilon =2$ and $\tan \theta =0.03$ for ${\mathrm{SiO}}_2$. The dashed line in (d) is the additive dielectric contribution of the ${\mathrm{SiO}}_2$ substrate with a thickness of $1.5\mu \mathrm{m}$, $\epsilon =4.44$ and $\tan \theta ={10}^{-3}$. (e) Measured ${\mathrm{\Gamma }}_{\mathrm{NCF}}$ as function of $f$ for different $d$ as in Fig. 2 over Au and (f) over ${\mathrm{SiO}}_2$. Datasets are offset for better visibility by $3\times {10}^{-15}\mathrm{kg}{\mathrm{s}}^{-1}$ each for Au and $5\times {10}^{-15}\mathrm{kg}{\mathrm{s}}^{-1}$ each for ${\mathrm{SiO}}_2$.