[1]$#1$

# A knob to tune the Casimir-Lifshitz force with gapped metals

M. Boström<EMAIL_ADDRESS>Centre of Excellence ENSEMBLE3 Sp. z
o. o., Wolczynska Str. 133, 01-919, Warsaw, Poland  M. R. Khan
<EMAIL_ADDRESS>Centre of Excellence ENSEMBLE3 Sp. z o. o.,
Wolczynska Str. 133, 01-919, Warsaw, Poland  H. R. Gopidi  Centre of
Excellence ENSEMBLE3 Sp. z o. o., Wolczynska Str. 133, 01-919, Warsaw, Poland
I. Brevik  Department of Energy and Process Engineering, Norwegian University
of Science and Technology, NO-7491 Trondheim, Norway  Y. Li  Department of
Physics, Nanchang University, Nanchang 330031, China  Institute of Space
Science and Technology, Nanchang University, Nanchang 330031, China  C.
Persson  Centre for Materials Science and Nanotechnology, Department of
Physics, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway
Department of Materials Science and Engineering, KTH Royal Institute of
Technology, SE-100 44 Stockholm, Sweden  O. I. Malyi
<EMAIL_ADDRESS>Centre of Excellence ENSEMBLE3 Sp. z o. o.,
Wolczynska Str. 133, 01-919, Warsaw, Poland

(September 3, 2024)

###### Abstract

The Casimir-Lifshitz interaction, a long-range force that arises between
solids and molecules due to quantum fluctuations in electromagnetic fields,
has been widely studied in solid-state physics. The degree of polarization in
this interaction is influenced by the dielectric properties of the materials
involved, which in turn are determined by factors such as band-to-band
transitions, free carrier contributions, phonon contributions, and exciton
contributions. Gapped metals, a new class of materials with unique electronic
structures, offer the potential to manipulate dielectric properties and,
consequently, the Casimir-Lifshitz interaction. In this study, we
theoretically investigate the finite temperature Casimir-Lifshitz interaction
in La3Te4-based gapped metal systems with varying off-stoichiometry levels. We
demonstrate that off-stoichiometric effects in gapped metals can be used to
control the magnitude and, in some cases, even the sign of Casimir-Lifshitz
interactions. We predict measurable corrections due to stoichiometry on the
predicted Casimir force between a La3Te4 surface and a gold sphere, attached
to an atomic force microscopy tip.

##  I Introduction

Solid-state physics textbooks teach us about the Casimir-Lifshitz interaction
Milton (2001); Bordag  _et al._ (2009); Ninham and Lo Nostro (2010); Sernelius
(2018), as a long-range force arising between solids and molecules due to
quantum fluctuations in the electromagnetic fields Casimir (1948);
Dzyaloshinskii  _et al._ (1961); Parsegian and Ninham (1969). This interaction
causes the solids or molecules to become polarized when they are close to each
other, with the degree of polarization directly influenced by the dielectric
properties of the materials Richmond and Ninham (1971); Richmond  _et al._
(1973); Lamoreaux (1997); Boström and Sernelius (2000); Bordag  _et al._
(2000); Sernelius and Boström (2001); Høye  _et al._ (2003a); Milton (2004);
Lamoreaux (2004); Chang  _et al._ (2012); Klimchitskaya  _et al._ (2009);
Sushkov  _et al._ (2011); Boström  _et al._ (2018); Esteso  _et al._ (2019);
Klimchitskaya  _et al._ (2022); Nesterenko (2022). This relationship indicates
that by manipulating the dielectric properties through external means, it is
possible to make an impact on the Casimir-Lifshitz interaction directly. The
effects of such manipulations can be directly measured using atomic force
microscopy Ducker  _et al._ (1991) or even be employed in the development of
functional devices.

As electronic structure theory has advanced, it has become clear that a
material’s dielectric properties are determined by several factors: (i) band-
to-band transitions between occupied and unoccupied states, which play a
significant role across all frequency ranges; (ii) free carrier contributions,
typically found in metallic systems at low frequencies; (iii) the phonon
contribution; and (iv) exciton contributions, with the latter two primarily
limited to the low-frequency range. This understanding implies that by
identifying ways to influence factors (i) through (iv), we can effectively
tune the dielectric properties of materials and, consequently, the Casimir-
Lifshitz interaction. Therefore, the primary challenge lies in discovering
materials with tunable dielectric properties.

Table 1: Parametrization of the average dielectric function of continuous
media, $\varepsilon(i\xi)$, at imaginary frequencies for La3-xTe4

as calculated with first-principles calculations and a damping coefficient
($\Gamma$) set to 0.2 eV. The $\omega_{j}$ modes are given in $\rm eV$. The
largest difference between fitted and calculated $\varepsilon(i\xi)$ is
0.04$\%$.

modes ($\omega_{j}$) | $C_{j}$ for different La3-xTe4 compounds |
---|---|---
|  La3Te4 |  La2.92Te4 |  La2.83Te4 |  La2.75Te4 |  La2.67Te4
0.0203 |  5.6047 |  24.2875 |  0.0 |  0.0 |  0.0
0.0362 |  15.3667 |  55.9154 |  0.0 |  0.0 |  0.0
0.0694 |  27.1114 |  98.6417 |  1.2411 |  0.4796 |  0.0008
0.1325 |  35.8341 |  72.487 |  42.8899 |  19.6292 |  0.0
0.2085 |  93.1702 |  47.7692 |  42.1523 |  25.6095 |  0.0
0.4297 |  6.5149 |  7.3256 |  5.479 |  2.4194 |  0.0095
0.8328 |  2.9159 |  1.447 |  0.7619 |  0.4956 |  0.0421
1.9083 |  2.4749 |  2.9666 |  3.022 |  3.3971 |  3.1313
3.2388 |  4.6107 |  4.3332 |  4.464 |  4.5286 |  4.8426
5.2955 |  2.7231 |  2.8949 |  2.7872 |  2.6545 |  2.4639
8.9753 |  0.7087 |  0.5961 |  0.6287 |  0.6727 |  0.7464
18.2815 |  0.1959 |  0.2782 |  0.2486 |  0.205 |  0.1581
23.0355 |  0.0695 |  0.0069 |  0.022 |  0.0474 |  0.0716
42.0922 |  0.0 |  0.0057 |  0.0042 |  0.0018 |  0.0

Recently, gapped metals have emerged as a new class of materials possessing
unique electronic structures Malyi  _et al._ (2019); Zhang  _et al._ (2015);
Ricci  _et al._ (2020); Zhang  _et al._ (2016); Matsuishi  _et al._ (2003);
May  _et al._ (2008). These compounds set themselves apart from both metals
and insulators, as they possess a Fermi level within the conduction (or
valence) band, resulting in a high intrinsic concentration of free carriers
with an internal band gap between their primary band edges. Such materials
have attracted significant attention in connection with transparent conductors
Zhang  _et al._ (2016), thermoelectrics May  _et al._ (2008), and electrides
Matsuishi  _et al._ (2003). What makes these materials special is that they
can develop off-stoichiometry (within the same parental structure) due to the
decay of conducting electrons (holes) to acceptor (donor) states (Fig. 1a,b)
formed by intrinsic defect formation Malyi  _et al._ (2019); Khan  _et al._
(2023a); Malyi and Zunger (2020); Malyi  _et al._ (2020); Zunger and Malyi
(2021). This is different from the traditional metals and insulators, where
defect formation is usually limited to high temperatures and is primarily
driven by an increase in configurational entropy Zunger and Malyi (2021). This
situation is unique because different off-stoichiometry levels for gapped
metals can be achieved, each exhibiting distinct dielectric properties. For
example, La3Te4 \- an n-type gapped metal (i.e., one with its Fermi level in
the primary conduction band) - can be experimentally synthesized Ramsey  _et
al._ (1965); Delaire  _et al._ (2009); Li  _et al._ (2022) (simply by change
synthesis conditions) across a range of phases from La3Te4 to La2.66Te4 with
their properties tunable from metallic to insulating Khan  _et al._ (2023a) as
schematically shown in Fig. 1c. Motivated by the above, we study here
theoretically the finite temperature Casimir-Lifshitz interaction between
different systems involving La3Te4-based gapped metals, demonstrating how off-
stoichiometry can be used as a knob to tune such long-range interaction.

Figure 1:  (Color online) Origin of off-stoichiometry in La3Te4 compound.
Formation of acceptor vacancy in (a) insulator and (b) n-type gapped metal.
Here, EF and E${}_{g}^{int}$ correspond to the Fermi level and internal band
gap between the principal valence band maximum and conduction band minimum.
(c) Schematic crystal structure for La-Te system depicting metallic state with
free electrons in conduction bands and insulator state of La3-xTe4 phase due
to presence of acceptor vacancy.

##  II Methods

###  II.1 Computational details and dielectric functions

To compute the dielectric properties of gapped metals, we perform first-
principles calculations using the Perdew-Burke-Ernzerhof exchange-correlation
(PBE XC) functional Perdew  _et al._ (1996) within the VASP framework Kresse
and Hafner (1993); Kresse and Furthmüller (1996a, b); Kresse and Joubert
(1999). Our analysis focuses on five distinct La3-xTe4 compounds, previously
identified in our earlier work Khan  _et al._ (2023a). For each system, we
calculate the dielectric properties, considering only the Drude contribution
and interband transitions. We employ $\Gamma$-centered Monkhorst-Pack k-grids
Monkhorst and Pack (1976) with 20,000 points per reciprocal atom for the
calculations of direct band transitions and plasma frequencies and introduce a
subtle Lorentzian broadening of 0.01 eV in the Kramers-Kronig
transformationLandau and Lifshitz (2013). To include the Drude term in the
optical properties, we utilize the kram code Blaha  _et al._ (2020, 1990,
2001), setting the damping coefficient ($\Gamma$) to 0.2 eV, and additionally
investigate the influence of the $\Gamma$ parameter on our findings. The other
details on computational parameters can be found elsewhere Khan  _et al._
(2023a, b). Although we are unable to directly compare our calculated
dielectric functions with corresponding experimental measurements (no data are
available), it is important to highlight that the used methods can describe
sufficiently well the experimental trends related to transparency and coloring
across a range of gapped metals. Malyi  _et al._ (2019); Khan  _et al._
(2023c)

The quantity related to forces follows from the imaginary
($\varepsilon_{i}^{\prime\prime}$) part of the dielectric function:

$\varepsilon_{i}(i\xi_{m})=1+\frac{2}{\pi}\int_{0}^{\infty}d\omega\frac{\omega\varepsilon_{i}^{\prime\prime}(\omega)}{\omega^{2}+\xi_{m}^{2}},\,{i=1,2,3}$
(1)

where the Matsubara frequency is $\xi_{m}=2\pi kTm/\hbar$, and the subscript i
indicates the medium. As seen in Fig. 2 the curves show strong dependence for
the dielectric function on off-stoichiometry for La3-xTe4 (to be specific:
La3Te4 (with E${}_{g}^{int}$ = 1.22 eV), La2.92Te4 (with E${}_{g}^{int}$ =
1.16 $\pm$ 0.05 eV), La2.83Te4 (with E${}_{g}^{int}$ = 1.16 $\pm$ 0.05 eV),
La2.75Te4 (with E${}_{g}^{int}$ = 1.16 $\pm$ 0.05 eV), La2.67Te4 (with
E${}_{g}^{int}$ = 1.13 eV)) going from a metallic to insulator behavior. In
the Fig. 2 b we see for low-frequencies the dependence for the dielectric
function of La3Te4 on the $\Gamma$ parameter for La3Te4. In particular, we
present the ratio of the dielectric function for La3Te4 with different
$\Gamma$ to the corresponding values with La3Te4 with $\Gamma$ = 0.2 eV. This
highlights that it is only at low frequencies the $\Gamma$ parameter
influences the dielectric function. To use the calculated dielectric functions
for Casimir-Lifshitz interaction, we also develop the parametrization of the
average dielectric function (Table 1) with 14-mode oscillator model Malyi  _et
al._ (2016):

$\varepsilon(i\xi)=1+\sum_{j}\frac{C_{j}}{1+(\xi/\omega_{j})^{2}},$ (2)

where $\omega_{j}$ are characteristic frequencies and $c_{j}$ are proportional
to the oscillator strengths.

Figure 2:  (Color online) (a) Dielectric functions for imaginary frequencies
from top to bottom for La3Te4, La2.92Te4, La2.83Te4, La2.75Te4, La2.67Te4. (b)
We see in the low-frequency region ratio of the dielectric function for La3Te4
with different $\Gamma$ (damping coefficient also known as dissipation
parameter) to the corresponding values with La3Te4 with $\Gamma=0.2$ eV. The
results include a superposition of interband transitions and Drude (free
electron) contributions, both calculated using DFT.

To describe the Casimir interaction in systems amenable to experimental
measurement, we additionally compute the dielectric properties of gold. These
calculations are performed using PBEsol Perdew  _et al._ (2008) XC functional
combined with an effective Hubbard correction of $U=3.0$ eV on the
$d$-orbitals according to Dudarev et al. Dudarev  _et al._ (1998). Here, the
calculations are performed for 1 atom per primitive cell, the k-space
summation involved an $80\times 80\times 80$ k-mesh and a Gaussian smearing of
0.05 eV. The resulting permittivity for gold agrees with the experimental data
presented in Ref. Zorić  _et al._ (2011). The plasma frequency was calculated
to 9.6 eV and we used that with $\Gamma=0.05$ eV when calculating the Drude
contribution.

###  II.2 Theory of the Casimir Force

Let $F(L)$ be the free energy per unit surface area between planar surfaces,
and $f(L)$ be the force between a gapped metal (medium 1) and a sphere with
radius $R$ (medium 3). For $L<<R$ the force between a sphere and a planar
surface can be deduced from the free energy between two planar surfaces using
the so-called proximity force approximation Sernelius (2018). This
approximation is used to estimate the force between a sphere and a planar
surface based on the interaction-free energy between two half-planes.
Sernelius (2018) The sphere may be gold, a gapped transparent conductor, or an
air bubble. The intervening medium 2 may be a diluted gas with
$\varepsilon_{2}(i\xi_{m})\sim 1$, or water with dielectric functions given in
the literature Fiedler  _et al._ (2020); Luengo-Marquez  _et al._ (2022). The
free energy and force can be written as Bordag  _et al._ (2009); Sernelius
(2018)

$F(L)=\frac{f(L)}{2\pi
R}=\frac{k_{B}T}{2\pi}{\sum_{m=0}^{\infty}}{}^{\prime}\int\limits_{0}^{\infty}dq\,q\sum_{\sigma}\ln(1-r_{\sigma}^{21}r_{\sigma}^{23}\mathrm{e}^{-2\kappa_{2}L}),$
(3)

where $\sigma=\rm TE,TM$, and the prime in the sum above indicates that the
first term ($m$ = 0) has to be weighted by $1/2$. The Fresnel reflection
coefficients between surfaces $i$ and $j$ for the transverse magnetic (TM) and
transverse electric (TE) polarizations are given by

$r_{\rm
TE}^{ij}=\frac{\kappa_{i}-\kappa_{j}}{\kappa_{i}+\kappa_{j}}\,;\,\,\,\,\,r_{\rm
TM}^{ij}=\frac{\varepsilon_{j}\kappa_{i}-\varepsilon_{i}\kappa_{j}}{\varepsilon_{j}\kappa_{i}+\varepsilon_{i}\kappa_{j}}\,.$
(4)

Here $\kappa_{i}=\sqrt{{q}^{2}+\varepsilon_{i}\xi_{m}^{2}/c^{2}}$, with
$i=1,2,3$ and the Matsubara frequency is defined above. Notably, as the
dielectric constant is finite for gapped metals unless we set $\Gamma=0$, the
transverse electric reflection coefficient between non-magnetic gapped metals
and dilute gas goes to zero ($r_{\rm TE}^{21}=0\quad as\quad\xi\to 0$). As a
comparison, we also consider what happens when we assume perfect metallic
behavior for the transverse electric reflection coefficients in the zero
frequency limit Boström and Sernelius (2000); Sernelius and Boström (2001);
Klimchitskaya  _et al._ (2022) ($r_{\rm TE}^{21}r_{\rm TE}^{23}=1\quad
as\quad\xi\to 0$). The alternative limiting behavior leads to the “perfect
metal correction” (PMC) to the interaction between real metal surfaces (when
$L<<R$) Boström and Sernelius (2000); Sernelius and Boström (2001);
Klimchitskaya  _et al._ (2022),

$\Delta F^{\rm PMC}(L)=\frac{\Delta f^{PMC}(L)}{2\pi
R}=\frac{-k_{B}T\zeta(3)}{16\pi L^{2}},$ (5)

where $\zeta(3)$ is a Riemann zeta function. This could be the case if the
dielectric function for La3Te4 is not finite as $\xi\rightarrow 0$, and goes
to infinite at least as $1/\xi^{2}$. The energy correction factor,
$\eta(L,T)$, is defined Sernelius (2018) as the ratio of the calculated
Casimir force and the corresponding Casimir force between a perfectly
conducting sphere with radius $R$ and a perfectly conducting plane (when
$L<<R$) at zero temperature Casimir (1948),

$F_{C}(L)=\frac{f_{C}(L)}{2\pi R}\cong\frac{-\pi^{2}\hbar c}{720L^{3}},$ (6)

$\eta(L,T)=\frac{f(L)}{f_{C}(L)}=\frac{F(L)}{F_{C}(L)}.$ (7)

We will match our results against an ideal case (relevant to any metal
surfaces behaving as ideal plasmas Bordag  _et al._ (2009)). For two metallic
surfaces interacting across air, it is thus relevant for comparison to include
the “perfect metal correction” by adding Eq. (5) and Eq. (3) and divide the
sum with Eq. (6). The corrected ratio is,

$\eta^{*}=\frac{f(L)+\Delta f^{PMC}(L)}{f_{C}(L)}=\frac{F(L)+\Delta
F^{PMC}(L)}{F_{C}(L)}.$ (8)

###  II.3  A note on the effect of dissipation

The dissipation is included in our paper via a damping term $\Gamma$ in the
Drude dispersion relation, thus not from microscopic models. To see this from
a broader perspective, it may be worthwhile to start from the general case
where the temperature dependence of $\Gamma$ is also taken into account. We
have then

$\varepsilon(i\xi,T)=1+\frac{\omega_{pl}^{2}}{\xi[\xi+\Gamma(T)]},$ (9)

where $\xi$ is the imaginary frequency. If one at first ignores impurities,
one may here make use of the Bloch-Grüneisen formula Gray (1972) for the
temperature variation of the electrical resistivity $\rho$, the latter being
proportional to $\Gamma$. This formula implies, among other things, that
$\varepsilon(i\xi,T)$ is actually higher when $T$ is low than at room
temperature if the frequencies are lower than about
$10^{14}\leavevmode\nobreak\ $rad/s, i.e., of the same order as the first
Matsubara frequency at room temperature (cf. the discussion on these points by
Høye et al. Høye  _et al._ (2003b); Brevik  _et al._ (2005)). Now, from
numerical estimates, it turns out that the influence of the temperature on
$\Gamma$ is modest. A more important factor in this context is the presence of
impurities in the metal. The existence of these makes the resistivity $\rho$,
and thus also $\Gamma$, constant at low temperatures and low frequencies.
Consequently, we can put
$\varepsilon(i\xi,T)\rightarrow\varepsilon(i\xi)\propto 1/\xi$ when
$\xi\rightarrow 0$. Now one must recognize that for practical purposes it is
not $\varepsilon$ itself that is the central quantity, but rather the
combination

$\xi^{2}[\varepsilon(i\xi)-1],$ (10)

which has to go to zero as $\xi\rightarrow 0$. This relationship is satisfied
by the simple Drude ansatz for the dispersion equation, and the Casimir theory
for metals becomes consistent. The expression (10) implies that the
contribution to the Casimir force from the TE zero mode vanishes, and also
that the Nernst theorem becomes satisfied (i.e., that the free energy depicted
as a function of $T$ has a zero slope as $T\rightarrow 0$). Cf. again the
mentioned references by Høye et al. Høye  _et al._ (2003b); Brevik  _et al._
(2005).

##  III Results

###  III.1 A pair of identical La3-xTe4 surfaces

To demonstrate the effect of off-stoichiometry on the Casimir-Lifshitz force,
we first consider the interaction between gapped metal planar surfaces
interacting across the air as a function of interplane distance (Fig. 3a). At
short separations, that is, when the finite velocity of light can be
approximated as infinite, the product of reflection coefficients goes as

$\frac{(\varepsilon_{1}-\varepsilon_{2})^{2}}{(\varepsilon_{1}+\varepsilon_{2})^{2}}.$
(11)

From this part of the full expression, two things can be deduced: (1) the
force between identical surfaces is attractive, and (2) the closer the ratio
in Eq. (11) is to 1 (i.e. the more metallic), the stronger the attraction.

The Casimir-Lifshitz free energy between gapped metal surfaces at finite
temperatures is predicted to deviate strongly from the T= 0 K ideal metal
Casimir interaction. (Note that the long-range thermal asymptote decays slower
than the zero-temperature Casimir asymptote, leading to an increase of the
ratio with increasing separations). As can be seen in Fig. 3b-d, off-
stoichiometry effects for gapped metals can be used as an effective knob to
induce 10-40 % changes in the magnitude of the interaction. Notably, in the
separation range where forces are typically measured, the effects are of the
same magnitude as any potential corrections from “the ideal metal” (plasma
model) plate zero frequency transverse electric contribution. The force in
Fig. 3b increases in magnitude as the surfaces become more metallic (up to a
limit where the product of the reflection coefficient in a specific frequency
range is close to one). The results shown in Fig. 3d is the corresponding case
when the ideal metal or "plasma model" approximation Bordag  _et al._ (2009)
is used for the zero frequency term.

Another important observation is that interaction between different metallic
${\rm La}_{3-x}{\rm Te}_{4}$ surfaces is also different (i.e., dependent on
$x$). This behavior can be understood on the electronic structure theory level
as increasing the off-stoichiometry results in a change of free carrier
concentration (each La vacancy removes three electrons from the principal
conduction band), reducing the free carrier contribution to
$\varepsilon(i\xi)$. Hence, even though multiple La3-xTe4 surfaces are
metallic, they substantially differ in dielectric functions simply due to
differences in free carrier concentration.

As noted above, the dielectric properties of materials are defined by the
superposition of different contributions. While one can explicitly calculate
the band-to-band transition, the calculation of free carrier contribution to
dielectric function relies on the Drude model. We described in Fig. 2 the
effect on the low-frequency tail of the dielectric function from changes in
the choice of damping coefficient in the Drude model ($\Gamma$). The static
dielectric constant rises to higher values with a reduced $\Gamma$. However,
in Fig. 3c, we show how slight the energy correction factor is on the choice
of damping coefficient for the specific case of a pair of La3Te4 surfaces.
Since the gapped metal surfaces have large but finite dielectric constants,
one would expect the “perfect metal correction” to be purely hypothetical.
However, future optical and force measurements will enable us to distinguish
better how to accurately model the dielectric functions at extremely low
frequencies.

###  III.2 Gold sphere in air near La3-xTe4 surfaces

As is well known, all ordinary metals have a finite static conductivity. The
Drude model describes the optical and dielectric properties quite well for
small frequencies. The dielectric function in the Drude model is

$\varepsilon\left(\omega\right)=1+\frac{{i\sigma\left(\omega\right)}}{\omega}=1-\frac{{\omega_{pl}^{2}}}{{\omega\left({\omega+i\Gamma}\right)}}.$
(12)

Setting the damping parameter $\Gamma$ zero, the plasma model is obtained. The
damping parameter has a real physical origin, and is the result of scattering
of the carriers against lattice imperfections. At finite temperature,
processes with phonons emitted or absorbed further contribute. Boström and
Sernelius Boström and Sernelius (2000); Sernelius and Boström (2001) found
that the damping parameter has a dramatic effect on the Casimir interaction
between gold surfaces at separations where the finite temperature is
important. The plasma model actually predicts a result that coincides with
that of the classic Casimir gedanken experiment between two perfectly
reflecting half-spaces, while the Drude model predicts that this result is
reduced by a factor of two. In the limit of large $L\rightarrow\infty$ (while
still demanding $\frac{L}{R}<<1$), the ratio for two interacting metallic
surfaces would, within the PMC, become asymptotically,

$\eta^{*}\sim\frac{90Lk_{B}T\zeta(3)}{\pi^{3}\hbar c},$ (13)

while $\eta$ in the same limit has half this magnitude. For all cases
considered, the ratios go (for large $L$) $\propto Lk_{B}T/(\hbar c)$. This
means, as is well known, that the ratio increases linearly with separation for
large $L$. Some experiments favor the Drude model Lamoreaux (1997, 2004);
Sushkov  _et al._ (2011) while most appear to favor the plasma model Chang
_et al._ (2012); Klimchitskaya  _et al._ (2009); Bordag  _et al._ (2009);
Klimchitskaya  _et al._ (2022). The materials we consider are evaluated by
density functional theory to have large, but finite, values for the zero
frequency permittivity (but should go to infinity if $\Gamma\rightarrow 0$ ).
They should hence have a free energy behavior not observed experimentally by,
for example, Mohideen and collaborators for metal surfaces Chang  _et al._
(2012); Klimchitskaya  _et al._ (2009); Bordag  _et al._ (2009); Klimchitskaya
_et al._ (2022). Rather it is expected to be closer to the experimental
observations by Lamoreaux and his collaborators Lamoreaux (1997, 2004);
Sushkov  _et al._ (2011) (and even more to the case of doped silicon
surfaces). If future experiments would prove this to be wrong, further
research must be prompted into how to model the low-frequency part of the
dielectric functions for gapped metals within density functional theory.

Typically, experimental measurements of Casimir forces are conducted using
atomic force microscopy (AFM), where a specialized sphere (e.g., Au) is
attached to the AFM tip. Because of this, herein, we also consider another
proposed experimental setup shown in Fig. 4, where we consider Au sphere
attached to an atomic force microscope tip, interacting with a planar surface
of different La3-xTe4 surfaces at different separation. We present our result
in terms of the ratio between the calculated force and the corresponding
hypothetical force between a perfectly conducting plane and a perfectly
conducting sphere (in the limit of zero temperature). Very similar systems,
but with different material combinations, have been carefully studied by some
of the world’s best groups in force measurements. Notably, as we clearly
demonstrate, both off-stoichiometry and “perfect metal correction” effects are
large enough to be within the few percentage measurement accuracy claimed in
the 0.1 $\mu\,m$ range in several experimental labs (for instance by Mohideen
and co-workers) Lamoreaux (1997); Harris  _et al._ (2000); Decca  _et al._
(2003); Le Cunuder  _et al._ (2018); Sushkov  _et al._ (2011); Klimchitskaya
_et al._ (2022); Sushkov  _et al._ (2011); Chang  _et al._ (2012); Xu  _et
al._ (2022).

###  III.3 Air bubble in liquid water near La3-xTe4 surfaces

One interesting prediction from theory and force measurements is that these
forces can be repulsive Dzyaloshinskii  _et al._ (1961); Richmond and Ninham
(1971); Richmond  _et al._ (1973); Milling  _et al._ (1996); Lee and Sigmund
(2002); Feiler  _et al._ (2008); Munday  _et al._ (2009); van Zwol and
Palasantzas (2010); Tabor  _et al._ (2011), and even change sign Elbaum and
Schick (1991); Dou  _et al._ (2014). When reflecting on the expression for the
Casimir-Lifshitz interaction, it is clear that repulsion occurs whenever
$\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}$ for a broad range of finite
Matsubara frequencies. This was well known to Dzyaloshinskii, Lifshitz and
Pitaevskii Dzyaloshinskii  _et al._ (1961). A remarkable point discussed by
Elbaum and Schick Elbaum and Schick (1991) is that the dispersion forces
(including Sernelius (2018), e.g., van der Waals, Casimir-Polder, Lifshitz and
Casimir interactions) can change sign when for a range of “high” frequencies
(small separations) $\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}$, while
for low frequencies (large separations)
$\varepsilon_{2}>\varepsilon_{1}>\varepsilon_{3}$. The origin of this effect
is the relevant reflection coefficients combined with the exponential factor
$e^{-2\sqrt{{q}^{2}+\varepsilon_{2}\xi_{m}^{2}/c^{2}}L}$. At very large
separations (the factor $\sqrt{\varepsilon_{2}}L\xi_{m}/(qc)$ should be small
or of the order unity to result in a significant contribution to the
interaction), the finite velocity of light means only the zero frequency term
contributes. If $\varepsilon_{2}(0)>\varepsilon_{1}(0)>\varepsilon_{3}(0)$,
then long-range attraction follows.

We demonstrate in Fig. 5 that the Casimir-Lifshitz force between an air bubble
in water near a gapped metal surface can, via control of the off-stoichiometry
and separation, at large separations change from repulsion to attraction. We
propose that further studies on how to tune more effectively the transition
from repulsion to attraction-based trapping of gas bubbles in liquids must
include other effects including hydration, dissolved gases, surface charges,
and ion-specific double layer forces within the DLVO theory and beyond Ninham
and Lo Nostro (2010); Kunz (2009).

Figure 3:  (Color online) (a) Schematic scheme over the set-up. (b) Energy
correction factor ($\eta(L,T)$) between equal gapped metal surfaces across air
at T=300 K. (c) $\eta(L,T)$ for two equal La3-xTe4 varying the $\Gamma$. (d)
Same as in (b) but adding the “perfect metal correction” to the free energy
for the metallic systems. The Drude damping parameter is here introduced on a
phenomenological level. The influence from dissipation does not significantly
change the strength of the interaction. In Fig. 3d, the reported change is
however different, since it arises from the "perfect metal correction" (the
meaning of the zero frequency limit given is explained in Fig. 4).  Figure 4:
(Color online) Schematic set up and the $\eta(L,T)=f(L)/f_{C}$ at T=300 K for
a gold sphere interacting with (i) La3Te4, (ii) La3Te4 with PMC, i.e. $\xi=0$
transverse electric "Perfect Metal Correction", or (iii) La2.67Te4 surface.
The limiting value ($\varepsilon\rightarrow 1/\omega^{2}$) refers to the
slowest possible decay in the low-frequency limit (i.e. not the actual zero
frequency limit) in which the zero frequency TE mode would be expected to
contribute. Figure 5:  (Color online) Casimir force at T=273.16 K between an
air bubble in cold water Luengo-Marquez  _et al._ (2022) near (i) La3Te4 with
$\Gamma=0.2$ eV, (ii) La3Te4 with $\Gamma=0.01$ eV, and (iii) La2.67Te4. The
sizes of the bubbles vary depending on the system involved from nanobubbles,
micron-sized bubbles to milli bubbles.

##  IV Conclusions

As dimensions and distances relevant for nanomachines go down in size, it
becomes increasingly important to control the sign and magnitude of short-
range van der Waals/Casimir-Lifshitz interactions Chan  _et al._ (2001);
DelRio  _et al._ (2005); Munday  _et al._ (2009) and torques Somers  _et al._
(2018); Thiyam  _et al._ (2018). We observe that in the literature one finds
that similar analyses have been performed for conductive oxides, phase-
changing materials, chiral materials, magnetic material, Weyl semimetals,
graphene, topological insulators, and many other systems. We refer the readers
to the review by Woods and collaborators Woods  _et al._ (2016). Herein, we
investigate the effects of off-stoichiometry on the Casimir-Lifshitz force
between La3-xTe4 surfaces in various experimental setups. The results show
that off-stoichiometry affects significantly impact the interaction between
different La3-xTe4 surfaces. In particular, we propose that these forces can
be influenced to more than 10-40% by manipulating the off-stoichiometry of
gapped metal and tuning it from metallic to insulating behavior (this
phenomenon is due to the fact that La3Te4 gapped metal possesses a distinctive
electronic structure resulting in the reduction of free carrier concentration
by the formation of La vacancies). The study of this new material category
indicates a roadmap for how to enhance or reduce Casimir-Lifshitz
interactions. As we show, for gas bubbles in liquid water near a gapped metal,
tuning off-stoichiometry effects can even change the sign of the long-range
part of the Casimir-Lifshitz interactions. Our main conclusion is that off-
stoichiometry in gapped metals can be used as a knob to tune long-range
interactions. In a longer perspective, the use of controlled quantum switches
for fluid systems, together with an exploitation of off-stoichiometry effects,
may be a promising way to investigate the Drude/plasma controversy.

###### Acknowledgements.

The authors thank the "ENSEMBLE3 - Centre of Excellence for nanophotonics,
advanced materials and novel crystal growth-based technologies" project (GA
No. MAB/2020/14) carried out within the International Research Agendas
programme of the Foundation for Polish Science co-financed by the European
Union under the European Regional Development Fund, the European Union’s
Horizon 2020 research and innovation programme Teaming for Excellence (GA. No.
857543), and European Union’s Horizon 2020 research and innovation programme
(grant No. 869815), for support of this work. We gratefully acknowledge
Poland’s high-performance computing infrastructure PLGrid (HPC Centers: ACK
Cyfronet AGH) for providing computer facilities and support within
computational grant no. PLG/2022/015458. We also acknowledge access to high-
performance computing resources via NAISS, provided by NSC and PDC, as well as
NOTUR, provided by Sigma2.

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