††thanks: Present address: Institute for Solid State Physics, University of
Tokyo, Kashiwa, Chiba 277-8581, Japan

# Thermodynamics of transition to BCS-BEC crossover superconductivity in
FeSe1-xSx

Y. Mizukami1 Email<EMAIL_ADDRESS>M. Haze2 O. Tanaka1 K.
Matsuura1 D. Sano2 J. Böker3 I. Eremin3 S. Kasahara2 Y. Matsuda2 T. Shibauchi1
1Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba
277-8561, Japan
2Department of Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
3Institut fur Theoretische Physik III, Ruhr-Universitat Bochum, D-44801
Bochum, Germany

###### Abstract

The BCS-BEC crossover from strongly overlapping Cooper pairs to non-
overlapping composite bosons in the strong coupling limit has been a long-
standing issue of interacting many-body fermion systems. Recently, FeSe
semimetal with hole and electron bands emerged as a high-$T_{\rm c}$
superconductor located in the BCS-BEC crossover regime, owing to its very
small Fermi energies. In FeSe, however, an ordinary BCS-like heat-capacity
jump is observed at $T_{\rm c}$, posing a fundamental question on the
characteristics of the BCS-BEC crossover. Here we report on high-resolution
heat capacity, magnetic torque, and scanning tunneling spectroscopy
measurements in FeSe1-xSx. Upon entering the tetragonal phase at $x>0.17$,
where nematic order is suppressed, $T_{\rm c}$ discontinuously decreases. In
this phase, highly non-mean-field behaviors consistent with BEC-like pairing
are found in the thermodynamic quantities with giant superconducting
fluctuations extending far above $T_{\rm c}$, implying the change of pairing
nature. Moreover, the pseudogap formation, which is expected in BCS-BEC
crossover of single-band superconductors, is not observed in the tunneling
spectra. These results illuminate highly unusual features of the
superconducting states in the crossover regime with multiband electronic
structure and competing electronic instabilities.

##  I I. INTRODUCTION

The crossover between weak-coupling BCS theory of superconductivity and
strong-coupling BEC describes the fundamentals of quantum bound states of
particles, including information on the superconducting critical temperature
($T_{c}$) and the possible pseudogap formation melo08 ; randeria14 . In the
strong-coupling regime, the pair-formation temperature $T_{\rm pair}$ and the
actual superconducting transition temperature $T_{\rm c}$ are distinctly
separated (Fig. 1(a)). Between $T_{\rm pair}$ and $T_{\rm c}$, the so-called
preformed Cooper pairs exist, which give rise to large superconducting
fluctuations and possibly a depletion of the low-energy density of states
(DOS), namely the pseudogap. Almost all experimental studies on the crossover
physics have been performed in ultracold atomic systems for the past decades
randeria14 , because of the difficulty in tuning the attractive interaction
between electrons in solids. Therefore, it is extremely challenging to realize
such strongly-coupled pairs in electron systems. In underdoped high-$T_{\rm
c}$ cuprates, the pseudogap formation has been discussed in terms of the BCS-
BEC crossover chen , but recent experiments support a phase transition of
different kinds at the onset of pseudogap and the relevance of BCS-BEC
crossover in cuprates remains unresolved Keimer .

Recently the superconducting semimetal FeSe hsu08 ; boehmer13 ; shibauchi20
with very small hole and electron Fermi surfaces is found to exhibit large
ratios of the superconducting energy gap $\Delta$ and Fermi energy
$\varepsilon_{\rm F}$, $\Delta/\varepsilon_{\rm F}\approx 0.3$-1.0 kasahara14
; terashima14 ; rinott17 , which appears to place FeSe in the crossover regime
(Fig. 1(a)). Indeed, in high-quality single crystals with long mean free paths
of electrons kasahara14 , large superconducting fluctuations have been
reported from the torque magnetometry, providing support for the presence of
fluctuated Cooper pairs created above $T_{\rm c}$ in the crossover regime
kasahara16 . These observations underline the importance of the BCS-BEC
crossover physics both in the superconducting and normal states of FeSe. On
the other hand, scanning tunneling spectroscopy (STS) measurements do not
observe the pseudogap formation above $T_{\rm c}$ hanaguri19 .

Figure 1: Phase diagrams of BCS-BEC crossover in single-band systems and
multiband FeSe1-xSx superconductors. (a) Theoretical phase diagram of BCS-BEC
crossover generally obtained for single-band systems melo08 ; randeria14 . The
temperature $T$ is normalized by the Fermi temperature $T_{\rm F}$ and the
strength of attraction is given by dimensionless coupling constant $1/(k_{\rm
F}a_{s})$. Below the pairing temperature $T_{\rm pair}$, bosonic pairs form,
while the superconducting coherence is acquired when the condensation occurs
at $T_{\rm c}$. In the weak-coupling BCS limit, $\Delta/\varepsilon_{\rm F}$
is much smaller than unity, and the coherence length $\xi$ is much longer than
the interparticle distance ($\sim 1/k_{\rm F}$) and $T_{\rm pair}$ is very
close to the actual $T_{\rm c}$. In the BCS-BEC crossover regime
$\Delta/\varepsilon_{\rm F}\sim 1$, $T_{\rm c}/T_{\rm F}$ reaches a maximum,
and preformed Cooper pairs are expected to exist in an extended temperature
region between $T_{\rm pair}$ and $T_{\rm c}$. (b) Experimentally determined
$T$-$x$ phase diagram of FeSe1-xSx. The superconducting transition temperature
$T_{\rm c}$ (red squares) is determined by the present heat capacity
measurements using small crystals. The nematic transition temperature (blue
diamonds) determined by transport measurements hosoi16  is also plotted. The
abrupt change in $T_{\rm c}$ indicates a significantly different
superconducting ground states (SC1 and SC2) divided by the nematic end point
at $x_{\rm c}\approx 0.17$.

These results imply that the superconductivity in FeSe is not simply described
by a picture of the conventional BCS-BEC crossover. An important aspect which
has not been taken into account in the previous studies is the effects of
multiband electronic structure, which may have other tuning factors of the
crossover in addition to $\Delta/\varepsilon_{\rm F}$. In fact, Fermi surface
of FeSe is composed of separated hole and electron pockets with different
values of $\Delta/\varepsilon_{\rm F}$ kasahara14 ; terashima14 ; coldea17 .
It has been pointed out that in the multiband systems the developments of
superconducting fluctuations and pseudogap formation are sensitively affected
by the interband coupling strength salasnich18 , indicating that the normal
and superconducting properties associated with the crossover in multiband
systems can be different from those in the single band systems chubukov16 . It
is quite important, therefore, to clarify how the superconducting properties
evolve when the multiband electronic structure is tuned.

Here, we focus on FeSe1-xSx, where the orthorhombic (nematic) phase transition
at 90 K in FeSe can be tuned by isovalent sulphur (S) substitution. The
nematic transition that distorts the Fermi surface is completely suppressed at
a critical concentration $x_{\rm c}\approx 0.17$ hosoi16 ; Licciardello19 .
The superconductivity persists in the tetragonal (non-nematic) phase above
$x_{\rm c}$. In the case of applying hydrostatic pressure, the suppression of
nematic order is accompanied by the pressure-induced antiferromagnetic order
matsuura17 . In contrast, S substitution suppresses the nematic order without
inducing the antiferromagnetism that significantly changes the Fermi surface
through band folding even when crossing $x_{c}$. This enables us to
investigate the evolution of superconducting state in the BCS-BEC crossover
with little influence of antiferromagnetic fluctuations. Moreover, the
observations of quantum oscillations up to $x\approx 0.19$ coldea19
demonstrate that the S ions do not act as strong scattering centers.

Figure 2: Heat capacity in FeSe1-xSx superconductors. (a) The schematic of
experimental set up. A Cernox small thermometer chip is used as a heater and
large current pulses are applied. (b)-(i) $T$-dependence of electronic heat
capacity divided by $T$, $C_{\rm e}/T$ in FeSe1-xSx single crystals at low
temperatures for $x=0$ (b), $x=0.05$ (c), $x=0.10$ (d), $x=0.13$ (e) in the
orthorhombic phase and $x=0.18$ (g), $x=0.20$ (h), $x=0.21$ (i) in the
tetragonal phase. (f) $T$-dependence of $C_{\rm e}/T$ for $x=0.16$ near the
nematic end point. The data (orange dots) are fitted with a weighted sum
(black line) of the $x=0.13$ orthorhombic data (dashed line) and $x=0.18$
tetragonal data (dashed-dotted line) with adjusted $T_{\rm c}$ values.

##  II II. HEAT CAPACITY

Through the high-resolution heat capacity, magnetic torque, and STS
measurements on high-quality single crystals, we investigate the evolution of
superconducting and normal state properties with $x$ in FeSe1-xSx. In Fig.
1(b), we show the phase diagram obtained by the heat capacity measurements.
The drastic change in $T_{c}$ and superconducting fluctuations are found
across $x_{c}$. To ensure the sample homogeneity, we used very small crystals
(10-190 $\mu$g) for the thermodynamic measurements, and employed the long
relaxation method with a home-made cell (Fig. 2(a)) designed for small single
crystals. Figures 2(b)-(e) depict the $T$-dependence of electronic heat
capacity divided by $T$, $C_{\rm e}/T$, for $x=0$, 0.05, 0.10 and 0.13. Here
we subtract the phonon contribution, which shows $T^{3}$-dependence, as
confirmed by the measurements in the normal state at high fields
(Supplementary Materials). In Fig. 2(b), $C_{\rm e}/T(T)$ in FeSe ($x=0$)
exhibits an ordinary BCS-like jump at $T_{\rm c}=9$ K, consistent with the
previous studies on larger samples lin11 ; yang17 ; hardy19 . As the
temperature is lowered below $T_{\rm c}$, $C_{\rm e}/T$ decreases with
decreasing slope. Below $\sim 3$ K, $C_{\rm e}/T$ shows $T$-linear dependence,
which is consistent with the strongly anisotropic superconducting gap
kasahara14 ; hardy19 ; sprau17 . Similar $C_{\rm e}/T(T)$ is observed in
orthorhombic phase $x=$0.05, 0.10, and 0.13 with $T_{\rm c}$ of 9.7, 10 and
9.5 K, respectively (Figs. 2(c), (d) and (e)).

Upon entering the tetragonal phase at $x>x_{\rm c}$, $C_{\rm e}/T$ exhibits a
dramatic change not only in the superconducting state but also in the normal
state. As represented by $x$ = 0.20 in Fig. 2(h), $C_{\rm e}/T$ increases with
increasing slope in the normal state when approaching $T_{\rm c}$ from high
temperatures. In addition, $C_{\rm e}/T$ exhibits a kink at $T_{\rm c}$,
without showing a discontinuous jump. Such a continuous $T$-dependence at
$T_{\rm c}$ is reminiscent of the BEC transition in the free Bose gas systems
pethick08  although it is too simple to employ the free Bose gas model to our
system. We note that similar enhancement of $C_{\rm e}/T$ in the normal state
is obtained in the recent calculations for the strongly interacting Fermi
systems near the unitary limit in the BCS-BEC crossover regime Wyk16 . As the
temperature is lowered below $T_{\rm c}$, $C_{\rm e}/T(T)$ for $x=0.20$
decreases with increasing slope, in stark contrast to $C_{\rm e}/T$($T$) in
the orthorhombic crystals as shown in Figs. 2(b)-(e). Below $T_{\rm c}$,
$C_{\rm e}/T$ decreases nearly in proportion to $T^{1/2}$ with large residual
$C_{\rm e}/T$ in the limit $T\rightarrow 0$, which is indicative for a large
DOS at zero energy. The large residual DOS is consistent with the recent
thermal conductivity and STS measurements sato18 ; hanaguri18 . Similar
$C_{\rm e}/T(T)$ is also observed for $x$ = 0.18, and 0.21 in the tetragonal
phase as shown in Figs. 2(g), and (i). These results together with the sharp
change in $T_{c}$ across $x_{\rm c}$ demonstrate that the superconducting
properties in the tetragonal phase are drastically different from those in the
orthorhombic phase.

We stress that the observed peculiar $C_{\rm e}/T(T)$ in tetragonal phase does
not stem from chemical inhomogeneity within the small sample for the following
reasons. First, topographic image in STM of FeSe1-xSx crystals reveal no
evidence for the segregation of S atoms, indicating an excellent homogeneity
hanaguri18  (see also Figs. 3(a) and (b)). Second, the observation of the
quantum oscillations of the crystals in the same batch coldea19  demonstrates
a long mean free path of the carriers, ensuring the high quality of our
crystals. Third, according to the elemental mapping of the energy dispersive
X-ray spectroscopy (EDX) at different scales of distance (see Fig.S1), the
typical spatial variation of the composition $\Delta x$ is $\sim$ 0.01, and
there are no discernible segregations or large inhomogeneity of the chemical
compositions which can explain our observation in $C_{\rm e}/T(T)$. Fourth, we
measured the heat capacity on two different crystals for $x=0.20$(see Fig.
S2), and two other concentrations $x=0.18$ and 0.21(Figs. 2(g), and (i)), and
confirmed the reproducibility of the behavior. We also note that our results
are consistent with the previous data of the crystal with much larger volume
within the error bar (see Fig. S2). Fifth, most importantly, there is no
concentration of the sample, which shows the superconducting transition
between $\approx 4$ K and $\approx 7$ K in the phase diagram of FeSe1-xSx.
This is supported by $C_{\rm e}/T(T)$ of the crystal at $x\approx 0.16$, which
is located in the very vicinity of $x_{\rm c}$. As shown in Fig. 2(f), $C_{\rm
e}/T$ at $x\approx 0.16$ exhibits two well separated peaks at $T_{\rm c1}=4$ K
and $T_{\rm c2}=7$ K, indicating a phase separation (due to the small spatial
variation of $\Delta x\sim 0.01$) at $x_{c}$. This $T$-dependence is well
reproduced by the sum of $C_{\rm e}/T$ of $x=0.13$ ($T_{\rm c}=9.5$ K) and
0.18 ($T_{\rm c}=4.0$ K), assuming that both contribute equally. Here, for the
fitting, $T_{\rm c}$ values are shifted slightly. These results indicate a
discontinuous change of $T_{\rm c}$ at $x_{\rm c}$. Such a discontinuous
change in the superconducting properties is consistent with the jump in
$\Delta$ values at $x_{\rm c}$ recently reported by STS studies hanaguri18 .
The observed enhancement of $C_{\rm e}/T$ above $T_{\rm c}$ for the sample in
tetragonal phase, therefore, can be attributed to an intrinsic electronic
property.

##  III III. SCANNING TUNNELING MICROSCOPY/SPECTROSCOPY

To elucidate whether the depletion of DOS associated with the pseudogap
formation occurs, the STS measurements on the crystal of $x=0.25$ are
performed. By counting the number of S atoms in the STM topographic image, we
determined $x$ value accurately (Fig. 3(a)). Figure 3(b) shows the line
profile of the normalized conductance along the line AB depicted in Fig. 3(a).
Spectra show little variation within 10 nm scale, demonstrating the uniform
spatial distribution of the superconducting gap. We also note that the gap
value is consistent with the previous report hanaguri18 . Figure 3(c) depicts
the STS spectra normalized by the conductance above the superconducting gap in
a wide temperature range. Large residual DOS, which is more than half of the
normal state value outside the superconducting gap ($\Delta\approx 1$ meV) is
observed even at $T=0.4$ K ($T/T_{\rm c}=0.1$). This large residual DOS is
consistent with the large $C_{\rm e}/T$ at low temperatures. Remarkably, STS
spectrum becomes nearly energy independent at high temperatures above $\sim 5$
K, showing no signature of the gap formation. This is also seen by the
$T$-dependence of the gap depth plotted in the upper panel of Fig. 3(d). The
superconducting gap closes at $\approx 5$ K below which the $T$-dependence of
magnetization shows a rapid decrease as shown in the lower panel of Fig. 3(d).

Figure 3: Scanning tunneling spectroscopy in the tetragonal phase. (a)
Topographic STM image (10 mV, 100 pA) of FeSe0.75S0.25 surface over a 10 nm by
10 nm area. (b) A line profile of the normalized conductance along the AB
shown in a. The data are taken at 2 K. (c) Normalized tunneling conductance
spectra taken at several temperatures. The spectra are vertically shifted for
clarity. The tip is stabilized at $V$ = 6 mV and $I$ = 1 nA. (d) Upper panel
shows the $T$-dependence of gap depth, which is defined by the difference
between normalized conductance values at $V$ = 0 and 6 mV. The red dashed line
is a guide for the eye. Lower panel shows the $T$-dependence of magnetization
measured at a zero field cool condition.

Nearly energy-independent DOS at $\varepsilon_{\rm F}$ observed by STS above
$T_{\rm c}$ indicates that the observed enhancement of $C_{\rm e}/T$ above
$T_{\rm c}$ in tetragonal phase is not caused by the DOS effect. Therefore, it
is natural to consider that the enhancement of $C_{\rm e}/T$ stems from the
superconducting fluctuations. Indeed, similar enhancement of $C_{\rm e}/T$
toward $T_{c}$ due to the strong fluctuations has been previously reported in
other iron-based superconductors pribulova09 ; serafin10 ; welp11 . In order
to discuss the effect of the superconducting fluctuations in FeSe1-xSx
quantitatively, we extract the superconducting contribution $C_{\rm sc}$,
which is obtained by subtracting the normal state electronic contribution
$\gamma T$ from $C_{\rm e}$ for $x$=0 (blue circles in Fig. 4(a)) and $x$=0.20
(red circles in Fig. 4(b)). We compare $C_{\rm sc}$ with the conventional term
of the mean field Gaussian fluctuations inderhees88 (Supplementary Materials).
Obviously, the heat capacity contribution that significantly exceeds $C_{\rm
gauss}$ can be seen for $x=0.20$. This extra heat capacity is observable up to
$t\sim 0.8$. In Fig. 4(c), we display the $x$-dependence of $\Delta C/\gamma
T_{\rm c}$, where $\Delta C$ is the height of $C_{\rm sc}$ at $T_{\rm c}$. For
$x<x_{\rm c}$, the magnitude of $\Delta C/\gamma T_{\rm c}$ is close to the
BCS weak coupling value (1.43 for $s$-wave and 0.94 for $d$-wave). On the
other hand, $\Delta C/\gamma T_{\rm c}$ for $x>x_{\rm c}$ is reduced far below
the mean-field BCS value. As a consequence of unusual suppression of $\Delta
C/\gamma T_{c}$, the low-$T$ $C_{\rm e}/T$ is largely enhanced through the
entropy balance. Figure 4(d) depicts the $x$-dependence of the $C_{e}/T$ at
$T/T_{c}$ = 0.1 taken from Fig. 2. Remarkable enhancement of $C_{e}/T$ at
$x_{c}$ indicates the fundamental difference of the low-energy excitations
across $x_{c}$. We note that recent NMR measurements for $x>x_{c}$ report no
splitting of the spin-echo signal in the crystal from the same batch wiecki18
; kuwayama20 , which excludes the microscopic phase separation of
superconducting and non-superconducting regions.

Figure 4: Comparisons of superconducting transition properties between the
orthorhombic and tetragonal phases. (a) The superconducting contribution to
the electronic heat capacity $C_{\rm sc}=C_{\rm e}-\gamma T$ as a function of
reduced temperature $t\equiv\frac{T-T_{\rm c}}{T_{\rm c}}$ for $x=0$ (($A$))
and $x=0.20$ (($B$)). The conventional Gaussian fluctuation contribution
$C_{\rm gauss}$ is estimated (solid lines).(c), (d) $x$ dependence of $\Delta
C/\gamma T_{\rm c}$ (($C$)), and $C_{e}/T$ at $T/T_{c}$ = 0.1 (($D$)). (e),
(f) Entropy calculated from $C_{\rm e}/T(T)$ in the normal state ($S_{\rm n}$)
and superconducting state ($S_{\rm n}$) as a function of $t$ for $x=0$ (($E$))
and $x=0.20$ (($F$)). $S_{\rm n}$ is estimated from the high-field data where
superconductivity is suppressed. The entropy for $x=0.20$ is linearly
extrapolated below 1.0 K. (g) $x$ dependence of the relative entropy
difference between zero and high fields $(S_{\rm n}-S_{\rm n})/S_{\rm n}$ at
$T=1.03T_{\rm c}$. (h) The $T$-dependence of the anisotropy of the magnetic
susceptibility $\Delta\chi=\chi_{c}-\chi_{ab}$ for several fields in $x$ =
0.20. Each data is obtained by fitting the field-angle dependence of the
torque $\tau$($\theta$). The inset shows the $\tau$($\theta$) at 1.0 T and 41
K as the typical signal of torque, where the red markers and the solid black
line show the raw data and fitting curve, respectively.

##  IV IV. MAGNETIC TORQUE

The anomalous $C_{\rm e}/T$ in the tetragonal phase is reflected by the
entropies in the normal and superconducting states, $S_{\rm n}$ and $S_{\rm
s}$, respectively, as shown in Fig. 4(e) and (f). For $x=0$, the
$T$-dependence of $S_{\rm s}$ shows a kink at $T_{\rm c}$, which is typical
for the second-order superconducting transition. For $x=0.20$, on the other
hand, owing to the absence of the jump in $C_{\rm e}/T$ at $T_{\rm c}$, no
kink anomaly is observed in the $T$-dependence of $S_{s}$. Moreover, the
entropy is significantly suppressed from $S_{n}$ even above $T_{\rm c}$. In
Fig. 4(f), we show the $x$ dependence of the lost entropy normalized by
$S_{\rm n}$, $(S_{\rm n}-S_{\rm s})/S_{\rm n}$, just above $T_{\rm c}$,
$T=1.03T_{\rm c}$. Upon entering the tetragonal phase, an abrupt enhancement
of $(S_{\rm n}-S_{\rm s})/S_{\rm n}$ can be seen, indicating a drastic change
of the superconducting fluctuations associated with the preformed Cooper
pairs.

To obtain further insight into the superconducting fluctuations in the
tetragonal phase, the magnetic torque was measured by using micro-cantilevers.
The torque can sensitively detect the diamagnetic response through the
anisotropy of magnetic susceptibility kasahara16 . By measuring the field
angular variation of torque as a function of polar angle $\theta$ from the $c$
axis, the anisotropy $\Delta\chi=\chi_{c}-\chi_{ab}$ is determined, where
$\chi_{c}$ and $\chi_{ab}$ are the magnetic susceptibilities along the $c$
axis and in the $ab$ plane, respectively. As shown in the inset of Fig. 4(h),
$\tau(\theta)$ shows an almost perfect sinusoidal curve,
$\tau(\theta)\propto\Delta\chi\sin\theta$. Although $\Delta\chi(T)$ at high
fields is almost independent of $T$ and $H$, it exhibits strong $T$ and
$H$-dependence at low fields as shown in Fig. 4(h). It is well settled that
the fluctuation induced diamagnetic susceptibility of most superconductors
including multiband systems can be well described by the standard Gaussian-
type (Aslamasoz-Larkin, AL) fluctuation susceptibility $\chi_{\rm
AL}$(Supplementary Materials). Here we focus on the $H$-dependent diamagnetic
susceptibility at low fieldskasahara16 . Then, $\Delta\chi$ at low fields is
much larger than the $|\chi_{\rm AL}|$ which is added to high field 7 T data.
This implies that the superconducting fluctuations in FeSe1-xSx are distinctly
different from those in conventional superconductors, supporting the
fluctuation effects observed in heat capacity measurements. We stress that
although the multi-gap superconductivity may lead to a small value of $\Delta
C/\gamma T$, it cannot give rise to the non-mean-field behaviors observed in
heat capacity and magnetic susceptibility above $T_{c}$.

##  V V. DISCUSSION AND CONCLUSION

In FeSe, while large superconducting fluctuations that well exceed the
Gaussian fluctuations are observed, the $C_{\rm e}/T$ around $T_{c}$ does not
exhibit a significant deviation from the mean-field behavior. This suggests
that tetragonal FeSe1-xSx are closer to the BEC regime. However, with
increasing S composition, the volume of the Fermi surface increases, as
reported by quantum oscillation experiments coldea19 , and $\Delta$ is
reduced, according to STM measurements hanaguri18 . Then
$\Delta/\varepsilon_{F}$ is expected to become smaller with increasing S
concentration, which is opposite to the tendency of approaching the BEC regime
hashimoto20 . This suggests that there is another important factor, which has
not been duly taken into consideration.

A key factor that controls the BCS-BEC crossover in FeSe1-xSx is the multiband
character. As shown previously chubukov16 ; hanaguri19 , in FeSe the strong
interband interaction between electron and hole pockets prevents the splitting
of the $T_{\rm pair}$ and the $T_{c}$, determined by the superfluid stiffness.
We note that this strong interband interaction is also an important ingredient
in the orbital-selective scenario sprau17 ; yu18 ; hu18 ; nica21  and
therefore we expect that even taking orbital selectivity into account, the
main result, i.e. mean-field like jump of the specific heat will remain the
same. At the same time, much less is known on the evolution of the ratio of
the interband versus intraband interactions upon approaching $x_{c}$ in
FeSe1-xSx. Furthermore, recent experimental Yi2019  and theoretical Rhodes2020
; Steffensen2020  works pointed out an interesting possibility of the
substantial non-local nematic order on the $d_{xy}$ orbital and/or inter-
orbital-nematicity between $d_{xy}$ and $d_{yz}$ orbitals in FeSe. The most
important consequence of this phenomenon would be an additional Lifshitz
transition in FeSe1-xSx upon S substitution, which occurs close to $x_{c}$.
This in turn implies the appearance of the incipient band near $x_{c}$ (see
for example Fig.6 in Ref.Rhodes2020 ). On general grounds, the presence of the
incipient band would enhance the tendency towards the BCS-BEC crossover. In
addition, given nearly equal mixture of the $s$\- and $d$-wave symmetry
components of the superconducting gap in FeSe hashimoto18  to express such
anisotropic gap, it is natural to assume that they are competing in the
tetragonal phase of FeSe1-xSx. In particular, the ratio of the pairing
interactions in the $d$-wave and $s$-wave channels $|g^{d}_{\rm
ee}|/g^{s}_{\rm eh}$ may, in addition to the incipient band, be another key
parameter of the BCS-BEC crossover as follows from our model calculations (see
Fig. S4). The pair-formation temperature $T_{\rm pair}$ and condensation
temperature $T_{\rm c}$ can be split more with increasing the ratio of
intraband and interband interactions $|g^{d}_{\rm ee}|/g^{s}_{\rm eh}$ (see
Fig. S4(b)) without changing $\varepsilon_{\rm F}$. This quantitatively
explains the appearance of strong coupling superconductivity for $x>x_{c}$
with no apparent enhancement of $\Delta/\varepsilon_{F}$. A remarkable and
unexpected feature is the absence of pseudogap above $T_{\rm c}$ in the STS
spectra, despite the presence of giant superconducting fluctuations in
thermodynamic quantities. It should be noted that the relationship between the
preformed pairs and the pseudogap formation is still elusive even in the
ultra-cold atoms randeria14 . In addition, the onset temperature of the
pseudogap may differ from that of fluctuations Wyk16 . Moreover, it has been
pointed out that the pseudogap phenomena in multiband systems are markedly
altered from those in the single band systems. For instance, in multiband
systems, the pseudogap formation is less prominent by the reduction of its
onset temperature tajima20 . Quite recently, the BEC-like dispersion of the
Bogoliubov quasiparticles has been reported in the tetragonal FeSe1-xSx by
using ARPES measurements hashimoto20 . Although the direct comparison of the
ARPES data with the present thermodynamic results is not straightforward,
because ARPES measures only the portion of hole Fermi pocket, it provides
spectroscopic evidence of the BCS-BEC crossover. The ARPES measurements also
report the presence of distinct pseudogap in the hole pocket. However, we
point out that this is not an apparent contradiction to the present STM
results. It has been suggested that the DOS of electron pocket is much larger
than that of hole pockets. As STM measures the DOS integrated over all bands,
STM is insensitive to the pseudogap formation in the hole pocket. Therefore,
the STM spectra, combined with the ARPES results, suggest that the pseudogap
formation is also orbital dependent.

It still, however, cannot fully explain the reduction of the gap magnitudes
found experimentally and further mechanisms may be needed such as influence of
magnetic fluctuations which would further reduce the superfluid stiffness
Simard2019 , a Lifishitz transition near the orthorhombic to tetragonal
transition Rhodes2020 , or nematic fluctuations. In FeSe1-xSx, the nematic
fluctuations are peaked at $q_{\rm nem}\approx 0$ because its ordered state
breaks the rotational symmetry while preserving translational symmetry.
Enhanced nematic fluctuations hosoi16  play an important role for the normal
state properties Licciardello19 ; reiss20 , particularly at $x\sim x_{c}$
Licciardello19 . It has also been pointed out theoretically that nematic
fluctuations strongly influence the superconductivity maier14 ; lederer17 .
Therefore, it is tempting to consider that the nematic fluctuations and
incipient bands in FeSe1-xSx upon approaching a tetragonal phase can enhance
the pairing interaction, leading to the system to approach the BEC regime.
Further theoretical and experimental studies are required to uncover whether
the nematic fluctuations and multi-band nature affect the crossover physics.

Our thermodynamic studies have revealed a highly non-mean-field behavior of
superconducting transition in FeSe1-xSx when the nematic order of FeSe is
completely suppressed by S substitution. This is consistent with the recent
ARPES measurements which report the BEC-like dispersion of the Bogoliubov
quasiparticles in the tetragonal regime. Our findings demonstrate that
FeSe1-xSx offers a unique playground to study the superconducting properties
in the BCS-BEC crossover regime of multiband systems.

##  VI ACKNOWLEDGMENTS

The authors thank A. Carrington, L. Malone, P. Walmsley, K. Ishida, K. Sugii,
T. Taen, and T. Osada for experimental assistance. We also thank Y. Tada, H.
Ikeda, K. Adachi, Y. Ohashi, and D. Inotani for valuable discussion. This work
has been supported by KAKENHI (Nos. JP20H02600, JP20K21139, JP19H00649,
JP18H05227, JP18K13492) and Grant-in-Aid for Scientific Research on Innovative
Areas “Quantum Liquid Crystals” (No. JP19H05824) from JSPS.

##  VII APPENDIX A: MATERIALS AND METHODS

Crystal growth and characterization The single crystals of FeSe1-xSx were
grown by the chemical vapor transport technique as described in Refs.
boehmer13 ; hosoi16 . The composition $x$ is determined by the energy
dispersive X-ray spectroscopy.

Heat capacity The heat capacity of the crystals is measured using long
relaxation method wang01 ; taylor07 , whose experimental setup is illustrated
in the inset of Fig. 1(a). A single bare chip of Cernox resistor is used as
the thermometer, heater and sample stage, which is suspended from the cold
stage by silver-coated glass fibers in order that the bare chip has weak
thermal link to the cold stage, and electrical connection for the sensor
reading. The mass of the samples measured in this study is 81, 41, 171, 137,
179, 50, 107, 192, and 10 $\mu$g for $x$ = 0, 0.05, 0.1, 0.13, 0.16, 0.18,
0.20#1, 0.20#2, and 0.21, respectively. The samples are mounted on the bare
chip using Apiezon N grease. The heat capacity of the crystals are typically
obtained by subtracting the heat capacity of bare chip and grease from the raw
data.

Torque The torque measurements are performed using micro-cantilevers. The
small single crystals with typical size of 150 $\mu$m$\times$100
$\mu$m$\times$10 $\mu$m are mounted on the tip of micro cantilevers with small
amount of Apiezon N grease. The magnetic field is applied inside the
$ac$($bc$) plane by vector magnet, and the field-angle dependence of torque is
obtained by rotating the whole cryostat within $ac$($bc$) plane. Magnetic
torque $\tau$ can be expressed as $\tau$ = $\mu_{0}V\bm{M}\times\bm{H}$, where
$\mu_{0}$ is vacuum permeability, $V$ is sample volume, $\bm{M}$ is
magnetization, and $\bm{H}$ is external magnetic field. Here the $\bm{H}$ is
rotated within the $ac$ ($bc$) plane, and $\theta$ is polar angle from the $c$
axis. In this configuration, the $\tau$ is expressed as
$\tau_{2\theta}(T,H,\theta)$ =
$\frac{1}{2}\mu_{0}H^{2}V\Delta\chi$sin($\theta$). Here $\Delta\chi$ =
$\chi_{c}$ \- $\chi_{ab}$ is anisotropy of magnetic susceptibility between
$ac$($ab$) plane and $c$ axis.

Scanning tunneling microscopy/spectroscopy The scanning tunneling
microscopy/spectroscopy measurements have been performed with a commercial low
temperature STM system. Fresh and atomically flat surfaces are obtained by
$in$-$situ$ cleavage at liquid nitrogen temperature in ultra high vacuum. All
conductance spectra and topographic images are obtained by a PtIr tip.
Conductance spectra are measured by a lock-in technique with a modulation
frequency 997 Hz and modulation voltage 200 $\mu$V. The backgrounds of raw
conductance data are normalized by curves which are derived by 2nd order
polynomial fitting using outside superconducting gaps.

Theoretical calculation The theoretical calculations were done within a
simplified interacting two-band model in two dimensions, employed previously
chubukov16  with hole and electron pockets with small Fermi energies [Fig. 8].
We assume superconductivity due to repulsive interactions. While the dominant
interband interaction favors Cooper-pairing in the $s$-wave symmetry channel,
the $d$-wave projected intraband interaction yields attraction in the d-wave
channel. We obtained $T_{pair}$ from the solution of the linearized mean-field
gap equations, whereas Tc in each case is estimated from superfluid stiffness.
Details of the calculations are given in the Supplemental Information.

##  VIII APPENDIX B: ENERGY DISPERSIVE X-RAY SPECTROSCOPY (EDX) ANALYSIS

Figure 5: Energy Dispersive X-ray spectroscopy (EDX) analysis on tetragonal
FeSe0.80S0.20 samples in comparison with FeSe. (a) Elemental mapping of Fe
(green dots), Se (yellow dots), and S (red dots) at 500 $\mu$m scale for $x$ =
0. (b)-(f) Elemental mapping of Fe, Se, and S for $x$ = 0.20 at the scale of
500 $\mu$m((b)), 50 $\mu$m((c)), 5 $\mu$m((d)), 500 nm((e)), and 50 nm((f)),
respectively.

Energy Dispersive X-ray spectroscopy (EDX) analysis is conducted on tetragonal
$x$ = 0.20 samples in comparison with $x$ = 0 to investigate the spatial
homogeneity of the sulfur content. For $x$ = 0, sulfur intensity is almost
same as the background outside the sample position in Fig. 5(a), showing that
there is no sulfur substituted into the sample as expected. Figures 5(b)-(f)
show the elemental distribution of $x$ = 0.20 at the scale of 500 $\mu$m, 50
$\mu$m, 5 $\mu$m, 500 nm, and 50 nm, respectively. Each data shows S intensity
with the much thicker color than $x$ = 0 although there is some small darker
areas in Fig. 5(b), and (c) which originate from the large impurity attached
on the crystal surface and surface roughness in a macroscopic scale and are
not from the inhomogeneity of chemical composition. The data at every length
scale show the uniform distribution of sulfur which is almost comparable to
iron and selenium, indicating that there is no discernible segregation and
inhomogeneity down to the mesoscopic scale $\simeq$10 nm. We quantitatively
investigate the sulfur content $x$ at the several area inside the sample and
the typical variation of $x$ is $\simeq$0.01, demonstrating that such
variation of composition cannot explain the results of specific heat and
torque measurements.

##  IX APPENDIX C: COMPARISON WITH PREVIOUS HEAT CAPACITY DATA

Figure 6: Temperature dependence of heat capacity in tetragonal FeSe1-xSx
samples. (a), (b) Temperature dependence of electronic heat capacity divided
by temperature for $x$ = 0.20 #1 ((a)) and #2 ((b)) in comparison with the
data in Refs. sato18 .

The heat capacity in the tetragonal FeSe1-xSx has been reported previously by
Y. Sato et al., using quasi-adiabatic methodsato18 . We plot our $C_{e}$/$T$
data for two $x$ = 0.20 samples in Figs. 6(a), (b) together with the data of
Ref.sato18 . The two samples in this work exhibit almost identical temperature
dependence and absolute values, indicating the reproducibility of the heat
capacity data in $x$ = 0.20 with better resolution in our studies. Although
the absolute value of our data is slightly smaller than the data in Ref.sato18
, our data is within the error range of previous data, and both data are
basically consistent up to $\simeq$ 6 K which is the highest temperature in
the previous reports.

##  X APPENDIX D: UPPER CRITICAL FIELD $H_{C2}$ AND ESTIMATION OF GAUSSIAN
FLUCTUATION TERM

Figure 7: Heat capacity in magnetic field for FeSe1-xSx $x$ = 0 and 0.20 and
$x$-dependence of upper critical field. (a), (b) Temperature dependence of
electronic heat capacity divided by temperature in magnetic field for $x$ = 0
((a)) and $x$ = 0.20 #2 ((b)). The magnetic field is applied along $c$-axis
direction for both samples. (c) Temperature dependence of upper critical field
$H_{c2}$($T$) determined from the heat capacity in magnetic field. The
$H_{c2}$($T$) for $H$ // $ab$ in $x$ = 0.20 and the $H_{c2}$($T$) for $H$ //
$c$ in $x$ = 0.10 are also plotted in the same panel.

The field $H$ dependence of heat capacity is measured in the perpendicular
field $H//c$ for $x$ = 0.0, and 0.10, and in the perpendicular and parallel
field $H//c,ab$ for $x$ = 0.20. The $C_{e}$/$T$ in $H//c$ for several $H$
values are shown for $x$ = 0.0 and 0.20 in Figs. 7(a), (b), respectively. The
heat capacity jump due to the superconducting transition becomes broader with
increasing $H$ for both S concentrations. This behavior is possibly due to the
enhanced fluctuations in the magnetic field as discussed in
YBa2Cu3O7-δoverend94 . In zero field, we observe the large superconducting
fluctuations in heat capacity in $x$ = 0.20. The additional heat capacity due
to the superconducting fluctuations just above $T_{c}$ is suppressed by
applying magnetic field, and then the $C_{e}$/$T$ approaches to the normal
state value. This fact indicates that the additional heat capacity responds
sensitively to the magnetic field, and does not stem from the incorrect
estimation of electronic heat capacity in the normal state when we subtract
the lattice contribution. From the $T_{c}$ defined as the peak temperature in
$C_{e}$/$T$ under field, we obtain the $H$-$T$ phase diagram of $x$ = 0.0,
0.10 and 0.20 as shown in Fig. 7(c). In $x$ = 0.0 and 0.10 the $T$-dependence
of upper critical field $H_{c2}$ is consistent with previous reportsmuratov17
; abdel15 , while the $H_{c2}$ for $\mu_{0}H>12$ T cannot be determined from
our measurements. The $H_{c2}$ in $x$ = 0.20 shows small value compared to $x$
= 0.0, reflecting the lower $T_{c}$. From the $H_{c2}(0)$ estimated by the WHH
relation $H_{c2}(0)=0.69T_{c}|dH_{c2}/dT|_{T=T_{c}}$werthamer66 , we obtain
the coherence length $\xi_{ab}=13.5$ nm, $\xi_{c}=4.1$ nm for $x$ = 0.20
through $H_{c2}(0)=\Phi_{0}/(2\pi\xi_{ab}^{2})$, and
$\Phi_{0}/(2\pi\xi_{ab}\xi_{c})$ for $H//c$, and $H//ab$, respectively.

The contribution of the mean field Gaussian fluctuations to the heat capacity
inderhees88  is given by $C_{\rm gauss}=C^{+}t^{-0.5}$, where $C^{+}=k_{\rm
B}/(8\pi\xi_{ab}^{2}\xi_{c}$) and $t\equiv\frac{T-T_{\rm c}}{T_{\rm c}}$ is
the reduced temperature, and $\xi_{ab}$ and $\xi_{c}$ are in-plane and out-of-
plane coherence lengths at $T=0$, respectively. The dashed lines in Figs. 4(a)
and (b) represent the contribution of Gaussian fluctuations obtained by using
$\xi_{ab}=5.5$ nm, $\xi_{c}=1.5$ nm for $x=0$ kasahara16  and $\xi_{ab}=13.5$
nm, $\xi_{c}=4.1$ nm for $x=0.20$ (Fig. 7).

The Gaussian-type (Aslamasoz-Larkin, AL) fluctuation contribution in
susceptibility is given by

$\chi_{\rm AL}\approx\frac{2\pi^{2}}{3}\frac{k_{\rm B}T_{\rm
c}}{\Phi_{0}^{2}}\frac{\xi_{ab}^{2}}{\xi_{c}}t^{-\frac{1}{2}}$ (1)

in the zero-field limit. Here $\Phi_{0}$ is the flux quantum. In the multiband
case, the behavior of $\chi_{\rm AL}$ is determined by the shortest coherence
length of the main band, which governs the orbital upper critical field. As
the diamagnetic contribution $\chi_{\rm AL}$ is expected to become smaller in
magnitude at higher magnetic fields, $|\chi_{\rm AL}|$ yields an upper bound
for the conventional Gaussian-type amplitude fluctuations.

##  XI APPENDIX E: CALCULATIONS OF PAIRING INSTABILITY AND PAIRING CHANNEL
BASED ON THE TWO BAND MODEL

Figure 8: Calculation results of pairing and condensation temperature, and
intensity of pairing channels based on the two band model. (a) Schematic
picture of the assumed electronic structure with hole(blue curve) and
electron(red curve) bands. The $E^{h}_{F}$ and $E^{e}_{F}$ indicate the Fermi
energy of the hole and electron band, respectively. (b) Calculated pairing
temperature $T_{\rm pair}$ and condensation temperature $T_{\rm c}$ in units
of $E_{F}^{h}$ as a function of the ratio of intraband to interband
interactions in a two-band model. (c) Calculated superconducting gaps in units
of $E_{F}^{h}$ for $s$-wave and $d$-wave channels in the electron (e) and hole
(h) bands as a function of the ratio of intraband to interband interactions.

To make an estimate of the splitting between the Cooper-pair formation
temperature, $T_{pair}$, and the actual superconducting transition
temperature, $T_{c}$, we followed Ref.chubukov16  and consider a simplified
interacting two-band model in two dimensions with hole and electron pockets
with small Fermi energies ($E_{F}^{h}=20$meV and $E_{F}^{e}=10$meV) as
depicted in the Fig. 8(a). As we are interested in the evolution of the FeSe
system as a function of the Sulfur doping we assumed the system is
orthorhombic. However, once the intraband interaction in the $d$-wave
interaction dominates and the $s-$wave component of the gap is small the
results can be easily extrapolated to the purely tetragonal system. The
Hamiltonian reads

$\displaystyle\begin{split}\hat{H}=&\sum_{\textbf{k}\alpha\sigma}\xi_{\textbf{k}\alpha}c^{\dagger}_{\alpha\textbf{k}\sigma}c_{\alpha\textbf{k}\sigma}\\\
&+\sum_{\textbf{k},\textbf{k}^{\prime},\alpha,\alpha^{\prime}}U_{\alpha\alpha^{\prime}}(\textbf{k}\textbf{k}^{\prime})c^{\dagger}_{\alpha\textbf{k}\uparrow}c^{\dagger}_{\alpha-\textbf{k}\downarrow}c_{\alpha^{\prime}-\textbf{k}^{\prime}\downarrow}c_{\alpha^{\prime}\textbf{k}^{\prime}\uparrow}\end{split}$

where $\alpha\in\\{\mathrm{e},\mathrm{h}\\}$, and
$\xi_{\textbf{k}\mathrm{e}}$, $\xi_{\textbf{k}\mathrm{h}}$ are the electron
and hole energy dispersions separated by the large momentum, respectively as
shown in the Fig. 8(a). Assuming superconductivity due to repulsive
interaction in the $A_{1g}$ and $B_{2g}$ symmetry channels we write the
interaction terms as follows

$\displaystyle U_{\mathrm{eh}}(\textbf{k},\textbf{k}^{\prime})$
$\displaystyle=$ $\displaystyle
U^{s}_{\mathrm{eh}}+U^{d}_{\mathrm{eh}}\phi(\textbf{k})\phi(\textbf{k}^{\prime})$
$\displaystyle U_{\alpha\alpha}(\textbf{k},\textbf{k}^{\prime})$
$\displaystyle=$ $\displaystyle
U^{d}_{\alpha\alpha}\phi(\textbf{k})\phi(\textbf{k}^{\prime})$

with
$\phi(\textbf{k})=\cos(2\varphi)=\frac{k_{x}^{2}-k_{y}^{2}}{k_{x}^{2}+k_{y}^{2}}$.
Here, we further assume that the inter-band repulsion drives $s^{\pm}$-wave
symmetry of the superconducting order parameter, while interaction in the
$d$-wave channel is mostly intra-band. We define superconducting order
parameters as

$\displaystyle\Delta_{\alpha}(\textbf{k})=\sum_{\textbf{k}^{\prime},\alpha\prime}U_{\alpha\alpha\prime}(\textbf{k},\textbf{k}^{\prime})\langle
c_{\alpha^{\prime}-\textbf{k}^{\prime}\downarrow}c_{\alpha^{\prime}\textbf{k}^{\prime}\uparrow}\rangle$
(2)

and perform a mean-field decoupling to find the mean-field gap equations

$\displaystyle\begin{split}\Delta^{s}_{\mathrm{e}}=&-g^{s}_{\mathrm{eh}}\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\tanh\left(\frac{E_{\mathrm{e}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{e}}(\textbf{k})}\\\
\Delta^{s}_{\mathrm{h}}=&-g^{s}_{\mathrm{eh}}\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\tanh\left(\frac{E_{\mathrm{h}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{h}}(\textbf{k})}\\\
\Delta^{d}_{\mathrm{e}}=&|g^{d}_{\mathrm{ee}}|\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\cos^{2}(\varphi)\tanh\left(\frac{E_{\mathrm{e}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{e}}(\textbf{k})}\\\
&-g^{d}_{\mathrm{eh}}\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\cos^{2}(\varphi)\tanh\left(\frac{E_{\mathrm{h}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{h}}(\textbf{k})}\\\
\Delta^{d}_{\mathrm{h}}=&|g^{d}_{\mathrm{hh}}|\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\cos^{2}(\varphi)\tanh\left(\frac{E_{\mathrm{h}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{h}}(\textbf{k})}\\\
&-g^{d}_{\mathrm{eh}}\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\cos^{2}(\varphi)\tanh\left(\frac{E_{\mathrm{e}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{e}}(\textbf{k})}.\\\
\end{split}$

The dimensionless coupling constants are now given by dimensionless
$g_{\alpha\alpha\prime}=N_{0}U_{\alpha\alpha\prime}$ with the density of
states in two dimensions $N_{0}=\frac{m}{2\pi}$, $\Lambda=1$eV $\gg E_{F}^{h}$
is the high energy cut-off and
$E_{\alpha}(\textbf{k})=\sqrt{\xi_{\alpha}^{2}+\left[\Delta^{s}_{\alpha}+\Delta^{d}_{\alpha}\cos(\varphi)\right]^{2}}$
are the energy dispersion of the Bogoliubov quasiparticles. Note that while
the inter-band term between electron and hole pockets is assumed to be
repulsive, the intraband $g^{d}_{\mathrm{ee}}<0$ and $g^{d}_{\mathrm{hh}}<0$
are attractive in the $d$-wave channels. In case of
$\Delta_{\alpha}/E_{\mathrm{F}\alpha}\sim 1$, where $E_{\mathrm{F}\alpha}$ is
the Fermi energy of each band, we need to renormalize the chemical potential
assuming the total number of particles is conserved. The equation determining
the particle number is given by

$\displaystyle\begin{split}E_{\mathrm{F_{e}}}-E_{\mathrm{F_{h}}}=&-\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\\\
&\left(\frac{\xi_{e}(\textbf{k})\tanh\left(\frac{E_{\mathrm{e}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{e}}(\textbf{k})}+\frac{\xi_{h}(\textbf{k})\tanh\left(\frac{E_{\mathrm{h}}(\textbf{k})}{2T}\right)}{2E_{\mathrm{h}}(\textbf{k})}\right)\end{split}$
(3)

and it has to be solved self-consistently with Eqs.(LABEL:GapEuqation). The
pair building temperature $T_{\mathrm{pair}}$ at which electrons form Cooper
pairs can be obtained from the condition that the determinant in

$\displaystyle
0=-\left(\begin{array}[]{cccc}1&g^{s}_{\mathrm{eh}}\Pi^{s}_{h}&&\\\
g^{s}_{\mathrm{eh}}\Pi^{s}_{e}&1&&\\\
&&1-|g^{d}_{\mathrm{ee}}|\Pi^{s}_{e}&g^{d}_{\mathrm{eh}}\Pi^{d}_{e}\\\
&&g^{d}_{\mathrm{eh}}\Pi^{d}_{e}&1-|g^{d}_{\mathrm{hh}}|\Pi^{s}_{h}\end{array}\right)\left(\begin{array}[]{c}\Delta^{s}_{\mathrm{e}}\\\
\Delta^{s}_{\mathrm{h}}\\\ \Delta^{d}_{\mathrm{e}}\\\
\Delta^{d}_{\mathrm{h}}\end{array}\right)$ (12)

vanishes and

$\displaystyle\Pi^{s}_{\alpha}$
$\displaystyle=\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\tanh\left(\frac{\xi_{\alpha}}{2T_{\mathrm{ins}}}\right)}{2\xi_{\alpha}}$
(13) $\displaystyle\Pi^{d}_{\alpha}$
$\displaystyle=\int_{0}^{2\pi}\frac{d\varphi}{2\pi}\int_{0}^{\Lambda}d\epsilon\frac{\cos^{2}(2\varphi)\tanh\left(\frac{\xi_{\alpha}}{2T_{\mathrm{pair}}}\right)}{2\xi_{\alpha}}.$
(14)

In the usual BCS case the phase fluctuations are costly and
$T_{\mathrm{c}}\approx T_{\mathrm{pair}}$. In our case due to smallness of the
Fermi energies, the condensation of pairs may happen at lower temperature,
$T_{\mathrm{c}}\leq T_{\mathrm{pair}}$. We estimate
$T_{\mathrm{c}}\approx\frac{\pi}{2}\rho_{\mathrm{s}}$ where
$\rho_{\mathrm{s}}(T=0)$ is the superfluid-stiffness following Ref. chubukov16
. Note that in two dimensions, the superconducting transition temperature
$T_{c}\sim\rho_{s}(T_{c})$ (see, e.g., Refs.Pokrovsky79 ; Beasley79 ). The
interplay between $T_{c}$ and $T_{pair}$ depends on the ratio
$\rho_{s}(T=0)/T_{pair}$. If this ratio is large, the superfluid stiffness
rapidly increases below Tpair. In this situation, $T_{c}=T_{pair}$ minus a
small correction, i.e., the phases of bound pairs order almost immediately
after the pairs develop (phase fluctuations cost too much energy). If
$\rho_{s}(T=0)/T_{pair}$ is small, $\rho_{s}(T)$ increases slowly below Tpair
and $T_{c}$ is of order $\rho_{s}(T=0)$. In Ref.chubukov16  its expression was
extended to the multiband case where $\rho_{\mathrm{s}}$ is given by
$\rho_{\mathrm{s}}\approx\rho_{\mathrm{e}}+\rho_{\mathrm{h}}$,

$\displaystyle\rho_{\mathrm{e}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{2\pi}d\varphi\int_{0}^{\Lambda}d\epsilon\Delta_{\mathrm{e}}^{2}\frac{\epsilon}{E_{\mathrm{e}}(\textbf{k})^{3}}$
(15) $\displaystyle\rho_{\mathrm{h}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int_{0}^{2\pi}d\varphi\int_{0}^{\Lambda}d\epsilon\Delta_{\mathrm{h}}^{2}\frac{\epsilon}{E_{\mathrm{h}}(\textbf{k})^{3}}.$
(16)

We solved Eqs. (LABEL:GapEuqation) together with Eq. (3) for the mixed
$s+d$-wave gaps in the orthorhombic state at $T=0$ and used them as an input
parameter to calculate $T_{\mathrm{c}}$. The Cooper-pair formation
temperature, $T_{\mathrm{pair}}$ is found from the condition that the
determinant in Eq. (12) vanishes and the results are shown in the Fig. 8(b) in
units of $E_{Fh}$. The pair-formation temperature $T_{\rm pair}$ and
condensation temperature $T_{\rm c}$ are calculated as a function of the ratio
of intraband and interband interactions $|g^{d}_{\rm ee}|/g^{s}_{\rm eh}$
shown in Fig. 8(b). The two temperatures split and the difference grows with
increasing $|g^{d}_{\rm ee}|/g^{s}_{\rm eh}$. Fig. 8(c) shows the evolution of
the gap symmetry as a function of $|g^{d}_{\rm ee}|/g^{s}_{\rm eh}$ also in
units of $E_{Fh}$, and we can see the dominant $d$-wave character for each
band with large $|g^{d}_{\rm ee}|/g^{s}_{\rm eh}$ regime. Observe that for
smaller ratio of $|g^{d}_{ee}|/g^{s}_{eh}$ the pairing symmetry is a pure
$s^{+-}$ driven by inter-band interactions. In this case both gaps are equal
in magnitude, but with the larger gap at the band with smaller $E_{F}$. With
increasing intra-band coupling in the d-wave gaps $\Delta^{d}_{e}$ and
$\Delta^{d}_{h}$ grow while the s-wave gaps become smaller. Since we assume
$|g^{d}_{ee}|\gg g^{d}_{eh}$, $\Delta^{d}_{e}$ and $\Delta^{d}_{h}$ differ in
magnitude with the larger gap at the band with larger $E_{F}$.

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# SnoW: Serverless n-Party calls over WebRTC

Thomas Sandholm<EMAIL_ADDRESS>
###### Abstract

We present a novel WebRTC communication system capable of hosting multi-party
audio and video conferencing sessions without a media server. We implement
various communication models based on the needs and capabilities of the
communicating parties, and show that we can construct the equivalent of Mesh,
SFU, and MCU WebRTC networks in our peer-to-peer architecture. In our
evaluation we conclude that using a limited number of merged streams can
improve the user experience significantly, in particular if low-resource
devices are involved.

##  1 Introduction

WebRTC [1] has become the de facto standard for video-conferencing both on the
Web and in native applications across stationary as well as mobile platforms.

Putting the low-level encoding and rendering functionality in an open-source,
multi-platform runtime, while allowing developers to fully control the user
interface, signaling plane and call setup logic has turned out to be a winning
combination.

There are numerous cookbooks and tutorials on how to set up 2-party calls
online, and the W3C-standardized JavaScript API available in all modern
browser runtimes today is tailor-made to the 2-party peer-to-peer use case.
The simplicity of this design facilitates use in 2-party calls, but it has
also lead to a common misconception that WebRTC can only be used with 2-party
calls.

Although, this restriction is clearly not true as demonstrated by numerous
commercial as well as open-source communication tools, developers who want to
integrate video-conferencing in their applications are, in practice, hard
pressed to avoid using a complex and costly media server backend-dependency to
their solution.

Media server providers are also quick to point out that WebRTC requires a
signaling server and a media relay server already, so putting it all in one
black box called media server that also does the multiplexing of streams is
not a stretch.

Signalling servers (e.g. WebSocket [2] servers) and relay servers (e.g. TURN
[3] providers) are however, significantly less complex to set up and manage,
and also require less server resources than a multiplexing media server.

To unleash 3+-party Web conferencing use cases, without the added burden of
deploying a media server, we have designed a series of models that allow
n-party calls to be coordinated, multiplexed, merged and integrated
efficiently in a peer-to-peer environment akin to the 2-party model of WebRTC
peer connections. Coordination is done in standard JavaScript within the call,
making it trivial to upgrade a 2-party solution to an n-party solution.

As a proof of concept we have also implemented our approach in a home-care
application where medical staff can communicate with patients who are in their
homes. One of the critical requirements of this application is that the media
streams need to be contained in a fully controlled, secure local environment,
i.e. signalling servers and TURN servers are set up to be dedicated for the
application, as opposed to using some pre-existing Cloud service. Furthermore,
streams cannot for regulatory reasons leave the country.

This paper is organized as follows: we first give an overview of related work
on WebRTC architectures beyond 2-party calls in Section 2. We then discuss the
use case that motivated this work in Section 3. We describe our proposed
n-party serverless models and our open-source implementation of them,
Serverless n-party calls over WebRTC (SnoW), in Section 4. In Section 5 we
provide both quantitative and qualitative evaluations of our models, and then
finally we conclude with lessons learned and recommendations in Section 6.

##  2 WebRTC

WebRTC comprises a set of specifications of the JavaScript API offered by
compliant browser runtimes to set up connections, including ICE [4] and JSEP
[5]. A typical WebRTC session also interacts with a variety of HTML5 features
such as getUserMedia [6], and video elements. WebRTC defines the SDP [7]
contents in offers and answers during a session establishment. Finally WebRTC
defines how STUN and TURN can be used to provide connectivity between session
peers.

The most powerful feature of WebRTC is that all modern browsers support the
data plane and rendering engine out of the box, which has made it a popular
technology for Web app integrations. Since its inception it has spread beyond
the browser and is now also frequently used in native iOS and Android apps as
well as in non-browser runtimes such as Go 111https://github.com/pion/webrtc.

It is notable that WebRTC does not define the signalling plane of the session,
i.e. how offers, answers and ice candidates are communicated between
connection endpoints. Furthermore, media multiplexing and forwarding are not
defined beyond peer connections, i.e. 2-party video and audio calls. Most
systems use some kind of real-time bus architecture like WebSockets to
communicate WebRTC messages between peers and to allow them to rendezvous or
call each other.

3+-party calls hence have to use some proprietary protocol to establish
sessions. These protocols generally follow three types of architectures: Mesh,
MCU, and SFU, discussed next.

###  2.1 Mesh

In a WebRTC Mesh network every peer connects to every other peer in a call. So
in a 3-party call every peer has to maintain two peer connections. The local
video and audio need to be streamed over two connections concurrently. This
model is the easiest to implement as it does not require any media server, but
it scales poorly beyond 4-6 party calls. The poor scalability stems from
streams being duplicated on the network but also from separate rendering
stacks consuming resources such as memory and CPU in the local browser.

To overcome these shortcomings media server are often deployed in large multi-
party systems. Media servers come in two flavors, MCU and SFU.

###  2.2 MCU

A Multipoint Control Unit (MCU) [8], 222aka Multi Conference Unit or Multi
Communication Unit in WebRTC is a central media server component that takes
streams from all parties in a call and then merges or multiplexes them to
produces a single stream with all video and audio combined that is sent to all
the participants. In a MCU network the participants do not use up more
resources than in a 2-party call, but the server system that runs the MCU
needs to have both excellent network bandwidth and powerful compute resources,
such as CPU and memory. In a large system it can be very costly to operate an
MCU. Another drawback of the MCU model is that the layout of streams is
determined in the backend and the clients have no control over stream layout,
which may be problematic, e.g. when a phone switches between portrait and
landscape mode.

###  2.3 SFU

A Smart Forwarding Unit (SFU) [9] addresses the layout problem and instead of
the clients receiving a single merged stream, they receive a single stream
with multiple tracks, two for each participant in the call (audio, video).
Hence, each client can decide on how to display the individual streams, and
decisions can be made dynamically. A typical example of an SFU feature is to
display all video feeds as thumbnails except the one feed that is from the
person currently speaking or presenting content. The SFU model is also
typically offered in a centralized powerful media server.

Note that there are open source MCU/SFU offerings [10] [11], but they still
need to be provisioned on powerful compute resources, and be maintained and
integrated into applications.

In our work we explore using Mesh, MCU and SFU concepts without the need to
deploy a media server (only TURN and signaling servers). We believe these
capabilities could empower seamless WebRTC application integrations in multi-
party use cases as well.

There has been an effort to integrate an MCU into the browser runtime in the
past [12], but the work involved taking an open source MCU and an open source
browser and implementing a one-off custom integration. Our goal is to do
everything in standard browsers, just using JavaScript.

MCUs and SFUs were envisioned to improve scalability of video conferencing
sessions [13] [8] and our goal remains the same, while adding the requirement
to do media merging and coordination in a peer-to-peer architecture.

###  2.4 STUN and TURN

Many clients are behind firewalls and NATs, in particular in enterprise
environments. Hence peer connections cannot be established directly between
call participants. STUN servers allow some traversal by determining whether
there are public IP addresses that allow routing into a client, a technique
sometimes referred to as hole punching [14]. Some NATs do not allow this kind
of traversal and discovery. For those environments the only remaining option
is to relay the stream through a third party. TURN was designed for this
purpose and allows a client behind a NAT/Firewall to allocate a publicly
accessible IP and PORT to consume streams on a TURN server. TURN servers may
be deployed anywhere in a network but must be accessible by all participant in
a call and are hence often deployed on the public Internet. Like MCU and SFU
servers, a TURN server also needs to have very high inbound and outbound
bandwidth, but does not consume other resource to the same extent as a media
service as it is only concerned with routing streams. Although there are
popular open-source TURN servers available that are easy to install, there are
very few open server deployments available, as they are typically deployed as
part of a paid video conferencing offer. STUN servers are much easier to
deploy and consume less resources as they only provide discovery services, so
a large number of them are available free of use in different geographies to
reduce latency 333https://ourcodeworld.com/articles/read/1536/list-of-free-
functional-public-stun-servers-2021.

Due to cost, integration, and maintenance overhead many WebRTC systems only
deploy TURN servers and a WebSocket server with 2-party calls. This model is
also attractive since it is very easy to change the application logic in
JavaScript run in clients, and browsers already have built in support for
WebRTC TURN interactions (e.g. authentication, allocation, relay), as part of
the ICE protocol that is run when a WebRTC session is established.

##  3 Use Case

Our use case is a Swedish medical IT company 444https://itacih.se/, itACiH,
providing coordination and communication support for remote, at-home patient
care [15]. Patients have native Android tablet clients and hospital staff use
Web browsers. 2-party calls have been a popular feature between staff, and
between staff and patients.

Due to the sensitivity of the communication streams, a 3rd party media server
may not be deployed outside of the country. As a result itACiH only hosts
local TURN servers and provides the signalling server to establish calls.

The staff would like to have the ability to invite in interpreters and close
relatives when talking to a patient in a 3-party call. They may also want to
pull in multiple staff and multiple relatives in the future.

A media server is seen both as an operational burden, a negative cost impact,
as well as a security liability. Two-party calls have also been in operation
for close to a decade and itACiH would like to enable 3-party calls without
much architectural or other intrusive changes to the existing mission critical
production systems.

##  4 Serverless Multi-party Models

In this section we discuss different models to go beyond 2-party calls without
the need of a media server. Five models are discussed: 3-party Mesh, 3-party
SFU, 3-party MCU, 3-party 2-node MCU, and n-party MCU. Although the first four
models may be generalized beyond three parties, they were designed
specifically to meet the resource constraints of serverless calls where
coordination is done entirely in JavaScript browser run-times. With these
models resource, network as well as signalling complexity may become issues
when more parties are involved. To address that challenge we propose a fifth
model n-party MCU for serverless calls with more parties.

The MCU models rely on merging audio and video streams in JavaScript 555We
have for example used the library at https://github.com/t-mullen/video-stream-
merger. The audio may be multiplexed with the HTML5 AudioContext API and audio
destinations. The video can be constructed by mapping streams and drawing them
onto an HTML5 canvas, which then can be sampled at a given frame rate to
produce a merged stream. This kind of audio and video merging is simply
referred to as stream merging below. It may at first glance seem like an
inefficient way to multiplex streams, but with the maturity that comes with
wider adoption of WebRTC browser runtimes and HTML5 features over recent years
combined with the continuous improvements in hardware processing speeds it can
be done seamlessly on modern PCs, such as laptops and desktops today.

###  4.1 MESH: 3-party Mesh

As we alluded to above, in this model (see Figure 1) every participant needs
to maintain two peer connections. The process of establishing a 3-party Mesh
call proceeds as follow:

  1. 1.

(Steps 1-2) An initiator starts a regular 2-party call with a 2nd party,
including sending it an SDP offer and receiving an answer, as well as
exchanging ICE candidates.

  2. 2.

(Steps 4) When the call is established, a 3rd party is selected and an SDP
attribute is sent to the 2nd party indicating it should expect a call from the
3rd party soon that it needs to accept.

  3. 3.

(Steps 4-5) The initiator starts a regular 2-party call with the 3rd party,
including sending an SDP offer extended with an attribute specifying that it
should call up the 2nd party when the call with the initiator has been
established.

  4. 4.

(Steps 6-7) The 3rd party starts a regular 2-party call with the 2nd party,
including sending it an SDP odder and receiving an answer. The 2nd party
accepts the call as it was informed by the initiator.

Figure 1: MESH model

When only a single party hangs up the other parties may continue the call.
However, to follow the semantics of an initiator controlling the call, we only
allow the call to continue if one of the remaining parties is the initiator.

###  4.2 SFU: 3-party SFU

In the case where all parties are Web clients the above model is the most
flexible as all streams may be laid out at will, but if one party is a native
client it could be a high effort to add the 3-party Mesh semantics on the new
platform. To this end we provide a model (see Figure 2) where one party can
simply run the 2-party call establishment but instead of just receiving a
remote stream with a single audio and a single video track, it receives two
audio and two video tracks corresponding to the other participants. In this
case the native client can still render the layouts at will but the call setup
process does not change.

  1. 1.

(Steps 1-2) An initiator starts a regular 2-party call with a 2nd party,
including sending it an SDP offer and receiving an answer, as well as
exchanging ICE candidates.

  2. 2.

(Steps 3-4) When the call is established the initiator establishes a call with
a 3rd party, e.g. patient tablet, where both the local stream tracks as well
as the 2nd party streams are added

  3. 3.

(Steps 5-6) When the call to the 3rd party is established, the initiator adds
the remote stream from the 3rd party to a new connection with the 2nd party,
who does not send any stream in return.

Figure 2: SFU model

###  4.3 MCU: 3-party MCU

In the model just described the native client would still need to be capable
of extracting the additional tracks in the stream received. If the client
should be capable of participating in 3-party calls without any changes to the
2-party model the streams may be merged instead of adding tracks to them as
the call with the 3rd party is established (see Figure 3).

  1. 1.

(Steps 1-2) An initiator starts a regular 2-party call with a 2nd party,
including sending it an SDP offer and receiving an answer, as well as
exchanging ICE candidates.

  2. 2.

(Steps 3-4) When the call is established the initiator establishes a call with
the 3rd party (e.g. patient) where both the local stream tracks as well as the
2nd party streams are merged into a single stream which is shared with the 3rd
party.

  3. 3.

(Steps 5-6) When the call to the 3rd party is established, the initiator adds
the remote stream from the 3rd party to a new connection. The 2nd party does
not need to send any media back on this second connection from the initiator.

Figure 3: MCU model

###  4.4 MCUTwo: 3-party, 2-node MCU

The 3-party MCU model assumes there is only one party that wants to run the
simpler 2-party model. If both parties that the initiator communicates with
only wants to run a simple 2-party call the call can be established as follows
(see also Figure 4):

  1. 1.

(Step 0) The initiator creates two merged streams and adds its local tracks to
both of them

  2. 2.

(Step 1-2) An initiator starts a regular 2-party call with a 2nd party,
including sending it an SDP offer and receiving an answer, as well as
exchanging ICE candidates. The stream that is shared is not the local stream
of the initiator but the first merged stream.

  3. 3.

When the call is established the remote stream of the 2nd party is added to
and merged into the 2nd merged stream.

  4. 4.

(Step 3-4) The initiator now makes a call to the 3rd party (e.g. patient) and
adds the second merged stream to the peer connection.

  5. 5.

When the call with the 3rd party is established the stream sent by the 3rd
party is added to the first merged stream maintained by the initiator, and
thus automatically shared with the 2nd party.

In this model the initiator receives all streams individually and can display
them att will. The 2nd and 3rd party participants will however receive a pre-
merged stream with a fixed layout determined by the initiator.

Figure 4: MCUTwo model

###  4.5 MCUMulti: n-party MCU

Maintaining a separate merged stream for each participant to avoid merging its
local stream is resource intensive and does not scale. Hence in the case of
four or more parties we provide a model where the initiator maintains a single
merged stream that will be shared with all participants (see Figure 5) . For
each additional participant the initiator proceeds as follows.

  1. 1.

(Steps 1,3,5) The initiator starts a regular 2-party call with the n-th party,
including sending it an SDP offer and receiving an answer, as well as
exchanging ICE candidates.

  2. 2.

As stream to share the initiator passes the current merged stream

  3. 3.

(Steps 2,4,6) When the call is established the remote stream received is added
to the merged stream and is thus automatically picked up by all participants
in the call already.

Figure 5: MCUMulti model

The initiator is still in full control over the layout and could change it
dynamically, e.g. based on who is speaking, but all participants see the same
layout, and all participants see themselves in the layout too. In all models
presented above the initiator does most of the work to establish the call, and
should thus be on a good network with a good amount of compute resources
available. We note, however, that in contrast to a traditional SFU and MCU
architectures each call has its own coordinator and the burden of multiplexing
media is thus shared across calls.

##  5 Evaluation

We set up 3 PCs on the same LAN, two connected to Wi-Fi and one connected to
Ethernet. The PC connected to Ethernet is a low-cpu NUC running Ubuntu, and
the Wi-Fi PCs are two MacBook Pros (2016 and 2022). We let the latest MacBook
Pro always be the call initiator and the NUC be the final participant to join
(to emulate a patient tablet). I.e. the NUC will get the merged stream in both
MCU and MCUTwo, and the second stream on the same connection in SFU.

We started a connection and then waited one minute after call initiation to
start collecting WebRTC Statistics 666https://developer.mozilla.org/en-
US/docs/Web/API/WebRTC_Statistics_API. We then collected mean and standard
deviation for the following minute for four reported metrics of the inbound-
rtp report type. This includes both video and audio metrics, however only
minimal sound was emitted during the calls, so the video stream is dominant in
terms of metric impact. The reported metrics include: packetsLost,
jitterBufferDelay, totalInterFrameDelay, and jitter. We then took the average
across all three PCs across 5 runs and compared the improvement of the
different models over the corresponding values for the Mesh model. The
improvement is computed as $\frac{mesh-metric}{mesh}$. A negative value hence
means the model did worse in that metric than the Mesh model. We repeated this
experiment twice on two consecutive days with 24h in between to see if the
trends were stable. The results including the improvement tuple over the two
days can be seen in Table 1.

Table 1: Quantitative Evaluation. Model | Metric | Improvements
---|---|---
SFU | packetsLost | -.69,-.31
| jitterBufferDelay | 5.3,4.3
| totalInterFrameDelay | .16,.07
| jitter | .72,.80
MCU | packetsLost | -.38,-.15
| jitterBufferDelay | 5.7,4.3
| totalInterFrameDelay | .12,.09
| jitter | .78,.76
MCUTwo | packetsLost | 1.2,.35
| jitterBufferDelay | 4.4,3.5
| totalInterFrameDelay | .00,-.05
| jitter | .72,1.1
MCUMulti | packetsLost | -.35,-.24
| jitterBufferDelay | 3.5,3.1
| totalInterFrameDelay | -.02,-.03
| jitter | .83,.58

We also qualitatively evaluated the user experience of the calls and noted
three degrading factors, primarily observed on the resource constrained NUC.
These problems were furthermore observed when the NUC was made initiator and
hence required more processing. Delays, denote a high latency between a
movement and the rendering of that movement in the HTML5 video element
displayed in a browser; High CPU denotes a high usage of CPU that could cause
the call to fail; and Slow Connect, denoting the fact that some models need to
wait for pre-requisite connections to be established before connecting to a
remote peer, and hence the time from the initiator starting a call until all
peers being connected shows a delay correlated with the time it takes to set
up one connection (ICE negotiation). The result of the qualitative evaluation
can be seen in Table 2.

Table 2: Qualitative Evaluation. | Delays | High CPU | Slow Connect
---|---|---|---
Mesh | YES | YES | NO
SFU | NO | NO | YES
MCU | NO | NO | YES
MCUTwo | NO | YES | NO
MCUMulti | NO | NO | NO

The measured statistics for all the calls and PCs are shown in Figures
6,7,8,9. Note that the first point in the series of three is always from the
initiator of the call and the last point is from the NUC. Points 1-15 are from
the calls measured during the first day and points 16-30 are from the second
day. The shaded areas represent half a standard deviation above and below the
mean.

From these results and from the quantitative and qualitative evaluation
summaries, we can observe a number of trends.

  * •

The SFU amd MCU* models have lower jitter and jitter buffer delay than Mesh

  * •

The MCUTwo model has fewer packets lost than the other models

  * •

The MCU and SFU models have lower total inter frame delay than the other
models

A recommendation purely based on these performance-related findings could
hence be to make use of the MCUMulti model when there are devices with
constrained processing power involved in the call, and to use MCUTwo if the
initiator is on a powerful device. When the network is slow the SFU and MCU
models should be avoided as it could take a long time to establish the call.

Note that the SFU model has the advantage that the receivers can all decide on
how to layout the remote streams, just as in the Mesh case, whereas in the MCU
case only the first two callers can decide on the layout. In the MCUTwo and
MCUMulti models only the initiator of the call can decide on how to layout the
stream. Of course the merged streams could be mapped back to something like a
canvas on the receiver end too and then only the relevant area can be picked
and sized appropriately. This approach, however, creates a brittle layout
dependency between the merging party (the call initiator) and the call
recipient.

Figure 6: Packets Lost Figure 7: Jitter Buffer Delay Figure 8: Total Inter
Frame Delay Figure 9: Jitter

To easily reproduce these results and test the models in different
environments we provide the model implementations as well as experiment code
as open source at:
https://github.com/patompa/snow.

##  6 Discussion

This work has focused on 3-party calls and alternatives to mesh networks when
no media server is available.

Mesh calls as well as the SFU and MCU models presented here can clearly also
be extended to more than 3 parties, but with varying complexity.

The models utilizing merged streams are easiest to extend as they do not
require additional signalling. A merged stream may be shared with participants
before any streams have been added, making the MCUMulti model very easy to
implement for an unlimited number of parties. You simply share the merge in
the offer and then add the stream received in the answer to the merge when the
connection has been established.

The MCUTwo model is also easy to extend as a new merge is created by the
initiator for each new participant containing all the other participants on
the call. However, maintaining a merge stream is resource demanding and this
model like the traditional mesh model would scale poorly. Even with three
parties, the coordinator would need a good amount of CPU and memory resources
to maintain an acceptable user experience.

The SFU and MCU models are the most complex to expand and they already use
additional streams to avoid re-negotiating connections. Extending these models
to more than 3 parties would probably warrant re-negotiation as maintaining an
extra connection incurs too much overhead.

Signalling in a Mesh model can be quite complex as all parties need to connect
to all other parties. For instance, when a new party joins all existing
parties would need to either accept a call from the new party or send it an
offer. For more than 3 parties this could also lead to significant overhead in
establishing the call.

Although the SFU model provides additional flexibility on the receiving end in
terms of layout decisions, a fully merged stream could also be visualized in
custom ways if the layout is kept simple, e.g. side-by-side, or top-bottom
with fixed widths and heights for all streams. Re-aligning the individual
streams could however again demand local resource and may not work well on
small mobile units like smart phones.

We note that the model that merges all streams (MCUMulti) will echo back the
speaker’s audio, and hence a poor network connection with delays or a poor
echo cancellation speaker and headset combination could impact the user
experience. In our tests this impact was minor, but one solution would be to
use the technique from the MCUTwo model where different media merges are kept
for each peer. The videos could be merged in one stream but audio may be
merged separately and one could also mute or remove audio that is not
participating in the discussion dynamically using speaker detection (which is
also possible in pure JavaScript with WebRTC).

For our itACiH use case we implemented the 3-party Mesh model (MESH) between
staff, the 3-party MCU (MCU) model between 2 staff members and a patient, and
the 3-party, 2 node MCU model between a staff member and two parties among
patients, relatives and interpreters (MCUTwo).

The 3-party SFU (SFU) and n-party MCU (MCUMulti) models are considered for
future deployments.

##  7 Conclusion

In conclusion we have shown that the decision regarding which multi-party
WebRTC model to choose not only depends on networking conditions and hosting
costs, but also on local device resources. Furthermore, there are ways to
mitigate these challenges with merged streams and serverless architectures.

As long as the initiator of a call is a reasonably powerful PC, like a modern
laptop, with good network connectivity, e.g. stable Wi-Fi, it can serve as a
simple multi-party media coordinator without the need to deploy expensive and
intrusive MCU or SFU media servers. In full spirit of the peer-to-peer design
of the WebRTC protocol itself, we also believe our solution would scale better
and be more reliable as there is no single point of failure or a single pipe
the data traffic needs to be routed through.

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