Strong suppression of near-field radiative heat transfer by superconductivity in NbN

Near-field (NF) radiative heat transfer (RHT) over vacuum space between closely spaced bodies can exceed the Planck's far-field (FF) values by orders of magnitude. A strong effect of superconductivity on NF RHT between plane-parallel thin-film surfaces of niobium (Nb) was recently discovered and discussed in a short paper [T. Králík et al., Phys. Rev. B 95, 060503 (2017)]. We present here an extensive set of experimental results on NF as well as FF RHT for geometrically identical samples made of niobium nitride (NbN), including a detailed discussion of the experimental setup and errors. The results with NbN show more precise agreement with theory than the original experiments with Nb. We observed a steep decrease of the heat flux at the transition to superconductivity when the colder sample (absorber) passed from the normal to the superconducting (SC) state $(T_{\mathrm{c}}\approx 15.2\mathrm{K})$, corresponding to an up to eightfold contrast between the normal and SC states. This differs dramatically from the situation in the FF regime, where only a weak effect of superconductivity was observed. Surprisingly, the contrast remains sizable even at high temperatures of the hot sample (radiator) with the characteristic energy of radiation far above the SC energy gap. We explain the maximum of contrast in heat flux between the normal and SC states, found at a distance about ten times shorter than the crossover distance between NF and FF heat flux, being $d\approx 1000/T\left[\mu \mathrm{m}\right]$. We analyze in detail the roles of transversal electric (TE) and magnetic (TM) modes in the steep decrease of heat flux below the SC critical temperature and the subsequent flux saturation at low temperatures. Interestingly, we expose experimentally the effect of destructive interference of FF thermal radiation in the vacuum gap, which was observable at temperatures below the absorber superconducting transition.

Figure 1

Resistivity dependences on the temperatures of the absorber and radiator NbN samples. We determine the critical temperatures at half the maximum value of residual resistivity: For the absorber $T_{\mathrm{c}1}=(15.25\pm 0.01)\mathrm{K}$, and for the radiator $T_{\mathrm{c}2}=(15.21\pm 0.01)\mathrm{K}$. Decrease from $95\%$ down to $5\%$ of residual resistivity value occurs within a temperature interval of about 120 mK; see the inset. We measured the resistivity by a four-point probe at the samples’ center.

Figure 2

Scheme of the EWA measurement chamber core, containing the cold (absorber at temperature $T_1$) and hot (radiator at temperature $T_2$) samples with a mechanism for plane-parallelism control and a mount limiting the samples’ planarity deformation due to thermal stress. The intersample distance $d$ (i.e., the “vacuum gap”) is determined via capacitance measurement. Adapted from Ref. [4].

Figure 3

Measured radiative heat power $P$ transferred over the $d\approx 9$-μm-wide vacuum gap between the plane-parallel NbN films, dependent on the absorber temperature $T_1$. The absorber passes from the normal to the SC state at $T_1=T_{\mathrm{c}1}=15.25\mathrm{K}$, as seen for several radiator temperatures $T_2>T_{\mathrm{c}}$ between 16 and 50 K.

Figure 4

Measured radiative heat power $P$ dependent on the vacuum gap size $d$ between the plane-parallel NbN films. The radiator is in the normal state at $T_2=20\mathrm{K}$. The absorber is at temperatures $T_1=5\mathrm{K}$, 9.8 K (SC state), and 15.3 K (normal state). Note to the point at $d=200\mu \mathrm{m}$, $T_1=15.3\mathrm{K}$: data points $P<0.5\mu \mathrm{W}$ might be affected by random background changes leading to errors exceeding $10\%$ (see Sec. 3b).

Figure 5

Measured radiative heat power transferred over the vacuum gap dependent on the radiator temperature $T_2$ passing the SC transition at $T_2=T_{\mathrm{c}2}=15.21\mathrm{K}$. The absorber is in the SC state at constant $T_1=9.8\mathrm{K}$. The error bars show the $P$ resolution derived from the temperature resolution of $50\mu \mathrm{K}$, Eq. (13).

Figure 6

Heat flux dependent on the temperature difference $T_2\text{–}{T}_1$ between the radiator and absorber measured for a set of radiator temperatures $T_2>T_{\mathrm{c}}$ for vacuum gap widths $d\approx 5.5\mu \mathrm{m}$ (a), $\approx 9\mu \mathrm{m}$ (b) and $\approx 12.4\mu \mathrm{m}$ (c). At a constant radiator temperature $T_2>T_{\mathrm{c}}$, the absorber state was varied from normal metallic to superconducting (data points from left to right in the plot). Dashed lines plot theoretical dependences for selected temperatures $T_2$. Theoretical “envelopes” of the data correspond to varying $T_2$ at constant $T_1\approx T_{\mathrm{c}}=15.25\mathrm{K}$ (upper envelope), $T_1=5\mathrm{K}$ (lower envelope), and $T_1=0\mathrm{K}$ (dot-dashed line).

Figure 7

The measured (data points) and theoretical (solid lines) contrast $C$, Eq. (14), in the NF regime dependent on the radiator temperature $T_2$ at three distances $d$ between the NbN layers.

Figure 8

Thermal conductivity $K=q/\left(T_2\text{–}{T}_1\right)$ of the vacuum gap dependent on the absorber temperature $T_1$ measured at a set of fixed radiator temperatures $T_2$ for three vacuum gap widths: $d\approx 5.5\mu \mathrm{m}$ (a), $d\approx 9\mu \mathrm{m}$ (b), and $d\approx 12.4\mu \mathrm{m}$ (c).

Figure 9

The mutual emissivity [heat flux $q/q_{\mathrm{bb}}$ “normalized to the blackbody”, Eq. (17)] dependent on the product $T_{2}d$ for both samples in the normal state. Data are collected from measurements of $q\left(d\right)$ and $q\left(T_1\right)$ dependences, where the remaining two parameters $T_1$, $T_2$ and $T_2$, $d$, respectively, are fixed. Error bars show the uncertainty in $T_{2}d$ corresponding to intersample distance $d$ uncertainty of 0.5 μm [cf. Eq. (13), last term].

Figure 10

The same as in Fig. 9 but depending on $d$ instead of $T_{2}d$, for the absorber at $T_1=5\mathrm{K}$ (superconducting) while the radiator is in the normal state at various temperatures $T_2$ ranging from 18 to 45 K.

Figure 11

Spectrum of modified vacuum gap NF transmissivity $M^{\mathrm{NF}}$, Eq. (7), dependent on the transferred heat flux frequency $\omega$ at various absorber temperatures $T_1$ for $d=5.4\mu \mathrm{m}$ (a) and $d=12.4\mu \mathrm{m}$ (b). Notice that the TE and TM components of $M^{\mathrm{NF}}$ are shown for radiator temperatures $T_2>T_{\mathrm{c}}$, where $M$ practically does not depend on $T_2$. For comparison, we show also the (monotonous) oscillator energy $E$ dependences (multiplied by a constant) for three radiator temperatures.

Figure 12

Heat flux $q$ over the vacuum gap $d=5.4\mu \mathrm{m}$ (a) and $d=12.4\mu \mathrm{m}$ (b) between the NbN sample layers (experiment—points, theory—lines). The radiator is in the normal state at $T_2=20\mathrm{K}$, while the absorber undergoes the SC transition at $T_{\mathrm{c}1}=15.25\mathrm{K}$. NF heat fluxes from the radiator at $T_2$ to absorber at $T_1$ (labeled with “+”) and vice versa (“−”) are distinguished (their difference gives the total NF heat flux). Notice that the FF total heat flux (dotted line) is small at these intersample distances $d$ throughout the $T$ range considered.

Figure 13

Theoretical (lines) and experimental (points) mutual emissivity $e=q/q_{\mathrm{BB}}$ depending on the product $T_{2}d$, for both samples normal. The radiator temperatures are $T_2=20\mathrm{K}$ and 30 K, while the absorber is at $T_1=15.3\mathrm{K}$. The theoretical heat flux curve is decomposed to the TE and TM modes of both the NF and FF contributions.

Figure 14

The same as in Fig. 13 for the radiator at $T_2=20\mathrm{K}$ (normal) and absorber at $T_1=9\mathrm{K}$ (superconducting). The open circle point corresponds to the classical FF heat flux. The remaining points (full green circles) are measured.

Figure 15

Full spectra $EM$, Eqs. (6) and (7), of the FF heat flux TM mode at a distance $d=50\mu \mathrm{m}$ and $200\mu \mathrm{m}$, are compared with those corresponding to a large vacuum gap of $d=2\mathrm{mm}$. The destructive interference apparent at $d=50\mu \mathrm{m}$ is more effective in the SC state (b) in which the radiation is “more monochromatic” due to the energy gap cutoff below ${\omega }_g\approx 8\times {10}^{12}\mathrm{rads}/\mathrm{s}$. Wavelengths at maxima of FF TM spectral heat flux over vacuum gap $d=2000\mu \mathrm{m}$ are $\lambda \approx 250\mu \mathrm{m}$ (a) and $\lambda \approx 160\mu \mathrm{m}$ (b).