Second-order calculation of the local density of states above a nanostructured surface

We have numerically implemented a perturbation series for the scattered electromagnetic fields above rough surfaces, due to Greffet, allowing us to evaluate the local density of states to second order in the surface profile function. We present typical results for thermal near fields of surfaces with regular nanostructures, investigating the relative magnitude of the contributions appearing in successive orders. The method is then employed for estimating the resolution limit of an idealized near-field scanning thermal microscope.

Figure 1

(Color online) Profile function (39) with parameters as employed in our model calculations, height $h=5\text{nm}$, width $w=30\text{nm}$, and inverse smoothing length $\zeta ={10}^9{\text{m}}^{-1}$ (full line), together with the reference profile (40) (dashed). In either case, the dielectric properties of the sample are given by the Drude model with parameters for gold at 300 K.

Figure 2

(Color online) Zeroth-, first-, and second-order contributions to the (a) electric and to the (b) magnetic parts of the LDOS above a gold surface structured with the nanobar (39) as sketched in Fig. 1, for $\omega ={10}^{14}{\text{s}}^{-1}$ at an observation height of 10 nm above the base substrate plane. Note the different scales.

Figure 3

(Color online) As Fig. 2, but for a sample consisting of gallium nitride (GaN).

Figure 4

(Color online) As Fig. 2, for a gold sample with the reference profile (40).

Figure 5

(Color online) Ratios of the zeroth-, first-, and second-order contributions to the (a) electric and to the (b) magnetic LDOS for $\omega ={10}^{14}{\text{s}}^{-1}$ at an observation point $x=0$ above the center of the bar (39).

Figure 6

(Color online) (a) Observation distance above the one-bar gold sample described by Eq. (39) (1 bar), and above the sample with two parallel bars specified by Eq. (51) (2 bars), determined such that the second-order LDOS for $\omega ={10}^{14}{\text{s}}^{-1}$ constantly keeps that value which is attained for the distance $a_{\text{eff}}=20\text{nm}$ far away from the bars. (b) Definition of the quantities $d$ and $\Delta d$ employed for discussing the resolution of an ideal NSThM.

Figure 7

(Color online) Ratio $\Delta d/d$ for three different values of $a_{\text{eff}}$ (20, 30, and 40 nm), as functions of the bar separation $s$.