#  Light-induced large and tunable valley-selective Hall effect in a
centrosymmetric system

Naoya Arakawa<EMAIL_ADDRESS>The Institute of Science and
Engineering, Chuo University, Bunkyo, Tokyo, 112-8551, Japan  Kenji Yonemitsu
The Institute of Science and Engineering, Chuo University, Bunkyo, Tokyo,
112-8551, Japan  Department of Physics, Chuo University, Bunkyo, Tokyo
112-8551, Japan

###### Abstract

We propose that a large and tunable valley-selective Hall effect can be
realized in a centrosymmetric system via light-induced breaking of inversion
and time-reversal symmetries. This is demonstrated in graphene driven by
bicircularly polarized light, which consists of a linear combination of left-
and right-handed circularly polarized light with different frequencies. We
also show that our Hall conductivity is two orders of magnitude larger than
the maximum value obtained in noncentrosymmetric systems, and that the main
valley can be switched by tuning a phase difference between the left- and
right-handed circularly polarized light. Our results will enable us to
generate and control the valley-selective Hall effect in centrosymmetric
systems.

Introduction.Electron systems can get a valley degree of freedom with broken
inversion symmetry. In epitaxial or bilayer graphene Epi-graphene ; Bilayer-
graphene  and some transition-metal dichalcogenides TMD-valley , electrons at
the K and K′ points in the momentum space have the valley degree of freedom.
These systems are noncentrosymmetric because their lattices break inversion
symmetry; this symmetry holds in centrosymmetric systems. The valley degree of
freedom can be used to realize valleytronics phenomena. An example is the
valley-selective Hall effect, in which the current from one of two valleys
causes the charge current perpendicular to an electric field. This was
experimentally observed in MoS2 VHE-exp1  or bilayer graphene VHE-exp2  with
resonant circularly polarized light (CPL), which excites an electron at one
valley Valley-CPL-Graphene ; Valley-CPL-TMD1 ; Valley-CPL-TMD2 ; Valley-CPL-
TMD3 . Another example is a valley-contrasting Hall effect VHE-theory , in
which the electrons around different valleys generate opposite currents
perpendicular to the electric field. This is distinct from the valley-
selective Hall effect because in the latter the contribution from one valley
is negligible. Since the valley degree of freedom can be utilized in similar
ways to the spin degree of freedom in spintronics, valleytronics has opened
new phenomena and applications utilizing it.

Bicircularly polarized light (BCPL) will provide a new way for generating and
controlling the valley degree of freedom. BCPL is generated by a linear
combination of left- and right-handed CPL Bicirc-NatPhoto  and described by
$\mathbf{A}_{\textrm{BCPL}}(t)=(A_{x}(t)\ A_{y}(t))^{T}$ [e.g., see Figs. 1(a)
and 1(b)], where

$\displaystyle A_{x}(t)+iA_{y}(t)=A_{0}e^{i\Omega t}+A_{0}e^{-i(\beta\Omega
t-\theta)}.$ (1)

If $A_{0}$ is strong enough to be treated nonperturbatively, BCPL can break
time-reversal symmetry and inversion symmetry Bicirc-Oka ; Bicirc-PRL .
Therefore, BCPL could generate the valley degree of freedom. To treat such
nonperturbative effects, we need to consider a system driven by BCPL using the
Floquet theory Floquet1 ; Floquet2 . Since the Hamiltonian in this theory can
be changed by varying parameters of $\mathbf{A}_{\textrm{BCPL}}(t)$, BCPL
could also control the valley degree of freedom. However, a possibility of a
BCPL-induced valley-selective or valley-contrasting Hall effect remains
unexplored.

Figure 1:  (a), (b) Trajectories of $\mathbf{A}_{\textrm{BCPL}}(t)$ per period
$T_{\textrm{p}}=2\pi/\Omega$ for $\beta=2$ and $3$. The red, yellow, and blue
lines correspond to those at $\theta=0$, $\frac{\pi}{2}$, and $\pi$,
respectively. (c) The set-up for the valley-selective Hall effect in graphene
driven by BCPL. The system, graphene, is driven by the pump field, the field
of BCPL, and is weakly coupled to the heat bath. The charge current
perpendicular to the probe field is generated.  Figure 2:  (a)–(c) The
momentum dependences of $\epsilon_{+}^{(\textrm{eff})}(\mathbf{k})$ and
$\epsilon_{-}^{(\textrm{eff})}(\mathbf{k})$ obtained in the high-frequency
expansion for $\beta=2$ at $\Omega=10t_{\textrm{NN}}$, $u=0.4$ and $0.8$, and
$\theta=0$, $\frac{\pi}{2}$, and $\pi$. The momenta at the symmetric points
K′, $\Gamma$, K, and M are $(\frac{8\pi}{3\sqrt{3}}\ 0)^{T}$, $(0,0)^{T}$,
$(\frac{4\pi}{3\sqrt{3}}\ 0)^{T}$, and $(\frac{\pi}{\sqrt{3}}\
\frac{\pi}{3})^{T}$, respectively. Here $(\frac{8\pi}{3\sqrt{3}}\ 0)^{T}$ and
$(0,0)^{T}$ are equivalent to $(-\frac{4\pi}{3\sqrt{3}}\ 0)^{T}$ and
$(\frac{4\pi}{\sqrt{3}}\ 0)^{T}$, respectively. (d)–(f) The momentum
dependences of $B_{\alpha}^{z}(\mathbf{k})$ obtained in the high-frequency
expansion for $\beta=2$ at $\Omega=10t_{\textrm{NN}}$ and $\theta=0$,
$\frac{\pi}{2}$, and $\pi$ with $(\alpha,u)=(+,0.4)$, $(-,0.4)$, $(+,0.8)$,
and $(-,0.8)$. The insets show the results for $u=0.8$ on a smaller scale.

Here we demonstrate the large and tunable valley-selective Hall effect in
monolayer graphene driven by BCPL. Using a high-frequency expansion Highw1 ;
Highw2  of the Floquet theory, we show that inversion symmetry can be broken
only for even $\beta$ in Eq. (1), whereas time-reversal symmetry can be broken
for any $\beta$. We also show the effects of BCPL on the energy bands and the
Berry curvatures. Then, using the Floquet linear-response theory Eckstein ;
Oka-PRB ; Tsuji ; Mikami ; NA-FloquetSHE ; NA-linear , we show that the
valley-selective Hall effect can be realized for $\beta=2$ at $\theta=0$ or
$\pi$. Our Hall conductivity is much larger than the maximum value known so
far, and the main valley can be switched by tuning $\theta$. Our results
indicate that BCPL can be used to generate and control the valley-selective
Hall effect in centrosymmetric systems. This work will open the door to the
valley-selective Hall effect in centrosymmetric systems.

Model.Graphene driven by BCPL [Fig. 1(c)] is described by the Hamiltonian,

$\displaystyle H=H_{\textrm{s}}(t)+H_{\textrm{b}}+H_{\textrm{sb}},$ (2)

where $H_{\textrm{s}}(t)$ is the system Hamiltonian for graphene with the BCPL
field $\mathbf{A}_{\textrm{BCPL}}(t)$, $H_{\textrm{b}}$ is the bath
Hamiltonian, and $H_{\textrm{sb}}$ is the system-bath Hamiltonian. We have
treated the nonperturbative effects of the BCPL field as the Peierls phase
factors of the kinetic energy Oka-PRB ; Mikami ; NA-FloquetSHE ; NA-linear .
In addition to $H_{\textrm{s}}(t)$, we have considered $H_{\textrm{b}}$ and
$H_{\textrm{sb}}$ Tsuji ; Mikami ; NA-FloquetSHE ; NA-linear , where the bath
is Büttiker-type HeatBath1 ; HeatBath2  and in equilibrium at temperature $T$.
The main effect of these Hamiltonians is to induce the damping $\Gamma$ Tsuji
; Mikami ; NA-FloquetSHE , which could be used to realize a nonequilibrium
steady state under heating Heating1 ; Heating2  induced by BCPL. Throughout
this paper, we set $\hbar=k_{\textrm{B}}=c=a_{\textrm{NN}}=1$, where
$a_{\textrm{NN}}$ is the length between nearest neighbor sites.

Inversion or time-reversal symmetry breaking.–We begin with symmetry breaking
induced by BCPL. Using the high-frequency expansion Highw1 ; Highw2  of the
Floquet theory, we obtain an effective Hamiltonian for describing the
nonperturbative effects of off-resonant BCPL,

$\displaystyle
H_{\textrm{eff}}=\sum_{\mathbf{k}}\sum_{a,b=A,B}\sum_{\sigma=\uparrow,\downarrow}\bar{\epsilon}_{ab}(\mathbf{k})c_{\mathbf{k}a\sigma}^{\dagger}c_{\mathbf{k}b\sigma},$
(3)

where

$\displaystyle\bar{\epsilon}_{AA}(\mathbf{k})=-\bar{\epsilon}_{BB}(\mathbf{k})=\Delta_{\textrm{BCPL}}+K_{AA}^{(\textrm{eff})}(\mathbf{k}),$
(4)
$\displaystyle\bar{\epsilon}_{AB}(\mathbf{k})=\bar{\epsilon}_{BA}(\mathbf{k})^{\ast}=\epsilon_{AB}^{(\textrm{eff})}(\mathbf{k}).$
(5)

(For the derivation, see the Supplemental Material SM .) Here
$\Delta_{\textrm{BCPL}}$ is the staggered sublattice potential Graphene-ISB-
potential ; Kane-Mele ; Bicirc-Oka ,
$\epsilon_{AB}^{(\textrm{eff})}(\mathbf{k})=\sum_{\mathbf{R}=\mathbf{R}_{0},\mathbf{R}_{1},\mathbf{R}_{2}}t_{\textrm{eff}}^{(AB)}(\mathbf{R})e^{-i\mathbf{k}\cdot\mathbf{R}}$,
and
$K_{AA}^{(\textrm{eff})}(\mathbf{k})=\sum_{\mathbf{R}=\pm\mathbf{R}^{\prime}_{0},\pm\mathbf{R}^{\prime}_{1},\pm\mathbf{R}^{\prime}_{2}}K_{\textrm{eff}}^{(AA)}(\mathbf{R})e^{-i\mathbf{k}\cdot\mathbf{R}}$,
where $t_{\textrm{eff}}^{(AB)}(\mathbf{R})$ and
$K_{\textrm{eff}}^{(AA)}(\mathbf{R})$ are the nearest-neighbor and next-
nearest-neighbor hopping integrals, respectively. Note that
$\mathbf{R}_{0}=(0\ 1)^{T}$, $\mathbf{R}_{1}=(-\frac{\sqrt{3}}{2}\
-\frac{1}{2})^{T}$, $\mathbf{R}_{2}=(\frac{\sqrt{3}}{2}\ -\frac{1}{2})^{T}$,
$\mathbf{R}^{\prime}_{0}=(\sqrt{3}\ 0)^{T}$,
$\mathbf{R}^{\prime}_{1}=(-\frac{\sqrt{3}}{2}\ \frac{3}{2})^{T}$, and
$\mathbf{R}^{\prime}_{2}=(-\frac{\sqrt{3}}{2}\ -\frac{3}{2})^{T}$ (see Fig. 1
of the Supplemental Material SM ). Equations (3)–(5) show that BCPL not only
modifies the nearest-neighbor hopping integrals, but also induces the
staggered sublattice potential and the next-nearest-neighbor hopping
integrals. The latter quantities depend on $\beta$ [see Eqs. (34) and (35) in
the Supplemental Material SM ]: for even $\beta$, $\Delta_{\textrm{BCPL}}$ is
finite and $K_{\textrm{eff}}^{(AA)}(\mathbf{R})$ has real and imaginary parts;
for odd $\beta$, $\Delta_{\textrm{BCPL}}$ is zero and
$K_{\textrm{eff}}^{(AA)}(\mathbf{R})$ is pure-imaginary. As specific cases, we
show them for $\beta=2$ and $3$ in $u=eA_{0}\ll 1$ (for their derivation, see
the Supplemental Material SM ):

$\displaystyle\Delta_{\textrm{BCPL}}=\begin{cases}-6K_{1}(u,\theta)\
(\beta=2),\\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ (\beta=3),\end{cases}$ (6)
$\displaystyle
K_{\textrm{eff}}^{(AA)}(\pm\mathbf{R}_{l}^{\prime})=\begin{cases}\mp
iK_{0}(u)+K_{1}(u,\theta)\ \ \ \ \ (\beta=2),\\\ \mp
i[K_{0}^{\prime}(u)+K_{l+1}^{\prime}(u,\theta)]\ (\beta=3),\end{cases}$ (7)

where $K_{0}(u)=\frac{\sqrt{3}t_{\textrm{NN}}^{2}}{4\Omega}u^{2}$,
$K_{1}(u,\theta)=\frac{t_{\textrm{NN}}^{2}}{4\Omega}u^{3}\cos\theta$,
$K_{0}^{\prime}(u)=\frac{\sqrt{3}t_{\textrm{NN}}^{2}}{4\Omega}(u^{2}-\frac{3}{4}u^{4})$,
$K_{1}^{\prime}(u,\theta)=\frac{\sqrt{3}t_{\textrm{NN}}^{2}}{16\Omega}u^{4}\cos\theta$,
$K_{2}^{\prime}(u,\theta)=\frac{t_{\textrm{NN}}^{2}}{16\Omega}u^{4}[\sin\theta-\sin(\theta-\frac{4\pi}{3})]$,
and
$K_{3}^{\prime}(u,\theta)=-\frac{t_{\textrm{NN}}^{2}}{16\Omega}u^{4}[\sin\theta-\sin(\theta-\frac{2\pi}{3})]$.
Therefore, BCPL can break inversion symmetry only for even $\beta$ because it
is broken by $\Delta_{\textrm{BCPL}}$ and the real parts of the next-nearest-
neighbor hopping integrals. Furthermore, since time-reversal symmetry is
broken by the imaginary parts of the next-nearest-neighbor hopping integrals
Haldane ; Kitagawa ; Mikami , BCPL can break it for any $\beta$. Note that the
$l$-independent $K_{\textrm{eff}}^{(AA)}(\pm\mathbf{R}_{l}^{\prime})$’s for
$\beta=2$ preserve $C_{3}$ rotational symmetry, whereas the $l$-dependent ones
for $\beta=3$ break it.

Figure 3:  (a), (b) The $u$ dependences of $\sigma_{yx}^{\textrm{C}}$ obtained
in the Floquet linear-response theory for $\beta=2$ or $3$ at
$\Omega=8t_{\textrm{NN}}$ and $\theta=0$, $\frac{\pi}{2}$, and $\pi$. Here
$u=eA_{0}$ is dimensionless. The dotted lines correspond to a quantized value
$-2e^{2}/h$. (c)–(h) The momentum dependences of
$\sigma_{yx}^{\textrm{C}}(\mathbf{k})$ obtained in the Floquet linear-response
theory for $\beta=2$ at $\Omega=8t_{\textrm{NN}}$ with $(\theta,u)=(0,0.2)$,
$(0,0.4)$, $(0,0.6)$, $(\pi,0.2)$, $(\pi,0.4)$, and $(\pi,0.6)$. The symmetric
points are the same as those used in Fig. 2.

Valley degeneracy lifting.The above differences between the effects of BCPL
for even and odd $\beta$ lead to a difference in the valley degeneracy. The
energy dispersion of Eq. (3) is given by

$\displaystyle\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k})=\pm\sqrt{[\Delta_{\textrm{BCPL}}+K_{AA}^{(\textrm{eff})}(\mathbf{k})]^{2}+|\epsilon_{AB}^{(\textrm{eff})}(\mathbf{k})|^{2}}.$
(8)

We estimate $\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k})$’s at the K and K′
points for $\beta=2$ and $3$ in $u\ll 1$; the results for $\beta=2$ are
$\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}})=\pm|3\sqrt{3}K_{0}(u)-9K_{1}(u,\theta)|$
and
$\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}^{\prime}})=\pm|3\sqrt{3}K_{0}(u)+9K_{1}(u,\theta)|$,
where $\mathbf{k}_{\textrm{K}}=(\frac{4\pi}{3\sqrt{3}}\ 0)^{T}$ and
$\mathbf{k}_{\textrm{K}^{\prime}}=(\frac{8\pi}{3\sqrt{3}}\ 0)^{T}$; those for
$\beta=3$ are
$\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}})=\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}^{\prime}})$.
Therefore, the valley degeneracy can be lifted for even $\beta$ due to a
combination of breaking both time-reversal symmetry and inversion symmetry,
whereas it is preserved for odd $\beta$.

The energy difference between the two valleys,
$\Delta_{\textrm{valley}}=|\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}})-\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k}_{\textrm{K}^{\prime}})|$,
for $\beta=2$ can be controlled by changing $u$ and $\theta$. Figures
2(a)–2(c) show the $\theta$ dependence of
$\epsilon_{\pm}^{(\textrm{eff})}(\mathbf{k})$ numerically calculated for
$\beta=2$ at $\Omega=10t_{\textrm{NN}}$ and $u=0.4$ and $0.8$. (For details of
the numerical calculations, see the Supplemental Material SM .) At $\theta=0$
and $\pi$, $\Delta_{\textrm{valley}}$ is finite and increases with increasing
$u$ [see Figs. 2(a) and 2(c)]. This is because $K_{0}(u)$ and
$|K_{1}(u,\theta)|$ increase with increasing $u$. The valley which has the
larger gap can be switched by changing $\theta$ from $0$ to $\pi$ or vice
versa [compare Figs. 2(a) and 2(c)]. Meanwhile, at $\theta=\frac{\pi}{2}$,
$\Delta_{\textrm{valley}}=0$, i.e., the valley degeneracy holds [see Fig.
2(b)]. This is because $K_{1}(u,\theta)\propto\cos\theta$.

Valley-selective Hall effect.We turn to the nonperturbative effects of BCPL on
the Berry curvatures for the driven system described by Eq. (3) for $\beta=2$.
The Berry curvature for the upper or lower band, $B_{+}^{z}(\mathbf{k})$ or
$B_{-}^{z}(\mathbf{k})$, is given by

$\displaystyle
B_{\pm}^{z}(\mathbf{k})=-i\frac{v_{\pm\mp}^{x}(\mathbf{k})v_{\mp\pm}^{y}(\mathbf{k})-v_{\pm\mp}^{y}(\mathbf{k})v_{\mp\pm}^{x}(\mathbf{k})}{[\epsilon_{+}^{(\textrm{eff})}(\mathbf{k})-\epsilon_{-}^{(\textrm{eff})}(\mathbf{k})]^{2}},$
(9)

where
$v_{\alpha\beta}^{\nu}(\mathbf{k})=\sum_{a,b=A,B}(U_{\mathbf{k}}^{\dagger})_{\alpha
a}v_{ab}^{(\textrm{eff})\nu}(\mathbf{k})(U_{\mathbf{k}})_{b\beta}$
($\alpha,\beta=+,-$ and $\nu=x,y$), $(U_{\mathbf{k}})_{a\alpha}$ is the
unitary matrix to diagonalize Eq. (3), and
$v_{ab}^{(\textrm{eff})\nu}(\mathbf{k})=\frac{\partial\bar{\epsilon}_{ab}(\mathbf{k})}{\partial
k_{\nu}}$. Figures 2(d)–2(f) show the $\theta$ dependences of
$B_{+}^{z}(\mathbf{k})$ and $B_{-}^{z}(\mathbf{k})$ numerically calculated for
$\beta=2$ at $\Omega=10t_{\textrm{NN}}$ and $u=0.4$ and $0.8$. (For details of
the numerical calculations, see the Supplemental Material SM .) If the valley
degeneracy is lifted (i.e., $\theta$ is $0$ or $\pi$), the Berry curvatures at
the two valleys are different in magnitude. The valley which gives the largest
contribution to the Berry curvatures can be switched by changing $\theta$ from
$0$ to $\pi$ or vice versa. These results suggest that BCPL could induce the
valley-selective Hall effect and switch its dominant valley by changing
$\theta$. Furthermore, by changing $\theta$ from $0$ or $\pi$ to
$\frac{\pi}{2}$ or vice versa, a crossover between the valley-selective and
valley-independent Hall effects could be induced. However, even with broken
inversion symmetry, the Berry curvatures of the upper or lower band at the two
valleys have the same sign, which means the absence of the valley-contrasting
Hall effect.

The results shown above are consistent with the time-averaged anomalous Hall
conductivity (AHC) $\sigma_{yx}^{\textrm{C}}$ calculated in the Floquet
linear-response theory Eckstein ; Oka-PRB ; Tsuji ; Mikami ; NA-FloquetSHE ;
NA-linear . Figure 3(a) or 3(b) shows $\sigma_{yx}^{\textrm{C}}$ numerically
calculated at $\Omega=8t_{\textrm{NN}}$ for $\beta=2$ or $3$. (For details of
the numerical calculations, see the Supplemental Material SM .) For $\beta=3$,
$\sigma_{yx}^{\textrm{C}}$ is quantized in a similar way to that of graphene
driven by CPL Oka-PRB ; Mikami ; Kitagawa . This agrees with the analyses
using the high-frequency expansion because the effective Hamiltonian for
$\beta=3$ is qualitatively the same as that of graphene driven by CPL Mikami ;
Kitagawa . Meanwhile, for $\beta=2$, $\sigma_{yx}^{\textrm{C}}$ is quantized
at $\theta=\frac{\pi}{2}$, whereas that is reduced from the quantized value
for moderately large $u$’s at $\theta=0$ and $\pi$. The magnitude reduction in
$\sigma_{yx}^{\textrm{C}}$ for large $u$’s at $\theta=0$ or $\pi$ is
attributed to a drastic reduction in the contributions near one valley due to
a larger gap opening. In all the cases,
$\sigma_{yx}^{\textrm{C}}\approx-\sigma_{xy}^{\textrm{C}}$ is satisfied (see
Fig. 2 in the Supplemental Material SM ). (Because of this, it is reasonable
to call $\sigma_{yx}^{\textrm{C}}$ the anomalous Hall conductivity.) Since
such an antisymmetric part
$(\sigma_{yx}^{\textrm{C}}-\sigma_{xy}^{\textrm{C}})/2$ is finite only with
broken time-reversal symmetry Onsager1 ; Onsager2 , these results indicate
that time-reversal symmetry is broken in all the cases, which agrees with the
high-frequency expansion.

Then, Figs. 3(c)–3(h) show $\sigma_{yx}^{\textrm{C}}(\mathbf{k})$’s calculated
in the Floquet linear-response theory for $\beta=2$ at
$\Omega=8t_{\textrm{NN}}$ with $(\theta,u)=(0,0.2)$, $(0,0.4)$, $(0,0.6)$,
$(\pi,0.2)$, $(\pi,0.4)$, and $(\pi,0.6)$, where
$\sigma_{yx}^{\textrm{C}}(\mathbf{k})$ is defined as
$\sigma_{yx}^{\textrm{C}}=\sum_{\mathbf{k}}\sigma_{yx}^{\textrm{C}}(\mathbf{k})$.
For $u=0.2$, $\sigma_{yx}^{\textrm{C}}(\mathbf{k}_{\textrm{K}})$ and
$\sigma_{yx}^{\textrm{C}}(\mathbf{k}_{\textrm{K}^{\prime}})$ are almost the
same. Meanwhile, for $u=0.4$ and $0.6$, the main contribution at $\theta=0$ or
$\pi$ comes from the vicinity at the K or K′ point, respectively. These
results indicate that moderately strong BCPL can induce the valley-selective
Hall effect, and that the valley which gives the main contribution to this
Hall effect can be switched by changing $\theta$ from $0$ to $\pi$ or vice
versa. In contrast, $\sigma_{yx}^{\textrm{C}}(\mathbf{k}_{\textrm{K}})$ and
$\sigma_{yx}^{\textrm{C}}(\mathbf{k}_{\textrm{K}^{\prime}})$ are degenerate at
$\theta=\frac{\pi}{2}$ even for $u=0.4$ and $0.6$ (see Fig. 3 in the
Supplemental Material SM ); the same degeneracy holds for $\beta=3$ at
$\theta=0$, $\frac{\pi}{2}$, and $\pi$ (see Fig. 3 in the Supplemental
Material SM ). Therefore, by changing $\theta$ for $\beta=2$, the crossover
between the valley-selective and valley-independent Hall effects can be
induced. Since the valley degeneracy is lifted only without both time-reversal
symmetry and inversion symmetry, these results indicate that inversion
symmetry is broken for even $\beta$ at $\theta=0$ and $\pi$, which also agrees
with the high-frequency expansion.

Similar results are obtained at $\Omega=6t_{\textrm{NN}}$ (see Fig. 4 in the
Supplemental Material SM ), implying that the similar properties hold even in
the resonant case. Therefore, our valley-selective Hall effect could be
experimentally observed because for graphene driven by CPL, the results
obtained in the Floquet linear-response theory Oka-PRB ; Mikami  are
qualitatively reproducible in experiments using a smaller light frequency
Light-AHE-exp2 . Note that the Floquet linear-response theory can analyze the
off-resonant case and the resonant case, whereas the high-frequency expansion
can analyze only the former case. Our light is resonant or off-resonant if
$\Omega\leq W_{\textrm{band}}$ or $\Omega>W_{\textrm{band}}$, respectively,
where $W_{\textrm{band}}=6t_{\textrm{NN}}$ is the bandwidth of the nondriven
system.

Figure 4:  (a), (b) The momentum dependences of $B_{+}^{z}(\mathbf{k})$ and
$B_{-}^{z}(\mathbf{k})$ in the special cases with $K_{0}(u)=0$ and
$K_{1}(u,\theta)=0$. The symmetric points are the same as those used in Fig.
2.

Discussion.To understand why the valley-contrasting Hall effect is absent, we
compare the Berry curvatures calculated for $\beta=2$ using the high-frequency
expansion with those in two special cases. In these cases, the effective
Hamiltonian is given by Eq. (3) for $\beta=2$ with $K_{0}(u)=0$ or
$K_{1}(u,\theta)=0$, and the other parameters are the same as those used for
$\beta=2$ and $\Omega=10t_{\textrm{NN}}$ at $\theta=0$ and $u=0.4$. Since the
$K_{0}(u)$ term and $K_{1}(u,\theta)$ terms break time-reversal symmetry and
inversion symmetry, respectively [Eqs. (6) and (7)], the first case possesses
time-reversal symmetry, whereas the other possesses inversion symmetry. (The
first case is similar to that studied in Ref. VHE-theory, .) Note that for
$\beta=2$, $K_{1}(u,\theta)$ appears in $\Delta_{\textrm{BCPL}}$ and
$K_{\textrm{eff}}^{(AA)}(\pm\mathbf{R}_{l}^{\prime})$ [Eqs. (6) and (7)].
Figures 4(a) and 4(b) show the Berry curvatures numerically calculated in
these two cases. The Berry curvatures of the upper or lower band at the two
valleys are opposite in sign with time-reversal symmetry. Therefore, we
conclude that the valley-contrasting Hall effect is absent in graphene driven
by BCPL due to the broken time-reversal symmetry. Note that the difference
between the signs of the Berry curvatures at the two valleys without and with
time-reversal symmetry may be similar to that between a ferromagnet and an
antiferromagnet.

We compare our study with the relevant studies. The BCPL-induced symmetry
breaking has been partly clarified in some off-resonant cases at $\beta=2$
using the high-frequency expansion Bicirc-Oka ; Bicirc-PRL . Meanwhile, our
results obtained in the high-frequency expansion are applicable to any
$\beta$. Furthermore, our Floquet linear-response theory showed that these
results remain qualitatively unchanged at smaller light frequencies including
a resonant one for $\beta=2$ and $3$. Therefore, our results have a wider
applicability. Then, there is no previous study showing the valley-selective
Hall effect via the light-induced inversion symmetry breaking, although there
are many studies about valley-dependent properties of periodically driven
systems Circ2-Floquet-graphene ; BCPL-Floquet-graphene1 ; BCPL-Floquet-
graphene2 ; Valley-Abergel ; staticE-ISB1 ; staticE-ISB2 . Therefore, our
study is the first work demonstrating the BCPL-induced valley-selective Hall
effect.

We also comment on three advantages of our valley-selective Hall effect. In
the standard mechanism VHE-exp1 ; VHE-exp2 , the inversion symmetry is broken
by the lattice and the time-reversal symmetry is broken by resonant CPL.
Meanwhile, in our mechanism, these symmetries are both broken by BCPL.
Therefore, only our mechanism works in centrosymmetric systems. In addition,
our mechanism enables us to switch the main valley and induce the crossover
between the valley-selective and valley-independent Hall effects by tuning
$\theta$. Note that in the standard mechanism, the main valley can be switched
by changing the helicity of CPL. Then, our Hall conductivity, which is
$O(e^{2}/h)$ [e.g., see the value at $u=0.6$ and $\theta=0$ in Fig. 3(a)], is
two orders of magnitude larger than the maximum value obtained in the standard
mechanism VHE-exp2 . This comes from a special property of monolayer graphene
that the energy gaps at the two valleys are tunable solely by light, which
enables us to make the energy gap at one valley much larger than the other
with keeping the other small. Therefore, monolayer graphene driven by BCPL may
provide the best opportunity for the valley-selective Hall effect.

## References

  * (1) S. Y. Zhou et al., Substrate-induced bandgap opening in epitaxial graphene. Nat. Mater. 6, 770-775 (2007).
  * (2) T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Controlling the electronic structure of bilayer graphene. Science 313, 951 (2006).
  * (3) D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).
  * (4) K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen, The valley Hall effect in MoS2 transistors. Science 344, 1489 (2014).
  * (5) J. Yin et al., Tunable and giant valley-selective Hall effect in gapped bilayer graphene. Science 375, 1398 (2022).
  * (6) W. Yao, D. Xiao, and Q. Niu, Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B 77, 235406 (2008).
  * (7) T. Cao et al., Valley-selective circular dichroism of monolayer molybdenum disulphide. Nat. Commun. 3, 887 (2012).
  * (8) H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Valley polarization in MoS2 monolayers by optical pumping. Nature Nanotech. 7, 490–493 (2012).
  * (9) K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS2 by optical helicity. Nature Nanotech. 7, 494–498 (2012).
  * (10) D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics in graphene: Magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).
  * (11) O. Kfir et al., Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics. Nat. Photonics 9, 99 (2015).
  * (12) T. Nag, R.-J. Slager, T. Higuchi, and T. Oka, Dynamical synchronization transition in interacting electron systems. Phys. Rev. B 100, 134301 (2019).
  * (13) T. V. Trevisan, P. V. Arribi, O. Heinonen, R.-J. Slager, and P. P. Orth, Bicircular light Floquet engineering of magnetic symmetry and topology and its application to the Dirac semimetal Cd3As2. Phys. Rev. Lett. 128, 066602 (2022).
  * (14) J. H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979 (1965).
  * (15) H. Sambe, Steady states and quasienergies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7, 2203 (1973).
  * (16) A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective. New J. Phys. 17, 093039 (2015).
  * (17) M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139 (2015).
  * (18) T. Oka and H. Aoki, Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406(R) (2009).
  * (19) M. Eckstein and M. Kollar, Theory of time-resolved optical spectroscopy on correlated electron systems. Phys. Rev. B 78, 205119 (2008).
  * (20) N. Arakawa and K. Yonemitsu, Light-induced mirror symmetry breaking and charge transport. unpublished (under review in Phys. Rev. B).
  * (21) N. Tsuji, T. Oka, and H. Aoki, Nonequilibrium steady state of photoexcited correlated electrons in the presence of dissipation. Phys. Rev. Lett. 103, 047403 (2009).
  * (22) T. Mikami et al., Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators. Phys. Rev. B 93, 144307 (2016).
  * (23) N. Arakawa and K. Yonemitsu, Symmetry-protected difference between spin Hall and anomalous Hall effects of a periodically driven multiorbital metal. Commun. Phys. 6, 43 (2023).
  * (24) M. Büttiker, Small normal-metal loop coupled to an electron reservoir. Phys. Rev. B 32, 1846(R) (1985).
  * (25) M. Büttiker, Role of quantum coherence in series resistors. Phys. Rev. B 33, 3020 (1986).
  * (26) L. D’Alessio and M. Rigol, Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).
  * (27) A. Lazarides, A. Das, and R. Moessner, Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).
  * (28) See the Supplemental Material for the derivations of Eqs. (3), (6), and (7), the details of the numerical calculations, and additional numerical results.
  * (29) S. Gwo and C. K. Shih, Site-selective imaging in scanning tunneling microscopy of graphite: The nature of site asymmetry. Phys. Rev. B 47, 13059(R) (1993).
  * (30) C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
  * (31) F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015 (1988).
  * (32) T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels. Phys. Rev. B 84, 235108 (2011).
  * (33) L. Onsager, Reciprocal relations in irreversible processes. I.. Phys. Rev. 37, 405 (1931).
  * (34) L. Onsager, Reciprocal relations in irreversible processes. II.. Phys. Rev. 38, 2265 (1931).
  * (35) J. W. McIver et al., Light-induced anomalous Hall effect in graphene, Nat. Phys. 16, 38–41 (2020).
  * (36) A. Kundu, H. A. Fertig, and B. Seradjeh, Floquet-engineered valleytronics in Dirac systems. Phys. Rev. Lett. 116, 016802 (2016).
  * (37) M. S. Mrudul, Á. J.-Galán, M. Ivanov, and G. Dixit, Light-induced valleytronics in pristine graphene. Optica 8, 422 (2021).
  * (38) M. S. Mrudul and G. Dixit, Controlling valley-polarisation in graphene via tailored light pulses. J. Phys. B: At. Mol. Opt. Phys. 54, 224001 (2021).
  * (39) D. S. L. Abergel and T. Chakraborty, Generation of valley polarized current in bilayer graphene. App. Phys. Lett. 95, 062107 (2009).
  * (40) M. Tahir, Q. Y. Zhang, and U. Schwingenschlögl, Floquet edge states in germanene nanoribbons, Sci. Rep. 6, 31821 (2016).
  * (41) J. Mei, L. Shao, H. Xu, X. Zhu, and N. Xu, Photomodulated edge states and multiterminal transport in silicenelike nanoribbons, Phys. Rev. B 99, 045444 (2019).

##  I Acknowledgments

This work was supported by JST CREST Grant No. JPMJCR1901, JSPS KAKENHI Grant
No. JP22K03532, and MEXT Q-LEAP Grant No. JP-MXS0118067426.


# Lifetime of vertical giant soap films: role of the relative humidity and
film dimensions

Marina Pasquet  Université Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405, Orsay, France.  François Boulogne  Université Paris-Saclay,
CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.  Frédéric
Restagno  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405, Orsay, France.  Emmanuelle Rio∗ Université Paris-Saclay, CNRS,
Laboratoire de Physique des Solides, 91405, Orsay, France.

###### Abstract

We consider the lifetime of rectangular vertical soap films and we explore the
influence of relative humidity and both dimensions on the stability of large
soap films, reaching heights of up to 1.2 m. Using an automated rupture
detection system, we achieve a robust statistical measurement of their
lifetimes and we also measure the film thinning dynamics. We demonstrate that
drainage has a negligible impact on the film stability as opposed to
evaporation. To do so, we compare the measured lifetimes with predictions from
the Boulogne & Dollet model [1], originally designed to describe the
convective evaporation of hydrogels. Interestingly, we show that this model
can predict a maximum film lifetime for all sizes.

##  1 Introduction

In their performances, bubble artists daily make soap films of large sizes, as
evidenced by some world records such as the highest indoor bubble exceeding 10
meters high [2] or the largest outdoor and indoor bubbles exceeding
respectively 96 [3] and 21 [4] m3 in volume. The reported lifetimes of these
giant objects are generally of the order of a few seconds, but they depend
strongly on the composition of the soap solution [5], as well as on the
ambient air humidity.

Large soap films are also the subject of scientific studies [6, 7, 8, 9, 10,
11]. Experiments have been conducted on the impact of air humidity on the
stability of foams [12, 13] and on centimeter scale soapy objects, bubbles and
films [14, 15, 16]. They demonstrated the crucial role of humidity on the
lifetime of soapy objects, and some studies enlighten the complex role of the
nature of surfactants on evaporation [17, 18, 19].

To date, the thinning rate of surface bubbles is actually better understood
than the one of flat soap films, the rotational symmetry being helpful for a
quantitative description. More specifically, the thinning rate of the soap
film surrounding surface bubbles is influenced by drainage, and also by
evaporation when the atmosphere is not saturated in humidity. The question of
the rupture thickness and the nucleation mechanism of the holes in soap films
is still widely open, nevertheless it appears that it only widens the
distribution of lifetime around an average value fixed by the thinning rate.
Within this frame, the lifetime of surface bubbles can be predicted by
describing their thinning through drainage and evaporation [15, 16]. In
particular, the description of evaporation must take into account the natural
air convection [15].

The drainage of both surface bubbles and vertical flat films has been proven
to be due to marginal regeneration [20, 21, 22]: in the vicinity of the
meniscus, a groove develops [23], which destabilizes in thin patches whose
smaller apparent density [24] makes them rise. Since the interface can be
considered as incompressible [21], the consecutive sliding puzzle dynamics
[25] with rising thin patches and descending thick zones is at the origin of
the drainage [20, 21, 22].

For centimetric vertical flat films, the exact thinning law still lacks a
quantitative description. In this geometry, the marginal regeneration indeed
occurs not only at the bottom meniscus but also on the vertical borders. Two
gutters, close to the lateral menisci, allow a rapid rise of thin areas,
resulting from the destabilization of the lateral groove [20, 21]. This
contribution to drainage is not fully described in the literature.
Nevertheless, a previous study [14] has shown experimentally that for
thicknesses smaller than a few hundreds of nanometers, the evaporation becomes
predominant in the thinning rate. Concerning the lifetime of flat films,
Frazier et al [26] propose a measure for films of a few centimeters high for
solution containing a mixture of dish washing liquid and lubricant as a
function of humidity and lubricant concentration. They show that the addition
of lubricant increases the lifetime. Furthermore, they provide a measure of
the film lifetime with the atmospheric humidity, which increases drastically
above 80%.

Recently, we developed a setup allowing to study metric films, which can be
qualified as giant soap films [8]. We measured their thickness profile under
gravity and described their dynamics during their generation [11]. In this
paper, we use the same setup and measure the evolution of the lifetime of
giant films with ambient humidity and with their dimensions. We show that the
results are in agreement with a description, in which evaporation has a
primary role in thinning, allowing to understand the maximal lifetime for the
films.

##  2 Experimental setup

Figure 1: (a) Schematic representation of the experimental setup allowing
generation and automated lifetime measurements of giant soap films, with the
notations used in the article. A spectrometer measures the film thickness at a
given position. (b) Photograph of a static soap film of height $H=120$ cm and
width $w=70$ cm, made with a solution of Fairy diluted to 4 %. The
spectrometer is removed for the purpose of illustration.

To study the lifetime of giant soap films, we have used the experimental setup
presented in Fig. 1 (a) and a photograph of a studied film is proposed in Fig.
1 (b). In this section, we give the main characteristics of this setup for
which more details are available in a dedicated article by Mariot et al. [8].

The films are generated by pulling a horizontal wire out of a glass tank. For
each experiment, the tank is filled with 5 liters of solution made of 4 wt.%
dishwashing liquid (Fairy from Procter & Gamble, with an anionic surfactant
concentration between 15 and 30 %) with ultrapure water (resistivity greater
than 18.2 M$\Omega\cdot$cm). The wire is a fishing line with a diameter of
0.74 mm entrained by two lateral transmissions toothed belts and two stepping
motors located at the top of the device. The resulting wire velocity is
$V=100$ cm/s. The width $w$ of the generated films is varied by a factor of
two: we will study films with $w=35$ and $70$ cm. The height $H$ of the films
is in our study between 20 and 120 cm and four values are considered: $H=20$,
$40$, $80$, and $120$ cm.

The soap film lifetime is recorded automatically by light reflectivity. A
white light is shed at the bottom of the soap film and the reflection is
measured by a detector composed of three photoresistors. The soap film rupture
leads to a loss of light detection. This automation allows performing robust
statistical measurements of the lifetimes of the films, as we will see in the
following.

We also measure the film thickness $h$ locally, using a UV-VIS spectrometer
(Ocean Optics Nanocalc 2000) associated with an optical fiber of 400 $\mu$m
diameter at different heights and at an equal distance between the two belts
as shown in Fig. 1(a).

The humidity in the chamber of size 2.2 $\times$ 1.0 $\times$ 0.75 m3 where
the films are generated is controlled with two commercial humidifiers, which
allow having a homogeneous humidity in the chamber with a precision of 4 %:
one humidifier is placed at the top and the second is located at the bottom.
The humidity is measured throughout the experiments with a SHT25 sensor
(purchased from RS). Note that reaching atmospheric humidities higher than 80
% is challenging for such a large box. We thus choose to work between 40 and
80 %. This sensor also allows measuring the room temperature, which is about
22 ∘C during all the experiments.

##  3 Results

###  3.1 Experimental measurements

Figure 2: (a) Lifetime distributions measured for soap films of height $H=20$
cm and width $w=70$ cm, for different humidity conditions: $\text{RH}=39\ \%$
(in red) and $\text{RH}=63\ \%$ (in green). These histograms are obtained for
about 100 films. The corresponding normal distributions are superimposed. (b)
Average lifetimes measured for different humidity conditions (indicated by the
colormap). Two film widths are studied: $w=35$ cm (full symbols) and $w=70$ cm
(empty symbols). The error bars correspond to the standard deviation measured
for each cycle studied of a hundred films.

In this section, we present the results of automated film lifetime
measurements obtained from our device. To ensure statistical robustness, we
systematically measure the lifetimes with cycles of 100 films for each probed
height and humidity condition.

Two examples of histograms for films of height $H=20$ cm, corresponding to two
distinct relative humidity values ($\text{RH}=39\ \%$ and $\text{RH}=63\ \%$)
are given in Fig. 2 (a). It can be observed that the distributions are well-
defined for each humidity, and the solid lines represent the normal
distributions fitted over the data. As expected, the average lifetime
decreases as humidity decreases, varying on average from 22 seconds to 12
seconds in our example, when changing the relative humidity from 63 % to 39 %.

The dependence of the mean lifetime of the soap films on the height is given
in Fig. 2 (b). In this graph, two different widths are represented: $w=35$ cm
and $w=70$ cm. The first tendency that we observe is that for each height
probed, the lifetime decreases when RH decreases. The other important
observation is that the lifetime $\tau$ decreases when $H$ increases for a
given value of RH. We can also note a slight decrease of $\tau$ with $w$.

As explained in the introduction, we propose that the film lifetime is mainly
fixed by the thinning rate. This hypothesis is supported by the distributions
shown in Fig. 2 (a): the width of the distribution, which is due to the
stochastic rupture of the films is indeed smaller than the variation of the
mean value from one experiment to another. Therefore, we focus our analysis on
the thinning dynamics.

###  3.2 Thinning of giant films

Figure 3:  Evolution of the thickness $h$ of the soap films during and after
the generation at 1 m/s as a function of time ($w=70$ cm). (a) The
measurements are performed 13 cm above the liquid bath (at the center of the
films) at $\text{RH}=60\ \pm\ 4\%$. The dashed lines represent linear fits to
measure the film thinning velocity, in the linear regime $\mathcal{R}_{2}$. In
(b), the measurements are performed below the fishing line (at the center of
the films, $H=20$ cm) at two humidity values: $\text{RH}=39\ \%$ (in red) and
$\text{RH}=63\ \%$ (in green). The dashed line represents a linear fit to
measure the film thinning velocity, in the linear regime $\mathcal{R}_{2}$.
The slope is given in the plot.  Figure 4: Evolution of the time $t^{\ast}$ as
a function of the film height for different relative humidity values ($w=70$
cm). The humidity values given in the figure have an accuracy of $\pm\ 4\%$.
The colored points correspond to measurements that are performed at 13 cm
above the liquid bath (at the center of the films). The black points
correspond to the measurements of $t^{\ast}$ using the data measured under the
fishing line at $\text{RH}=42\ \%$ and/or $\text{RH}=60\ \%$. The symbols
correspond to those used in Figs. 2 and 3, with each film height represented
by a different marker.

Figure 3 shows thinning curves measured respectively at the bottom of the
films, 13 cm above the bath, for films of four different heights at $60$ %
humidity (Figure 3(a)), and at the top of the film below the fishing line of a
20 cm high static film, for two humidity values ($\text{RH}=39\ \%$ and
$\text{RH}=63\ \%$) (Figure 3(b)).

At the bottom of the film (Figure 3(a)), two regimes can be observed. During a
first regime, $\mathcal{R}_{1}$, the film thickness increases before reaching
a maximum and then, decreases almost linearly in time, which constitutes the
second regime $\mathcal{R}_{2}$. The first regime has been the subject of a
previous article [11], where we have drawn the following conclusions. During
and just after the generation, the film thickness profile is exponential, with
a characteristic length fixed by the balance between gravity and the
elasticity of the soap film. The thickness evolution in this first regime, is
linked to stress relaxation in the film following generation. The time
$t^{\star}$, at which the second regime starts, corresponding to the maximum
thickness, is plotted in color in Fig. 4.

We observe systematically that these giant soap films burst from the top (see
video in SI). We thus focus on the thinning dynamics at the top, in the
vicinity of the fishing line (Figure 3(b)). Here, the first regime
$\mathcal{R}_{1}$ corresponds to quite high thinning rate, whereas a linear
regime is also observed in the regime $\mathcal{R}_{2}$. The transition time
$t^{*}$, corresponding to the apparition of a linear regime, is plotted in
black in Figure 4 and is very close to the one extracted from the thinning
rate measured at the bottom of the film. Note that with the spectrometer, it
is difficult to measure thicknesses below 400 nm, so that our last measured
point in Figure 3 does not necessarily correspond to the moment of film
rupture. The thinning velocity in the second regime is of the order of tens of
nanometers per second and depends on the relative humidity and on the position
in the film.

##  4 Lifetime of giant soap films

The total lifetime $\tau$ of the films is the sum of the transition time
$t^{*}$ and the lasting time of the second regime $t_{\mathcal{R}_{2}}$. The
value of $t^{*}$ is the measured value, extracted from the experimental data,
since there is no theoretical prediction of its value. Fortunately, $t^{*}\leq
t_{\mathcal{R}_{2}}$ in many situations. Thus, we will use the measured value
of $t^{*}$ and develop a model to describe the thinning in the second regime,
and to predict $t_{\mathcal{R}_{2}}$.

###  4.1 A negligible drainage

In absence of evaporation the downward Poiseuille flow in a foam film due to
the gravitational field with the $z$-axis pointing downwards, writes [20]

$\frac{\partial h}{\partial t}=-\frac{\rho_{w}g}{4\eta}h^{2}\frac{\partial
h}{\partial z},$ (1)

where $\rho_{w}$ and $\eta$ are respectively the density and the dynamic
viscosity in the film and $g$ the gravitational acceleration constant. We can
note an important dependence on the thickness in $h^{3}$ in Eq. (1).
Nevertheless, the maximum film thickness observed in the regime
$\mathcal{R}_{2}$ is around 1 $\mu$m. In this situation, a simple scaling
analysis gives a thinning rate $\frac{\partial h}{\partial
t}\simeq-\frac{\rho_{w}g}{4\eta}h^{3}/H$. For the density and the viscosity of
water and $H=2$ m, the thinning rate is thus around $10^{-12}$ m/s, which is
negligible compared to the actual thinning rate raging from a few tens to a
hundred nanometers per second as shown in Figure 3.

This is why, as explained in the introduction, the drainage is attributed to
marginal regeneration. Nevertheless, in these giant soap films, we never
observed any development of marginal regeneration at the bottom meniscus.
Additionally, the lateral gutters, close to the meniscus, which are crucial in
centimetric films, are not visible, and their contribution is probably
negligible. Indeed, if they exist, their width is much smaller than the total
film width in contrast to centimetric films. Thus, we can rule out the
drainage due to marginal regeneration.

We finally propose to attribute the thinning in the linear regime
$\mathcal{R}_{2}$ to evaporation, in good agreement with the order of
magnitude of the thinning rate, about tens of nanometers per second [1]. In
the next section, we will estimate a value for $\tau_{\mathcal{R}_{2}}$ within
these hypotheses.

###  4.2 Prediction of the evaporation timescale

Figure 5: (a) Evolution of the measured film lifetime $\tau$ versus the
lifetime predicted by the Eq. (5). The colors correspond to the humidity RH
and are indicated using the vertical color bar. Four static film heights are
studied ($H=20$ cm, $H=40$ cm, $H=80$ cm and $H=120$ cm) as well as two
distinct film widths: $w_{0}=35$ cm (full symbols) and $w_{0}=70$ cm (empty
symbols). The black line corresponds to the equality between the axis. (b)
Identical graph with different colors, corresponding to the height of the
films $H$. The dotted lines correspond to a linear fit performed for all the
measured lifetimes for films of a given height $H$. The slopes are indicated
in the legend.

To estimate the lifetime of the films as a function of the humidity conditions
present in the chamber, we can estimate their evaporation rate. This
necessitates modeling the concentration gradient of vapor established from the
surface of the evaporating film to its equilibrium value, far from the
interface. For water-based evaporating surfaces larger than a typical scale of
about one centimeter, the convection due to the lower density of humid air
compared to dry air must be considered [27, 28, 1]. The transition toward a
convective evaporation is quantified by the Grashof number

$\text{Gr}\ =\ \dfrac{\mid\Delta\rho_{\text{a}}\mid}{\rho_{\text{a}},\infty}\
\dfrac{g}{\nu^{2}}\ \mathcal{L}^{3},$ (2)

where $\nu$ is the kinematic viscosity of air ($\nu=1.55\ 10^{-5}$ m2/s at 20
∘C), $\rho_{\text{a},\infty}$ is the density of the ambient air (away from the
evaporating interface), $\mid\Delta\rho_{\text{a}}\mid=\mid\rho_{\text{a,
sat}}-\rho_{\text{a},\infty}\mid$ is the difference in density between the air
at the surface and the ambient air, and $\mathcal{L}$ is the vertical
characteristic length over which the concentration gradient of vapor is
established. Natural convection must be taken into account when $\text{Gr}\gg
1$. A first determination of the Grashof numbers, using the Eq. (2) for an
intermediate humidity corresponding to $\text{RH}=60\ \%$, indicates that for
our problem: $\text{Gr}>10^{6}$. The convection flux due to the evaporation of
these large vertical films must therefore be taken into account.

We propose to use the calculation of the natural convection flow on a vertical
plate developed by Schmidt and Beckmann [29] adapted for the evaporation of
hydrogel slabs up to 20 cm high [1]. Their calculation of the local
evaporative flux for a vertical film, which assumes a laminar boundary layer,
leads to:

$j_{\text{evap}}\ \approx\ \widehat{c_{0}}\
\dfrac{D\left(c_{\text{sat}}-c_{\infty}\right)}{\rho_{w}}\
\dfrac{\text{Gr}^{1/4}}{\mathcal{L}},$ (3)

where $D$ is the water vapor diffusion coefficient in air ($D=2.44\ 10^{-5}$
m2/s at 20 ∘C), $\rho_{\text{w}}$ is the density of water, $c_{\text{sat}}$
and $c_{\infty}$ are respectively the mass concentrations of saturated vapor
and water vapor in the air ($c_{\text{sat}}=17$ g/m3 at 20 ∘C), and
$\widehat{c_{0}}$ is a numerically determined prefactor
($\widehat{c_{0}}\simeq 0.478$). This model is developed for centimetric
vertical films made of hydrogel, for which water evaporates in the atmosphere.
The characteristic size $\mathcal{L}$ on which the boundary layer is developed
corresponds to the height $H$ of the film.

Thus, the thinning rate due to evaporation writes

$\frac{\partial h}{\partial t}\ =\ 2\ j_{\text{evap}},$ (4)

where the factor of two takes into account the contribution of both liquid-air
interfaces of the films. As $j_{\rm evap}$ remains constant over time, this
equation is in agreement with the linear decrease of the film thickness
observed experimentally, which tend to confirm that evaporation is the main
thinning mechanism in this configuration.

Eq. (4) directly leads to a prediction for the evaporation time:

$\tau_{\text{evap}}\ =\ \dfrac{h^{\ast}\ -\ h_{\text{rupt}}}{2\
j_{\text{evap}}}$ (5)

with $h_{\text{rupt}}$ the thickness at which the films break. We consider
here the thickness at the top of the films, since this is where $h$ is minimum
and thus the films are more likely to rupture.

In the next section, we will compare the lifetime of the films with
$t^{*}+t_{\mathcal{R}_{2}}$ with $t^{*}$ measured experimentally (Figure 4)
and $t_{\mathcal{R}_{2}}=\tau_{\text{evap}}$.

###  4.3 Discussion

In Fig. 5 (a), we plot all the measurements made for different heights and
humidity conditions as a function of the lifetime given by Eq. (5), in the
case where $h_{\text{rupt}}=0$. This is the maximum time that can be predicted
for $\tau_{\text{evap}}$, considering that the films break when their
thickness is equal to zero. On this figure, several thousands of films are
represented and the solid line corresponds to a line of equation
$\tau=t^{*}+\tau_{\rm evap}$. This graph shows that the vast majority of the
measured lifetimes are below this limit.

This graph is shown again in Fig. 5 (b), with a different color scale, which
corresponds to the height of the films $H$. For the films of height $H=20$ cm,
we observed that the measured lifetimes are below but close to the theoretical
prediction (solid black line). This suggests that the model of Boulogne &
Dollet [1] can describe the evaporation time of these soap films as well. In
this case, the discrepancy between the prediction and measurements is
attributed to the rupture of the films when they are still at non-zero
thicknesses.

As the height of the films increases, the films tend to break systematically
at smaller time. For each film height probed, a dotted line shows the linear
fit of all lifetimes as a function of $t^{\star}+\tau_{\text{evap}}$. The
measured slope is given in the legend of the plot in figure 5 (b). We notice
that as the height of the films increases, the value of this slope decreases
and deviates from unity. This suggests that the main physical ingredients of
the model are still holding, but that the evaluation of the evaporation rate
is no longer suitable for higher films. The best candidate to explain this
mismatch is the value of the film length $\mathcal{L}$, which we took equal to
$H$ following the argument presented in reference [1]. Nevertheless, this
length seems to be smaller as the film height increases. As suggested by
Haaland and Sparrow, the boundary layer can undergo a destabilization due to
the difference of buoyancy with the surrounding atmosphere that causes a more
complex three-dimensional flow increasing the evaporation of the film [30].
Such a mechanism, that must be confirmed with dedicated experimental
investigations, would explain the difference between the prediction of the
model and the measured film thinning.

##  5 Conclusion

In this paper, we have presented a study of the stability of soap films having
a height between 20 cm and 1.20 m, generated at a fixed velocity of 1.0 m/s in
a humidity-controlled chamber. We have highlighted the influence of their
width and of the relative humidity on their stability: as expected and already
observed on other small soap systems [15], as the relative humidity RH
increases, the lifetime of these large static films is also increased.

To quantify this observation, we propose that evaporation is the predominant
mechanism. The convection must be considered in the description of the
evaporation of our films, since the Grashof numbers are considerable. We
therefore compared the lifetimes measured to the evaporation times predicted
by Boulogne & Dollet [1]. This gives a good prediction of the maximum value of
the lifetimes for all film sizes and a quantitative prediction for the 20 cm
films. Nevertheless, a full description of the films lifetime would
necessitate a comprehensive understanding of the first regime and thus of film
relaxation as well as a better understanding of the evaporation of vertical
liquid sheets at a meter scale. In particular, the regime at high humidity,
above 80 %, identified by Frazier et al [26] would certainly be very
interesting to explore. In addition, the recent findings on the cooling of
soap films induced by evaporation raises also the question of the significance
of the thermal effects in the convective regime, which must be addressed in
future studies [31, 32, 33].

## Acknowledgments

Funding from ESA (MAP Soft Matter Dynamics), CNES (through the GDR MFA) and
the ANR DRAINFILM is acknowledged. We thank Isabelle Cantat for fruitful
discussions and Sandrine Mariot for the design and development of the
experimental setup.

## References

  * [1] François Boulogne and Benjamin Dollet.  Convective evaporation of vertical films.  Soft Matter, 14:1665–1671, 2018.
  * [2] Graeme Denton.  Tallest free-standing soap bubble, 2020.
  * [3] Gary Pearlman.  Largest outdoor free floating soap bubble, 2015.
  * [4] Gary Pearlman.  Largest free floating soap bubble (indoors), 2019.
  * [5] Marina Pasquet, Laura Wallon, Pierre-Yves Fusier, Frédéric Restagno, and Emmanuelle Rio.  An optimized recipe for making giant bubbles.  The European Physical Journal E, 45(12):101, 2022.
  * [6] P Ballet and F Graner.  Giant soap curtains for public presentations.  European Journal of Physics, 27(4):951–967, jun 2006.
  * [7] Caroline Cohen, Baptiste Darbois Texier, Etienne Reyssat, Jacco H. Snoeijer, David Quéré, and Christophe Clanet.  On the shape of giant soap bubbles.  Proceedings of the National Academy of Sciences, 114(10):2515–2519, 2017.
  * [8] Sandrine Mariot, Marina Pasquet, Vincent Klein, Frédéric Restagno, and Emmanuelle Rio.  A new setup for giant soap films characterization.  The European Physical Journal E, 44(4):1–8, 2021.
  * [9] M. A. Rutgers, X. L. Wu, and W. B. Daniel.  Conducting fluid dynamics experiments with vertically falling soap films.  Review of Scientific Instruments, 72(7):3025–3037, 2001.
  * [10] Hamid Kellay and Walter I Goldburg.  Two-dimensional turbulence: a review of some recent experiments.  Reports on Progress in Physics, 65(5):845–894, apr 2002.
  * [11] M. Pasquet, F. Restagno, I. Cantat, and E. Rio.  Thickness profiles of giant soap films.  Physical Review Fluids, 8(3):034001, 2023.
  * [12] X. Li, R. Shaw, and P. Stevenson.  Effect of humidity on dynamic foam stability.  International Journal of Mineral Processing, 94(1):14 – 19, 2010\.
  * [13] Xueliang Li, Stoyan I. Karakashev, Geoffrey M. Evans, and Paul Stevenson.  Effect of environmental humidity on static foam stability.  Langmuir, 28(9):4060–4068, 2012.
  * [14] Lorène Champougny, Jonas Miguet, Robin Henaff, Frédéric Restagno, Francois Boulogne, and Emmanuelle Rio.  Influence of evaporation on soap film rupture.  Langmuir, 34(10):3221–3227, 2018.
  * [15] Jonas Miguet, Marina Pasquet, Florence Rouyer, Yuan Fang, and Emmanuelle Rio.  Stability of big surface bubbles: impact of evaporation and bubble size.  Soft Matter, 16(4):1082–1090, 2020.
  * [16] S. Poulain and L. Bourouiba.  Biosurfactants change the thinning of contaminated bubbles at bacteria-laden water interfaces.  Phys. Rev. Lett., 121:204502, Nov 2018.
  * [17] Victor K. La Mer and Thomas W. Healy.  Evaporation of Water: Its Retardation by Monolayers: Spreading a monomolecular film on the surface is a tested and economical means of reducing water loss.  Science, 148(3666):36–42, April 1965.
  * [18] S Poulain, E Villermaux, and L Bourouiba.  Ageing and burst of surface bubbles.  Journal of fluid mechanics, 851:636–671, 2018.
  * [19] Marina Pasquet, François Boulogne, Julien Sant-Anna, Frédéric Restagno, and Emmanuelle Rio.  The impact of physical-chemistry on film thinning in surface bubbles.  Soft Matter, 18:4536–4542, 2022.
  * [20] KJ Mysels, K. Shinoda, and S.Frankel.  Soap Films: Studies of their Thinning and a Bibliography.  Pergamon, New York, 1959.
  * [21] Jacopo Seiwert, Ronan Kervil, Soniraks Nou, and Isabelle Cantat.  Velocity field in a vertical foam film.  Phys. Rev. Lett., 118:048001, Jan 2017.
  * [22] J. Miguet, M. Pasquet, F. Rouyer, Y. Fang, and E. Rio.  Marginal regeneration-induced drainage of surface bubbles.  Physical Review Fluids, 6(10):L101601, 2021.
  * [23] A. Aradian, E. Raphael, and P-G De Gennes.  “marginal pinching” in soap films.  EPL, 55(6):834, 2001.
  * [24] N. Adami and H. Caps.  Capillary-driven two-dimensional buoyancy in vertical soap films.  EPL, 106(4):46001, 2014.
  * [25] Alice Gros, Adrien Bussonnière, Sanjiban Nath, and Isabelle Cantat.  Marginal regeneration in a horizontal film: Instability growth law in the nonlinear regime.  Phys. Rev. Fluids, 6(2):024004, 2021.
  * [26] Stephen Frazier, Xinyi Jiang, and Justin C. Burton.  How to make a giant bubble.  Phys. Rev. Fluids, 5:013304, Jan 2020.
  * [27] N. Shahidzadeh-Bonn, S. Rafaï, A. Azouni, and D. Bonn.  Evaporating droplets.  J. Fluid Mech., 549:307–313, 2006.
  * [28] Benjamin Dollet and François Boulogne.  Natural convection above circular disks of evaporating liquids.  Physical Review Fluids, 2(5):53501, may 2017.
  * [29] E. Schmidt and W. Beckmann.  Das temperatur- und geschwindigkeitsfeld vor einer wärme abgebenden senkrechten platte bei natürlicher konvektion.  Technische Mechanik und Thermodynamik, 1(11):391–406, 1930.
  * [30] S.E. Haaland and E.M. Sparrow.  Vortex instability of natural convection flow on inclined surfaces.  International Journal of Heat and Mass Transfer, 16(12):2355–2367, 1973.
  * [31] F. Boulogne, F. Restagno, and E. Rio.  Measurement of the temperature decrease in evaporating soap films.  Phys. Rev. Lett., 129:268001, 2022.
  * [32] F. Boulogne, E. Rio, and F. Restagno.  Evaporation-induced temperature gradient in a foam column.  Langmuir, 39(40):14256–14262, 2023.
  * [33] M. Corpart, F. Restagno, and F. Boulogne.  Measuring relative humidity from evaporation with a wet-bulb thermometer: the psychrometer.  American Journal of Physics, 92(1):36–42, 01 2024.