# Ordering and topological defects in social wasps’ nests

Shivani Krishna1,* Apoorva Gopinath1 and Somendra M. Bhattacharjee2

###### Abstract

Social insects have evolved a variety of architectural formations. Bees and
wasps are well known for their ability to achieve compact structures by
building hexagonal cells. Polistes wattii, an open nesting paper wasp species,
builds planar hexagonal structures. Here, using the pair correlation function
approach, we show that their nests exhibit short-range hexagonal order (no
long-range order) akin to amorphous materials. Hexagonal orientational order
was well preserved globally. We also show the presence of topological defects
such as dislocations (pentagon-heptagon disclination pairs) and Stone-Wales
quadrupoles, and discuss how these defects were organized in the nest, thereby
restoring order. Furthermore, we suggest the possible role of such defects in
shaping nesting architectures of other social insect species.

Keywords: nest architecture, dislocations, disclinations, paper wasps, short-
range order, Burgers vector

## Introduction

Animal nests and their architecture are crucial for the development of the
young, protection from predators, and storage of resources. The diversity and
complexity in architecture signify how these functions are fulfilled in
diverse habitats and environmental conditions. Such complexity is exemplified
by the nests of social insects (ants, bees, termites, and wasps) owing to
their ability to use or modify the surroundings, and create ordered and
plastic structures1, 2, 3. These nests have captivated mathematicians and
ecologists alike for their structural complexity and the mechanisms behind the
coordinated building4, 5, 6, 7. How do multiple individuals in a group or a
colony come together to construct these tiled layouts? The principle of
stigmergy explains this as the formation of patterns by emergence 8, including
self-organisation and self-assembly mechanisms 9, 10. In most species, the
construction rules depend on environmental (e.g., humidity, soil moisture) or
pheromone gradients1, 11. In this context, the relative importance of self-
organised12, 13 vs template-based interactions has been examined, and current
evidence points towards architectural mechanisms being an interactive effect
of both these interactions14. It is paramount to place these mechanisms in the
context of behavioural and cognitive processes to understand nest construction
comprehensively. The repertoires required for complex nest architectures are
often characterised by simplified clusters of direct steps. However, the
processes involved in planning construction in various environmental
conditions, detecting errors and their remediation, working around novel
obstructions, etc., require substantial cognitive abilities that outwit simple
algorithms15.

In bees and wasps, cells within the nest are usually circular or hexagonal.
Hexagonal tiling provides a compact structure and is proven to cover a planar
region with regular units of equal area whilst minimising the total perimeter
(honeycomb conjecture16). Such optimal utilisation of space and energy has
been proposed as the significant selection pressure for shaping these
structures 17, 18, 19, 15. Honeybees use wax to shape their cells, while most
wasp species use either mud or fibrous materials of plant origin for
construction. Though the cells are hexagonal in both cases, the construction
materials possess strikingly distinct properties. Therefore, the underlying
processes could be different. Construction of bees’ wax-based cells is
postulated to begin as an array of circles (laid over packed cylinders as a
base) that are modified to rounded hexagons purely by mechanical/thermodynamic
means20, 21. However, experimental work from nests of European honeybees
suggests that bees actively construct hexagonal cells by handling the wax and
controlling its temperature22. Given the plasticity of wax, any disruptions to
tiling in these structures can be rectified, thereby reducing the possibility
of overall defects in the nests. Hence the legend of honeycomb ordering. On
the other hand, nests made up of plant fibres such as those of paper wasps are
unlikely to follow similar modes of construction. The reduced plasticity makes
them interesting as ordered physical systems prone to topological defects. A
topological defect in systems of broken symmetry, like crystals, magnets,
liquid crystals, etc., is defined as a disruption of order throughout the
system in a way that defies restoration via any continuous deformation23, 24.
In contrast, defects such as vacancies, hilly terrain in a flat land, etc.,
are geometric defects as they affect order locally in their neighbourhood
without any signature far away from them. A key characteristic of topological
defects is that they cannot be repaired by local rearrangements alone but
instead require changes to the system globally. Geometric defects can be
amended by making changes locally at the scale of basic elements or
constituents25. Here, we refer to topological defects as those that occur and
affect at length scales beyond a single unit/element, such as a single cell
within a nest. The emergence and evolution of such topological defects in
paper wasps’ nests have never been studied. Some of the most common
topological defects in crystals and liquid crystals manifest as disclinations
and dislocations26, 27. Examples of such disclinations have been shown in
living systems such as protein coats of viruses 28 and insect corneal
nanostructures29, where insertion of pentagons into an array of packed
hexagons results in Gaussian curvature and breaks the sixfold rotational
symmetry. Similarly, the Stone-Wales defect, characterised by two pentagons
and two heptagons30, is a well-known defect in graphene, fullerene, and carbon
nanotubes31, 32.

Paper wasps of Polistes genus build nests hanging from one or more stalks. The
nest itself is made up of open hexagonal cells. Variations in nesting
architectures within paper wasps have been attributed to selection pressures
such as predation from ants or flying insects, insulation from heat, and
economic use of building material33, 18, 34, 35, 36. The diversity of nest
forms in paper wasps and their growth have been elegantly explained by a
simple set of rules by Karsai and Pénzes37. The model suggests that multiple
forms could be built by changing the weightage assigned to different parts of
the nest. However, by construction, these models describe regular hexagonal
structures only. The likelihood of these models explaining changes to regular
hexagonal tiling or within nest changes to order is small. In this paper,
analysing the nesting architecture of paper wasps (Polistes wattii), we ask
the following questions:

  * (a)

What is the nature of ordering (i.e., short-range vs long-range) and spatial
arrangement of cells within and between the nests? In the current context,
order describes the regularity in the tiling of the cells within the nests.
Short-range order (SRO) represents the arrangements of nearest neighbours in
the nest, while long-range order (LRO) represents the regularity over a longer
distance. More quantitatively, a nest is said to have long-range order if
there is a nonzero probability of finding the corners of two cells on the same
regular lattice, even for large separations between the cells.

  * (b)

What are the topological defects found in theses nests?

## Methods

The study was conducted within the campus of Ashoka University, Sonepat, India
(28.9457∘ N, 77.1021∘ E) located at an altitude of 224 m. The maximum
temperature in summers is 45-47∘C, and the lowest temperature in winters is
around 4-6∘C. Wasps of the genus Polistes have a widespread distribution and
are a well-studied example for the evolution of sociality and dominance
hierarchies in insects. It is a primitive and speciose genus with more than
200 species38. P.wattii is known to be distributed across central and South
Asia39. They are inactive during the winter, and nest building begins once
they emerge from hibernation. The construction of nests often lasts till late
summer. In our study, all the analysed P.wattii nests were from anthropogenic
habitats. Nesting height was typically within the range of 5-15 m. Images of
nests were taken in the summer of 2020 and 2021. Adult nests that were fully
constructed and completed the season were considered for analysis (typically
with at least 50 cells, Figure S4). These nests were photographed with a
reference scale. Images were further analysed by subtracting the background
and binarisation using ImageJ40. The positions of all the individual vertices
(corners shared by at least three cells) were marked on the images, and their
x, y coordinates were exported for subsequent analysis. Here, we have taken
the structure to be planar (see Results and Discussion for details). To avoid
the effect of boundary which includes incomplete cells, a few layers of
vertices along the edges were excluded. Apart from these, a few cells with
cylindrical projections where identifying the underlying vertices was
difficult were excluded from the final analyses. Overall, 25 nests were used
to characterise the degree of ordering. The presence of topological defects
was identified by inspecting the number of vertices for the cells within 35
different nests. We characterised the defects by identifying their basic
features (number of vertices and neighbouring cells) and calculating the
internal angles of the first neighbouring hexagonal layer from coordinates.
For a region without defects, we would expect these angles to be close to
120∘.

To quantify ordering and analyse the spatial distribution of cells, we used
pair correlation function $g(r)$ (henceforth PCF) based on Ripley’s K function
derivative method in the spatstat package with default smoothing parameter41.
The approach relies on the probability of finding a pair of vertices separated
by a distance $r$. This results in an average representation of the local
spatial neighbourhood at a distance $r$ from any given vertex. PCF of a
regular lattice exhibits sharp peaks at the lattice spacing distances
(resolved using a lower smoothing parameter than the default value). Such
sharp peaks indicate perfect ordering. On the other hand, an amorphous
structure will show some reduced degree of order; in particular, it would have
$g(r)$ approaching 1 for large values of $r$. The limiting value of 1
indicates no correlation between the positions of vertices when they are far
apart. In a PCF, the peak heights are generally related to the number of
neighbours. The first two peaks were discernible in nests of different sizes,
allowing us to focus on the location and width of these peaks for each of the
nests. The following function consisting of two Gaussians with amplitudes
$a_{1},a_{2}$ has been used to fit the first two peaks of the $g(r)$-vs-$r$
curve for each nest,

$\displaystyle g_{\rm
fit}(r)=a_{1}\,\exp\left({-\frac{(r-r_{1})^{2}}{2s_{1}^{2}}}\right)+a_{2}\,\exp\left({-\frac{(r-r_{2})^{2}}{2s_{2}^{2}}}\right),$
(1)

where $r_{1},s_{1}$ represent the location and the width of the first peak,
and $r_{2},s_{2}$ of the second peak. We also calculated the coefficient of
variation (CV) of location and width for the first two peaks to quantify the
extent of variation in order between nests.

We measured the individual cell areas and compared the variation between and
within the analysed nests. We also calculated the nearest neighbour distances
or bond lengths, i.e., distances between vertices (henceforth wall lengths).
Based on the the fitted values of Eq. 1, the wall length would be in the range
$r_{1}\pm 2s_{1}$. Information of translational ordering is given by the PCF,
and to obtain the orientational order, we defined cell-orientational order in
analogy with bond-orientational order in liquids. Cell-orientational order
parameter is a measure of the geometrical arrangement of the vertices of a
given cell around its cell centre42, 43. Cell-orientational order parameter
$\psi_{n}$ is given by a complex number defined at each cell centre as,

$\psi_{n}(\mathbf{X}_{j})=\frac{1}{N_{j}}\sum^{N_{j}}_{k=1}e^{in\theta_{jk}},$
(2)

where $N_{j}$ is the number of vertices of cell $j$ centered at
${\mathbf{X}_{j}}$, $\theta_{jk}$ represents the angle between the line
joining vertex $k$ to the centre and the chosen x-axis, and $n$ is an index
for orientational ordering (Figure 1).

Figure 1: Cell-orientation order parameter (Eq. 2). The angle $\theta_{jk}$ is
measured from an arbitrarily chosen x-axis through the centre as shown in (a).
For a regular hexagon with inner angles at $2\pi/3$, the orientation in (a)
has $\psi_{n}=1$ while in (b) $\psi_{n}=-1$. In general, $\psi_{n}$ is
orientation-dependent with $|\psi_{n}|$ less than or equal to 1\. For regular
hexagons, individual $\psi_{n}$ lies on the unit circle. The six-fold symmetry
of a regular hexagon is preserved by $\psi_{n}$ as it returns to its value
under a $\pi/3$ rotation of the hexagon around its centre.

For hexagonal ordering, we choose $n=6$. Note that $\psi_{n}$ neither depends
on the vertex numbering scheme nor the size of the hexagon. As a complex
number, $\psi_{n}=|\psi_{n}|e^{i\theta}$ lies on or within the unit circle in
the complex plane, and therefore, $\psi_{n}(\mathbf{X}_{j})$ can be
represented graphically by a vector at an angle $\theta$ with the x-axis at
the cell centre $\mathbf{X}_{j}$. The average $\langle\psi_{n}\rangle$ of
$\psi_{n}(\mathbf{X}_{j})$ over a large number of hexagons lies in the range
$0\leq|\langle\psi_{n}\rangle|\leq 1$ with the two extreme values representing
a random orientational arrangement ($\langle\psi_{6}\rangle=0$) and a perfect
hexagonal ordering ($\langle\psi_{6}\rangle=1$). We calculated the cell-
orientational order parameter for a simulated set of random central angles to
study the decay of $|\langle\psi_{6}\rangle|$ from the perfect value of 1
(note that $\langle\psi_{6}\rangle=0$ corresponds to independent hexagons
which do not form a regular lattice; details in Supplementary Information).
Analyses were performed by using Fortran (https://gcc.gnu.org) and R
software44.

## Results and Discussion

### Nature of ordering and spatial arrangement of cells in the nests

Figure 2: Pair correlation function as a function of $r$, i.e., the distance
between vertices in cm. The two curves depict a representative nest and a
regular hexagonal lattice. The regular hexagonal lattice has been constructed
with a wall length equal to the average wall length of the representative
nest. A schematic on the top shows a regular hexagonal lattice and the
vertices in the first, second, and third shells around any one vertex. Each
vertex is connected to three other vertices in the first shell.

The number of cells in the analysed nests ranged between 50 and 740. For each
nest, we characterised the nature of ordering by calculating pair correlation
function for the vertices. For a regular hexagonal lattice, sharp peaks (in
principle, $\delta$-function) occur at distances corresponding to radii of
different shells indicating the presence of both short and long-range order45.
As shown in Figure 2, for any vertex on the hexagonal lattice, the first,
second, and third neighbours lie on circular shells and the first 3 PCF peaks
reflect this regularity. We found that there is a substantial overlap between
the first and second peaks of a regular hexagonal lattice and those of a
typical nest. As $r$ increases, this overlap disappears. This pattern of the
PCF illustrates the SRO in the structure of nests. The presence of a peak in a
PCF confirms that any given vertex would have neighbouring vertices at
distances provided by the peak position. This is an example of SRO. Moreover,
the decaying envelope of the PCF to $g(r)=1$ is a signature of the absence of
LRO. The proximity of the third peak to the second peak is an inherent
characteristic of a hexagonal lattice, suggesting that the two shells are
closeby. However, the second shell and third shell are not resolved in the
case of nests, due to broad wall-length distribution. The absence of sharp
peaks and the missing proximity of second and third shells are signatures of
deviation from regularity. Indeed, the broad peaks in the nest PCF curves
indicate heterogeneity of cell sizes that impairs ordering over long
distances. Therefore, we conclude that similar to amorphous structures, nests
exhibit SRO but no LRO.

The PCF curves of nests shown in Figure 3(a) have been normalized such that
the first peak is at $r=1$ (non-normalized individual nests’ PCFs are shown in
Supplementary Information, Figure S1). When variations in translational order
of nests were analysed, the nest-to-nest variation was found to be restricted
to larger $r$ values. This is further corroborated by the low variance values
(coefficient of variation, CV) of location and width of the first peak
compared to the second peak (Figure 3(b), (c), (d)).

Figure 3: (a) Pair correlation function of vertices from 10 randomly chosen
nests, showing a significant overlap around the first and second peaks; beyond
the third peak, the positioning of vertices approaches random distribution
(approaching 1). The x-axis is scaled ($r/r_{0}$) such that the first peak is
at the same x-value for all nests. Each coloured line represents one nest.
Distribution of (b) locations and (c) widths obtained by Gaussian
approximation of first and second peaks of the pair correlation function of 25
nests. (d) Relative difference in coefficient of variation (CV) of locations
and widths of first and second peaks. Each CV value depicts variation within
the nest.

The structural features were analysed by characterizing the distribution of
wall lengths, cell areas, and cell orientational order (Figure 4). Figure 4(a)
depicts the extent of variation in wall lengths between the different nests.
The lengths varied within the range of 0.1 and 0.9 cm, with a median value of
0.3 cm. The cell areas were in the range of 0.07 to 0.85 sq.cm (median= 0.25
sq.cm, Figure 4(b)). The distribution of cell wall-lengths also fits well with
the values obtained from the Gaussian fit of the first peak of the PCF.
Furthermore, we analysed if the degree of deviation was greater within the
nest or between the nests. Percent CV values of cell areas ranged between 13
and 28 while the wall-lengths varied from 44 to 58%. The wall-length variation
was greater within the nests than between the nests. Cell wall length
distributions were analyzed for randomly chosen cells from central and
peripheral regions of wasp nests to test the assumption of planarity of the
nest. For a curved surface, one expects the wall lengths of peripheral cells
to be different from those in the central region. These distributions were
found to overlap (Figure S5). Therefore, we conclude that the curvature-
induced systematic distortion (if any) is indistinguishable from the wall-
length distribution of the cells.

Our analysis suggests that the orientation of the cells is fairly preserved
and aligns well with the nearest neighbours (Figure 4(c), (d)). The
orientational order parameter varied between the nests in the range of 0.1 and
1, with a median value of 0.85. Significant overlap in cell orientational
order parameter distributions of the nests suggests that the presence of
orientational order is true across the nests (Figure 4(c)). Percent CV values
varied between 16 and 31, indicating that variation within nests in cell
orientational order was smaller than the variation in wall lengths. Compared
to the case of a single cell with random internal lengths (see Supplementary
Information), the probability distribution of $\langle\psi_{6}\rangle$ in
Figure 4(c) is wider. We attribute this to the cooperative effect of building
a compact nest and yet becoming amorphous-like in the large scale limit. A
well-studied example of such cooperative-effect induced broadening is the
probability distribution of magnetism as one approaches the Curie temperature
of a magnet46. Taken together, the pair correlation function and distribution
of cell orientational order parameters suggest the presence of short-range
hexagonal order and an overall orientational order even though there is no
long-range order akin to amorphous materials.

Figure 4: Frequency distribution of (a) cell wall lengths, (b) cell areas, and
(c) cell orientational order parameter (dimensionless), depicting variation
between and within the nests. (d) Heat map showing $z={\rm Im}\;\psi_{6}$, the
imaginary part of $\psi_{6}$ of cell orientational order parameter of a
representative nest ($x$,$y$ in cm indicate centre of each cell).

### Topological defects

Hexagonal structures are prone to defects that emerge in a variety of ways. In
the analysed nests, we show the presence of topological defects in the form of
non-hexagonal cells. Akin to defects in most materials, these were found to
occur at low frequency (20% of the nests). In few nests of P. annularis47,
non-hexagonal cells were reported to induce the required curvature of nests.
However, the presence of octagons and higher-order defects has not been
reported.

Defects such as one pentagon or one heptagon amidst hexagons disrupt the
planarity of a honeycomb structure as shown in Figure 5(a,b)48. If structural
disorders such as topological defects are left uncorrected, they affect the
overall planarity of the nest. The missing link of a pentagon in a hexagonal
net necessarily requires the removal of a wedge of hexagons, producing a cone-
like structure as shown in Figure 5(a). Similarly, an opposite curvature can
be produced by a single heptagon (Figure 5(b)) with an extra link that
requires the insertion of a wedge. As these curvatures cannot be ironed out,
these are topological defects called disclinations that tend to disrupt the
orientational order of the lattice. A pair of the two opposite types do not
cancel each other but instead carry with them a line defect, which is called a
dislocation line. This dislocation line consists of an extra array of
hexagonal cells (Figure 5(c)). Furthermore, a dislocation is characterised by
drawing a loop in a region without defects and comparing it with a similar
loop that encloses it. The steps required to close the loop are characteristic
of the defect and independent of the size and shape of the loop. This
topological invariant called Burgers vector is shown in Figure 5(c).

Figure 5: Illustration of emergent structure that deviates from planarity with
the insertion of (a) a pentagon or (b) a heptagon (shading as a guide to the
eye). (c) A disclination pair is indicated on a wasp nest. A view of parallel
lines encompassing an additional hexagonal layer as they pass through this
pair, thereby leaving a scar on the lattice (dislocation line). A Burgers
vector obtained from a loop enclosing the dislocation is a topological
invariant. A loop not enclosing a defect is shown on the top right.
$\widehat{\bf e}_{1}$ and $\widehat{\bf e}_{2}$ are the basis vectors.

Figures 6 and 7 show the distinct types of such defects found within the
analyzed nests. Stone-Wales defect comprises of two pentagons and two
heptagons (Figure 6(b)), which typically arises by a simple 90∘ rotation of
the bond (wall) between two hexagons. As a pentagon-heptagon pair is a dipole,
Stone-Wales defect becomes equivalent to a disclination quadrupole (two
oppositely oriented dipoles). These nomenclatures are analogous to those of
electric charges. Though a local fault in construction can produce a Stone-
Wales defect(Figure 6(b)), it is still topological in nature as the distortion
propagates to the boundary. Such topological defects are known to occur
commonly in graphite and graphene49. For cells with defects that occur in
pairs or as groups, we measured the distribution angles of the neighbouring
vertices for upto one shell of nearest neighbours. For a regular hexagonal
lattice, these angles would be 120∘, and for the nest without defects, we
found that a set of randomly chosen cells subtended angles that ranged between
90∘ and 140∘. However, hexagonal cells adjoining the defect pairs such as
pentagon-heptagon showed angles that range from 99∘ to 151∘, which differed
from both the regular hexagonal lattice as well as hexagonal cells in the
nest. Figure 7(a) depicts a defect where an octagon is embedded within three
pentagons, four hexagons, and a heptagon. This is obtained by a 90∘ rotation
of the wall between three hexagons and a heptagon. Another defect that was
observed was an octagon with two pentagons on either side (Figure 7). This
complex changes to a four-hexagon configuration by a wall addition followed by
a split (change in the wall angles of the central wall). Such extended defects
are equivalent to pairs of dislocations, which do not have any associated
Burgers vector. These arise in scenarios where wasp(s) have constructed the
outline of cells, and the internal walls were added subsequently. All the
above-described defects were found to maintain the three-point vertices
(vertex connected by three walls).

Figure 6: Schematic representation of (a) a pentagon and a heptagon amidst
hexagons (dipole), and (b) Stone-Wales defect (pentagon-heptagon quadrupole)
or a dislocation dipole. The arrow points towards the representation of
hexagons when these defects are fixed. The possible modifications to the wall
are indicated above the arrow. Bottom panels show their outline in the wasp
nests. Figure 7: Schematic representation of (a) an octagon surrounded by a
heptagon and three pentagons, and (b) an octagon between two pentagons. The
arrows point towards the representations of hexagons when these defects are
fixed. The possible modifications to the wall are indicated above the arrow.
Bottom panels show their outline in the wasp nests.

A dislocation for a two-dimensional system consists of a pentagon-heptagon
pair (i.e., a pair of opposite disclinations). The presence of such a defect
necessarily entails an extra row of hexagons that vitiates the translational
order at large distances but not the bond-orientational order. Consequently, a
proliferation of dislocations can produce a state of bond-orientational order
without any translational order (a distinct phase of matter well known as the
hexatic phase). A liquid is formed when the dislocations break into
disclinations (independent pentagons and heptagons)50, which destroy the
residual bond-orientational order. For instance, when these defects are
induced by thermal energy, one sees a two-step melting process of a two
dimensional crystal. In colloidal crystals, dislocations or disclination
dominated phases can be produced by external agencies like laser. In short,
topological defects significantly influence a system’s behavior. In this
spirit, the disposition of the two-dimensional nests is ultimately determined
by the nature of the individual cells and the topological defects, which are
generated by the construction rules adopted by the wasps.

While pentagon-heptagon pairs were found to occur in nests, the configuration
where hexagons separate a pentagon and a heptagon was never observed in our
dataset. Interestingly, some of the described defects and their properties are
utilized by honeybees to attain curvature as well as to merge cells of
different sizes51, 52. Cells of different sizes are routinely observed in
honeycombs as larvae of drones and workers are of different sizes. Retaining
order with cells of different sizes that are being built independently by
different individuals poses a challenge, and the honeybees counter this by
local sensing and introducing non-hexagonal cells where required53, 52, 54.
The introduction of a pentagon or a heptagon adjacent to each other is only
possible by a regular set of local inspections carried out by wasps to
identify possible deviations from hexagonality. There are two alternatives to
explain the emergence of these defects that require further investigations.
These alternatives are as follows: a) As the angle of a regular heptagon is
128.5∘ is closer to that of a regular hexagon than a pentagon whose angle is
105∘, heptagons have arisen first, which are then fixed immediately by adding
a pentagon, and b) the higher number of pentagons overall suggests that
moulding cells with lesser material could have erroneously resulted in
pentagons that were subsequently fixed by inserting a heptagon. It is likely
that experience and learning play a crucial role in repair processes.
Experiments in P. fuscatus have shown the presence of an elaborate building
programme that is beyond a set of linear steps, including repairing cell walls
when damaged, adding pulp, and strengthening the stalk when required, which is
done by multiple inspections55. This suggests that deviations from 6-sidedness
are also possibly assessed via such inspections. However, it is unknown if
these inspections involve identifying the number of vertices, measuring the
internal angles or by Burgers vector.

## Conclusion

Our results suggest that the nests of Polistes wattii show orientational order
without any translational long-range order in the placement of the hexagonal
cells on a planar surface. However, the ordering is well preserved in the
immediate neighborhood of the cells (short-range order). Furthermore,
topological defects were identified in the form of dislocations and
disclinations (non-hexagonal cells) in the nests. We showed that these defects
were organized such that the planarity of the nests was maintained. We used
the approach of pair correlation function and topological methods to analyse
these structural features of wasp nests.

Two key features that were never violated in the nests were a) convexity of
individual cells and b) planar global structure. We postulate that wasps
conduct systematic localized inspections, detect changes in local symmetry or
order, and make changes to the adjoining cells to restore order. Going
further, we intend to test the role of cell size heterogeneity on the
emergence of topological defects and the relationship between the decay length
of pair correlation function with the size of the nests. Future studies could
place the nests in captivity and characterise the behaviours that precede and
succeed the construction of cells around the defects. The introduction of
defects experimentally in such captive nests would help understand the
response of individuals and the group. While our study establishes the
presence and repair of the defects, it is unclear if all the wasps equally
participate in this process or certain keystone individuals are efficient at
inspection and repair processes. These additional studies would therefore
allow us to understand the behavioral mechanisms encompassing the repair of
topological defects.

Data availability
The datasets used in the current study will be available from the
corresponding author on request.

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Acknowledgements
We thank Sreya Dey for taking up this idea in its pilot phase and initiating
the work. Thanks to Smruti Ranjan and Abdelsalam Gena for help with data
analysis. We thank Upasana Sengupta, Diana Michael, and Nageshwar Kumar for
help with collecting nests.

Author contributions
S.K. conceived and designed the study. A.G. and SK collected the data. S.K.
and S.M.B. performed and interpreted the analysis and drafted the manuscript.
All authors gave final approval for publication.

Funding
S.K. acknowledges intramural funding from Ashoka University. A.G. thanks
Science and Engineering Research Board, India (CRG/2019/003297) for
fellowship. S.M.B. thanks JC Bose Fellowship (SR/S2/JCB-71/2009) from Science
and Engineering Research Board, India.

Competing interests
The authors declare no competing interests.

See pages 1,2,3,4 of Wasp_Suppl.pdf


# Temperature dependence of microwave losses in lumped-element resonators made
from superconducting nanowires with high kinetic inductance

Ermes Scarano  Department of Applied Physics, KTH Royal Institute of
Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm, Sweden  Elisabet
Arvidsson  Department of Applied Physics, KTH Royal Institute of Technology,
Hannes Alfvéns väg 12, SE-114 19 Stockholm, Sweden  August Roos  Department of
Applied Physics, KTH Royal Institute of Technology, Hannes Alfvéns väg 12,
SE-114 19 Stockholm, Sweden  Erik Holmgren  Department of Applied Physics, KTH
Royal Institute of Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm,
Sweden  David Haviland<EMAIL_ADDRESS>Department of Applied Physics, KTH
Royal Institute of Technology, Hannes Alfvéns väg 12, SE-114 19 Stockholm,
Sweden

###### Abstract

We study the response of several microwave resonators made from
superconducting NbTiN thin-film meandering nanowires with large kinetic
inductance, having different circuit topology and coupling to the transmission
line. Reflection measurements reveal the parameters of the circuit and
analysis of their temperature dependence in the range
$1.76\text{\,}\mathrm{K}\text{/}$ extract the superconducting energy gap and
critical temperature. The lumped-element LC resonator, valid in our frequency
range of interest, allows us to predict the quasiparticle contribution to
internal loss, independent of circuit topology and characteristic impedance.
Our analysis shows that the internal quality factor is limited not by thermal-
equilibrium quasiparticles, but an additional temperature-dependent source of
internal microwave loss.

##  I Introduction

Superconducting nanowires with large kinetic inductance are interesting for
applications in quantum technology due to their low loss and adjustable-
strength nonlinearity. Their small size and low capacitance open the
possibility of achieving high characteristic impedance, in some cases higher
than the quantum resistance $R_{Q}=h/4e^{2}$ =
$6.45\text{\,}\mathrm{k\SIUnitSymbolOhm}\text{/}$ [1, 2, 3, 4]. High kinetic
inductance nanowires are fabricated from thin films with a single process
step, resulting in compact, robust and reproducible resonant circuits in a
microwave band, e.g. $48\text{\,}\mathrm{GHz}\text{/}$. As lumped elements,
nanowire inductors provide a versatile platform for the design of quantum
devices, sensors and detectors. Here we explore the properties of such lumped
elements, with a particular focus on understanding microwave losses and the
role of thermal-equilibrium quasiparticles (QP).

Materials such as Nb [5], NbN [1, 6, 7], TiN [8, 9, 10], and NbTiN [11, 12,
13, 14, 15, 16, 17, 18] feature critical temperatures on the order of
$10\text{\,}\mathrm{K}\text{/}$, making them good candidates for applications
in devices such as superconducting single-photon detectors (SSPDs) [19],
Microwave Kinetic Inductance Detectors (MKIDs) [20, 21], Kinetic Inductance
Parametric Amplifiers (KIPAs) [18] and force transducers [22]. Such devices
are routinely operated above $1\text{\,}\mathrm{K}\text{/}$ where microwave
losses are influenced by thermal-equilibrium QPs , whose concentration
increases exponentially with temperature. At lower temperatures losses are
typically attributed to two-level systems (TLS) and out-of-equilibrium QPs.

Understanding microwave losses in high kinetic inductance materials requires
consideration of the resonator design and coupling to the transmission line
through which we measure its physical response. A lumped-element model allows
us to relate values of the individual circuit variables to the scattering
parameters of a reflection measurement. We study the temperature dependence of
these variables in the range $1.76\text{\,}\mathrm{K}\text{/}$, fitting two
models to the measured data: the traditional two-fluid model (TTF) [23] and
the model based on the Mattis-Bardeen (MB) equations [24]. The temperature
dependence of the MB model accurately describes the behavior of NbTiN as a
strong-coupling, disordered BCS superconductor, allowing us to determine the
critical temperature $T_{c}$ and the energy gap $\Delta_{0}$. We analyze
resonators with both series and parallel circuit topologies, and with
different couplings to the microwave transmission line.

Combining the lumped-element circuit description and two-fluid model, we are
able to unambiguously quantify the contribution to microwave loss. As expected
for an accurate lumped-element model, our analysis shows that the QP loss, at
a given resonance frequency, is independent of the circuit topology and the
values of the lumped elements. Comparing the data to the prediction of the
model, we show that our resonators contain an additional temperature-dependent
loss contribution which is not explained by thermal-equilibrium QPs.

##  II Devices and models

We realize high-Q factor resonators in both series and parallel circuit
topologies by combining high kinetic inductance superconducting nanowires and
interdigital capacitors. Figure 1(a) shows a illustrative layout of the two
circuit topologies and Fig. 1(b) the lumped-element model of the kinetic
inductor. Figures 1(c)-(f) show scanning electron micrographs of the
individual lumped elements. The circuits are fabricated from a single layer of
NbTiN with thickness $t$ = $15\text{\,}\mathrm{nm}\text{/}$ deposited on a
$600\text{\,}\mathrm{nm}\text{/}$-thick layer of silicon nitride (SiN) on
silicon (Si) substrate. The resonators are either directly coupled to a
transmission line or coupled via a reactive element (capacitor or inductor,
depending on topology) to adjust the total quality factor and coupling
coefficient $\eta=\kappa_{\mathrm{ext}}/(\kappa_{\mathrm{ext}}+\kappa_{0})$,
where $\kappa_{\mathrm{ext}}$ is the loss rate to the external transmission
line, and $\kappa_{0}$ the loss rate internal to the resonator.

Figure 1:  (a) The device layout for a series resonator with inductive
coupling and a parallel resonator with capacitive coupling. Scanning electron
micrographs (SEM) of the components inside the colored boxes are shown in
(c)-(f). (b) Lumped-element model of the meandering nanowire kinetic inductor
consisting of two parallel channels: a temperature-dependent inductance
$L_{\mathrm{k}}(T)$, and a temperature-dependent resistance $R(T)$. (c) An SEM
image of the interdigital series capacitor. (d) An SEM image of the short
nanowire shunting inductor, for inductive coupling of the series resonator.
(e) An SEM image of an interdigital parallel capacitor for the parallel
resonator. (f) An SEM image of a meandering nanowire kinetic inductor used in
both designs.

Our description of the kinetic inductor as a single lumped-element is valid
for frequencies much lower than the first eigenmode of the meandering nanowire
[25], $\sim$24\text{\,}\mathrm{GHz}\text{/}$$ for the design shown in Fig.
1(f). Disordered superconductors such as NbTiN have low Cooper pair density
$n_{s}$ and large London penetration depth $\lambda_{\mathrm{L}}\propto
1/\sqrt{n_{s}}\approx$ $650\text{\,}\mathrm{nm}\text{/}$. In the regime of
small thickness and width $t,w\ll\lambda_{\mathrm{L}}$ the nanowire’s kinetic
inductance per unit length
$L_{\mathrm{k}}/\ell=\mu_{0}\lambda_{\mathrm{L}}^{2}/tw=m_{e}/2n_{s}e^{2}tw$
far exceeds its geometric inductance.

The measurement setup is described in Appendix A. We measure the reflection
coefficient,

$\Gamma(\omega)=\frac{Z_{\mathrm{ext}}-Z_{R}(\omega)}{Z_{\mathrm{ext}}+Z_{R}(\omega)},$
(1)

where $Z_{R}(\omega)$ is impedance of the resonator and $Z_{\mathrm{ext}}$ is
the impedance seen by the resonator. For a single port high-Q resonator,
$\Gamma(\omega)$ can be expressed as

$\Gamma(\omega)=\frac{\kappa_{0}-\kappa_{\mathrm{ext}}+i2(\omega-\omega_{0})}{\kappa_{0}+\kappa_{\mathrm{ext}}+i2(\omega-\omega_{0})},$
(2)

where the resonance frequency $\omega_{0}$, internal loss rate $\kappa_{0}$
and external loss rate $\kappa_{\mathrm{ext}}$, are the key figures-of-merit
that we use to compare different resonators.

The lumped-element model allows us to relate the resonator parameters to
individual circuit variables. For a series topology, we have [26],

$\omega_{0}=\frac{1}{\sqrt{LC}},~{}\kappa_{\mathrm{ext}}=\frac{\mathrm{Re}[Z_{\mathrm{ext}}]}{L},~{}\kappa_{0}=\frac{R_{s}}{L},$
(3)

and for parallel we have

$\omega_{0}=\frac{1}{\sqrt{LC}},~{}\kappa_{\mathrm{ext}}=\frac{1}{\mathrm{Re}[Z_{\mathrm{ext}}]C},~{}\kappa_{0}=\frac{1}{R_{p}C},$
(4)

where $R_{s}$ ($R_{p}$) is an effective resistance describing losses. Losses
in the lumped-element nanowire kinetic inductor are well modeled by a
temperature-dependent parallel resistance $R(T)$, as shown in Figure 1(b).
Thus $R_{p}=R(T)$ while $R_{s}=\real\left[R(T)\parallel i\omega
L_{\mathrm{k}}\right]=\omega_{0}^{2}L_{\mathrm{k}}^{2}/R(T)$.

##  III Two-fluid model

Quasiparticle losses in the lumped-element kinetic inductor can be described
by the two-fluid model of superconductivity [27]. This model invokes a complex
conductivity, with a real part describing dissipation due to QP current, and
an imaginary part describing the inertia of the condensate. The QP resistance
is inversely proportional to the normal carrier density
$R_{\mathrm{QP}}\propto 1/n_{n}$, and the kinetic inductance inversely
proportional to the density of pairs, $L_{\mathrm{k}}\propto 1/n_{s}$.

The concentration of both normal and superconducting carriers changes with
temperature, but the total density of charge carriers $n=n_{s}(T)+n_{n}(T)$ is
independent of temperature. With $n=n_{s}(0)=n_{n}(T_{c})$ we can explicitly
write the temperature scaling of $L_{\mathrm{k}}$ and $R_{\mathrm{QP}}$,

$\frac{L_{\mathrm{k}}(T)}{L_{\mathrm{k}}(0)}=\frac{n}{n_{s}(T)}\equiv\Lambda(T),$
(5)

and

$\frac{R_{\mathrm{QP}}(T)}{R_{\mathrm{N}}}=\frac{n}{n-n_{s}(T)}=\frac{\Lambda(T)}{\Lambda(T)-1},$
(6)

where $R_{\mathrm{N}}=R_{\mathrm{QP}}(T_{c})$ is the normal state resistance
of the kinetic inductor.

We fit both the TTF and MB models to the measured data. The former predicts a
quadratic dependence of the London penetration depth on temperature, leading
to a power-four scaling of the kinetic inductance [27],

$\Lambda_{\mathrm{TTF}}(T)=\frac{1}{1-(T/T_{c})^{4}},$ (7)

and latter relates the complex conductivity to the BCS energy gap. In the low
frequency limit $\hbar\omega\ll\Delta$, where pair breaking from the driving
field is negligible, the MB model predicts [28],

$\Lambda_{\mathrm{MB}}(T)=\frac{\Delta(0)}{\Delta(T)}\coth\left(\frac{\Delta(T)}{2k_{\mathrm{B}}T}\right).$
(8)

In the fitting procedure we use an expression which is a good approximation to
the temperature dependence of the superconducting energy gap of BCS
superconductors [29]:

$\frac{\Delta(T)}{\Delta(0)}=\sqrt{\cos\left(\frac{\pi}{2}\left(\frac{T}{T_{c}}\right)^{2}\right)}.$
(9)

##  IV Data analysis

Figure 2 shows the temperature dependence of the parameters of a series
resonator obtained by fitting Eqn. (2) to the measured $\Gamma(\omega)$ at
each temperature. The resonator consists of a single interdigital capacitor in
series with a $100\text{\,}\mathrm{nm}\text{/}$-wide nanowire kinetic
inductor, directly coupled to the transmission line [see inset of Fig. 2(a)].
We take the capacitor to be independent of temperature. For the directly
coupled resonator we can analytically relate the resonator parameters to the
lumped-element circuit variables, allowing us to express the temperature
dependence of the resonance frequency

$\omega_{0}(T)=\frac{\omega_{0}(0)}{\sqrt{\Lambda(T)}}.$ (10)

We use Eqn. (10) to fit the TTF and MB models with $\Lambda(T)$ given by Eqns.
(7) and (8), respectively. The fits are shown in Fig. 2(a) where the MB model
shows excellent agreement to the data, with critical temperature $T_{c}$ =
$9.45\text{\,}\mathrm{K}\text{/}$ and energy gap ratio
$\Delta_{0}/k_{\mathrm{B}}T_{c}=1.85$. From this fit we conclude that NbTiN
behaves as a strong coupling “dirty” BCS superconductor [30].

The temperature dependence of the external loss rate
$\kappa_{\mathrm{ext}}(T)$ is well explained with the same
$\Lambda_{\mathrm{MB}}(T)$ found in the fit of $\omega_{0}(T)$ by adjusting
only one scaling factor $\kappa_{\mathrm{ext}}(0)$ [see Fig. 2(b)],

$\kappa_{\mathrm{ext}}(T)=\frac{\real[Z_{\mathrm{ext}}]}{L_{\mathrm{k}}(0)\Lambda(T)}=\frac{\kappa_{\mathrm{ext}}(0)}{\Lambda_{\mathrm{MB}}(T)}.$
(11)

Here we point out that
$\kappa_{\mathrm{ext}}\approx$30\text{\,}\mathrm{MHz}\text{/}$\gg\kappa_{0}$
over most of the temperature range studied. The low quality factor of this
over-coupled resonator makes the determination of $\Gamma(\omega)$ sensitive
to details of $Z_{\mathrm{ext}}$. For example, standing waves from small
impedance mismatches in the transmission line give a frequency dependent
$Z_{\mathrm{ext}}(\omega)$ which may vary significantly over the bandwidth of
the resonator, increasing the uncertainty in the determination of the circuit
variables.

To explain the temperature dependence of the internal loss rate
$\kappa_{0}(T)$ we first consider only the QP contribution, i.e.
$R=R_{\mathrm{QP}}(T)$. The series topology gives

$\kappa_{0}^{\mathrm{QP}}=\frac{R_{s}}{L_{\mathrm{k}}}=\frac{\omega_{0}^{2}(T)L_{\mathrm{k}}(T)}{R_{\mathrm{QP}}(T)}.$
(12)

One arrives at the same expression for the parallel topology, where
$\kappa_{0}^{\mathrm{QP}}=(R_{\mathrm{QP}}(T)C)^{-1}$. Combining Eqns. (5),
(6) and (12) with the relation between the zero-temperature kinetic inductance
and the normal-state resistance [27]

$\Delta_{0}=\frac{\hbar R_{\mathrm{N}}}{\pi L_{\mathrm{k}}},$ (13)

we arrive at an expression for the internal loss rate due to QPs,

$\kappa_{0}^{\mathrm{QP}}=\frac{\omega_{0}^{2}(0)\hbar}{\pi\Delta_{0}}\left(\frac{\Lambda(T)-1}{\Lambda(T)}\right).$
(14)

Equation (14) tells us that, for a given resonance frequency, the QP loss rate
is independent of the resonator’s characteristic impedance
$Z_{c}=\sqrt{L_{\mathrm{k}}/C}$ and circuit topology.

We plot Eqn. (14) in Fig. 2(b), where we see that the measured $\kappa_{0}$ is
nearly an order of magnitude larger than $\kappa_{0}^{\mathrm{QP}}$. This
implies that there are additional sources of microwave dissipation, e.g.
radiative losses (not into the transmission line) or temperature-dependent
dielectric losses in the insulating layer [31]. We model these as additional
resistive channels in parallel with the kinetic inductor,

$\frac{1}{R(T)}=\frac{1}{R_{\mathrm{QP}}(T)}+\frac{1}{R_{d0}}+\frac{1}{R_{d}e^{\beta/T}}.$
(15)

The parameter $R_{d0}$ describes residual losses at zero temperature, and the
parameters $R_{d}$ and $\beta$ describe a thermally activated loss mechanism.
Figure 2(b) shows a best fit with the values of all variables given in Table
1.

Figure 2:  (a) Temperature dependence of the resonance frequency $\omega_{0}$
of a directly coupled series $LC$-resonator, with fits using the Mattis-
Bardeen (MB) model and the traditional two-fluid model (TTF). The inset
displays the equivalent circuit diagram. (b) External $\kappa_{\mathrm{ext}}$
and internal $\kappa_{0}$ decay rates with fits using the MB model. The lowest
solid green line traces the predicted internal losses due to the QP
contribution in the two-fluid model $\kappa_{0}^{\mathrm{QP}}$.

For the case of a series resonator directly connected to a transmission line,
the value of $L_{\mathrm{k}}=\kappa_{0}(0)/\real\left[Z_{\mathrm{ext}}\right]$
in Table 1 is found by assuming $\real\left[Z_{\mathrm{ext}}\right]$ =
$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{/}$. As previously mentioned, this
assumption is not very reliable and furthermore the strongly over-coupled
resonator leads to intrinsic uncertainty in the determination of
$\kappa_{0}(T)$ [32]. More accurate analysis can be made closer to critical
coupling where $\kappa_{\mathrm{ext}}\approx\kappa_{0}$. Critical coupling is
difficult to achieve with direct connection because typically
$\real\left[Z_{R}(\omega_{0})\right]\ll$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{/}$$.
Adding a reactive circuit element allows us to tune the external impedance
closer to critical coupling while increasing total quality factor.

### Reactively coupled resonators

Figure 3:  (a) Temperature dependence of the resonance frequency $\omega_{0}$
of an inductively coupled series $LC$-resonator, with a fit using the Mattis-
Bardeen (MB) model. The inset displays the equivalent circuit diagram. (b)
External $\kappa_{\mathrm{ext}}$ and internal $\kappa_{0}$ decay rates with
fits using the MB model. The lowest solid green line traces the predicted
internal losses due to the QP contribution in the two-fluid model
$\kappa_{0}^{\mathrm{QP}}$.

The series resonator is reactively coupled to the transmission line with an
inductive shunt $L_{s}$ to ground, realized by the short section of nanowire
in the illustrative layout of Fig. 1(a), and shown schematically in the inset
of Fig. 3(a). This circuit is the dual of the more common capacitively-coupled
parallel resonator, which we discuss later. The shunt inductance $L_{s}$
modifies the external load impedance as

$\displaystyle Z_{\mathrm{ext}}$ $\displaystyle=$ $\displaystyle 50\parallel
i\omega L_{s}=\frac{\omega^{2}L_{s}^{2}50+i\omega
L_{s}50^{2}}{50^{2}+\omega^{2}L_{s}^{2}}$ (16) $\displaystyle\approx$
$\displaystyle\frac{\omega^{2}L_{s}^{2}}{50}+i\omega L_{s},$

with the approximation valid when $\omega
L_{s}\ll$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{/}$$, which is the case for
the range of frequency and temperature studied here. With $L_{s}\ll
L_{\mathrm{k}}$ the resonator parameters become

$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{C(L_{\mathrm{k}}+L_{s})}},$ (17)
$\displaystyle\kappa_{\mathrm{ext}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}^{2}L_{s}^{2}}{50(L_{\mathrm{k}}+L_{s})},$ (18)
$\displaystyle\kappa_{0}$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}L_{\mathrm{k}}^{2}}{R(L_{\mathrm{k}}+L_{s})}.$
(19)

where $R$, $L_{\mathrm{k}}$ and $L_{s}$ are temperature-dependent. Assuming
$L_{s}$ and $L_{\mathrm{k}}$ have the same temperature scaling, Eqn. (18)
would predict $\kappa_{\mathrm{ext}}$ to be independent of temperature.

The above expressions capture the qualitative behavior with temperature, but
they are not accurate enough for fitting. Rather, we fit Eqn. (1) to the
measured $\Gamma(\omega)$ at all temperatures to obtain the circuit variables,
without the approximations and assumptions stated above. Here we fit only the
MB model, as the fit does not converge with the TTF model. The results are
given in Table 1 and the corresponding resonator parameters for this fit are
shown by the solid lines in Fig. 3. As expected the fit shows a nearly
constant $\kappa_{\mathrm{ext}}\approx$1\text{\,}\mathrm{MHz}\text{/}$$. In
the range of temperature studied $\kappa_{0}(T)$ crosses
$\kappa_{\mathrm{ext}}$, with the resonator going from over-coupled at low
temperature to under-coupled at higher temperature. The analysis spans a range
of coupling coefficient $0.25<\eta<0.9$, where determination of the internal
losses from the measured scattering parameters is most accurate. As for the
directly coupled series resonator, we find the theoretically predicted
$\kappa_{0}^{\mathrm{QP}}$ to be far lower than that determined from the
measured scattering parameters.

Figure 4:  (a) Temperature dependence of the resonance frequency $\omega_{0}$
of a parallel $LC$-resonator, with fits using the Mattis-Bardeen (MB) model
and the traditional two-fluid model (TTF). The inset displays the equivalent
circuit diagram. (b) External $\kappa_{\mathrm{ext}}$ and internal
$\kappa_{0}$ decay rates with fits using the MB model. The lowest solid green
line traces the predicted internal losses due to the QP contribution in the
two-fluid model $\kappa_{0}^{\mathrm{QP}}$.

We also studied the capacitively-coupled parallel $LC$-resonator, with a
layout illustrated in Fig. 1(a) and shown schematically in the inset of Fig.
4(a). The coupling capacitance $C_{c}$ transforms the external load admittance
as

$Y_{\mathrm{ext}}=\frac{1}{Z_{\mathrm{ext}}}=\frac{1}{50+1/i\omega
C_{c}}=\frac{\omega^{2}C_{c}^{2}50+i\omega
C_{c}}{\omega^{2}C_{c}^{2}50^{2}+1}.$ (20)

A high-Q factor requires a coupling capacitance such that near resonance
$\omega C_{c}\ll 1/$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{/}$$. In this
case, the external admittance becomes

$Y_{\mathrm{ext}}=\omega^{2}C_{c}^{2}50+i\omega C_{c}.$ (21)

The real part reduces the external decay rate, increasing the quality factor,
while the imaginary part adds to the resonator capacitance. For $C_{c}\ll C$
there is a slight shift of the resonance frequency. The resonator parameters
become

$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{L_{\mathrm{k}}(C+C_{c})}},$ (22)
$\displaystyle\kappa_{\mathrm{ext}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}^{2}C_{c}^{2}50}{C+C_{c}},$ (23)
$\displaystyle\kappa_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{R(C+C_{c})}.$ (24)

with $L_{\mathrm{k}}$ and $R$ depending on temperature.

|  $L_{\mathrm{k}}$ ($\mathrm{nH}\text{/}$) |  C ($\mathrm{fF}\text{/}$) |  $L_{s}$ ($\mathrm{nH}\text{/}$) |  $C_{c}$ ($\mathrm{fF}\text{/}$) |  $R_{\mathrm{N}}$ ($\mathrm{k\SIUnitSymbolOhm}\text{/}$) |  $R_{d}$ ($\mathrm{k\SIUnitSymbolOhm}\text{/}$) |  $R_{d0}$ ($\mathrm{M\SIUnitSymbolOhm}\text{/}$) |  $\beta$
---|---|---|---|---|---|---|---|---
Fig. 2† |  238 |  5.44 |  — |  — |  1720 |  514 |  23 |  11
Fig. 3 |  107 |  10.9 |  0.205 |  — |  786 |  805 |  83 |  15
Fig. 4 |  2.38 |  805 |  — |  13.8 |  5.85 |  1.73 |  1.7 |  16
Table 1:  Best-fit parameters for the solid lines shown in the respective
figures. †The directly coupled series resonator of Fig. 2 had low quality
factor and was strongly over-coupled, leading to larger uncertainty.

For the the capacitively-coupled parallel $LC$-resonator, four circuit
variables define the three resonator parameters. The fit requires prior
knowledge of at least one circuit variable. The data in Fig. 4(a) shows the
best fit of $\omega_{0}(T)$ using the TFF and MB models. Again the MB model
fits better than the TFF model, and we find that the measured internal loss
rate $\kappa_{0}(T)$ exceeds the expected QP contribution by nearly an order
of magnitude, as shown in Fig. 4(b). We also show the best fit of
$\kappa_{0}(T)$ to the loss model for $R(T)$ given in Eqn. (15) with best-fit
values given in Table 1. Here we note that the residual resistance $R_{d0}$
for the parallel resonator is almost three orders of magnitude smaller that
that for the series configuration, consistent with an increased contribution
from dielectric losses due to the much larger resonator capacitance $C$.

##  V Conclusion

In a temperature range where thermal-equilibrium QP contributions to the
internal losses cannot be neglected, we analyzed the resonance frequency and
loss rates of microwave resonators made from superconducting meandering
nanowires with high kinetic inductance. We showed how microwave measurements
of the resonator reflection coefficient in a relatively narrow temperature
interval $1.76\text{\,}\mathrm{K}\text{/}$ provide sufficient information to
determine the critical temperature and superconducting energy gap. Quantifying
these parameters is essential as they, for a given resonance frequency,
uniquely determine the QP contribution to losses and therefore the maximum
achievable quality factor at any given temperature, irrespective of the
resonators characteristic impedance and circuit topology.

We studied three lumped-element circuit topologies with different coupling to
the transmission line. For each we show that the Mattis-Bardeen model provides
a consistently good description of the temperature dependence of the data,
with no topology-specific additional assumption. Combined with a proper
characterization of the measurement setup, our analysis allows us to quantify
the zero-temperature kinetic inductance and normal-state resistance of the
nanowire.

Our analysis showed the presence of additional internal microwave losses, not
explained by thermal-equilibrium QPs. A comparison between the measured and
predicted QP losses for different circuit topologies provides information
about the origin of these additional losses. Compared to the series case, the
reduced residual resistance of a parallel topology points to the presence of
substrate dielectric losses, consistent with the much larger area of the
interdigital capacitor.

Our study provides tools for those interested in circuit design with high
kinetic inductance, including alternative approaches to achieve a desired
coupling to the transmission line. From our analysis we conclude that there is
significant room for increasing the quality factor of microwave resonators
realised with high kinetic inductance nanowires. Future work should analyze
nanowires of different materials, on different substrates, or suspended in
vacuum. We hope that the methods and analysis presented here will inspire such
future studies.

## Data Availability Statement

All raw and processed data, as well as supporting code for data processing and
figure generation, are available on Zenodo [33].

## Acknowledgments

The European Union Horizon 2020 Future and Emerging Technologies (FET) Grant
Agreement No. 828966 — QAFM and the Swedish SSF Grant No. ITM17–0343 supported
this work. We thank the Quantum-Limited Atomic Force Microscopy (QAFM) team
for fruitful discussions: T. Glatzel, M. Zutter, E. Tholén, D. Forchheimer, I.
Ignat, M. Kwon, and D. Platz.

## Conflict of Interest

The authors have no conflicts to disclose.

##  Appendix A Measurement setup

Figure 5:  Schematic of the cryogenic measurement setup which is inserted into
a variable temperature physical properties measurement system (DynaCool,
Quantum Design). The excitation signal is attenuated at low temperature and
passed to the sample through a directional coupler. The reflected signal is
amplified with a cryogenic low-noise amplifier (Low Noise Factory LNF-LNC4_8E
with $40\text{\,}\mathrm{dB}\text{/}$ of gain) and a room-temperature
amplifier (Mini-Circuit ZX60-83LN-S+ with $21\text{\,}\mathrm{dB}\text{/}$ of
gain).  Figure 6:  Scattering parameters of the directional coupler. The
coupling corresponds to the scattering parameter $S_{31}$ while the isolation
is given by $S_{41}$. The ratio $D=S_{41}/S_{31}$ yields the directivity of
the directional coupler.

The measurements employ a digital microwave synthesis and analysis platform
operated as a vector network analyzer. The frequency response of the resonator
is determined from a transmission measurement through a
$20\text{\,}\mathrm{dB}\text{/}$ directional coupler, as shown in Fig. 5. To
extract the reflection coefficient $\Gamma(\omega)$ of the resonator, we need
to determine the scattering matrix of the directional coupler,

$\begin{pmatrix}V_{1}^{-}\\\ V_{2}^{-}\\\ V_{3}^{-}\\\
V_{4}^{-}\end{pmatrix}=\begin{pmatrix}0&T&C&I\\\ T&0&I&C\\\ C&I&0&T\\\
I&C&T&0\\\ \end{pmatrix}\begin{pmatrix}V_{1}^{+}\\\ V_{2}^{+}\\\ V_{3}^{+}\\\
V_{4}^{+}\end{pmatrix},$ (25)

where $T\equiv S_{21}\approx 1$ is the transmission coefficient, $C\equiv
S_{31}$ is the coupling and $I\equiv S_{41}$ is the isolation. The measured
signal is the amplified voltage wave $V_{4}^{-}$ propagating out of port 4.
Given $V_{3}^{+}=\Gamma(\omega)V_{3}^{-}$ and $V_{2}^{+}=V_{4}^{+}=0$,

$V_{4}^{-}=V_{1}^{+}(I+C\Gamma(\omega)).$ (26)

The drive $V_{\mathrm{RFout}}$ is connected to port 1 of the directional
coupler through a total attenuation $A=$30\text{\,}\mathrm{dB}\text{/}$$ and
the readout $V_{\mathrm{RFin}}$ from port 4 passes through a total gain
$G=$61\text{\,}\mathrm{dB}\text{/}$$. For $V_{1}^{+}=AV_{\mathrm{RFout}}$ and
$V_{\mathrm{RFin}}=GV_{4}^{-}$,

$V_{\mathrm{RFin}}=GAV_{\mathrm{RFout}}C(D+\Gamma(\omega)),$ (27)

where $D=I/C$ is the directivity. Figure 6 shows the measured coupling
$C\equiv S_{31}$ and isolation $I\equiv S_{41}$ as a function of frequency.

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