# Measuring the magnetic dipole transition of single nanorods by spectroscopy
and Fourier microscopy

Reinaldo Chacon1 Aymeric Leray1 Jeongmo Kim2 Khalid Lahlil2 Sanro Mathew1
Alexandre Bouhelier1 Jong-Wook Kim2 Thierry Gacoin2 Gérard Colas des Francs1
<EMAIL_ADDRESS>1 Laboratoire Interdisciplinaire Carnot
de Bourgogne (ICB), UMR 6303 CNRS, Université Bourgogne Franche-Comté, 9
Avenue Savary, BP 47870, 21078 Dijon Cedex, France  2Physique de la Matière
Condensée, CNRS UMR 7643, Ecole Polytechnique, 91128 Palaiseau, France

###### Abstract

Rare-earth doped nanocrystals possess optical transitions with significant
either electric or magnetic dipole characters. They are of strong interest for
understanding and engineering light-matter interactions at the nanoscale with
numerous applications in nanophotonics. Here, we study the
5D${}_{0}\rightarrow^{7}$F1 transition dipole vector in individual NaYF4:Eu3+
nanorod crystals by Fourier and confocal microscopies. Single crystalline host
matrix leads to narrow emission lines at room temperature that permit to
separate Stark sublevels resulting from the crystal field splitting. We
observe a fully magnetic transition and low variability of the transition
dipole orientation over several single nanorods. We estimate the proportion of
the dipole transitions for the Stark sublevels. We also determine an effective
altitude of the rod with respect to the substrate. The narrow emission lines
characteristic of NaYF4:Eu3+ ensure well-defined electric or magnetic
transitions, and are thus instrumental for probing locally their
electromagnetic environment by standard confocal microscopy.

##  I Introduction

Light-matter interaction in the optical regime is principally of electric
nature because the magnetic contribution is orders of magnitude weaker.
Therefore, engineering optical magnetic dipole (MD) opens novel design rules
for developing metamaterials [1] and optical antennas [2]. Examples are
artificial MDs created by specific ring-like geometries [3, 4, 5, 6]. Of
particular interest in this context are rare-earth ions because they naturally
have MD optical transitions [7] that can be manipulated by the crystalline or
molecular hosting environment.

Rare-earth doped crystals have original spectroscopic properties and extensive
theoretical and experimental efforts have been devoted to investigate their
distinctive luminescence features. They are widely produced for solid state
lasers or as phosphors for lightening [8], but several other promising
applications are also envisioned. Since some emission lines are very sensitive
to the local temperature, rare-earth doped nanocrystals have been used to
measure the internal temperature of tumor cells during thermally induced
apoptosis [9] or used as temperature sensor scanned above an electronic
circuit [10]. Luminescent nanocrystals are also considered as active probes
for scanning near-field optical microscopy delivering a signal proportional to
the local electric field [11]. Moreover, some transitions present a magnetic
dipole activity used for probing the magnetic near-field [12, 13]. Finally
lifetime measurements of the electric dipole (ED) and MD allowed transitions
in rare-earth doped nanocrystal were implemented to experimentally measure the
local density of electromagnetic states (LDOS) [14, 15, 16, 17]. As a
consequence, these luminescent nanoparticles constitute a family of nanoprobes
extremely versatile and a complete understanding and characterization of these
emission lines are a prerequisite to unlock the full potential of
applications.

In this work, we are studying a specific transition of NaYF4:Eu3+
nanocrystals. The spectrum of trivalent europium ions strongly depends on the
site symmetry and are thus sensitive to the electronic structure. For
instance, the 5D${}_{0}\rightarrow^{7}$F0 transition is related to the
coordination number of the Eu3+ and the intensity of the robustness of the
5D${}_{0}\rightarrow^{7}$F1 transition may be used as a reference for
spectroscopic calibration [18]. This 5D${}_{0}\rightarrow^{7}$F1 transition
has a mainly MD character [18]. In 1941, Freed and Weissman analyzed wide
angle emission and concluded to the multipole nature of this line for randomly
oriented emitters in solution [19]. In 1980, Kunz and Lukosz measured the
lifetime in thin films of Eu3+ doped benzoyltrifluoroacetone chelate and
quantified the magnetic/electric ratio of the transition and found a
proportion of $80\%$ MD and $20\%$ ED in the spectral range $\lambda=592\pm 5$
nm [20]. Last, in 2012 Taminiau et al. adapted the method of Freed and
Weissman by using Fourier microscopy and obtained that the transition is
$91\pm 2\%$ MD for Y2O3:Eu3+ doped thin film [7]. All these samples involve
randomly oriented emitters but in rare-earth doped nanocrystals, the single
crystalline host matrix leads to ED or MD optical transitions that exhibit
strong polarization properties directly linked to the crystal orientation [21,
22]. The dipole transition has a fixed orientation with respect to the rod
axis and constitutes a vectorial nanoprobe. Also of strong interest here, we
use a single crystalline NaYF4 host matrix for the Eu3+ ions, limiting
multiphonon relaxation channels and ensuring well separated transition lines
[16]. Indeed, the 5D${}_{1}\rightarrow^{7}$F3 is an ED transition with an
emission wavelength ($\lambda=585$ nm) close to the
5D${}_{0}\rightarrow^{7}$F1 MD transition ($\lambda=592$ nm). Narrow lines are
thus needed to carefully characterize the multipole nature of the transitions.
The recorded data are correctly interpreted by considering a fully MD
transition. We measure the MD orientation and quantify the contribution of MD
transitions towards the Stark sublevels resulting from the crystal field
splitting of the otherwise degenerated 7F1 level. The work is organized as
follows. In Sec. II, we show the experimental measurements and modeling of the
spectroscopic properties of individual europium doped single crystalline NaYF4
nanorods. We record spectrally-resolved polarized emission diagram and deduce
the nature of the 5D${}_{0}\rightarrow^{7}$F1 transition as well as its dipole
moment orientation. In Sec. III, we measure the rod emission by Fourier plane
microscopy and discuss the contribution of the MDs within the spectral range.

##  II Optical transition dipole vector

###  II.1 Electric and magnetic dipole optical transitions

Figure 1: Schematic representation of the confocal optical microscope. The
Europium ions are excited by a continuous-wave laser emitting at a wavelength
of 392 nm and focused by a microscope objective (40$\times$, NA=0.6 or
60$\times$ NA=1.49, Nikon). The luminescence is collected by the same
objective and is separated from the excitation laser by a dichroic mirror and
routed either to an avalanche photodiode (APD) for confocal imaging or to a
spectrometer. The polarisation of the emitted light is analyzed placing a
polarizer after the sample (”analyzer”).

Europium doped single crystalline NaYF4:Eu3+ nanorods with an average diameter
of 90 nm and a length of 1500 nm are synthesized as described in Ref. [23, 24]
and are randomly dispersed on a quartz substrate (see appendix A). The
emission of single nanorod is acquired with a home made epi-confocal
microscope, depicted in Fig. 1. Depending on the measurement, we use either a
low numerical objective (NA = 0.6) or a high NA lens (NA =1.49). A power of
$236\text{\,}\mathrm{\SIUnitSymbolMicro W}$ at the focal point is necessary to
generate a detectable luminescence signal. To be sure that emission spectra
are originating from nanorods, a confocal image of the luminescence is first
acquired by scanning the sample with a piezoelectric stage and collecting the
light with a fiber coupled APD, see Fig. 2(b). The center of the nanorod is
then positioned to the focal area and an emission spectrum is recorded, see
Fig. 3(a). Scanning electron micrographs (SEM) of the measured rods are
acquired a posteriori and only single rod measurements are kept in the
analysis, see Fig. 2.

Figure 2: (a) SEM image of a NaYF4:Eu3+ single nanorod with a diameter of 90
nm and a length of 1500 nm. (b) Confocal image of the luminescence emitted by
the same rod.

The photoluminescence spectra of a single rod is characterized by distinct
emission lines, see Figure 3(a) corresponding to transitions from level
${}^{5}D_{0}$ or ${}^{5}D_{1}$ to ${}^{7}F_{J}$ with $J=$0…6. Each transition
has a degeneracy of $2J+1$, that can be lifted due to the crystal field
perturbation splitting of the ${}^{7}F_{J}$ levels into different Stark
sublevels [Figure 3(b)] [18]. According to the Judd-Ofelt model [25], all
these transitions correspond to ED except for the
${}^{5}D_{0}\rightarrow^{7}F_{1}$ transition (peak at 592 nm), which behaves
as a MD. Nanorods exhibit the hexagonal $\beta$-NaYF4 structure with the
c-axis corresponding to the main axis of the rods. The symmetry group of the Y
site is C3h. We thus expect two emission lines for the
${}^{5}D_{0}\rightarrow^{7}$F1 transition and one emission line for the
${}^{5}D_{0}\rightarrow^{7}$F2 transition [18]. However, because of the larger
ionic radius of Eu3+ compare to Y3+, a symmetry breaking down to a lower Cs
symmetry occurs in presence of doping Eu3+ at Y sites [26]. Therefore,
crystal-field splitting lifts the degeneracy of the Stark sublevels and we
observe three emission lines for the ${}^{5}D_{0}\rightarrow^{7}$F1 MD
transition but resolve three of the five emission lines for the
${}^{5}D_{0}\rightarrow^{7}$F2 ED transition. In the following, we use
confocal microscopy to study the ${}^{5}D_{0}\rightarrow^{7}$F1 MDs
orientation within individual rods. We measure the orientation of the dipole
transition that presents a fixed angle with respect to the c-axis.

Figure 3: (a) Spectrum of a single NaYF4:Eu3+ nanorod (obtained without the
analyzer, and with an objective NA=0.6). (b) Energy diagram of trivalent
Europium. The ion has a $4f^{6}$ configuration with optical transitions
occurring between the levels ${}^{5}D_{1}$ or ${}^{5}D_{0}$ and ${}^{7}F_{J}$.
Each level has a degeneracy of $2J+1$ split by the crystalline field into
different Stark levels.

###  II.2 Polarized emission diagram

####  II.2.1 Model of the luminescence emitted by rare-earth doped nanorods

##### Nanocrystal emission

We consider a single rod with its c-axis aligned with the substrate/air
interface (the general case is derived in the appendix B). The ED or MD
moments associated to a transition line have a fixed angle $\alpha$ with
respect to the c-axis as illustrated in Fig. 4. We assume a MD in the
following, but similar expressions hold for the electric field emitted by an
ED (see appendix C). Due to random distribution of emitters among the Y host
sites of the matrix, the rod emission is modeled by the incoherent emission of
dipolar emitters presenting a fixed orientation with respect to the c-axis
(angle $\alpha$), leaving free the angle $\psi$ in the plane perpendicular to
the c-axis. The dipole vector is ${\bf
p}=p_{0}(\cos\alpha,\sin\alpha\sin\psi,\sin\alpha\cos\psi)$ with
$\psi\in[0;2\pi[$.

Figure 4: Orientation of the dipole transition moment for a rod on the
substrate. The emission is modeled by dipolar emitters oriented at a fixed
angle with respect to the c-axis (cone of semi-angle $\alpha$).

The magnetic field scattered at ${\bf r}$ by a dipole located at ${\bf r_{0}}$
can be expressed thanks to the magnetic Green’s tensor ${\bf Q}$

$\displaystyle{\bf H}({\bf r})=\mu_{0}\omega_{0}^{2}{\bf Q}({\bf r},{\bf
r_{0}},\omega_{0})\cdot{\bf p}$ (1)

where $\omega_{0}$ refers to the angular frequency of the oscillating dipole,
corresponding to the emission line. The detected intensity in the medium of
optical index $n_{3}$ follows $I({\bf r})=\left|{\bf H}({\bf
r})\right|^{2}/n_{3}$. Finally, for a random orientation $\psi$ in the plane
perpendicular to the c-axis, the detected intensity is the incoherent emission
of all the emitting dipoles so that it can be expressed as (symbol $\propto$
means proportional to)

$\displaystyle{\bf I}({\bf r})$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
n_{3}}\int_{0}^{2\pi}\left|{\bf H}({\bf
r})\right|^{2}d\psi\propto\left(\left|Q_{xx}\right|^{2}+\left|Q_{yx}\right|^{2}+\left|Q_{zx}\right|^{2}\right)\cos^{2}\alpha$
$\displaystyle+\left(\left|Q_{xy}\right|^{2}+\left|Q_{yy}\right|^{2}+\left|Q_{zy}\right|^{2}\right)\frac{1}{2}\sin^{2}\alpha+\left(\left|Q_{xz}\right|^{2}+\left|Q_{yz}\right|^{2}+\left|Q_{zz}\right|^{2}\right)\frac{1}{2}\sin^{2}\alpha$

that is equivalent to the incoherent emission of three orthogonal dipoles

$\displaystyle{\bf p}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos\alpha\\\ 0\\\ 0\end{pmatrix}\,,{\bf
p}_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ \sin\alpha\\\
0\end{pmatrix}\,,{\bf p}_{3}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\\\
\sin\alpha\end{pmatrix}\,.$

##### Paraxial approximation

For a detector located on the confocal optical axis, just after an analyzer,
the asymptotic expansion of the Green’s tensor leads to the polarized emission
(see appendix C for details):

$\displaystyle I(\Phi_{A})=A\sin^{2}\Phi_{A}+B\cos^{2}\Phi_{A}$ (3)

where $\Phi_{A}$ refers to the angle of the analyzer with respect to the rod
c-axis. $A=(\sin^{2}\alpha)/2,B=\cos^{2}\alpha$ for an electric dipole, and
$A=\cos^{2}\alpha,B=(\sin^{2}\alpha)/2$ for a magnetic dipole. So, the emitter
angle with respect to the c-axis can be estimated from the fitting parameters
$A,B$ of the measured polarized diagrams discussed later in §II.2.2. Numerical
simulations (not shown) reveal that this expression also correctly reproduces
the polarized diagram detected with a low NA objective. Note that an isotropic
emission occurs for the magic angle $\alpha_{iso}=$
atan$(\sqrt{2})=$54.7\text{\,}\mathrm{\SIUnitSymbolDegree}$$ corresponding to
$A=B$.

####  II.2.2 Analysis of the polarized emission

Table 1: Angles $\alpha$ with respect to the c-axis of magnetic dipoles deduced from spectral analysis of single NaYF4:Eu3+ nanorods. Two models are considered: the paraxial approximation (parax., Eq. 3) and the full model (full, Eq. 23) with NA=0.6. The uncertainties with a 95% confidence interval are estimated from the fits. The mean and the standard deviation are also indicated in the last rows. | MD1 angle | MD2 angle | MD3 angle
---|---|---|---
Rod | parax. | full | parax. | full | parax. | full
1 | 65.7$\pm 0.1^{\circ}$ | 67.6$\pm 0.3^{\circ}$ | 70.0$\pm 0.2^{\circ}$ | 73.3$\pm 0.3^{\circ}$ | 37.3$\pm 0.3^{\circ}$ | 35.9$\pm 0.4^{\circ}$
2 | 66.0$\pm 0.1^{\circ}$ | 68.1$\pm 0.3^{\circ}$ | 69.0$\pm 0.3^{\circ}$ | 73.3$\pm 0.5^{\circ}$ | 37.2$\pm 0.3^{\circ}$ | 35.7$\pm 0.3^{\circ}$
3 | 66.6$\pm 0.1^{\circ}$ | 69.4$\pm 0.7^{\circ}$ | 69.7$\pm 0.2^{\circ}$ | 73.5$\pm 0.2^{\circ}$ | 38.0$\pm 0.2^{\circ}$ | 36.3$\pm 0.6^{\circ}$
4 | 65.3$\pm 0.2^{\circ}$ | 67.4$\pm 0.4^{\circ}$ | 67.7$\pm 0.2^{\circ}$ | 70.9$\pm 0.3^{\circ}$ | 38.4$\pm 0.9^{\circ}$ | 36.9$\pm 0.9^{\circ}$
5 | 65.0$\pm 0.1^{\circ}$ | 66.8$\pm 0.2^{\circ}$ | 68.3$\pm 0.1^{\circ}$ | 71.3$\pm 0.1^{\circ}$ | 38.3$\pm 0.2^{\circ}$ | 36.7$\pm 0.3^{\circ}$
mean | 65.7∘ | 67.9∘ | 68.9∘ | 72.5∘ | 37.8∘ | 36.3∘
std | 0.6∘ | 1.0∘ | 1.0∘ | 1.3∘ | 0.6∘ | 0.5∘

The emission of single nanorod are acquired with the home made epi-confocal
microscope described in section II.1 and equipped with the low NA objective to
stay within the paraxial approximation discussed above. The emission spectra
are recorded by varying the analyzer angle between 0∘ and 360∘ with a step of
10∘ and are shown in Fig. 5(a).

Figure 5: (a) Luminescence specra of single NaYF4:Eu3+ nanorod for different
analyzer angles. The 5D${}_{0}\rightarrow^{7}$F1 magnetic transitions are
highlighted in gray and zoomed in (b) for an analyzer angle of 0∘. Spectral
unmixing is performed by the fitting experimental data with a sum of three
Gaussians. Three magnetic dipoles can be isolated, they are noted MD1 (in red,
gaussian width 1.2 nm), MD2 (in green, gaussian width 0.8 nm) and MD3 (in
blue, gaussian width 1.5 nm). The normalized polarization diagrams of MD1, MD2
and MD3 are shown with dots in the polar plots (c1), (c2) and (c3),
respectively. For each dipole, a normalization is performed on the maximum
emission of single nanorod. The lines correspond to fits using the full model.

Let us focus on the 5D${}_{0}\rightarrow^{7}$F1 magnetic transitions occurring
between 588 nm and 598 nm (gray region). In this wavelength range, three main
peaks are clearly visible. They are reproduced in Fig. 5(b). For unmixing the
proportion of each transition, the spectra are fitted with a sum of three
Gaussian functions, represented in red, green and blue. We observe relatively
narrow lines ($\sim 1$ nm) thanks to the single crystalline NaYF4 host matrix,
so that the three MD transitions can be well separated at room temperature and
a full understanding of the polarization diagram can be proposed. Based on the
fitting amplitudes obtained for every analyzer angle, which are proportional
to the total integral for a Gaussian, the normalized polarization diagrams of
each of the transitions can be plotted. They are indicated with open circles
in Fig. 5(c). Note that the luminescence emitted by magnetic transitions 1 and
2 (noted MD1 and MD2) have a preferred polarization aligned with the c-axis,
whereas the luminescence from magnetic transition 3 (MD3) is emitted
perpendicularly to this axis. As already shown by Kim et al. [22], the
orientation angle of the nanorod can now be determined and the angle of each
magnetic dipole (from the c-axis) can be extracted by using the paraxial
approximation (see Eq. 3).

We determine all these parameters (dipole angle and nanorod angle) by fitting
the experimental points with the full theoretical model detailed in appendix
C, see Eq. 23. For rod #4, whose recorded spectra are displayed in Fig. 5(a),
the angles between each dipoles and the c-axis obtained in the paraxial
approximation are equal to 65.3$\pm$0.2∘ for MD1 (in red), 67.7$\pm$0.2∘ for
MD2 (in green) and 38.4$\pm$0.9∘ for MD3 (in blue). With the full model these
angles become: 67.4$\pm$0.4∘ for dipole 1, 70.9$\pm$0.3∘ for MD2 and
36.9$\pm$0.9∘ for dipole MD3. The difference between the angles deduced from
the paraxial approximation and the full model remains small (less than 3∘).

This procedure has been applied to several individual nanorods. The results
are summarized in Table 1. For all nanorods, the small deviation between the
values extracted from the paraxial approximation and the full model confirms
that the paraxial theory is valid for a numerical aperture lower than 0.7
[27]. The small standard deviation (less than 1.5∘) indicates a minor
dispersion of the angles. We conclude that the deduced dipole angles do not
dependent on the chosen nanorod because these angles reflect the dipolar
geometry resulting from the crystal field splitting of sublevels.

##  III Fourier plane leakage radiation microscopy

The characterization of a single nanorod dipolar emission may also be
performed with an other method called Fourier plane microscopy [28, 29, 7].
Unlike spectroscopic measurement based on a low NA detection, this method uses
a high numerical aperture oil immersion objective (NA 1.49, Nikon) and a
camera adequately positioned in a conjugate Fourier plane of the microscope.
The same continuous laser of wavelength 392 nm was used and a power of
$287.5\text{\,}\mathrm{\SIUnitSymbolMicro W}$ (at the focal point) is used to
excite Europium ions. The luminescence is separated from the excitation laser
with the same dichroic mirror, passed through a rotating analyzer and is
detected with a scientific CMOS camera (Andor Zyla) placed in the Fourier
plane. To achieve a good signal to noise ratio, we use a binning of $4\times
4$ and an integration times of 15 s. Two distinct band pass filters are used:
from 586.8 to 593.6 nm (noted BP1) selecting equivalently MD1 and MD2
(proportions $p_{1}\approx p_{2}\approx 50\%$), and from 591 nm to 598.6 nm
(noted BP2) selecting predominantly MD3 ($p_{3}\approx 88\%$) compared to MD2
($p_{2}\approx 12\%$).

Typical emission patterns from a single nanorod with BP1 are shown in Fig.
6-(a1), (a2) and (a3) for three orientations of the analyzer. We observe two
distinct lobes of higher intensity at an angle near the critical angle
$\theta_{c}$, with $\theta_{c}=43.3^{\circ}$ for an air-quartz interface.
These two lobes slightly rotate with the analyzer angle (see video 1) and they
are almost in the same direction as the rod axis, which is in agreement with
the polarization emission plots discussed in Fig. 5-(c1) and (c2). A
normalization of the intensity is performed on the maximal emission measured
for all analyzer angles.

These experimental Fourier images can be fitted with the full theoretical
model detailed in appendix C (see Eq. 22). The calculated Fourier plane image
are shown in 6-(b1), (b2) and (b3), respectively. The agreement between the
experimental and simulated Fourier images is very good both on the overall
distribution of the intensity and the angular position of the lobes as
confirmed by the intensity profiles in Fig. 6-(c1), (c2) and (c3). From the
simulated Fourier planes; we are in position to determine the angles between
the magnetic dipoles MD1, MD2 and the c-axis as well as the proportion
$p_{1}$. For rod #4 for instance, we find angles of 67.4∘ and 70.8∘ for MD1
and MD2 respectively and a proportion of 50%. These angles are almost
identical to those obtained with the spectroscopic analysis (see previous
section).

Figure 6: Fourier imaging of single NaYF4:Eu3+ nanorod for 586.8-593.6 nm wavelength range. Experimental Fourier images acquired with analyzer angle of 0∘, 60∘ and 120∘ are shown respectively in a1, a2 and a3. The white circle with larger diameter indicates the maximum collection angle corresponding to the numerical aperture of the objective (NA=1.49). We also represent the circle indicating the critical angle for an air-quartz interface (corresponding to NA=1.0). These experimental images are compared to the theoretical ones shown in b1, b2 and b3 for an analyzer angle of respectively 0∘, 60∘ and 120∘. For each analyzer angle, we have represented in c1, c2 and c3, the experimental (in solid blue line) and theoretical (in dashed red line) intensity profiles corresponding to the white dashed lines in a) and b).  Table 2: Angles of magnetic dipoles deduced from Fourier images of single NaYF4:Eu3+ nanorods. Two band pass filters are considered: BP1 (586.8-593.6 nm) and BP2 (591.0-598.6 nm). Due to their large spectral widths, the emission of two dipoles are simultaneously collected. The proportion of MD1 (noted $p_{1}$), the proportion of MD3 (noted $p_{3}$) and the distance $z_{0}$ to the substrate are also estimated. The mean and the standard deviation are indicated in the last rows. | BP1 | BP2 |
---|---|---|---
Rod | MD1 | MD2 | $p_{1}$ | MD2 | MD3 | $p_{3}$ |  $z_{0}$ (nm)
1 | 68.2∘ | 73.6∘ | 0.49 | 73.6∘ | 36.2∘ | 0.87 | 65.6
2 | 67.9∘ | 70.6∘ | 0.50 | 72.9∘ | 35.1∘ | 0.89 | 50.1
3 | 69.6∘ | 73.6∘ | 0.50 | 73.4∘ | 36.6∘ | 0.88 | 61.4
4 | 67.4∘ | 70.8∘ | 0.50 | 69.9∘ | 35.3∘ | 0.90 | 57.8
5 | 66.5∘ | 71.2∘ | 0.50 | 71.2∘ | 36.9∘ | 0.88 | 63.2
mean | 67.9∘ | 72.0∘ | 0.50 | 72.2∘ | 36.0∘ | 0.88 | 59.6
std | 1.1∘ | 1.5∘ | 0.01 | 1.6∘ | 0.8∘ | 0.01 | 6.0
Figure 7: Video1. Fourier imaging of single NaYF4:Eu3+ nanorod for 586.8-593.6
nm wavelength range. Left) Experimental and right) calculated Fourier plane
images as a function of analyzer angle from 0∘ to 180∘ by step of 30∘.

We have applied the same procedure for analyzing the Fourier images acquired
with the second band pass filter (BP2) shown in Fig. 8-(a1), (a2) and (a3).
The two lobes rotate with the analyzer angle (see video 2) and are now
perpendicular to the c-axis, as expected from Fig. 5(c3). By fitting these
experimental images with the full theoretical model, we find angles of 69.9∘
and 35.3∘ for MD2 and MD3 respectively and a MD3 proportion of 90%. Again,
these angles are close to those obtained in the previous section.

The results obtained for several individual nanorods are summarized in Table
2. The angles are very similar to those obtained with full model used in
conjunction of the spectral analysis, confirming that this method is robust
for determining dipolar emission angles. The variability of the angle
measurements is small and within the experimental error, estimated to 2∘ [30],
confirming that it does not depend on the chosen rod but reveals the local
site symmetry of Europiums ions.

We emphasize that we fully explain the recorded Fourier images considering a
fully magnetic transition but taking into account the contribution of the two
MD transitions within the detected spectral range. Differently, several works
considered randomly oriented emitters in solution or thin films and observe
that the MD transition could present an electric dipole character [19, 20, 7],
as discussed in the introduction. The ED contribution could originate from e.g
the perturbation of the selection rules depending on the host matrix or from
the contribution of the neighbor 5D${}_{1}\rightarrow^{7}$F3 ED line [31].
Further investigations are necessary to conclude. For instance, low
temperature measurement would help separating the emission lines.

Furthermore, Fourier imaging performed with a high NA objective collects light
emitted towards angles larger than the critical angle. Under this condition,
the distance between the nanorod and the substrate (noted $z_{0}$) is critical
and should be considered as an additional fitting parameter in the model. As
indicated in Table 2, we find an effective height of 59.6 nm which is in
agreement with the dimension of the nanorod. This effective height is due to
the finite diameter of the rod and consequently an integrated signal pickup
over a finite rod volume. More precisely, the dipolar emission above the
critical angle originates from the evanescent coupling into the substrate
$e^{-\kappa_{1}z_{0}}\;,\kappa_{1}=\sqrt{\epsilon_{3}\sin^{2}\theta-\epsilon_{1}}$
for $\theta>\theta_{c}$ (see Eq. 17). Therefore, high NA Fourier microscopy
gives access to the effective height of the rod onto the substrate. It
represents the ponderation by the evanescent coupling over the rod height and
can be used to extract information on the distance between the rod and the
substrate.

Figure 8: Fourier imaging of single NaYF4:Eu3+ nanorod for 591.0-598.6 nm
wavelength range. Experimental Fourier images acquired with analyzer angle of
0∘, 60∘ and 120∘ are shown respectively in a1, a2 and a3. The white circle
with larger diameter indicates the maximum collection angle corresponding to
the numerical aperture of the objective (NA=1.49). We also represent the
circle indicating the critical angle for an air-quartz interface
(corresponding to NA=1.0). These experimental images are compared to the
theoretical ones shown in b1, b2 and b3 for an analyzer angle of respectively
0∘, 60∘ and 120∘. For each analyzer angle, we have represented in c1, c2 and
c3, the experimental (in solid blue line) and theoretical (in dashed red line)
intensity profiles corresponding to the white dashed lines in a) and b.
Figure 9: Video 2. Fourier imaging of single NaYF4:Eu3+ nanorod for
591.0-598.6 nm wavelength range. Left) Experimental and right) calculated
Fourier plane images as a function of analyzer angle from 0∘ to 180∘ by step
of 30∘.

##  IV Conclusion

In summary, we fully characterized the magnetic dipole moment associated to
the 5D${}_{0}\rightarrow^{7}$F1 transition in individual NaYF4:Eu3+
nanocrystals by two different approaches. We determine the orientation of the
three magnetic dipoles for the Stark sublevels originating from the crystal
field degeneracy breaking. The orientation of the dipole moment is an
intrinsic property of the synthesized nanorods and can be used for measuring
their absolute orientation [32, 33]. In addition, since the optical transition
properties are fully characterized and presents low variability from one rod
to another, such nanocrystals could be used for mapping of the magnetic field
and LDOS near nanophotonics structures, notably metamaterials and nano-
antennas. In particular, fluorescence lifetime of rare-earth doped
nanoparticles probes the electric or magnetic LDOS in complex environnement
[14, 15, 16, 17]. Random orientation of the dipolar emitters is generally
considered so that lifetime measures LDOS without information on the mode
polarization. We expect additional information on the mode polarization
considering the fixed dipole moment we characterized in NaYF4:Eu3+ single
crystalline nanorods. This will be investigated in a future work. Using a high
NA objective to pick up frustrated evanescent wave contribution of the dipolar
emission into the substrate, we also determine an effective height of the rod
dipole moment. The sensitivity of the Fourier imaging with respect to the
altitude of the rod is also of interest to calibrate near-field optical
measurement. It is worth mentioning that using a single crystalline NaYF4 host
matrix is a key parameter in this work since it leads to very narrow and well
separated emission lines at room temperature clarifying the role of the Stark
sublevels in the luminescence signal. Another significant advantage is the
ability to synthesize this compound as nanorods with very good control of size
and shape, thus allowing to easily characterize and control the spatial
orientation of the crystal and the crystalline axis, for full engineering of
their spectroscopic properties at the nanoscale level.

###### Acknowledgements.

We gratefully acknowledge Frédéric Herbst from the technological platform
ARCEN Carnot for SEM imaging of single nanorods. The platform ARCEN Carnot is
financed by the Région de Bourgogne Franche-Comté and the Délégation Régionale
à la Recherche et à la Technologie (DRRT). This work is supported by the
French Investissements d’Avenir program EUR-EIPHI (17-EURE-0002) and French
National Research Agency (ANR) project SpecTra (ANR-16-CE24-0014).

##  Appendix A Synthesis of single crystalline NaYF4:Eu3+nanorods

45 mmol (1.8 g) of NaOH in 6 mL of water were mixed with 15 mL of ethanol
(EtOH) and 30 mL of oleic acid (OA) under stirring. To the resulting mixture
were selectively added 0.95 mmol (288 mg) of Y(Cl)3,6H2O, 0.05 mmol (18 mg) of
Eu(Cl)3,6H2O and 10.2 mmol (377mg) of NH4F dissolved in 4 mL of water. The
solution was then transferred into a 75 mL autoclave and heated at
$200\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 24 h under stirring. After
cooling down to ambient temperature, the resulting nanoparticles were
precipitated by addition of 50 mL of ethanol, collected by centrifugation, and
washed three times with water and ethanol. They were finally dried under
vaccum and recovered as a white powder. For the optical experiments, a
functionalization by ligand exchange is needed to ensure the good dispersion
of the particles in water. About 20 mg of the NaYF${}_{4}@$oleic acid NPs are
dispersed in 2 ml of an aqueous 0.2 M sodium citrate solution. After
sonication and centrifugation, the process of ligand exchange is repeated two
more times. Finally, particles are washed three times with water and ethanol,
and dispersed with sonication in pure water. Sample for optical
characterizations are then prepared by spin-coating a droplet of this rod
solution (concentration 0.014% vol.) on a quartz substrate at 2000 rpm during
60s.

##  Appendix B Arbitrary rod orientation

For arbitrary orientation of the rod on the substrate, we define the angle
$\beta$ with respect to the normal to the substrate ($\beta=\pi/2$ for a rod
c-axis along the substrate). The dipole moment associated to a transition line
presents a fixed angle $\alpha$ with respect to the c-axis, and is expressed
${\bf p}=p_{0}{\cal{\bf
R}}_{y}(\beta)(\sin\alpha\cos\psi,\sin\alpha\sin\psi,\cos\alpha)$, where
${\cal{\bf R}}_{y}$ refers to the rotation around the y-axis. The magnetic
field scattered at ${\bf r}$ by a dipole located at ${\bf r_{0}}$ can be
expressed thanks to the magnetic Green’s tensor ${\bf Q}$

$\displaystyle{\bf H}({\bf r})=\mu_{0}\omega_{0}^{2}{\bf Q}({\bf r},{\bf
r_{0}},\omega_{0})\cdot{\bf p}$ (4)

and the incoherent emission of all the emitting dipole (for $\psi\in[0;2\pi[$)
can be expressed

$\displaystyle{\bf I}({\bf r})=\frac{1}{2\pi n_{3}}\int_{0}^{2\pi}\left|{\bf
H}({\bf
r})\right|^{2}d\psi\propto\left(\left|Q_{xx}\right|^{2}+\left|Q_{yx}\right|^{2}+\left|Q_{zx}\right|^{2}\right)\left(\frac{1}{2}\cos^{2}\beta\sin^{2}\alpha+\sin^{2}\beta\cos^{2}\alpha\right)$
(5)
$\displaystyle+\left(\left|Q_{xy}\right|^{2}+\left|Q_{yy}\right|^{2}+\left|Q_{zy}\right|^{2}\right)\frac{1}{2}\sin^{2}\alpha+\left(\left|Q_{xz}\right|^{2}+\left|Q_{yz}\right|^{2}+\left|Q_{zz}\right|^{2}\right)\left(\frac{1}{2}\sin^{2}\beta\sin^{2}\alpha
s+\cos^{2}\beta\cos^{2}\alpha\right)$
$\displaystyle+2Re\left(Q_{xx}G_{xy}+Q_{yx}G_{xz}+Q_{zx}Q_{zz}\right)\times\left(-\frac{1}{2}\cos\beta\sin\beta\sin^{2}\alpha+\sin\beta\cos\beta\cos^{2}\alpha\right)$

that is equivalent to the incoherent emission of three orthogonal dipoles

$\displaystyle{\bf p}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\sin\beta\cos\alpha\\\ 0\\\
\cos\beta\cos\alpha\end{pmatrix}\,,{\bf
p}_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ \sin\alpha\\\
0\end{pmatrix}\,,{\bf
p}_{3}=\frac{1}{\sqrt{2}}\begin{pmatrix}-\cos\beta\sin\alpha\\\ 0\\\
\sin\beta\sin\alpha\end{pmatrix}\,.$

Analogous expressions hold for the electric field emitted by an ED.

##  Appendix C Far-field ED and MD scattering

Figure 10: Notation for the scattered dipolar field.

###  C.1 Electric dipole

####  C.1.1 Electric field Green’s tensor

The electric field scattered at the position ${\bf r}$ from an ED located at
${\bf r}_{0}$ is

$\displaystyle{\bf E}({\bf r})=\frac{k_{0}^{2}}{\epsilon_{0}}{\bf G}({\bf
r},{\bf r_{0}},\omega_{0})\cdot{\bf p}$ (6)

where ${\bf G}$ is the electric field Green’s tensor. For $k_{0}r\gg 1$, far-
field appromixation holds and [34]

$\displaystyle{\bf G}({\bf r},{\bf
r}_{0})\underset{r\to\infty}{\sim}\frac{n_{3}}{n_{1}}\frac{e^{ik_{3}r}}{r}\cos\theta
e^{iw_{1}z_{0}}{\bf g}$

with
$k_{1}=n_{1}k_{0}\;,k_{3}=n_{3}k_{0}\;,w_{1}=(k_{1}^{2}-k_{3}^{2}\sin^{2}\theta)^{1/2}\;,[Im(w_{1})\geq
0]$ and

$\displaystyle g_{xx}$ $\displaystyle=$ $\displaystyle
t_{p}\cos\theta\cos^{2}\varphi-\frac{k_{1}}{w_{1}}t_{s}\sin^{2}\varphi$
$\displaystyle g_{xy}$ $\displaystyle=$ $\displaystyle
t_{p}\cos\theta\cos\varphi\sin\varphi+\frac{k_{1}}{w_{1}}t_{s}\cos\varphi\sin\varphi$
$\displaystyle g_{xz}$ $\displaystyle=$
$\displaystyle\frac{k_{3}}{w_{1}}t_{p}\cos\theta\sin\theta\cos\varphi$
$\displaystyle g_{yx}$ $\displaystyle=$ $\displaystyle g_{xy}$ $\displaystyle
g_{yy}$ $\displaystyle=$ $\displaystyle
t_{p}\cos\theta\sin^{2}\varphi-\frac{k_{1}}{w_{1}}t_{s}\cos^{2}\varphi$
$\displaystyle g_{yz}$ $\displaystyle=$
$\displaystyle\frac{k_{3}}{w_{1}}t_{p}\cos\theta\sin\theta\sin\varphi$
$\displaystyle g_{xz}$ $\displaystyle=$ $\displaystyle-
t_{p}\sin\theta\cos\varphi$ $\displaystyle g_{yz}$ $\displaystyle=$
$\displaystyle-t_{p}\sin\theta\sin\varphi$ $\displaystyle g_{yz}$
$\displaystyle=$ $\displaystyle-\frac{k_{3}}{w_{1}}t_{p}\sin^{2}\theta$

$t_{s}$ ($t_{p}$) refer to the fresnel air/glass transmission coefficient for
TE (TM) polarized light.

####  C.1.2 Polarized emission

The emission polarization is analysed by confocal microscopy. For numerical
simulations, we first decompose the electric field in TE and TM polarization.

$\displaystyle{\bf E}=E_{TM}{\bf u}_{p}+E_{TE}{\bf u}_{s}$ (7)

where we define the unitary vectors ($k_{p}^{2}=k_{x}^{2}+k_{y}^{2}$)

$\displaystyle{\bf u}_{p}=\frac{1}{k_{3}k_{p}}\begin{pmatrix}k_{x}w_{3}\\\
k_{y}w_{3}\\\ -k_{p}^{2}\end{pmatrix}\;;{\bf
u}_{s}=\frac{1}{k_{p}}\begin{pmatrix}k_{y}\\\ -k_{x}\\\ 0\end{pmatrix}$ (8)

After refraction on the reference sphere (see Fig. 10), the components of the
electric field are

$\displaystyle E_{fx}$ $\displaystyle=$ $\displaystyle-
E_{TM}\cos\varphi+E_{TE}\sin\varphi$ (9) $\displaystyle E_{fy}$
$\displaystyle=$ $\displaystyle-E_{TM}\sin\varphi-E_{TM}\cos\varphi$ (10)

Finally, if we note $\Phi_{A}$ the angle of the analyzer with respect to the
x-axis, the polarized intensity in the Fourier plane is expressed as

$\displaystyle
I(\theta,\varphi)=\frac{n_{3}}{|cos\theta|}\lvert\cos(\Phi_{A})E_{fx}+\sin(\Phi_{A})E_{fy}\rvert^{2}$
(11)

where the apodization factor $1/|\cos\theta|$ ensures the energy conservation.
In addition, the polarization diagram is obtained after integration over all
the numerical aperture ($\theta_{NA}=asin(NA/n_{3})$).

$\displaystyle
I_{pol}(\Phi_{A})=\int_{\pi}^{\pi+\theta_{NA}}d\theta\int_{0}^{2\pi}d\varphi
I(\theta,\varphi)|\cos\theta\sin\theta|$ (12)

For detector located on the confocal optical axis ($\theta=\pi,\varphi=0$),
$t_{p}=t_{s}$ and the incoherent emission of the three orthogonal dipoles

$\displaystyle{\bf p}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos\alpha\\\ 0\\\ 0\end{pmatrix}\,,{\bf
p}_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ \sin\alpha\\\
0\end{pmatrix}\,,{\bf p}_{3}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\\\
\sin\alpha\end{pmatrix}\,.$

simplifies to

$\displaystyle I_{pol}(\Phi_{A})$ $\displaystyle\propto$ $\displaystyle
A\sin^{2}\Phi_{A}+B\cos^{2}\Phi_{A}\;,\text{with}$ (13) $\displaystyle A$
$\displaystyle=$ $\displaystyle\frac{\sin^{2}\alpha}{2}$ (14) $\displaystyle
B$ $\displaystyle=$ $\displaystyle\cos^{2}\alpha$ (15)

###  C.2 Magnetic dipole

For a magnetic dipole, we have the analogous expressions

$\displaystyle{\bf H}({\bf r})=\mu_{0}\omega^{2}{\bf Q}({\bf r},{\bf
r_{0}},\omega_{0})\cdot{\bf p}$ (16)

with the magnetic Green tensor

$\displaystyle{\bf Q}({\bf r},{\bf
r}_{0})\underset{r\to\infty}{\sim}\epsilon_{3}\frac{e^{ik_{3}r}}{r}\cos\theta
e^{iw_{1}z_{0}}{\bf q}$ (17) $\displaystyle q_{xx}$ $\displaystyle=$
$\displaystyle
t_{s}\cos\theta\cos^{2}\varphi-\frac{k_{1}}{w_{1}}t_{p}\sin^{2}\varphi$
$\displaystyle q_{xy}$ $\displaystyle=$ $\displaystyle
t_{s}\cos\theta\cos\varphi\sin\varphi+\frac{k_{1}}{w_{1}}t_{p}\cos\varphi\sin\varphi$
$\displaystyle q_{xz}$ $\displaystyle=$
$\displaystyle\frac{k_{3}}{w_{1}}t_{s}\cos\theta\sin\theta\cos\varphi$
$\displaystyle q_{yx}$ $\displaystyle=$ $\displaystyle q_{xy}$ $\displaystyle
q_{yy}$ $\displaystyle=$ $\displaystyle
t_{s}\cos\theta\sin^{2}\varphi-\frac{k_{1}}{w_{1}}t_{p}\cos^{2}\varphi$
$\displaystyle q_{yz}$ $\displaystyle=$
$\displaystyle\frac{k_{3}}{w_{1}}t_{s}\cos\theta\sin\theta\sin\varphi$
$\displaystyle q_{xz}$ $\displaystyle=$ $\displaystyle-
t_{s}\sin\theta\cos\varphi$ $\displaystyle q_{yz}$ $\displaystyle=$
$\displaystyle-t_{s}\sin\theta\sin\varphi$ $\displaystyle q_{yz}$
$\displaystyle=$ $\displaystyle-\frac{k_{3}}{w_{1}}t_{s}\sin^{2}\theta$

The TE/TM decomposition follows

$\displaystyle{\bf H}=H_{TM}{\bf v}_{p}+H_{TE}{\bf v}_{s}$ (18)

with the unitary vectors

$\displaystyle{\bf v}_{s}=\frac{1}{k_{3}k_{p}}\begin{pmatrix}k_{x}w_{3}\\\
k_{y}w_{3}\\\ -k_{p}^{2}\end{pmatrix}\;;{\bf
v}_{p}=\frac{1}{k_{p}}\begin{pmatrix}-k_{y}\\\ k_{x}\\\ 0\end{pmatrix}\;.$
(19)

After refraction on the reference sphere (see Fig. 10), the components of the
magnetic field are

$\displaystyle H_{fx}$ $\displaystyle=$ $\displaystyle-H_{TE}\cos\varphi-
H_{TM}\sin\varphi$ (20) $\displaystyle H_{fy}$ $\displaystyle=$
$\displaystyle-H_{TE}\sin\varphi+H_{TE}\cos\varphi$ (21)

Finally, the polarized intensity in the Fourier plane is expressed as

$\displaystyle
I(\theta,\varphi)=\frac{1}{n_{3}|\cos\theta|}\lvert-\sin(\Phi_{A})H_{fx}+\cos(\Phi_{A})H_{fy}\rvert^{2}\hskip
21.33955pt$ (22)

The polarized emission recorded in the confocal microscope of numerical
aperture NA is

$\displaystyle
I_{pol}(\Phi_{A})=\int_{\pi}^{\pi+\theta_{NA}}d\theta\int_{0}^{2\pi}d\varphi
I(\theta,\varphi)|\cos\theta\sin\theta|\hskip 21.33955pt$ (23)

In the paraxial approximation, the incoherent emission of the three orthogonal
dipole defined above simplifies to

$\displaystyle I_{pol}(\Phi_{A})$ $\displaystyle=$ $\displaystyle
A\sin^{2}\Phi_{A}+B\cos^{2}\Phi_{A}\;,\text{with}$ (24) $\displaystyle A$
$\displaystyle=$ $\displaystyle\cos^{2}\alpha$ (25) $\displaystyle B$
$\displaystyle=$ $\displaystyle\frac{\sin^{2}\alpha}{2}$ (26)

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# Superconducting boron doped nanocrystalline diamond microwave coplanar
resonator

Jerome A. Cuenca<EMAIL_ADDRESS>Thomas Brien  Soumen Mandal  Scott
Manifold  Simon Doyle  Adrian Porch  Georgina M. Klemencic  Oliver A. Williams
School of Physics and Astronomy, Cardiff University, Cardiff, Wales, CF24 3AA,
UK  School of Engineering, Cardiff University, Cardiff, Wales, CF24 3AA, UK

###### Abstract

A superconducting boron doped nanocrystalline diamond (B-NCD) coplanar
waveguide resonator (CPR) is presented for kinetic inductance ($L_{k}$) and
penetration depth ($\lambda_{\rm{L}}$) measurements at microwave frequencies
of 0.4 to 1.2 GHz and at temperatures below 3 K. Using a simplified effective
medium CPR approach, this work demonstrates that thin granular B-NCD films
($t\approx$ $500\text{\,}\mathrm{nm}$) on Si have a large penetration depth
($\lambda_{\rm{L}}\approx 4.3$ to $4.4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$), and therefore an associated high kinetic inductance
($L_{k,\square}\approx$ 670 to 690 pH/$\square$). These values are much larger
than those typically obtained for films on single crystal diamond which is
likely due to the significant granularity of the nanocrystalline films. Based
on the measured Q factors of the structure, the calculated surface resistance
in this frequency range is found to be as low as $\approx$ 2 to
$4\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}$ at $T<2$ K,
demonstrating the potential for granular B-NCD for high quality factor
superconducting microwave resonators and highly sensitive kinetic inductance
detectors.

###### keywords:

Boron doped diamond, superconducting coplanar resonator, nanocrystalline
diamond, granularity.

††journal: Journal of LaTeX Templates

##  1 Introduction

Boron doped diamond (BDD) is electrically conducting at concentrations in
excess of $10^{20}$ cm-3[1, 2] and superconducts at low temperatures in excess
of $10^{21}$ cm-3, with an observed critical onset temperature ($T_{c}$)
ranging between 3 to 5 K at zero field and a high type-II upper critical field
($H_{c}$) of up to 8 T[3, 4, 5, 6, 7, 8, 9]. BDD is typically grown using
chemical vapour deposition (CVD) either on single crystal diamond substrates
(B-SCD) or on Si to produce heavily granular nanocrystalline diamond films
(B-NCD). These films can be made into superconducting quantum interference
devices (SQUIDs) as demonstrated by Kageura et al. on B-SCD[8] and in earlier
reports by Mandal et al. on B-NCD[10, 11, 7]. In order to understand the
effective areas of BDD SQUIDs and also utilise BDD for other superconducting
applications, such as microwave devices, further understanding of the magnetic
penetration depth ($\lambda_{\rm{L}}$), is needed. For B-SCD, moderate
penetration depths have been observed ($\lambda_{\rm{L}}\approx$ 0.2 to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$)[4, 12]. However, for granular
B-NCD, a recent study by Oripov et al. has demonstrated very large penetration
depths ($\lambda_{\rm{L}}\approx 2$ to $4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$) in the microwave frequency range, thereby enabling highly sensitive
microwave kinetic inductance ($L_{k}$) detectors[13]. The origin for the large
penetration depths in B-NCD is most likely related to the granularity of the
superconductor which is observed for other materials; for example, non-
granular aluminium films exhibit very small penetration depths
($\lambda_{\rm{L}}\approx$ 10 to $30\text{\,}\mathrm{nm}$[14]), however
granular aluminium films have much larger values ($\lambda_{\rm{L}}\approx$
0.3 to $1.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$[15]).

A method for investigating the penetration depth and kinetic inductance of
materials is by using the coplanar waveguide resonator (CPR) method[16, 17].
This approach involves patterning the superconductor into a planar microwave
structure with a known resonant frequency and attributing differences to the
designed frequency to the kinetic inductance, and therefore the penetration
depth. Since high quality factor resonators can be fabricated, this approach
offers a highly sensitivity measurement. Additionally, the CPR method is very
convenient for CVD films since a CPR only requires one side of a substrate to
patterned without the need for any additional ground planes on the underside.
The CPR method is less-widely reported for superconducting BDD, despite being
a well-known approach for understanding $L_{k}$ in a wide variety of materials
including aluminium[18, 19, 20], niobium nitride[16] and yttrium barium copper
oxide[21, 22, 23].

In this work the seemingly high $L_{k}$ and $\lambda_{\rm{L}}$ of granular
B-NCD is investigated using the microwave CPR method. Section 2 details a
simplified analytical model for the approximation of $L_{k}$ in a CPR using an
effective medium. Section 3 briefly demonstrates a finite element model (FEM)
to identify the resonant modes and implement the effective medium approach.
Section 4 details the device fabrication and characterisation process.
Finally, Sections 5 and 6 details the experimental results, including Raman
spectroscopy, scanning electron microscopy (SEM) and microwave power
transmission spectra.

##  2 Theory

Figure 1: 3D render schematic of the device layout (inset not to scale). ‘P1’
denotes port for the microwave signal lines and ‘GND’ denotes the ground
planes.

A microwave CPR is a well understood structure, with a designed resonant
frequency related to the electrical length of the transmission line and its
phase velocity. For a half-wavelength ($\lambda/2$) resonator with an infinite
number of integer modes, the analytical frequencies are given by[24, 25]:

$f_{r}=\frac{nc}{2l\sqrt{\mu_{\rm{r,eff}}\varepsilon_{\rm{r,eff}}}}=\frac{n}{2l\sqrt{L_{\rm{0}}C_{\rm{0}}}}$
(1)

where $f_{r}$ is the resonant frequency in Hz, $c$ is the speed of light in
m/s, $l$ is the length of the transmission line in m, $\lambda$ is the
wavelength in m, $n$ is the integer mode number, $\mu_{\rm{r,eff}}$ and
$\varepsilon_{\rm{r,eff}}$ are the effective relative permeability and
permittivity of the medium, respectively and $L_{\rm{0}}$ and $C_{\rm{0}}$
denote the total inductance and capacitance per unit length of the structure
in H/m and F/m, respectively. $\mu_{\rm{r,eff}}$ is assumed unity and
$\varepsilon_{\rm{r,eff}}$ is the average relative permittivity of the air
space region and the substrate
($\varepsilon_{\rm{r,eff}}\approx[\varepsilon_{\rm{r,sub}}+1]/2$, where
$\varepsilon_{\rm{r,sub}}$ is the relative permittivity of the substrate). In
the absence of any superconductivity, the only contribution to the stored
electromagnetic energy in the resonator is the per unit length inductance
($L_{g}$) and capacitance ($C_{g}$) of the structure, or the ‘geometric’
contributions, which through conformal mapping are estimated as[16]:

$\displaystyle L_{g}$
$\displaystyle=\frac{\mu_{0}}{4}\frac{K\left(k^{\prime}\right)}{K\left(k\right)}$
(2) $\displaystyle C_{g}$
$\displaystyle=4\varepsilon_{0}\varepsilon_{\rm{r,eff}}\frac{K\left(k\right)}{K\left(k^{\prime}\right)}$
(3) $\displaystyle k$ $\displaystyle=\frac{w}{w+2s}$ (4) $\displaystyle
k^{\prime}$ $\displaystyle=\sqrt{1-k^{2}}$ (5)

where $\mu_{0}$ and $\varepsilon_{0}$ are the permeability and permittivity of
free space in H/m and F/m respectively, $K$ is the complete elliptical
integral of the first kind, $w$ is the width of the centre conductor in m and
$s$ is the gap between the centre conductor and the ground plane in m. For a
superconductor however, as the temperature is varied and transitions from the
normal state into the superconducting state, the inertia of the migrating
Cooper pairs becomes non-negligible and $L_{k}$ adds to the total inductance
($L_{0}=L_{g}+L_{k}(T)$)[16]. This kinetic inductance is a function of the
magnetic penetration depth:

$L_{k}(T)=\mu_{0}\frac{\lambda_{\rm{L}}(T)^{2}}{dw}g(w,s,d)$ (6)

where $\lambda_{\rm{L}}(T)$ is the temperature dependent magnetic penetration
depth, $d$ is the film thickness, $g$ is the structural dependent geometric
factor[16]. The shift in frequency caused by $L_{k}$, can be modelled as a
change in the phase velocity by introducing an effective medium. In a simple
model, $\mu_{\rm{r,eff}}$ in (1) can be modelled using (2), (3) and (6):

$\mu_{\rm{r,eff}}(T)\approx\left[\frac{1}{4}\frac{K\left(k^{\prime}\right)}{K\left(k\right)}+\frac{\lambda(T)^{2}}{dw}g(w,s,d)\right]\times
4\frac{K\left(k\right)}{K\left(k^{\prime}\right)}$ (7)

Note that in the absence of any kinetic inductance (i.e. $\lambda(T)=0$),
$\mu_{\rm{r,eff}}=1$. Thus, the difference between the measured frequency
($f_{m}(T)$) and the analytical frequency ($f_{r}$) in (1) can be attributed
to $L_{k}$ and $\lambda(T)$. Additionally, the quality factor of the CPR can
be related to the effective surface resistance using the following[23]:

$R_{s,\rm{eff}}(T)=\mu_{0}\lambda(T)\frac{\pi
f_{m}(T)}{Q_{U}(T)}\frac{L_{k}+L_{g}}{L_{k}}$ (8)

where $R_{s,\rm{eff}}(T)$ is the temperature dependent surface resistance in
ohms and $Q_{U}$ is the unloaded quality factor.

The CPR can be terminated with either short or open circuits, to which
microwaves are coupled to the resonator using inductive or capacitive
coupling, respectively. There are numerous methods of probing the resonance
including using a feed-line or transmission directly through the resonator.
The feed line approach ‘taps’ off the microwaves to the CPR and is extremely
advantageous for multiplexing multiple resonators on the same chip[17].
However, for a sensitive readout, the coupling reactance needs to be
sufficiently large in order to observe the resonance minimum. The coupling
structures further load the resonator which results in a measured frequency
and $Q$ factor containing contributions from both the antennas and the
intrinsic resonance[24]. In the direct transmission approach, power
transmission only occurs at resonance when the device superconducts and
typically only one device can be measured. However assuming a high dynamic
range can be measured through the instrumentation, extremely weak coupling can
be achieved such that the measured properties are approximated to the
intrinsic resonance[26]. For simple material investigations, the latter
approach is favourable, as to minimise unknown variability caused by the
coupling structures for different materials. The CPR device used in this study
is an open circuit coupled $\lambda/2$ resonator shown in Figure 1, whereby a
meandering coplanar waveguide allows an electrical length of approximately
$l\approx$ 68.6 mm to be achieved on a $10\times 10$ mm2 substrate. Using (1)
and assuming $L_{k}=0$, $f_{r}\approx 867$ MHz).

##  3 Model

A simple 3D FEM solution was developed in COMSOL Multiphysics® to check the
resonant modes of the CPR meander design. Briefly, a 2D work-plane is used to
define the 10$\times$10 mm2 CPR lithography pattern. The work plane is then
sandwiched between two 10$\times$10$\times$0.5 mm3 domains, with the top
defined as air ($\varepsilon_{\rm{r}}=1$) and the bottom as Si
($\varepsilon_{\rm{r}}=11.7$, $\tan\delta=2\times 10^{-4}$[27]). The BDD
layers are simulated as infinitely thin perfect electric conductors (PEC) with
two multi-element lumped ports defined between the centre conductor pads and
the ground planes. External boundaries are defined as PEC boundaries. The
model is first run in the absence of any kinetic inductance
($\mu_{\rm{r,eff}}=1$) to obtain the unperturbed resonance. Subsequently, the
kinetic inductance is introduced ($L_{k}=0.1$ to
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro H}$/m) by modelling all domains with
the effective magnetic permeability defined in (7). An electromagnetic
eigenfrequency study was used to estimate the resonant mode frequencies and a
frequency domain study was used to estimate the S-parameters.

The FEM results are given in Figure 2, showing the calculated eigenfrequencies
in the gigahertz range. The field distributions clearly show the first, second
and third harmonics (868, 1736 and 2604 MHz). After introducing $L_{k}$ using
the effective medium in (7), the fundamental resonant frequency decreases to
as low as 336 MHz for the fundamental mode when $L_{k}$ is as large as
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro H}$/m or $\lambda_{\rm{L}}\approx$
$3.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for this structure with similar
fractional frequency shifts are observed for other modes.

##  4 Method

B-NCD films were grown in a Seki Diamond systems AX 6500 series MPCVD system.
High resistivity float-zone Si substrates ($\diameter$ = 2”, $t$ =
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$) were first seeded using a
nanodiamond colloid suspension[28, 29] and grown at 3.5 kW at 40 Torr with a
pyrometer measured substrate temperature of 755 ∘C for 3 hours. Deposition was
achieved in a gas mixture of CH4, trimethylboron (TMB) dilute in H2 in a total
flow rate of 500 sccm (3% CH4 and B/C ratio $\sim$12,800 ppm)[30].

The CPR device was patterned using a standard photolithography process
(solvent cleaning and AZ nLof 2020 resist recipe with an adhesion promoter). A
100 nm nickel mask was evaporated onto the B-NCD using an Edwards 306 physical
vapour deposition system with subsequent lift off. The samples were then
etched in an O2/SF6 inductively coupled plasma (ICP) using an Oxford
Instruments PlasmaPro 100 Cobra system. Scanning electron microscopy (SEM)
images were obtained using a Hitachi SU8200 ($10\text{\,}\mathrm{kV}$ at
$10\text{\,}\mathrm{\SIUnitSymbolMicro A}$) and Raman spectra was obtained
using a Horiba LabRAM HR Evolution ($\lambda=$ $532\text{\,}\mathrm{nm}$).
Temperature dependent resistance measurements were also carried out on a
separate piece of the same wafer using a Van der Pauw configuration in a
Quantum Design physical property measurement system from 2 to 300 K .

The CPR device was cooled using a custom-made cooling platform utilising a
Cryomech PT-407-RM pre cooler and two Chase Research Cryogenics sorption
fridges. The device measurement is achieved using a Rhode & Schwarz ZNB vector
network analyser (VNA) connected to an RF chain of attenuators with the return
signal boosted using low noise amplifiers from (Arizona State University and
L3 Narda-MITEQ). Power transmission data was obtained using the VNA (magnitude
$|S_{21}|^{2}$ and phase) and resonator parameter fitting was carried out
using a MATLAB script.

Figure 2: FEM results of (a) the electric (E) field distribution for $n=1,2$
and 3 and (b) the calculated resonant frequency as a function of $L_{k}$
(bottom axes) and $\lambda_{\rm{L}}$ (top axes) using the effective medium
model.  Figure 3: Microscope and SEM images of the B-NCD CPR device.  Figure
4: Raman spectra of B-NCD CPR device: (a) Wide survey spectra of the centre
conductor using 473, 532 and 660 nm and excitation wavelengths and (b) shows a
magnified region situated around the diamond ‘d’ band.

##  5 Results

Microscope and SEM images of the CPR device after ICP etching are shown in
Figure 3, showing the uninterrupted length of the transmission line structure
with the capacitive coupling gaps at the edges of the sample. The SEM images
show that in the gap regions, the B-NCD has been successfully etched which was
also corroborated using a probe station measurement. Raman spectra is given in
Figure 4, showing the first and second order bands of Si at $\sim$520 cm-1 and
$\sim$965 cm-1, respectively[31, 32, 33] and numerous signatures typically
associated with BDD at high dopant concentrations. The B-NCD bands are
labelled at $\sim$450 cm-1 (dA), $\sim$1200 cm-1 (dC), $\sim$1295 cm-1 (dB),
1332 cm-1 (d) and the wide region at $\sim$1500-1680 cm-1 (G)[34]. Figure 4(a)
shows that there is minimal variation in the spectra as a function of the
laser excitation wavelength, where typically the G band becomes prominent at
high laser wavelengths[31]. The magnified region given in Figure 4(b) shows
that the characteristic zone centre phonon line at 1332 cm-1 (line ‘d’) is
redshifted to lower Raman shifts which is a typical signature of highly boron
doped nan-ocrystalline films[34, 30, 35, 36, 37]. The low intensity G band for
all excitation wavelengths implies that while the film is nanocrystalline with
considerable grain boundaries, the non-diamond carbon concentration is low. In
the gap regions, there are no B-NCD signatures and only those from the
underlying Si.

Figure 5: Power transmission spectra of the B-NCD CPR at ($P_{\textrm{in}}=$ 0
dBm with 20 dB attenuation). (a) shows the wide $|S_{21}|^{2}$ spectra at 3 K
(grey) and 0.3 K (blue) with an FEM result for
$L_{k}\approx$0.93\text{\,}\mathrm{\SIUnitSymbolMicro H}$$/m (red dotted). (b)
to (c) show magnified views of the 3 modes from 1 K to 0.3 K and the fitted
$f_{m}$ and $Q_{L}$.

The power transmission characteristics of the CPR are shown in Figure 5. The
broadband spectra in Figure 5 (a) shows that at 3 K, the device fails to
resonate and only a background noise floor at $\approx-60$ dB is measurable.
As $T$ is lowered towards 300 mK, sharp resonances emerge at approximately
409, 813 and 1215 MHz, much lower than the designed frequencies of 868, 1736
and 2604 MHz, respectively. Using an effective kinetic inductance of
$L_{k}\approx$ $0.93\text{\,}\mathrm{\SIUnitSymbolMicro H}$/m, the FEM
frequency domain $|S_{21}|^{2}$ traces demonstrate a plausible correlation
with the experimental measurements. Additionally, the peak power transmission
is low, demonstrating weak coupling. Magnified views of each resonance as a
function of $T$ are shown in Figure 5 (b) to (d). Each of the spectra were
fitted to a Lorentzian function to extract the measured resonant frequency and
the loaded quality factor[38]:

$|S_{21}|^{2}=\frac{P_{0}}{1+4Q_{L}^{2}\left(\frac{f-f_{m}}{f_{m}}\right)^{2}}$
(9)

where $P_{0}$ is the peak power at resonance, $Q_{L}=Q_{U}(1-P_{0})$ is the
loaded quality factor and $f$ and $f_{m}$ are the frequency and the measured
resonant frequency, respectively in Hz. Note that for all resonances, the peak
transmitted power $P_{0}\ll 0$ dB such that $Q_{L}\approx Q_{U}$ where the
resonator is weakly coupled, thus mitigating the need for any de-embedding of
the coupling capacitance. This was further confirmed using power sweeps
whereby minimal shift in the resonant frequency was observed (not shown). The
resonant frequency increases as $T$ decreases and is observed to saturate as
$T<300$ mK with minimal back-bending. Nominal quality factors range from 4,000
to 6,000 at 300 mK, however, saturation was not observed owing to the limited
$T$ range. It is quite possible that the intrinsic Q was actually much higher
towards $T\rightarrow 0$ K.

Figure 6: Calculated superconducting properties of B-NCD CPR device. (a) shows
the per unit length $\lambda_{\rm{L}}$ and $L_{k}$ and (b) shows the van der
Pauw $R(T)$ (left) and the calculated $R_{s,\textrm{eff}}(T)$ (right).
Identification of $T_{c}$ is denoted by the divergence in resistance (blue
dotted)

Using (1) and (7), the estimated $L_{k}$ and $\lambda_{\rm{L}}$ are shown in
Figure 6(a). Based on these frequency shifts, the extracted kinetic inductance
is $\approx$ $1.25\text{\,}\mathrm{\SIUnitSymbolMicro H}$/m with an associated
penetration depth of $\approx$ $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at
1.75 K and decreases to $\approx$ $0.93\text{\,}\mathrm{\SIUnitSymbolMicro
H}$/m with an associated penetration depth $\approx$
$4.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at 300 mK. Since all harmonics are
linked by $n$, similar values were obtained across the modes. To estimate the
zero temperature penetration depth, temperature dependent models are required,
with one of the most typical approaches being the empirical two fluid model as
presented by Gorter and Casimir (GC):

$\lambda_{\rm{L}}(T)=\frac{\lambda_{0}}{\left[1-\left(\frac{T}{T_{c}}\right)^{\gamma}\right]^{\frac{1}{2}}}$
(10)

where $\lambda_{0}$ is the penetration depth at 0 K and $\gamma=4$, though
experimental measurements rarely exhibit this exponent depending on the type
of superconductor[39]. $T_{c}$ is estimated using the van der Pauw measurement
in Figure 6 (b), giving $T_{c}\approx 2.9$ K which is in agreement with B-NCD
films grown in the same reactor at similar conditions ($T_{c}\approx$ 2.4 to
4.5 K)[9, 40, 41]. Using this value and (10), the 0 K extrapolated values are
given in Table 1 for $\gamma=2.7$. Similarly, the temperature dependent
penetration depth can also be related to the superconducting gap energy
through the BCS low-temperature model:

$\lambda_{L}(T)=\lambda_{0}\sqrt{\frac{\pi\Delta_{0}}{2k_{B}T}}\exp{\left(\frac{-\Delta_{0}}{k_{B}T}\right)}+\lambda_{0}$
(11)

where $\Delta_{0}$ is the superconducting gap energy at 0 K and $k_{B}$ is the
Boltzmann constant. Figure 6(a) shows the fit to $\lambda_{\rm{L}}(T)$ using
the same $\lambda_{0}$ as (10), yielding $\Delta_{0}\approx$
$895\text{\,}\mathrm{\SIUnitSymbolMicro eV}$. Finally, the surface resistance
values obtained using (8) are given in Figure 6(b), demonstrating that
$R_{s,\rm{eff}}(T)$ decreases from 30 to
$50\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}$ at 1.75 K to 2 to
$4\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}$ at 300 mK.

Table 1: Calculated zero temperature kinetic inductance per unit length and
per square and penetration depth based on the CPR effective medium
model${}^{\textrm{a}}$

Mode, $n$ |  $L_{k,0}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro H}$/m) |  $L_{k\square}$ (pH/$\square$)${}^{\textrm{b}}$ |  $\lambda_{0}$ ($\text{\,}\mathrm{\SIUnitSymbolMicro m}$)
---|---|---|---
| $\pm 0.006$ | $\pm 4$ | $\pm 0.02$
1 | 0.92 | 671 | 4.33
2 | 0.93 | 681 | 4.36
3 | 0.94 | 687 | 4.38

${}^{\textrm{a}}$ Error assumes a conservative measurement uncertainty in
$f_{m}$ of $\pm 1$ MHz
${}^{\textrm{b}}$ Calculated by assuming $L_{k\square}=L_{k,0}\times w/l$

Table 2: Calculated zero temperature surface resistance and sheet resistance
based on the CPR effective medium model${}^{\textrm{a}}$

Mode, $n$ |  $R_{s,\rm{eff}}$ ($\mu\Omega$/m) |  $R_{s,\square}$ ($\text{\,}\mathrm{n\SIUnitSymbolOhm}$$/\square$)${}^{\textrm{b}}$
---|---|---
| $\pm 0.01$ | $\pm 0.007$
1 | 4.4 | 3.2
2 | 3.6 | 2.6
3 | 2.2 | 1.6

${}^{\textrm{a}}$ Error assumes a conservative measurement uncertainty in
$Q_{U}$ of $\pm 10$.
${}^{\textrm{b}}$ Calculated by $R_{s,\square}=R_{s,\rm{eff}}\times w/l$

##  6 Discussion

The results presented here suggest that using the B-NCD films have a
considerably large kinetic inductance and large penetration depths, far
greater than the theoretical London penetration depths of BDD
$\lambda_{0}=(\nicefrac{{m_{e}}}{{\mu_{0}ne^{2}}})^{1/2}\approx 168$ nm where
$m_{e}$ and $e$ are the electron mass and charge, respectively and
$n>10^{21}$cm-3. The penetration depths are more consistent with those
experimentally found in thin B-NCD films on Si substrates at a similar
temperature ranges; $\lambda_{\rm{L}}\approx$ 2 to
$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$[13], however, less consistent with
much thicker films or those measured on single crystal diamond substrates or
B-SCD; $\lambda_{\rm{L}}\approx$ 0.2 to $1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$[12, 4]. The results support the evidence that the granularity of the films
plays a large role in increasing the penetration depth, and therefore its
sensitivity for microwave kinetic inductance based devices. The laminar model
presented by Hylton and Beasley considers the effect of large and small grain
sizes on the effective penetration depth, and demonstrate that polycrystalline
materials in general have much larger effective penetration depths than single
crystal owing to the grain boundary separation of superconducting grains[42].

The temperature dependence of $\lambda_{\rm{L}}$ as described by (11) yields a
$\gamma$ exponent that tends towards that of an s-wave superconductor,
however, the exponent likely due to the heavily disordered and granular nature
of B-NCD[39]. The extracted superconducting conducting gap is large, however,
is of similar magnitude to those reported by other measurement techniques.
Ishizaka et al. obtained a gap of
$\Delta_{0}\approx$$780\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ by laser
excited photoemission spectroscopy, Sacépé et al. obtained
$\Delta_{0}\approx$285\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ by tunnelling
spectroscopy and finally, Oripov et al. obtained a gap
$\Delta_{0}\approx$924\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ through
microwave parallel plate resonator (PPR) [43, 44].

The calculated surface resistance is very low at
2-$4\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}$, and comparable to
measurements of YBCO at similar frequencies although at much lower
temperatures[45]. Additionally, the calculated values here are much lower than
those reported previously for B-NCD films in this frequency range, as is
apparent by the higher observed quality factors[13]. Since the presented
method and that of previous approaches using microwave PPR are both capable of
measuring to similarly low values of $R_{s}$[23, 46], the discrepancy may lie
in the material deposition approach, i.e. by either microwave plasma assisted
chemical vapour deposition (MP-CVD) or hot filament chemical vapour deposition
(HF-CVD). Grain boundaries are typically apparent in B-NCD films, irrespective
of the deposition methodology, however, for films grown using MP-CVD, very low
concentrations of non-diamond carbon are typically observed by the low G band
in Raman spectroscopy, in contrast to those by HF-CVD[34, 41, 47]. This result
suggests that non-diamond carbon microwave losses are potentially significant
at low temperatures. Regardless, the measured quality factor of this device
for B-NCD is still fairly low when compared to other material technologies
based on aluminium[48, 49] and niobium[50] at $Q_{U}>10^{4}$. The cause of
this is likely the substrate to which the B-NCD is deposited on; using the FEM
model to reduce the loss tangent to zero yields Q factors $>10^{5}$. High
resistivity float-zone Si substrates are known to have additional losses at
low $T$ owing to potential hopping conduction[27]. Additionally, boron
diffusion into the Si is likely to occur during the plasma assisted growth
process resulting in an additional dielectric loss from the substrate[51, 52,
53]. Further improvements could potentially be made using substrates to stop
boron diffusion, or more robust dielectrics that are less affected by
microwave H2 plasmas which are out of scope of this work.

##  7 Conclusion

This work demonstrates a measurement of the seemingly large penetration depth
and low surface resistance of B-NCD using a microwave CPR approach with an
effective medium model. Using a simple meander design with a weakly coupled
power transmission technique, the intrinsic superconducting properties of thin
B-NCD films can be obtained. This work demonstrates that coplanar microwave
devices made from B-NCD can be potentially used for highly sensitive kinetic
inductance detectors.

##  8 Acknowledgements

This project has been supported by Engineering and Physical Sciences Research
Council (EPSRC) under the GaN-DaME program grant (EP/P00945X/1) and the “A
Diamond Bridge to Phase Slip Physics” grant (EP/V048457/1 ). This project has
been supported by the European Research Council (ERC) Consolidator Grant under
the SUPERNEMS Project (647471). SEM was carried out in the clean-room of the
ERDF-funded Institute for Compound Semiconductors (ICS) at Cardiff University.

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