2022

[1]Jason T. Parker

[1]Mechanical Engineering, UC Berkeley, Hearst Ave., Berkeley, 94720, CA, USA
2]Ultramet, Montague St., Pacoima, 91331, CA, USA

# Enhanced Laboratory X-ray Particle Tracking Velocimetry With Newly Developed
Tungsten-Coated $O$(50 $\mu$m) Tracers

<EMAIL_ADDRESS>Jessica DeBerardinis<EMAIL_ADDRESS>Simo A. Mäkiharju<EMAIL_ADDRESS>*  [

###### Abstract

Tracer particles designed specifically for X-ray particle tracking and imaging
velocimetry (XPTV and XPIV) are necessary to widen the range of flows that can
be studied with these techniques. In this study, we demonstrate in-lab XPTV
using new, custom-designed $O$(50 $\mu$m) diameter tungsten-coated hollow
carbon spheres and a single energy threshold photon counting detector. To
explore the measurement quality enhancement enabled by the new tracer
particles and photon counting detector, a well understood Poiseulle pipe flow
is measured. The data show agreement with the analytical solution for the
depth-averaged velocity profile. The experiment also shows that the tungsten-
coated particles achieve higher contrast and are better localized than
previously available silver-coated particles, making faster and more precise
measurements attainable. The particles are manufactured with a readily
scalable chemical vapor deposition process.

We further show that laboratory XPTV is practical with currently available
energy-resolving photon counting detectors (PCDs), despite their presently
lower spatiotemporal resolution compared to scintillating detectors. This
finding suggests that energy-thresholding identification of different classes
of tracers is feasible, further motivating the exploration of the X-ray tracer
particle design space. The latest generation of PCDs are incorporating
multiple energy thresholds, and have higher count rate limits. In the near
future one could potentially expand on the work presented and track multiple
tracer species and scalar fields simultaneously.

###### keywords:

X-ray, Particle Image Velocimetry, Particle Tracking Velocimetry, Flow
Visualization, Tracer Particle, Photon Counting Detector

##  1 Introduction

Particle image velocimetry (PIV) and particle tracking velocimetry (PTV) at
visible wavelengths are quantitative flow visualization techniques capable of
resolving 2D and 3D flow fields that are readily compared to theory and
simulation (M. Raffel et al (2018)). With no probe to disturb the flow field,
these techniques are favored throughout the experimental fluid dynamics
community.

In a typical 2D PIV or PTV measurement of a flow, tracer particles are
illuminated with a laser sheet. Images of the tracer particle motion in the
fluid are captured by a high-speed camera. The tracer particles are ideally
small enough and density matched with fluid such that they follow the flow
accurately without influencing the flow itself. A primary limitation of PIV
and PTV is that they require optical access for the laser light to reach the
tracer particles, and for the scattered light to reach the camera sensor. For
many flows, either due to the opacity of the fluid itself or the surrounding
material, optical access is impossible at visible wavelengths. Metal foam
flows and natural soil flows, for example, must take place in opaque
containers and surroundings. Solid-liquid flows such as blood and silty water
can also be opaque to visible light. Even in multiphase flows where both
phases are individually transparent – such as air-water flows – refraction
across multiple moving, deforming interfaces makes the flow effectively
opaque. Standard flow visualization techniques struggle to yield data on these
flows, which together cover substantial swaths of fluid dynamics.

Photons in the X-ray range, however, can pass through many materials that are
opaque in the visible light wavelengths, and they have a refractive index near
unity. As a result, XPIV and XPTV can overcome the optical access limitations
of PIV and PTV. If XPIV and XPTV can be developed into practical techniques
with laboratory equipment, the impact would be significant. Three decades of
PIV and PTV algorithm development could be applied to previously inaccessible
systems such as biological flows (Park et al (2016); Antoine et al (2013); Kim
and Lee (2006); Jamison et al (2012); Krebs et al (2020)), multiphase flows
(Ganesh et al (2016); Mäkiharju et al (2013); Ganesh et al (2016); Mäkiharju
et al (2017); Yoon et al (2018)), and internal flows (Lappan et al (2020); Liu
et al (2021)). In fact, multiphase flow studies already occasionally use other
X-ray techniques for measuring the phase fraction.

Lee and Kim (2003) demonstrated XPIV for the first time at a synchrotron by
measuring the 2D-projected flow profile in a round pipe, and this work
provides a useful point of comparison for this study. Recently Ge et al
(2021), used XPIV at a synchrotron to investigate a cavitating shedding flow,
the kind of study that would benefit if it could be conducted in a laboratory.
In general, research to extend XPIV and XPTV (Dubsky et al (2009); Fouras et
al (2007)) has focused on work at synchrotrons. However, limiting XPIV and
XPTV to synchrotrons hinders the pace of research by imposing practical
limitations such as beam time, cost, and location. Furthermore, synchrotrons
typically have illuminated areas on the order of a few millimeters, which
constrains the experiment geometry and accessible parameter range. For
example, Ge et al (2021) could examine a larger domain with the larger
illuminated area of a laboratory source. Enabling XPTV at the laboratory-scale
is imperative for it to become a more widely useful fluid dynamics measurement
technique.

Although some progress has been made recently towards practical in-lab X-ray
PTV (Mäkiharju et al (2022); Parker and Mäkiharju (2022); Bultreys et al
(2022)), the technique’s applicability remains limited. Formidable obstacles
exist to obtaining a sufficient signal-to-noise ratio (SNR) given the dimness
of laboratory X-ray sources compared to synchrotrons, and the difficulty of
finding high-contrast particles that are also good flow tracers. Poelma (2020)
reviewed various techniques for measuring multiphase flows and also discussed
the challenges of performing laboratory XPIV and XPTV. For brevity, throughout
the rest of this paper, the discussion will focus on XPTV, although many of
the findings apply to XPIV as well.

In order to achieve particle image contrast, most previous in-lab XPTV
experiments have relied on particles that are either near order of magnitude
density mismatched with the fluid, are on the order of a millimeter in
diameter, or both (Lappan et al (2020); Lee et al (2009); Heindel et al
(2008)). However, most particles that are not neutrally buoyant bias the
velocity measurements and become unevenly distributed. Large particles, even
if they are nominally density matched, limit the spatial resolution and may
not trace the flow, particularly in large velocity gradients or if flow
features are on the order of magnitude of the particle size. Recently,
Mäkiharju et al (2022) investigated a creeping flow with 60 $\mu$m tracer
particles. In that study, we noted the significant limitations imposed by the
slow scan times that were necessary to achieve particle image contrast. As
part of that study, we identified improving the tracer particles to be of
paramount importance to laboratory XPTV.

XPTV imposes unique requirements on tracer particles that conflict with one
another. XPTV tracer particles must attenuate X-rays significantly more (or
less) than the ambient fluid to generate contrast, which favors large,
density-mismatched particles. At the same time, tracer particles must be
neutrally buoyant and have sufficiently low inertia to trace the flow
accurately, which favors small, density-matched particles. In addition, small
tracer particles maximize the spatial resolution of the measurement. Parker
and Mäkiharju (2022) developed and validated methods for simulating XPTV
images to predict the performance of existing and conceptual XPTV tracer
particles before investing in expensive custom tracer particle manufacturing.
These tools allow us to systematically balance the design criteria for XPTV
tracer particles. Thanks to modern manufacturing processes it is now feasible
to develop $O$(10 $\mu$m) prototype custom tracer particles for XPTV
applications.

In this paper we introduce new hollow carbon (C) tungsten-coated (W)
microsphere XPTV tracer particles (CW) that were designed using the simulation
tools in Parker and Mäkiharju (2022). We compare the performance of the CW
tracer particles to the performance of silver-coated hollow glass tracer
particles (AGSF-33), a previously available visible light PTV tracer particle.
Tracer particle performance is evaluated by measuring Poiseulle pipe flow with
2D-projected XPTV. Additioanlly, images of the tracer particles are captured
with a variable single-energy-threshold photon counting detector (PCD),
demonstrating that PCDs can be used for XPTV in the laboratory. PCDs can
attain a higher SNR than scintillating detectors. Using PCDs also opens up the
possibility of future experiments tracing multiple particle species, scalar
fields, or both with K-edge material detection.

The paper is organized as follows: in section 2 we discuss the methodology and
materials, section 3 discusses the properties of the CW and AGSF-33 tracer
particles, section 4 presents and discusses the results followed by the
conclusions in section 5.

##  2 Methodology

###  2.1 Experimental Setup

The experimental setup can be seen in figure 1. The source to detector
distance (SDD) and source to object distance (SOD) are 500 mm and 38 mm,
respectively. The SOD is measured from the X-ray emitting tungsten target
surface to the center of the pipe. The source emits a conical beam, which
geometrically magnifies objects in the field-of-view (FOV) by a factor of
approximately $M=13.1$. The detector has a panel size of 83.8 mm $\times$ 33.5
mm. Due to geometric magnification, the FOV in the object plane is 6.37 mm
$\times$ 2.55 mm at the central plane of the pipe.

The available quantity of the new custom particles made a closed loop
impractical. The tracer particles are seeded directly into the 6.35 mm (0.25
in.) inner diameter pipe by pouring a particle-glycerol mixture into the top
of the pipe at an angle, leaving room for the air in the pipe to exit. The
mixture is at equilibrium at 23.7 $\degree$C as measured by an Ambient Weather
WH31E temperature and humidity sensor adjacent to the X-ray source. A Harvard
Apparatus 11 Plus syringe pump pushes the glycerol at a constant flow rate
$Q=0.215$ mL/min. The 11 Plus syringe pump has a 0.9$\degree$ step rotation
for high accuracy flow rates. The Reynolds number of the flow,
$Re=Ud/\nu=0.0018$, is of comparable order of magnitude as Lee and Kim’s
experiment at a synchrotron, which, based on data they provide, is calculated
to be $Re_{LK}=0.0052$. Here, $U$ is the centerline velocity and $d$ is the
pipe inner diameter. The laminar development length for the current experiment
is $l=0.06ReD=0.69~{}\mu$m. Given that the pipe length upstream of the
measurement FOV is significantly longer than the entrance length, the flow can
be considered fully developed.

The pipe is mounted on a 3-axis linear motion system comprised of three Thor
Labs NRT150 stages. The NRT150 stages have an on-axis accuracy of 2 $\mu$m and
a minimum incremental motion of 0.1 $\mu$m. A Starrett No.98 machinist level
with a sensitivity of 0.024$\degree$ is used to ensure that the pipe is
mounted vertically.

The Dectris Pilatus3 PCD and YXLON FXE225.99 X-ray source are the same as used
in our prior study (Parker and Mäkiharju (2022)). The detector has a 1 mm
thick CdTe panel to capture photons in pixels of size 172 $\times$ 172 $\mu$m.
Sixteen tiles comprise the detector panel, leading to a 487 $\times$ 195 pixel
detection area. The Dectris Pilatus3 can achieve frame rates up to 500 Hz. Due
to the relatively dim X-ray source use, we chose a low flow speed such that
images captured with 15 ms exposure times would not exhibit motion blur, but
would have sufficient photon counts to achieve a usable SNR. Frame rates up to
about 70 Hz could be used based on this exposure time. For PTV, however, it is
preferable to have a notable particle shift between each frame, so we captured
images at 2.5 Hz to allow sufficient particle motion between frames. The
detector energy threshold is set to 20 keV to reduce photon scattering noise.
The X-ray source is operated at an acceleration voltage of 55 kV with a target
current of 500 $\mu$A.

Figure 1: The experimental setup (not to scale). A syringe pump pushes
glycerol and tracer particles through a vertically mounted pipe.

###  2.2 Image Processing Methods

####  2.2.1 Filtering and PIV Software

Raw particle images are corrected with a flat field correction map, inverted,
and then imported to LaVision DaVis 8.4 for filtering and PTV processing. We
use standard DaVis algorithms without modification to demonstrate that XPTV
data can be processed with existing algorithms. Details on the filters and
particle tracking settings can be found in appendix A.

####  2.2.2 Scaling and Uncertainty

Image pixel-to-mm scaling is done by identifying the pipe inner diameter in
the image. The pipe inner diameter is known to be 6.35 mm, and we count the
number of pixels required to traverse the inner diameter of the pipe. The time
step $\Delta$t is the time between the beginning of each exposure. For all of
the images collected in this study, $\Delta$t = 400 ms to ensure that the
particles travel sufficiently far between exposures for particle displacement
to be significantly larger than the uncertainty in localizing the particle
centroid. Given the 15 ms exposure time, the acquisition duty cycle for these
experiments is 3.75%.

As noted in section 1 and Mäkiharju et al (2022), the density mismatch between
fluid and particles can be a significant source of uncertainty for XPTV and
XPIV. Laboratory systems with dim X-ray sources are currently limited to slow
flows where the density mismatch has a more pronounced effect. Discussion of
the effects of buoyancy and flow tracing accuracy is reserved for section 3.

There are additional uncertainty contributions unique to XPIV and XPTV,
though. Most laboratory X-ray sources have conical beams, leading to geometric
magnification, as mentioned previously. The geometric magnification factor $M$
is a function of the distance from the source, so it changes along the beam
direction. Accordingly, the FOV will change over the depth of the experiment
geometry. In our experiment, the FOV changes from approximately 5.84 mm
$\times$ 2.33 mm at the front of the pipe to 6.90 mm $\times$ 2.76 mm at the
back. In all planes along the beam direction, the entire pipe is in the FOV.
Similarly, the motion of a particle image is a function of the magnification.
Images of particles closer to the source will appear larger and will move
faster than images of particles farther from the source, even when the
particles are the same size and have same exact velocity.

Consider, for example, two identical particles in a Poiseulle pipe flow, each
at the same radius, but one particle is closer to the source than the other,
as shown in figure 2. Being at the same radius, both particles will travel at
the same velocity. Since particle A is farther from the X-ray source, it will
be magnified less than particle B, and will appear to move less than particle
B even though it has traveled the same real distance for any given time
interval. Given that there is no information on particle depth in a
2D-projected measurement, the pixel-to-mm scaling assumes that all particles
are located in the central plane at $y=0$. Equation 1 gives the measured
displacement $\Delta z$ of a particle that has moved a real distance $\Delta
Z$. Recall that SOD is measured to the center of the pipe.

Figure 2: Particle A will be magnified less than particle B because it is
farther from the X-ray source. Particle A will also appear to move slower than
particle B.

$\Delta z=\frac{M(y)}{M(y=0)}\Delta Z=\frac{SOD\cdot\Delta Z}{SOD+y}$ (1)

The relative error, $\epsilon$, in the measured displacement versus the real
displacement will increase with particle distance $y$ from central plane and
is given by

$\epsilon=\frac{\lvert\Delta z-\Delta Z\rvert}{\Delta
Z}=1-\left(1+\frac{\lvert y\rvert}{SOD}\right)^{-1}.$ (2)

The velocity relative error is the same as equation 2. Note that the relative
error depends on SOD and domain size, but not directly on SDD. We denote the
ratio of the experiment depth $\delta=2y_{max}$ to the SOD as the
magnification aspect ratio. A good rule of thumb to avoid magnification error
is to keep the magnification aspect ratio below 0.2. The maximum relative
magnification error would then be 10%. In practice, SDD could affect the error
by modifying how many pixels a particle image occupies on detector,
contributing to the resolution error of localizing a particle centroid.

For this experiment, where the SOD is 38 mm and the maximum $y=3.2$ mm, the
relative magnification error never exceeds 9%, which occurs at the pipe wall
as the velocity approaches zero. At the center of the pipe, where the
calibration is defined, the relative magnification error is zero.
Experimenters with large depths of field should be cautious about the error
introduced by magnification effects. Using fully tomographic XPTV, such as in
Mäkiharju et al (2022), avoids issues with geometric magnification altogether.
Alternatively, if the tracer particles are sufficiently monodisperse, one
could determine the depth location based on the particle image size, correct
for the magnification error, or both.

###  2.3 Depth-Averaged Velocity Profile

In this study, 2D-projected images of flow in a pipe are captured. Particles
all along the X-ray beam direction (depth-wise direction $y$ in figure 2) of
the pipe are imaged at the same $x$-location. As a result, for particle
tracking velocimetry, one is not sampling a single value but rather the
distribution of velocities along the depth-wise direction of the pipe. Lee and
Kim (2003) compared the measured average velocity of the depth-wise
distribution to the analytical depth-averaged velocity profile (DAVP). Figure
3 depicts the depth-wise velocity variation in the pipe. Lee and Kim (2003)
assumed a uniform distribution of ideal tracer particles throughout the pipe,
and that all particles contribute to the distribution equally. Under these
conditions, the DAVP for a parallel-beam projection is simply the depth
integrated average velocity of the fluid, given by

$\begin{split}\left<u(x,y)\right>_{y}&=\frac{4Q}{3\pi
R^{2}}\left[1-\left(\frac{x}{R}\right)^{2}\right]=\frac{2}{3}u(r=x)\end{split}$
(3)

where $\left<u(x,y)\right>_{y}$ is the DAVP, $u(r)$ is the analytical solution
for Poiseulle pipe flow, $R$ is the pipe radius, and $Q$ is the volumetric
flow rate. Hidden in equation 3 is a cosine of the angle between the radius
and the $x$-direction on the detector. Since this cosine is unity for this
axisymmetric system with vertical alignment, it is omitted from equation 3 for
clarity. For consistency throughout the paper, the DAVP will be written as a
function of $x$, while the cross-sectional velocity profile will be written as
a function of radius $r$. The varying magnification due to a cone-beam instead
of a parallel beam is also not considered here since the magnification error
is relatively small.

Figure 3: The depth-wise direction is towards the detector. The analytical
velocity profile is depth-averaged.

It is worth noting that Lee and Kim compared their PIV measurement to the
DAVP, whereas we are comparing a PTV measurement to the DAVP. Fouras et al
(2007) points out that PIV calculates a modal average velocity, not the mean
average that Lee and Kim compared against. By comparing a PTV DAVP measurement
to the analytical DAVP, we omit any error associated with the discrepancy
between the distribution mode and mean.

##  3 Tracer Particle Properties

###  3.1 Silver-Coated Hollow Glass Spheres (AGSF-33)

Silver-coated hollow glass spheres are commonly used in visible wavelength PTV
applications, so they are readily available for use in XPTV. The silver
coating attenuates X-rays better than uncoated plastic or glass microspheres
owing to the relatively high mass attenuation of silver. Higher mass
attenuation elements such gold, silver, and tungsten achieve better image
contrast provided that the surrounding materials are comprised of lower mass
attenuation elements such as hydrogen, oxygen, and carbon. The opposite is
true for dense, heavy element fluids like liquid metals. In these experiments,
we use AGSF-33 silver-coated hollow glass spheres, a previously available off-
the-shelf tracer particle. AGSF-33 particles were previously shown to be
suitable XPTV tracer particles (Parker and Mäkiharju (2022); Mäkiharju et al
(2022)).

For these experiments, the AGSF-33 particles are sieved to between 45 and 53
$\mu$m to improve monodispersity. We assume a uniform distribution across this
size range. Although the AGSF-33 particles are on average matched with the
density of water, they, like most tracer particles, have a non-uniform density
within a batch. According to the manufacturer, Potters Beads, the particles
have density between 0.9 and 1.1 g/cm3. Density mismatches with the fluid
induce relative motion due to buoyancy, which can bias velocity measurements,
particularly in low speed flows if the velocity due to buoyancy is the same
order of magnitude as the flow itself. Although buoyancy-induced velocity is
often neglected in visible light PTV, early XPTV experiments will study low
speed flows where buoyancy-induced velocity should be considered. When the
relative velocity is low enough that $Re(u_{rel})\ll 1$, and Stokes drag is
appropriate, the terminal speed with which particles will rise or settle in a
fluid can be calculated from equation 4.

$u_{St}=d_{p}^{2}\frac{\rho_{fluid}-\rho_{particle}}{18\mu}g$ (4)

Here, $d_{p}$ is the particle diameter, $\rho_{i}$ is density of $i$, $\mu$ is
the dynamic viscosity of the fluid, and $g$ is the acceleration due to
gravity. The Stokes settling speed can be thought of as a bias error in the
velocity measurement. The Stokes settling speed is proportional to the
particle diameter squared, which incentivizes the use of smaller particles.
Smaller particles also have a lower Stokes number, which is discussed in
section 3.3. The trade off of using a smaller particle, however, is less X-ray
attenuation, which, according to the Beer-Lambert law, is exponentially
dependent on the particle size. These opposing design criteria are an example
of the challenges and opportunities of the XPTV tracer design space. Exploring
smaller tracer particle designs that incorporate composite- and single-element
hollow shells of high mass attenuation elements while remaining neutrally
buoyant is crucial for improving XPTV.

For a coated composite particle such as AGSF-33 the particle density
$\rho_{particle}$ is given based on the coating mass fraction $\gamma$ as

$\rho_{particle}=\frac{\rho_{pyrex}(r_{pyrex}^{3}-r_{air}^{3})+\rho_{air}r_{air}^{3}}{(1-\gamma)r_{silver}^{3}}$
(5)

where $r_{i}$ is the outer radius of material $i$. To obtain an approximation
of the effect of a particle-fluid density mismatch, we run Monte Carlo
simulations of the Stokes settling speed (JCGM (2008)). For simplicity, we
assume that the diameter and density are independent random variables. We take
the density to be normally distributed with a 0.033 g/cm3 standard deviation,
truncated at 0.9 g/cm3 and 1.1 g/cm3. We assume that Potters Beads’ quality
control has removed particles outside of this range. Randomly taking values
from these probability distributions, we simulate equation 4 $10^{7}$ times
for particles in water and glycerol each at 23.7 $\degree$C. The resulting
Stokes velocity distribution is shown in figure 4. In water, the distribution
mean is $\leq O$($\pm 0.01~{}\mu$m/s); the standard deviation 47 $\mu$m/s.
Using pure glycerol increases the fluid viscosity by an order of magnitude,
dramatically reducing the particle Stokes velocity spread, as seen in figure
4. In glycerol at 23.7$\degree$C, the AGSF-33 particles rise at an average
speed of 0.34 $\mu$m/s, with a standard deviation of 0.05 $\mu$m/s. It is
worth noting that a tighter density distribution would be possible using
sorting techniques such as floating and sinking separation. However, for
simplicity, the authors used the particles as-is to best emulate an off-the-
shelf product usage.

Figure 4: The probability distribution of the terminal speed magnitude for
sieved, uniformly distributed, 45-53$\mu$m AGSF-33 tracer particles in the
Stokes drag regime in water and pure glycerol at 23.7$\degree$C. The
distribution in water is effectively symmetric about zero; only the velocity
magnitude is shown here to accommodate a log-axis plot to highlight the water-
glycerol distribution contrast.

###  3.2 Tungsten-Coated Hollow Carbon Spheres (CW)

Compared to the AGSF-33 particles, the new CW particles conceived and
initially designed in the FLOW Lab and manufactured by Ultramet offer higher
X-ray image contrast. A tungsten coating has a higher mass attenuation
coefficient than silver, thereby generating greater contrast. In order to
maintain neutral buoyancy and maximize the coating thickness, the tungsten is
coated onto hollow carbon microspheres. Simulations using the Beer-Lambert
method in Parker and Mäkiharju (2022) show that the CW particles can achieve a
3.4$\times$ improvement in the SNR compared to the AGSF-33 particles on a
glass slide. The SNR improvement implies that CW particles will exhibit higher
contrast in a fluid as well. The diameter distribution of the tungsten-coated
hollow carbon spheres as-manufactured can be seen in figure 5. The
polydispersity shown in figure 5 is apparent in the scanning electron
microscope (SEM) image shown in figure 6. Figure 7 shows a nonspherical
particle with a broken coating, demonstrating the prototype nature of these
early CW particles. We take advantage of the broken coating to measure the
coating thickness. Sieving reduces the polydispersity, but in general
monodispersity in all properties will improve if there is wide demand and
manufacturers can fine-tune their scalable manufacturing processes for making
these new types of particles. In this study, the particles are sieved to the
same diameter range as the AGSF-33 particles to ensure a fair contrast
comparison.

Figure 5: The CW particle diameter distribution – measured using a non-aqueous
based dispersion (ISO 13320) with a Saturn DigiSizer – provided by Particle
Testing Authority. While the particles are polydisperse from the factory,
sieving and future manufacturing improvements can reduce the polydispersity.
Figure 6: An SEM image of the CW tracer particles used in these experiments
prior to sieving. Note the significant polydispersity prior to sieving. Figure
7: In this SEM image of a broken particle the coating thickness can be
measured. However, we find that the coating thickness varies significantly due
to the new manufacturing process not yet being fully refined. The particle
shown is also non-spherical, another challenge in manufacturing new tracer
particles.

We run the same settling speed Monte Carlo simulations for the CW particles as
for the AGSF-33 particles in section 3.1. Based on the observed settling
behavior and SEM images, the tungsten coating thickness is approximately 0.25
$\mu$m. However, in SEM images, some particles are observed to have a 0.5
$\mu$m thick coating or more, demonstrating the manufacturing variability of
the CW particles in these first batches. The average hollow carbon microsphere
density is $\left<\rho_{carbon}\right>=0.5767$ g/cm3. Using these data
alongside the diameter distribution in figure 5, we can estimate the CW
particles’ Stokes velocity distribution shown in figure 8. In water at
23.7$\degree$C, the CW particles settle at an average velocity of -200
$\mu$m/s with a standard deviation of 13.3 $\mu$m/s. While the mean CW
particle settling speed in water is much greater than the AGSF-33 particles,
in glycerol at 23.7$\degree$C, the average Stokes terminal speed of the CW
particles is 0.16 $\mu$m/s, with a standard deviation of 0.035 $\mu$m/s – a
much tighter and slower distribution than in water. While these first CW
particles are outperformed by AGSF-33 particles in traditional tracer particle
metrics like neutral buoyancy in water and polydispersity, as more
monodisperse carbon microspheres are sourced and the tungsten coating
thickness is better controlled, the traditional tracer particle metrics of CW
will become competitive with AGSF-33 in water as well.

Figure 8: The Stokes terminal speed probability distribution of the CW tracer
particles in water and pure glycerol.

####  3.2.1 CW Particle Manufacturing

The CW particles are manufactured by Ultramet using a chemical vapor
deposition (CVD) process to coat carbon microspheres with tungsten. The carbon
spheres are hollow, averaging a diameter of 61.74 $\pm$ 0.03 $\mu$m and a
density of 0.5767 $\pm$ 0.0009 g/cm3. Diameter and density are measured using
a non-aqueous based dispersion (ISO 13320) using a Saturn DigiSizer and
nitrogen pycnometry (ISO 12154), respectively, provided by Particle Testing
Authority. The tungsten coating is applied to the microspheres with a powder
bed CVD reactor. The coating thickness is varied by modification of the CVD
process conditions. The coated spheres are sorted based on their diameter
using various wire mesh sieves. These groups, where $d_{p}$ represents a
particles diameter, are: $d_{p}\in(0,38]$, $(38,45]$, $(45,53]$, $(53,63]$,
and $(63,250]~{}\mu$m. As mentioned in previous sections, CW particles with
$d_{p}\in(45,53]~{}\mu$m are used in this study to provide the best comparison
to the equivalently sized AGSF-33 particles.

###  3.3 Tracer Particle Contribution to Uncertainty

A summary of the settling speed Monte Carlo simulations for CW and AGSF-33
particles can be seen in table 1. Ideally, the Stokes terminal velocity is at
least an order of magnitude less than the characteristic velocity of the flow,
i.e., $u_{St}\ll U$. When $u_{St}\ll U$, the buoyancy induced velocity can be
neglected. Table 1 shows that this condition is satisfied for both particles
in glycerol.

Another source of error is the particle’s dynamic response to the flow. The
particle response time – the Stokes time – is an indicator of the tracer
particle’s ability to follow a flow accurately. Assuming the relative velocity
between the spherical particle and the surrounding fluid is such that the
$Re(u_{rel})\ll 1$, and the fluid acceleration is constant, the characteristic
response time of the particle is given by equation 6 (M. Raffel et al (2018)).

$\tau_{p}=d_{p}^{2}\frac{\rho_{p}}{18\mu}.$ (6)

The Stokes number, shown in equation 7, is the ratio of the Stokes time to the
characteristic time of the flow, $\tau_{f}$. For pipe flow, $\tau_{f}=D/U$,
where $D$ is the pipe diameter.

$St\equiv\frac{\tau_{p}}{\tau_{f}}=\frac{\rho_{p}d_{p}^{2}}{\rho_{f}D^{2}}\frac{Re}{18}$
(7)

A particle is said to follow the flow accurately when $St\ll 1$. Equation 7
shows that smaller diameter particles are needed to measure higher Reynolds
number flows, again motivating the need to explore new tracer particle designs
to expand the applicability of XPTV to more flows of interest. Table 1
presents the Stokes number of AGSF-33 and CW particles in both pure glycerol
and water at 23.7$\degree$C. Both particles have sufficiently small Stokes
numbers for the pipe flow investigated in this study.

| AGSF-33 | CW
---|---|---
$u_{St}/U$ (Glycerol) | 0.0013 | 0.00036
$u_{St}/U$ (Water) | 0.017 | 382
$St\times 10^{9}$ (Glycerol) | 5.2 | 6.1
$St\times 10^{9}$ (Water) | 6.6 | 7.7
Table 1: The Stokes number and terminal velocity ratio for the AGSF-33 and CW
tracer particles for flow $Re=0.0018$. Note that all dimensionless numbers are
much less than unity in glycerol, so the particles can be expected to
accurately trace the flow. AGSF-33 particles would also work as flow tracers
in water at these speeds, but the prototype CW particles considered here would
need to be used in flows faster than 2000 $\mu m/s$ to neglect buoyancy
effects in water.

##  4 Results and Discussion

###  4.1 Tracer Particle Measurement Comparison

The benefits of the CW tracer particles stem from their marked contrast
improvement. Figure 9 demonstrates the higher contrast of the CW tracer
particles compared to the AGSF-33. Figure 10 shows the SNR improvement of the
CW tracer particles over the AGSF-33, with SNR approximated as

$SNR\approx\frac{\lvert\left<I\right>_{r\leq
r_{p}}-\left<I\right>_{r_{p}<r\leq
3r_{p}}\rvert}{\sqrt{\left<I\right>_{r_{p}<r\leq 3r_{p}}}}$ (8)

where $I$ is the pixel intensity, $r$ is defined as the radius from the
particle centroid, $r_{p}$ is the particle radius, and $\left<\cdot\right>$
indicates mean. The average CW particle SNR is 47, while the average AGSF-33
particle SNR is 25, for a CW to AGSF-33 SNR ratio of 1.8. Equation 8 implies
that if other particles are within a particle diameter of the particle edge
(the definition we use for local), then the SNR calculation could be affected.
Figure 10 shows the general behavior of the AGSF-33 and CW tracer particle
SNR, but should not be taken as a high accuracy measurement of individual
particles’ SNR due to the effects of dense particle seeding and 2D-projection
imaging. A higher SNR improves particle localization, which yields higher
precision measurements. Figure 9 also shows the AGSF-33 particles clustering,
which we did not observe for the CW particles. Since the AGSF-33 particles
cluster, the actual SNR improvement by CW may in fact be higher than we
observed.

(a)

(b)

Figure 9: a) CW tracer particles and b) AGSF tracer particles imaged with a 15
ms exposure time, 500 mm SDD, 38 mm SOD, a 55 kV acceleration voltage, and a
500 $\mu$A target current. The CW tracer particles demonstrate higher contrast
than the AGSF-33 tracer particles for the same exposure time. Figure 10: The
measured SNR for the AGSF-33 and CW tracer particles. The CW SNR is shifted
higher as a result of the higher contrast compared to the AGSF-33. Note that
due to clustering of most of the AGSF-33 particles the true SNR of individual
AGSF-33 particles is likely lower than observed.

Figure 11 shows the DAVP measured with the AGSF-33 and CW tracer particles. We
use the PTV algorithms in LaVision DaVis 8.4, and the details of the
processing steps and settings can be found in appendix A. As mentioned
previously, the glycerol-particle mixture is poured into the pipe at an angle,
leaving room for air to escape the pipe. However, this resulted in particles
being pushed to one side by shear lift. Chipped or broken particles, which are
smaller, are more susceptible to these shear lift forces and are
preferentially pushed to one side. These broken particles are also more
buoyant than the Monte Carlo simulations in section 3 suggest. Indeed, an
artifact in the velocity profile at random locations could sometimes be
observed. We did not observe the same seeding distribution behavior with the
AGSF-33 particles, as they are mass produced via a more refined process and
broken particles have been more effectively removed. For a clear quantitative
comparison less sensitive to particle distribution bias we take the average
and standard deviation of both halves of the pipe combined to calculate a
projection of the radial velocity profile. Both the CW and AGSF-33 particles
recover the DAVP, as figure 11 shows. Close to the wall, the data do not go to
zero because particles have a finite diameter and do not stick to the wall.

Figure 11 also highlights how measurement variance is exacerbated by
2D-projected XPTV. Tracer particles at the front and back walls of the pipe
appear at the same $x$-location as particles at the center, so there is an
unavoidably wide spread in the measured velocities. Stereo or tomographic XPTV
methods are necessary to alleviate the issues with 2D-projection, and are
topics of ongoing research. Light sheets like those used in PTV are not
possible for XPTV. Most XPTV systems, including the one used in this study,
are based on transmitted light instead of scattered light, so even if optics
were available for in-lab XPTV, creating an X-ray light sheet would be not be
helpful.

There is, however, an alternative to multi-source-detector pair or rotating
systems for capturing 3D data. Provided that sufficiently monodisperse tracer
particles are used, one can recover 3D velocity information by tracking the
particle image size, which changes as a function of depth due to geometric
magnification. Such a system would only require one source-detector pair. One
could also filter for particle images of a certain size, effectively creating
an artificial 2D imaging plane akin to a laser sheet. XPTV tracer particles
with such a tight diameter tolerance are, to the best knowledge of the
authors, not presently available. Such tight tolerances for the particle
diameter may not be an insurmountable requirement, though.

Figure 11: The DAVP as measured by the CW and AGSF-33 tracer particles
compared against the analytical DAVP.

Figures 11 and 12 also show that CW tracer particles are more localizable due
to their higher contrast and they did not clump like the AGSF-33 tracer
particles. As a result, both the axial and $x$-component velocity measurements
with the CW tracer particles demonstrate less spread than the AGSF-33 tracer
particles. Better localizability is crucial because it reduces measurement
error in velocity and improves spatial resolution.

Figure 12: The measured $x$-component velocity, $v(x,y)$, for the CW and
AGSF-33 tracer particles. The CW tracer particles, being more localizable and
less clumped, exhibit smaller measurement variance compared to the AGSF-33.

Figure 13 shows the velocity vectors superimposed on an image. We note from
inspection of many such images that: 1) the PTV algorithm does not produce
spurious vectors a human observer would not create and 2) some particles that
a human would trace are not traced by the algorithm. These data could likely
be improved by optimizing the processing parameters or by using more advanced
PTV algorithms. Improving image quality by using higher contrast particles,
brighter sources, and better detectors will also help. These measurements
demonstrate that time-resolved, $O(1$ cm$)$ domain XPTV at nearly 70 Hz is
possible in the laboratory with both AGSF-33 and CW tracer particles. We
achieve this result in spite of a dim laboratory X-ray source by using a
single-threshold PCD and high-contrast tracer particles. Exploring the
performance limits of CW tracer particles is the subject of future work.
Brighter laboratory X-ray sources combined with advanced tracer particles will
enable even faster XPTV. Particle contrast is proportional to photon flux,
implying that with new X-ray sources that are 100 times brighter (Hemberg et
al (2003)), frame rates on the order of 1 kHz are possible.

(a)

(b)

Figure 13: The velocity vectors overlaid on the hundredth frame of the (a) CW
and (b) AGSF-33 footage. The greater variability of the AGSF-33 $x$-component
velocity due to variability of the apparent particle centroid location is
apparent, whereas the CW particles offer better contrast and are more robustly
localized. The color scale is inverted for PTV processing in DaVis.

Despite the need for future improvements in monodispersity and tungsten
coating thickness control, CW tracer particles outperform AGSF-33 tracer
particles in terms of image contrast and localizability while still measuring
a comparable DAVP with satisfactory agreement to the analytical solution. As
the tracer manufacturing process improves, CW tracer particles will enable the
measurement of higher speed flows, especially when combined with brighter
laboratory X-ray sources. The success and future improvement of the CW tracer
particles should motivate the community and seeding particle manufactures to
explore the XPTV tracer particle design space and push the boundaries of flows
that can be studied with this technique.

###  4.2 Benefits and Challenges of PCDs

For the same source flux, PCDs can achieve a higher SNR and less blurring than
scintillating detectors (Russo (2018)). PCDs directly detect photons,
dispensing with visible wavelength scatter and glow artifacts in
scintillators, which both contribute to blurring. Directly detecting photons
also avoids the thermal and electronic noise associated with CCD and CMOS
sensors. Despite these advantages, at this stage PCDs have a number of
drawbacks that require consideration.

Generally speaking, SNR increases monotonically with increasing source
brightness, enabling higher frame rates. For PCDs, however, the SNR can
rapidly degrade due to photon count pile-up. That is, as the number of photons
arriving at a pixel per unit time exceeds the circuit’s ability to distinguish
individual events, there ceases to be an increasing benefit to a brighter
source. Current PCD nominal count rate limits are $O(10^{6}$ \- $10^{7}$)
photons/pixel/second, often with some form of retriggering, software
correction, or both to mitigate the effects of pile-up. For the Dectris
Pilatus3 detector used here, the nominal count rate limit is $10^{7}$
photons/pixel/second when using retriggering and a count rate correction. We
turn retriggering and the correction off for these experiments, however,
because Dectris does not recommend using them with a polychromatic X-ray
source. They are calibrated at a synchrotron for use with a monochromatic
source. Unfortunately, most laboratory X-ray sources are polychromatic,
including the one used here. Without retriggering and the count rate
correction, the Dectris Pilatus3 can handle up to about $10^{6}$
photons/pixel/second. Once pile-up begins to dominate, SNR degradation is
observed and image artifacts such as detector tile boundaries become
uncorrectable. The count rate limit of PCDs is an important limitation on the
source brightness that bounds the achievable SNR.

A scintillating detector would not face the same limitations. X-rays are
turned into visible wavelength photons regardless of their arrival rate, and
the saturation limit for a CCD or CMOS sensor can often be adjusted. In
practice, then, a scintillating detector could achieve a higher SNR than a PCD
detector by enabling imaging with a brighter source than the PCD could handle.
Indeed, scintillating detectors are widely used for imaging at synchrotrons,
while PCDs are not placed directly in a synchrotron beamline. In practice,
then, scintillating detectors can currently achieve faster frame rates or
higher image quality than PCDs.

In addition to the temporal resolution advantage, scintillating detectors also
have a spatial resolution advantage. Pixel sizes on currently available PCDs
are relatively large – $O$(100 $\mu$m). Scintillating detectors offer much
smaller pixel sizes, achieving higher spatial resolution.

Why, then, use a PCD for X-ray flow visualization? First, although PCDs
currently suffer from count rate limitations, due to improving circuitry,
detector materials and reduced pixel sizes, future PCDs can be expected to
accommodate increasing count rates. When pile-up is not the limiting factor,
the theoretical SNR advantage of PCDs is apparent. Furthermore, despite the
current drawbacks, there are benefits to PCDs even now that in some cases may
outweigh the limitations. Chiefly, PCDs can directly measure the deposited
photon energy, in effect capturing “color” X-ray images. PCDs that are now
coming to market are able to resolve the incoming photon energy with two or
more thresholds, enabling K-edge material detection. One could then track
multiple tracers made of different materials, a process analogous to
fluorescing tracer particles with different colors in visible light flow
visualization. Additionally, one could observe the scalar mixing of two or
more phases while simultaneously measuring the fluid velocity field with
tracer particles, a difficult experiment with visible light techniques, and an
all but impossible one in opaque media or containers.

##  5 Conclusion

This study demonstrates quantitative laboratory XPTV by measuring the
2D-projected velocity profile of Poiseulle flow in a round pipe with two
different tracer particles. By capturing images of tracer particles in 15 ms
exposure times, we show the potential to speed up laboratory XPTV to nearly 70
Hz, a limit that can easily be extended with brighter sources and improved
detectors. We show the value of high-contrast tungsten-coated, hollow carbon
microsphere tracer particles designed specifically for quantitative XPTV. The
tungsten-coated CW tracer particles offer a contrast enhancement over the
previously available AGSF-33 tracer particles, making them easier to localize
and, in principle, allowing the same SNR with a shorter exposure time. As the
manufacturing process for the CW and other custom particles improves, their
utility as XPTV tracer particles can be expected to surpass AGSF-33.
Continuing exploration of the XPTV tracer particle design space is important
for developing application- and technique-specific tracers. For example, to
develop smaller particles or optimize the attenuation contrast for
application-specific fluids. The metrics used in this study, and methods
previously discussed in Parker and Mäkiharju (2022), can be used for testing
future tracer particle prototypes.

In addition to demonstrating the value of tracer particles developed
specifically for X-ray flow visualization, we explored the feasibility of
using PCDs for X-ray flow visualization. By showing that PCDs can acquire
laboratory XPTV images at speeds comparable to scintillating detectors, we
show that energy-thresholding material detection techniques are feasible for
future laboratory XPTV experiments. With PCDs improving rapidly, material
detection and other energy-thresholding techniques can improve tracer particle
detection beyond what is capable with scintillating detectors, opening up new
opportunities to use X-ray light to the experimentalist’s advantage.
Simultaneous multi-species and scalar field flow tracing experiments are
possible with energy-resolving PCDs, for example. As laboratory X-ray sources
get brighter, and PCDs get faster, these techniques will be of even greater
utility. The flexibility of the system that we propose lends itself to growing
alongside X-ray source and detector technology.

XPTV, XPIV, and related X-ray flow visualization techniques make possible the
study of flows that opacity and refraction make difficult or impossible to
observe with visible light. By demonstrating the potential of XPTV with PCDs
and advanced tracer particles in the laboratory, this promising technique can
proliferate in laboratories across the fluid dynamics community.

Acknowledgments We gratefully acknowledge the support of NSF EAGER award
#1922877 program managers Ron Joslin and Shahab Shojaei-Zadeh, and the
additional support provided by the Society of Hellman Fellows Fund.

We also acknowledge the contributions of Angel Rodriguez for his help setting
up the X-ray facility. We also are grateful for the contributions of Andrew
Kokubun who as an undergraduate researcher participated in the early stages of
this project. For advice in setting up the MCNP simulations we refer to the
authors also thank Prof. Massimiliano Fratoni.

##  Appendix A Image Processing

###  A.1 Image Pre-Processing

Flat-field correction is applied to all of the images. As part of this
correction, dead pixels are replaced with the median value of the surrounding
pixels. A flicker correction is also applied as the X-ray source intensity
exhibits an $O$(0.1%) variation at approximately 0.25 Hz and integer-multiple
frequency harmonics, which is presumed to be the controller update frequency
of the anode current feedback loop. The flicker correction involves
determining the 1st and 99th percentile pixel values for each image. Those
pixel values are then established as the saturation limits for their
respective image. The images are then inverted so that the particles are
bright on a dark background, which the Lavision DaVis processing algorithms
expects. After inversion, the images are converted into 16-bit images. Then,
the images are exported to undergo the following filtration and other
operations in DaVis:

  1. 1.

Mask the area outside of the region of interest.

  2. 2.

Subtract the average intensity of all frames.

  3. 3.

Spatial bandpass filter to features that are 3-6 pixels in length.

  4. 4.

Subtract 3,700 counts to isolate the particle images as positive integer
values in preparation for the next step. This value may change as a function
of particle SNR.

  5. 5.

Set any negative pixel value to 0.

###  A.2 PIV Processing

PIV is used as a pre-processing step to the PTV in order to establish an
approximate particle velocity and a search radius for the particle tracking
algorithm. The PIV settings chosen to provide a low spatial resolution
estimation of the fluid velocity are as follows:

  1. 1.

First pass: 64$\times$64 pixel window, 50% overlap, single pass.

  2. 2.

Second pass: 32$\times$32 pixel window, 50% overlap, double pass.

  3. 3.

Remove groups with $<$ 5 vectors.

  4. 4.

Allowable axial vector range: 3 to 19 pixels, encompassing range physically
possible given range of particle densities.

  5. 5.

Allowable x-component vector range: -4 to 4 pixels. Nominally zero, but due to
shifts in the detected centroid, the $x$-component velocity can be measured as
non-zero. This also allows for the $x$-axis of the images to potentially be
misaligned with tube axis.

###  A.3 PTV Processing

After the PIV processing step, DaVis’ particle tracking algorithm is called.
For each frame and tracked particle the centroid location, and $x$\- and
axial-velocity components are calculated. This data is imported to MATLAB to
bin the velocities by $x$-location and calculate the average axial velocity
component. The PTV settings are as follows:

  1. 1.

Particle size range: 3 to 12 pixels.

  2. 2.

Allowed vector range relative to reference (i.e., velocity from PIV): 10
pixels, while maximum displacement was 8 pixels.

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11institutetext: School of Science and Technology, Physics Division,
University of Camerino, Via Madonna delle Carceri, 9B, Camerino (MC), Italy
22institutetext: INFN-Sezione di Perugia, 06123 Perugia, Italy
33institutetext: School of Pharmacy, Physics Unit, University of Camerino, Via
Madonna delle Carceri, 9B, Camerino (MC), Italy

#  Giant amplification of Berezinskii-Kosterlitz-Thouless transition
temperature in superconducting systems characterized by cooperative interplay
of small-gapped valence and conduction bands

Giovanni Midei 1122<EMAIL_ADDRESS>Andrea Perali 33
<EMAIL_ADDRESS>
###### Abstract

Two-dimensional superconductors and electron-hole superfluids in van der Waals
heterostructures having tunable valence and conduction bands in the electronic
spectrum are emerging as rich platforms to investigate novel quantum phases
and topological phase transitions. In this work, by adopting a mean-field
approach considering multiple-channel pairings and the Kosterlitz-Nelson
criterion, we demonstrate giant amplifications of the Berezinskii-Kosterlitz-
Thouless (BKT) transition temperature and a shrinking of the pseudogap for
small energy separations between the conduction and valence bands and small
density of carriers in the conduction band. The presence of the holes in the
valence band, generated by intra-band and pair-exchange couplings, contributes
constructively to the phase stiffness of the total system, adding up to the
phase stiffness of the conduction band electrons that is boosted as well, due
to the presence of the valence band electrons. This strong cooperative effect
avoids the suppression of the BKT transition temperature for low density of
carriers, that occurs in single-band superconductors where only the conduction
band is present. Thus, we predict that in this regime, multi-band
superconducting and superfluid systems with valence and conduction bands can
exhibit much larger BKT critical temperatures with respect to single-band and
single-condensate systems.

Keywords: two-band superconductors, Berezinskii-Kosterlitz-Thouless
transition, critical temperature amplification, valence and conduction bands
systems

##  1 Introduction

Two dimensional multi-band superconductivity can generate rather interesting
physics Milosevic2015 , especially in the case of electronic systems with
valence and conduction bands both participating to the superconducting
condensate. In such configurations, complex redistribution of electrons occurs
between valence and conduction bands, leading to a new physics with respect to
single-band superconductors, such as density induced and band-selective BCS-
BEC crossover Andrenacci1999 , topological quantum phase transitions and
hidden criticalities Nozieres1999 ; sademelo ; Yue2022 ; Litak2012 .
Furthermore, at finite temperature, this phenomenon is also responsible for
the non-monotonic behavior of the superconducting gaps resulting in a
superconducting-normal state reentrant transition Midei2023 . A peculiar
feature about two dimensional systems regards the nature of the
superconducting transition. Indeed, in thin films the transition to the
superconducting state with decreasing temperature occurs in two stages: at
first, the finite amplitude of the order parameter forms around the mean-field
critical temperature $T_{MF}$ but without ordering of the phase, after that,
the true superconducting transition occurs with phase ordering at a lower
temperature, which is expected to belong to the universality class of the BKT
transition of the two dimensional XY model Berezinskii1972 ; Kosterlitz1973 ;
Kosterlitz1974 . In contrast to the power-law dependence of the coherence
length $\xi_{c}$ predicted in the Ginzburg-Landau theory Benfatto2000 ;
Mondal2011 , this transition is characterized by an exponential divergence at
the BKT critical temperature. In order to observe the BKT phenomenon, a
condition is that $d<<\xi_{c}$ being $d$ the sample thickness, though some
experiments have reported the BKT transition also outside the expected ranges
Chu2004 . In this regime, the transition to the normal state is driven by the
vortex-antivortex dissociation instability, which is connected with the
logarithmic dependence of the interaction energy on the separation between
vortices. This leads to the discontinuous jump of the phase stiffness $J$ from
a finite value right below $T_{BKT}$ to zero above it Nelson1977 ;
Minnhagen1987 . The value of $J$ at the BKT critical temperature can be
inferred via measurements of the London penetration depth or from the
nonlinear exponent of the I-V characteristics Venditti2019 ; Sharma2022 . The
study of the BKT transition in valence and conduction band systems can be
relevant for recently discovered 2D superconducting bilayer graphene systems
Zhou2022 ; Pantaleon2022 , where the energy shift between the conduction and
valence bands can be precisely tuned by an external electric field
perpendicular to the graphene layers, and electron-hole superfluid systems,
such as the double-bilayer graphene (2BLG) Perali2013 , consisting of two
conducting bilayer graphene sheets, one containing electrons and the other
holes. The two bilayer sheets are separated by an insulating barrier in order
to prevent tunneling between the sheets. This system can be studied by means
of a BCS mean-field theory by performing a standard particle-hole
transformation Conti2017 . In this way, the intra-band couplings create Cooper
pairs made up by a positively charged hole in the conduction/valence band of
the p-doped bilayer and an electron in the conduction/valence band of the
n-doped bilayer, while the pair-exchange couplings transfer electrons and
holes involved in the formation of Cooper pairs from the conduction band to
the valence band (and viceversa) of their respective graphene bilayer.
Furthermore, materials where charge density wave (CDW) and superconducting
orders coexist can be an interesting platform for studying valence and
conduction band configuration, such as underdoped cuprates Gabovich2009 ;
Gabovich2010 ; Arpaia2019 ; Perali1996  and transition metal dichalcogenides
Rossnagel2011 ; Neto2001 ; Kiss2007 . In fact, the presence of CDWs and their
fluctuations can modify the energy spectrum by opening (pseudo)gaps and at the
same time can mediate Cooper pairing, splitting single bands in two branches,
that in wave-vector space behave locally as valence (hole-like) and conduction
(electron-like) bands. Finally, in FeSe it is possible to tune the position of
the valence band with respect to the conduction band, and thus the carrier
density, through the chemical potential alignment with the trilayer graphene
substrate, where the local work function is spatially inhomogeneous Lin .
In this work, we have investigated the superconducting state properties of a
2D electronic system with valence and a conduction bands considering the case
of different intra-band and pair-exchange couplings, which is the typical
configuration in bilayer graphene superconductors and electron-hole multilayer
systems. We have obtained numerical results for the phase stiffness and the
BKT critical temperature as functions of the energy band gap, the intra-band,
and the pair-exchange couplings, that can be tuned in bilayer graphene systems
by applying external electric fields through metal gates and, in the case of
electron-hole multilayers, by tuning the insulating barrier width. We have
found that there exist a giant enhancement of the Kosterlitz-Thouless critical
temperature and a consequent shrinking of the pseudogap region, with a
different mechanism from the one found in Salasnich2019 , in the regime of
small energy gap between the bands and of low density of carriers in the
conduction band, with the latter being the optimal condition for observing
superfluidity in the electron-hole systems Perali2013 ; Conti2017 . Moreover,
we have found a minimum in the pseudogap region, in the intermediate-regime of
the pair-exchange couplings, shifting to the weak-coupling regime by reducing
the level of filling of the conduction band. We also made comparison with
single-band 2D superconducting systems, in which the phase stiffness is
directly proportional to the electron density, resulting in small values of
the BKT transition temperatures in the low density regime. It turns out that
the presence of the valence band act as a reservoir of electrons for the two-
band system, contributing constructively to the total stiffness of the system.
Thus, we predict that in this regime, multi-band superconducting and
superfluid systems with valence and conduction bands can have enhanced
Kosterlitz-Thouless critical temperatures with respect to single-band and
single-condensate systems.

The manuscript is organized as follow. In section 2 we describe the model for
the physical system considered and the theoretical approach for the evaluation
of the superconducting state properties. In section 3 we report our results.
The conclusions of our work will be reported in Section 4.

##  2 Model and Methods

We consider a two-dimensional (2D) two-band superconductor with a valence and
a conduction electronic band in a square lattice. This model can be applied to
2D electron-hole layered superfluids as well. The valence and the conduction
band have a tight-binding dispersion given, respectively, by Eqs. (1) and (2):

$\varepsilon_{1}(\mathbf{k})=2t[\cos(k_{x}a)+\cos(k_{y}a)]-8t-E_{g}$ (1)
$\varepsilon_{2}(\mathbf{k})=-2t[\cos(k_{x}a)+\cos(k_{y}a)]$ (2)

where $t$ is the nearest neighbour hopping parameter, $a$ the lattice
constant, $E_{g}$ the energy band-gap between the conduction and valence band.
and the wave-vectors belong to the first Brillouin zone $-\frac{\pi}{a}\leq
k_{x,y}\leq\frac{\pi}{a}$. The band dispersions are reported in Fig. 1.

Fig. 1: Band dispersions for the two-band system. The energy and the wave-
vectors are measured in units of $t$ and $a$, respectively. $E_{g}$ is the
energy gap between the valence and the conduction bands.

We assume that Cooper pairs formation is due to an attractive interaction
between opposite spin electrons. The two-particle interaction has been
approximated by a separable potential $V_{ij}(\mathbf{k},\mathbf{k^{\prime}})$
with an energy cutoff $\omega_{0}$, which is given by:

$V_{ij}(\mathbf{k},\mathbf{k^{\prime}})=-V_{ij}^{0}\Theta\Bigl{(}\omega_{0}-|\xi_{i}(\mathbf{k})|\Bigr{)}\Theta\Bigl{(}\omega_{0}-|\xi_{j}(\mathbf{k^{\prime}})|\Bigr{)}$
(3)

where $V_{ij}^{0}>0$ is the strength of the potential and $i,j$ label the
bands. $V_{11}^{0}$ and $V_{22}^{0}$ are the strength of the intra-band
pairing interactions (Cooper pairs are created and destroyed in the same
band). $V_{12}^{0}$ and $V_{21}^{0}$ are the strength of the pair-exchange
interactions (Cooper pairs are created in one band and destroyed in the other
band, and vice versa), so that superconductivity in one band can induce
superconductivity in the other band. We set $\hbar=1$ and $k_{B}=1$ throughout
the manuscript.

The same energy cutoff $\omega_{0}$ of the interaction for intra-band and
pair-exchange terms is considered, and is taken much larger than the bandwidth
so that the interaction can be considered contact-like. The terms
corresponding to Cooper pairs forming from electrons associated with different
bands (inter-band or cross-band pairing) are not considered in this work (see
Paredes2020 ). The term $\xi_{i}(\mathbf{k})=\varepsilon_{i}(\mathbf{k})-\mu$
in Eq. (3) is the energy dispersion with respect to the chemical potential
$\mu$. The index $i=1,2$ numerates the bands, where $i=1$ denotes the valence
band and $i=2$ the conduction band.

The superconducting state of the two-band system is examined within a mean
field theory. We consider the two-band case generalization of the
superconducting gap equation at finite temperature $T$

$\Delta_{i}(\mathbf{k})=-\frac{1}{\Omega}\sum_{j}\sum_{\mathbf{k^{\prime}}}V_{ij}(\mathbf{k},\mathbf{k^{\prime}})\frac{\tanh{\frac{E_{j}(\mathbf{k^{\prime}})}{2T}}}{2E_{j}(\mathbf{k^{\prime}})}\Delta_{j}(\mathbf{k^{\prime}})$
(4)

where
$E_{i}(\mathbf{k})=\sqrt{{\xi_{i}(\mathbf{k})}^{2}+{\Delta_{i}(\mathbf{k})}^{2}}$
are excitation branches in the superconducting state, and $\Omega$ is the area
occupied by the 2D system. Note that for a separable interaction of the form
of Eq. (3), the superconducting gaps assume the following expression:

$\Delta_{i}(\mathbf{k})=\Delta_{i}\Theta\Bigl{(}\omega_{0}-|\xi_{i}(\mathbf{k})|\Bigr{)}$
(5)

We point out that all the coupling configurations considered in this work led
to solutions of the gap equations in Eqs. 4 which are global minima of the
free energy and thus stable physical solutions for the two-gap superconducting
state, as discussed in Ref.Aase2023 . The total electron density of the two-
band system is fixed and it is given by the sum of the densities of the single
bands, $n_{tot}=\sum_{i}n_{i}$, that can vary instead. The electronic density
$n_{i}$ in the band ($i$) at a temperature T is defined as:

$n_{i}=\frac{2}{\Omega}\sum_{\mathbf{k}}\Bigl{[}{v_{i}(\mathbf{k})}^{2}f\big{(}-E_{i}(\mathbf{k})\big{)}+{u_{i}(\mathbf{k})}^{2}f\big{(}E_{i}(\mathbf{k})\big{)}\Bigr{]}$
(6)

where $f$ is the Fermi-Dirac distribution function. The BCS coherence weights
$v_{i}(\mathbf{k})$ and $u_{i}(\mathbf{k})$ are:

${v_{i}(\mathbf{k})}^{2}=\frac{1}{2}\Bigg{[}1-\frac{\xi_{i}(\mathbf{k})}{\sqrt{{\xi_{i}(\mathbf{k})}^{2}+{\Delta_{i}(\mathbf{k})}^{2}}}\Bigg{]}$
(7) ${u_{i}(\mathbf{k})}^{2}=1-{v_{i}(\mathbf{k})}^{2}$ (8)

The low-temperature physics of a 2D attractive Fermi gas turns out to be
different from that of a 3D one. This is a consequence of the MerminâWagner
theorem, which prohibits the spontaneous breaking of a continuous symmetry at
finite temperatures, allowing one to find off-diagonal long-range order only
at $T=0$ in two dimensions. However, in the low temperature phase of these
kind of 2D systems there exists a âquasiâlong range orderâ in which the phase
correlations decays algebraically, i.e. $\langle
e^{i(\theta(r)-\theta(0))}\rangle\sim|r|^{-\eta}$, where $\eta$ is a $T$ \-
dependent exponent and $\theta$ is the phase of the order parameter. This
algebraic decay of the correlation function is observed up to a finite
temperature $T_{KT}$, known as the Berezinskii-Kosterlitz-Thouless (BKT)
critical temperature, that separates the low-temperature from the high-
temperature phase. The high temperature phase is characterized instead by an
exponential decay of phase correlation $\langle
e^{i(\theta(r)-\theta(0))}\rangle\sim e^{-r/\xi}$, where $\xi$ is a
characteristic length of the system and depends on temperature.

The transition temperature can be determined through the relation

$T_{BKT}=\frac{\pi}{2}J_{TOT}(T_{BKT})$ (9)

where $J_{TOT}(T_{BKT})$ is the total phase stiffness at the Kosterlitz-
Thouless transition temperature, that in our case is given by the sum of the
stiffnesses of the single bands $J_{TOT}=\sum_{i}J_{i}$.
Following the approach used in Appendix A of Ref. Metzner2019 , the phase
stiffness of a superconductor can be computed from the current density induced
by an external electromagnetic field, given by

$j_{i\alpha}(\mathbf{q},\omega)=-\sum_{\alpha^{\prime}}K_{i\alpha\alpha^{\prime}}(\mathbf{q},\omega)A_{\alpha^{\prime}}(\mathbf{q},\omega)$
(10)

where $A_{\alpha}$ are the component of the vector potential in a gauge where
the scalar potential $\phi$ vanishes and $\nabla\cdot\mathbf{A}=0$,
$K_{i\alpha\alpha^{\prime}}(\mathbf{q},\omega)$ is the response function. The
index $\alpha$ refers to the Cartesian axis and $i$ is the band index. The
phase stiffness is related to the static limit of the response function,

$K_{i\alpha\alpha^{\prime}}=\lim_{\mathbf{q}\to
0}K_{i\alpha\alpha^{\prime}}(\mathbf{q},0)$ (11)

The response function in Eq. (10) is given by the sum of a paramagnetic and a
diamagnetic contribution,

$K_{i\alpha\alpha^{\prime}}=K_{i\alpha\alpha^{\prime}}^{dia}+K_{i\alpha\alpha^{\prime}}^{para}$
(12)
$K_{i\alpha\alpha^{\prime}}^{para}=\frac{2e^{2}}{V}\sum_{\mathbf{k}}f^{\prime}\big{(}E_{i}(\mathbf{k})\big{)}\varepsilon^{\alpha}_{i}(\mathbf{k})\varepsilon^{\alpha^{\prime}}_{i}(\mathbf{k})$
(13)
$K_{i\alpha\alpha^{\prime}}^{dia}=\frac{e^{2}}{V}\sum_{\mathbf{k}}\Biggl{[}1-\frac{\xi_{i}(\mathbf{k})}{E_{i}(\mathbf{k})}+\frac{2\xi_{i}(\mathbf{k})}{E_{i}(\mathbf{k})}f\big{(}E_{i}(\mathbf{k})\big{)}\Biggr{]}\varepsilon^{\alpha\alpha^{\prime}}_{i}(\mathbf{k})$
(14)

with
$\varepsilon^{\alpha}_{i}(\mathbf{k})=\partial{\varepsilon_{i}(\mathbf{k}})/\partial{k_{\alpha}}$,
and
$\varepsilon^{\alpha\alpha^{\prime}}_{i}(\mathbf{k})=\partial^{2}{\varepsilon_{i}(\mathbf{k})}/\bigl{(}\partial{k_{\alpha}}\partial{k_{\alpha^{\prime}}}\bigr{)}$
The off-diagonal elements ($\alpha\neq\alpha^{\prime}$) of the response
function vanish since the dispersions are symmetric in $k_{x}$ and $k_{y}$
because of the square symmetry of the considered lattice. Furthermore, the
response function does not depends on the direction, that is,
$K_{i\alpha}=K_{i}$ is independent of $\alpha$. The phase stiffness $J_{i}$ is
independent on the direction as well, and is given by

$J_{i}=\frac{K_{i}}{(2e)^{2}}$ (15)

The energies are normalized in units of the hopping parameter $t$ and the
dimensionless couplings $\lambda_{ij}$ are defined as
$\lambda_{ij}=NV_{ij}^{0}$, where $N=1/4\pi a^{2}t$ is the density of states
at the top/ bottom of the valence / conduction band, that coincide since the
density of states is not modified by the concavity of the band.

##  3 Results

In this section, we present results for the phase stiffness calculated at the
BKT transition temperature and the BKT transition temperature itself, that are
closely connected by the Kosterlitz-Nelson criterion in Eq.(9), for the two-
dimensional system made up by a valence and a conduction band. The
superconducting order parameters $\Delta_{1}$ and $\Delta_{2}$, which enter
the expressions for the total phase stiffness $J_{TOT}$ derived in Sec. 2, are
computed from Eqs.(4) coupled with Eq.(5), that gives the total density of the
system, considering a contact-type interaction ($\omega_{0}/t=20$).

(a)

(b)

(a) Fig. 3: Valence band $J_{1}$, conduction band $J_{2}$ and total phase
stiffness $J_{TOT}=J_{1}+J_{2}$ as functions of $E_{g}/t$ (a),
$\lambda_{12}=\lambda_{21}$ (b), and $\lambda_{22}=\lambda_{11}$ (c),
calculated at $T=T_{BKT}$. The values of the total density are
$a^{2}n_{tot}=2.005$ (top panels) and $a^{2}n_{tot}=2.20$ (lower panels). (a):
$\lambda_{12}=\lambda_{21}=0.3$, $\lambda_{22}=\lambda_{11}=0.5$; (b):
$E_{g}/t=0.0$; $\lambda_{22}=\lambda_{11}=0.5$; (c): $E_{g}/t=0.0$;
$\lambda_{12}=\lambda_{21}=0.3$.

We report the phase stiffness of single bands and the total phase stiffness as
functions of the band-gap energy $E_{g}$ in Figs.2(a). The presence of the
valence band contributes to enhance the total stiffness of the system,
especially in the region of small band-gap energy where the transfer of
electrons from the valence to the conduction band is favoured. The consequent
presence of the holes in the valence band contributes constructively to

(a)

(b)

(c)

Fig. 4: Phase diagram of the system in the temperature $T/t$ vs band-gap
energy $E_{g}/t$ (a), pair-exchange couplings $\lambda_{12}=\lambda_{21}$ (b)
and intra-band couplings $\lambda_{22}=\lambda_{11}$ (c) planes, for different
values of the total density. (a): $\lambda_{12}=\lambda_{21}=0.3$,
$\lambda_{22}=\lambda_{11}=0.5$; (b): $E_{g}/t=0.0$;
$\lambda_{22}=\lambda_{11}=0.5$; (c): $E_{g}/t=0.0$;
$\lambda_{12}=\lambda_{21}=0.3$.

the total stiffness of the system, adding up to the stiffness of the
conduction band electrons and furthermore, there is a boost in the conduction
band stiffness itself due to the electrons coming from the valence band. When
the band-gap energy becomes too large the transfer of electrons is reduced,
and superconductivity cannot exploit the presence of a proximate valence band,
with the two condensates being essentially decoupled. In this regime, when the
density of the conduction band is small (top panel of Fig.2(a))
superconductivity is strongly suppressed, since very few holes and electrons
that can form Cooper-pairs are present in valence and conduction bands,
respectively, thus leading to a suppression of the total phase stiffness
$J_{TOT}$. When the density in the conduction band is higher (bottom panel of
Fig.2(a)), for large $E_{g}$ superconductivity is sustained only by the
condensate in the conduction band, which is the only significant condensate in
the system. This means that the enhancement in the total phase stiffness due
to the valence band condensate is lost, since in absence of holes, the filled
valence band will not create any stiffness and will be inactive for
superconductivity. Thus, in this regime the system is acting as a single
component condensate. In Figs.2(b) we report the phase stiffness of the single
bands and the total phase stiffness as functions of pair-exchange couplings
$\lambda_{12}=\lambda_{21}$. For higher values of $\lambda_{12}=\lambda_{21}$
the total phase stiffness is enhanced, since the pair-exchange interactions
facilitate the mixing of electrons between the two bands. For small values of
the conduction band electron density (top panel of Fig.2(b)), even in the
limit of zero pair-exchange interactions the presence of the valence band is
enough to enhance the total stiffness with respect to a system having only the
conduction band. In fact,

(a)

(b)

(c)

Fig. 5: Ratio between the mean-field critical temperature $T_{MF}$ and the
Berezinskii-Kosterlitz-Thouless critical temperature $T_{BKT}$ as a function
of $E_{g}/t$ (a), $\lambda_{12}=\lambda_{21}$ (b), and
$\lambda_{22}=\lambda_{11}$ (c), for different values of the total density.
(a): $\lambda_{12}=\lambda_{21}=0.3$, $\lambda_{22}=\lambda_{11}=0.5$; (b):
$E_{g}/t=0.0$; $\lambda_{22}=\lambda_{11}=0.5$; (c): $E_{g}/t=0.0$;
$\lambda_{12}=\lambda_{21}=0.3$.

the transfer of electrons from the valence band to the conduction band is not
only due to the pair-exchange but to all the channels of the interaction,
including also the intra-band channels. Moreover, since we are dealing with a
system at finite temperature, thermal effects can also contribute to the
redistribution of the elctrons between the two bands. However, for higher
values of the conduction band electron density (lower panel of Fig. 2(b)), for
zero pair-exchange couplings the intra-band channels alone are not enough to
transfer the electrons in the conduction band, and the valence band condensate
in not superconducting. Also in this regime the system is behaving as a single
component condensate, with only the conduction band contributing to the total
phase stiffness $J_{TOT}$. When the pair-exchange couplings are turned on, the
Cooper pairs transfer from valence band to conduction band can start, leaving
holes in the valence band, that in this way can host a superconducting
condensate and give its contribution to the total stiffness. The physics is
very similar to the previous case in the conduction band low density regime,
when the intra-band couplings are tuned (top panel of Fig. 3(a)). For higher
values of the conduction band electron density, (bottom panel of Fig. 3(a)),
even though the stiffness of the conduction band condensate is again the only
relevant contribution to the total stiffness in the weak-coupling regime, the
stiffness of the valence band condensate is not completely zero as in the
previous case, due to the presence of finite pair-exchange interactions that
couple the two condensates. In the intermediate- and strong- coupling regime,
the stiffness of the conduction band condensate is decreasing, but the total
stiffness is sustained by the valence band condensate stiffness, that is
increasing instead.

The behavior of the phase stiffness described until now is reflected into the
BKT transition temperature. In Fig. 4(a) the mean-field and the Kosterlitz-
Thouless critical temperature are reported as functions of the band-gap energy
$E_{g}$. Note that the mean-field critical temperature $T_{MF}$ has been
evaluated from the linearized form of Eqs.(4) in the limit of vanishing
superconducting gaps. While the mean-field critical temperature remains finite
for low filling of the conduction band when $E_{g}$ becomes large, the
Kosterlitz-Thouless critical temperature is strongly suppressed since the
total phase stiffness is going to zero in this regime. Conversely, when
$E_{g}$ is small, there is a giant enhancement in the BKT critical temperature
due to the valence and conduction band condensates additive contributions to
the total system stiffness.

(a)

(b)

(c)

Fig. 6: Ratio between the Berezinskii-Kosterlitz-Thouless critical temperature
of the system with ($T_{BKT}$) and without ($T_{BKT_{1b}}$) valence band, as a
function of $E_{g}/t$ (a), $\lambda_{12}=\lambda_{21}$ (b), and
$\lambda_{22}=\lambda_{11}$ (c), for different values of the total density.
(a): $\lambda_{12}=\lambda_{21}=0.3$, $\lambda_{22}=\lambda_{11}=0.5$; (b):
$E_{g}/t=0.0$; $\lambda_{22}=\lambda_{11}=0.5$; (c): $E_{g}/t=0.0$;
$\lambda_{12}=\lambda_{21}=0.3$.

In Fig.4(b) the mean-field and the Kosterlitz-Thouless critical temperature
are reported as functions of the pair-exchange couplings $\lambda_{12}$. Both
are enhanced for increasing $\lambda_{12}$, since the pair-exchange coupling
induce the transfer of electrons from the valence band to the conduction band.
However, while $T_{BKT}$ is almost insensitive to the level of filling of the
conduction band, for small pair-exchange couplings values $T_{MF}$ is not. The
situation is reversed when the intra-band couplings are tuned, as shown in
Fig.4(c), with $T_{MF}$ that is insensitive to the level of filling of the
conduction band while, in the weak-coupling regime $T_{BKT}$ is not, resulting
in a suppression of superconductivity for decreasing values of the conduction
band electron density.

Having a general overview on the phase diagram of the system, we now focus on
the pseudogap region. In Fig. 5(a), we show the ratio between the mean-field
critical temperature $T_{MF}$ and the BKT transition temperature $T_{BKT}$ as
a function of the energy band gap $E_{g}$. The ratio is almost constant in the
region of small energy gap and for low density of carriers in the conduction
band, while for larger $E_{g}$ the ratio is increased, evidencing an
enhancement of the pseudo-gap region due to the decoupling of the two
condensates. For higher values of carrier density, the behavior of the ratio
is almost insensitive to the value of $E_{g}$ since the decoupling between the
condensates occurs also for increasing number of electrons in the conduction
band. In Fig.5(b), we show the ratio between the mean-field critical
temperature $T_{MF}$ and the BKT transition temperature $T_{BKT}$ as a
function of the pair-exchange couplings $\lambda_{21}=\lambda_{12}$. In this
case for large values of the total density the ratio exhibits a non-monotonic
behavior in the intermediate-coupling regime with the presence of a minimum in
the pseudo-gap region. For larger values of the couplings the ratio keeps
increasing as occurs in the conventional BCS theory, since $T_{BKT}$ saturates
while $T_{MF}$ is going to infinity. For smaller values of the total density
instead, the ratio is always monotonically increasing for all the values of
the $\lambda_{21}=\lambda_{12}$ and the minimum in the pseudo-gap region is
shifting toward the weak-coupling regime by further reducing the density. The
ratio between $T_{MF}$ and $T_{BKT}$ as a function of the intra-band couplings
in Fig.5(c) instead, is always monotonically increasing for the different
levels of filling of the conduction band, so that the minimum of the pseudogap
region is always in the weak-coupling regime.

The next step is to study in detail the effect of the valence band on the BKT
critical temperature. In Fig. 6(a) we show the ratio $T_{BKT}/T_{BKT_{1b}}$
between the BKT transition temperature of the two-band system and the BKT
transition temperature of a system made up by a conduction band only, as a
function of the band-gap energy. In the region of small $E_{g}$, the two
condensate positively interfere and the BKT critical temperature is
dramatically enhanced in the regime of low density, when the valence band is
present with respect to the system made up by a conduction band only. The
enhancement in the Kosterlitz-Thouless critical temperature can be observed
also when the pair-exchange coupling is tuned, as shown in Fig.6(b).
Surprisingly, we found that the amplification of $T_{BKT}$ with respect to the
single band case is important even when the pair-exchange couplings are very
small, for low levels of filling of the conduction band. This strong
cooperative effect between the valence and conduction bands condensates avoids
the suppression of the BKT transition temperature for low density of carriers,
that occurs in the single band and single gap superconductors. In Fig.6(c), we
show the ratio $T_{BKT}/T_{BKT_{1b}}$ between the BKT transition temperature
of our two-band system and the BKT transition temperature of the system made
up by a conduction band only, as a function of the intra-band couplings. We
have found again an amplification in the BKT critical temperature, which
becomes huge in the regime of low filling of the conduction band. Moreover, in
the weak-coupling regime the amplification presents a minimum, after which is
monotonically increasing in the intermediate- to strong-coupling regime.

##  4 Conclusions

In this work, we have studied the BKT transition in a 2D superconducting
electronic system with valence and conduction bands separated by a tunable
energy gap. The electrons form Cooper pairs in the s-wave channel by
interacting through an attractive potential with a large energy cutoff, which
has been considered to model electronic interactions or electronic bands with
relative small bandwidths. We have analyzed the behaviour of the phase
stiffness and of the BKT critical temperature as functions of the energy band
gap, intra-band, and pair-exchange couplings for different levels of filling
of the conduction band. We have found a giant enhancement of the BKT critical
temperature in the regime of small energy gap between the bands and of small
density of carriers in the conduction band.

Alongside with the BKT transition temperature amplification, the pseudogap
region between the mean-field temperature scale and the BKT transition
temperature is suppressed in the conduction band low density regime by tuning
the energy gap between the bands to small values, while is left unchanged for
higher density values. By looking at the pair-exchange couplings instead, the
pseudogap region has a non-monotonic behavior showing the presence of a
minimum in the intermediate coupling regime, that shifts to the weak-coupling
regime by reducing the filling of the valence band.

The pair-exchange and the intra-band couplings, favoured by small energy band
gaps, induce the transfer of electrons from the valence band to the conduction
band. The consequent presence of the holes in the valence band contributes
constructively to the stiffness of the total system, adding up to the
stiffness of the conduction band electrons that is boosted by the valence band
electrons as well. In the absence of holes, the filled valence band will not
create any stiffness and will be inactive for superconductivity. This strong
cooperative effect avoids the suppression of the BKT transition temperature
for low density of carriers, that occurs in single-band superconductors where
only the conduction band is present.

Despite the simplified nature of the pairing potential, our work gives a
qualitative insight on the BKT transition in valence and conduction bands 2D
superconducting systems and electron-hole superfluid systems, pointing toward
optimal parameter ranges for amplification of the BKT transition temperature.

##  5 Acknowledgments

This work has been supported by PNRR MUR project PE0000023-NQSTI.

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