Dynamical band gap tuning in Weyl semi-metals by intense elliptically polarized normal illumination and its application to $8-Pmmn$ borophene
V. G. Ibarra-Sierra J. C. Sandoval-Santana A. Kunold Gerardo G. Naumis
cond-mat.mes-hall




















The Dynamical-gap formation in
Weyl semimetals
modulated by intense elliptically polarized light 
is addressed through the solution
of the time-dependent Schrödinger equation
for the Weyl Hamiltonian
via the Floquet theorem.
The time-dependent wave functions and the quasi-energy spectrum
of the two-dimensional Weyl Hamiltonian under
normal incidence of elliptically polarized electromagnetic   
waves are obtained using a non-perturbative approach.
In it, the Weyl equation is reduced to an ordinary
second-order differential Mathieu equation.
It is shown that the stability conditions of the Mathieu functions
are directly inherited by the wave function
resulting in a quasiparticle spectrum
consisting of bands and gaps determined by dynamical diffraction and 
resonance conditions between the electron and the electromagnetic wave.
Estimations of the electromagnetic field intensity and frequency,
as well as the magnitude of the generated gap are obtained
for the 8-Pmmn phase of borophene.
We provide with a simple method that enables to predict the formation
of dynamical-gaps of unstable wave functions and their magnitudes.
This method can readily be adapted to other Weyl semimetals.



  Keywords
Weyl semi-metals; dynamical gap; Mathieu equation 





Dynamical band gap tuning in Weyl semi-metals by intense elliptically
polarized normal illumination and its application to 8-Pmmn borophene.

    Gerardo G. Naumis
    December 30, 2023
==============================================================================================================================================





§ INTRODUCTION


Recently, the so-called
Dirac and Weyl materials
have received considerable attention due to their
possible implementation into next-generation electronic devices 
<cit.>.
The main physical properties of these materials were observed for the 
first time in graphene, an allotrope of carbon 
consisting of a monolayer of atoms in a honeycomb lattice with an 
electron linear dispersion near the Dirac points. As a result
of this characteristics,
the charge carriers in graphene behave like massless Dirac fermions 
<cit.>.
Thereafter, a wide variety of two-dimensional materials with similar
properties has been discovered <cit.>.
Examples of these are: silicene
<cit.>,
germanene <cit.>,
stanene <cit.> and
artificial graphene <cit.>.
Borophen, a two dimensional allotrope of boron, also
falls in this category.
The chemical similarity between boron and carbon atoms
has triggered the search for stable two-dimensional
boron structures and synthesis techniques to
produce them<cit.>.
Since their theoretical prediction <cit.>,
many different allotropes of borophene have been
experimentally confirmed <cit.>.
Among its many different phases,
the orthorhombic 8-Pmmn is one of the most energetically stable
structures <cit.>,
having a ground state energy lower than its analogs 
<cit.>.
Borophene, in contrast with graphene, shows a highly anisotropic 
crystalline structure, which causes high optical anisotropy
and transparency
<cit.>.
It is thus a strong candidate for flexible
electronics, display technologies and in the design
of smart windows where minimal photon absorption and reflection
are required
<cit.>.


Despite the many useful and fascinating properties
of graphene, borophene and Weyl materials in general,
their lack of an electronic 
band gap has stimulated the search for either other
two-dimensional materials with
semiconducting properties or techniques to induce them artificially.
Among other proposals to circumvent this problem,
one of the most promising ideas is generating a light
induced dynamical-gap. As the electromagnetic field
is a periodical function of time, this technique has been
termed Floquet gap engineering.
High intensity electromagnetic waves interacting with
graphene have  been studied using perturbative approaches 
<cit.>. However, it has
been shown that light induces a renormalization of the electronic
spectrum of Dirac materials not captured by simple perturbation
techniques <cit.>.
In this regard, borophene brigns interesting possibilities
to study the light-matter interaction in Weyl semimetals due
to its asymmetric spectrum.
As graphene, borophene has a honeycomb lattice with
two nonequivalent sublattices. However, its
peculiar structure give rise to a tilted anisotropic cone
in the vicinity of the Dirac points
<cit.>, as opposed to
graphene whose spectrum is completely isotropic in K space.

Recently, the formation of energy gaps in borophene subject
high-intensity linearly
polarized light was studied beyond the
perturbative approximation <cit.>.
It was found that borophene, when interacting
with light, acquires a complex band structure
from the stability conditions of the solutions of the Mathieu
differential equation.
Among other effects, the interaction with light
produces a gap in the vicinity of borophene's Dirac point.
The effects of an intense circularly polarized
electromagnetic field have, nevertheless, not been discussed
for borophene yet.



In this paper, we address the general problem of a particle that obeys the
Weyl Hamiltonian subject to an intense elliptically polarized 
electromagnetic field.
As it is schematized in Fig. <ref>,
our solutions can be applied to the particular case of electrons in
borophene under a strong elliptically polarized field. 
We report the wave functions, the quasi-energy spectrum,
and the magnitude of the dynamic gap opening.
The case of linearly polarized light,
addressed by us previously <cit.>,
is proven to be fundamentally different from the elliptically
polarized one studied in this work.
Our analysis mainly focuses on the stability
and instability of the time-dependent wave functions.  
The results presented here display an interesting interplay between the tilted
anisotropy and the relative orientation of the
light-polarization ellipse.
Moreover, we show that the gaps
may be tuned by changing
the orientation of the elliptical polarization profile of light.

The paper is organized as follows. In Sec. <ref> we 
introduce the low-energy effective two-dimensional Weyl Hamiltonian
under an arbitrary electromagnetic field.
Subsequently, in Sec.  <ref>, we
determine the time-dependent wavefunction of electrons in borophene subject to an 
elliptically polarized electromagnetic field.
In this same section we analyze the stability of the solutions inherited from
Mathieu functions in the strong electromagnetic field or long wavelength 
regimes.
We workout the time-dependent wave functions and the solutions' stability 
chart. To get an insight into the gap structure,
in  Sec.  <ref>,
the stability and instability regions
are projected onto the tilted Dirac cones
of the free Weyl electrons.
In Sec. <ref>,
we extract the quasi-energy spectrum from the
time-dependent wave function and prove that it
consistently shows a similar gap structure to
that of the projected chart.
Finally, we summarize and conclude in Sec. <ref>.



§ WEYL ELECTRONS SUBJECT TO ELECTROMAGNETIC
FIELDS




 §.§ The Weyl Hamiltonian

The single-particle low-energy effective Weyl Hamiltonian
is given by 
<cit.>,

    Ĥ=μ(v_tP̂_yσ̂_0
      +v_xP̂_xσ̂_x +v_yP̂_yσ̂_y),

where P̂_x and P̂_y are the momentum operators, 
σ̂_i are the  Pauli matrices,
and σ̂_0 is the  2× 2 identity matrix.
For 8-Pmmn borophene,
the three velocities in the Weyl
Hamiltonian (<ref>)
are v_x=0.86v_F, v_y=0.69v_F and v_t=0.32v_F in
units of the Fermi velocity v_F=10^6 m/s <cit.>.
The two Dirac points 𝐤=±𝐤_D are given
by the valley index μ=± 1. The first term in
Eq. (<ref>) gives rise to the tilting
of the Dirac cones and the last ones correspond
to the kinetic energy.

The previous Hamiltonian results in the energy dispersion relation
<cit.>

    E_η,k^μ = μħ v_t k_y+ηϵ,

where 

    ϵ=ħ√(v_x^2 k_x^2+v_y^2k_y^2).

The corresponding free Weyl electron wave function is,

    ψ_η, k^μ(𝐫)
      = μexp(ik· r)/√(2)[
      [         1; ηexp(i Θ) ]],

where η= ± 1 is the band index,
Θ= tan^-1 (v_yk_y/v_xk_x) and the two-dimensional
momentum vector is given by 𝐤=(k_x,k_y). 



 §.§ The Weyl Hamiltonian in the presence of
an electromagnetic wave


Now we consider a charge carrier described by the Weyl Hamiltonian
subject to an electromagnetic wave that propagates along a direction
perpendicular to the surface of the crystal.
From Eq. (<ref>) and using the
minimal coupling we obtain, 

    Ĥ= ( 
    [            v_tΠ̂_y v_xΠ̂_x- i v_yΠ̂_y; v_xΠ̂_x+ i v_yΠ̂_y            v_tΠ̂_y ]),

where Π̂=P̂-eA, with
A=(A_x,A_y)
being the vector potential of the incident electromagnetic 
wave. Calculations are considerably simplified by choosing
a gauge in which the vector potential is only a function of time.
The Schrödinger equation for charge carriers in a Weyl semimetal
is thus given by

    Ĥ(𝐫,t)Ψ(r,t)
    = i ħ∂/∂ tΨ(r,t),

where, in the two dimensional spinor
Ψ(𝐫,t)
 =(Ψ_A(𝐫,t),Ψ_B(𝐫,t))^⊤,
A and B label the two sublattices.

To deduce the explicit form of the wave function Ψ(𝐫,t)
from Eq. (<ref>)
we make the following ansatz

    Ψ(𝐫,t)
    =exp(i𝐤·𝐫)Φ(t),

where Φ(t)= (Φ_A(t),Φ_B(t))^⊤.
Substituting (<ref>) reduces
Eq. (<ref>) into

    H(t)Φ(t)=iħd/dtΦ(t),

where the matrix H(t) is defined in
Appendix <ref>.

The diagonal terms of H(t) can be lifted
by explicitly adding a time-dependent phase to the wave function

    Φ(t)=exp[-i/ħ∫^t ds
      α_𝐤(s)]χ(t),

with α_𝐤(t)=ħ v_t k_y-e v_t A_y(t)
and χ(t)=(χ_A(t),χ_B (t))^⊤.
Following the procedure shown in the Appendix <ref>,
Eq. (<ref>) can be recast in the form
of a second-order ordinary differential equation as

    d^2/dt^2χ(t)+F(t)χ(t)=0,

where the function F(t) is defined by

    F(t)=-i/ħσ̂·dS/dt
    +1/ħ^2[H(t)]^2,

with [H(t)]^2
=|κ_v-S|^2,
κ_v=ħ(v_x k_x, v_y k_y)
and S=e(v_x A_x, v_y A_y).
In the last expression, the vector κ_v
is the directional energy flux of the electrons, and the components
of S represent the work done by the electromagnetic wave along
the x and y directions.



§ ELLIPTICALLY POLARIZED WAVES



Let us now study the case of an elliptically polarized electromagnetic wave
characterized by the vector potential

    A= 1/Ω(E_xcos(Ω t),E_ysin(Ω t)),

where E_x and E_y are constants and
Ω is the frequency of the electromagnetic wave.
The vector potential (<ref>)
corresponds to the electric field
E=-∂A/∂ t
=(E_x sin(Ω t),-E_y cos(Ω t)).

Rewriting Eq. (<ref>)
in terms of the phase

    ϕ=Ω t,

yields the Hill equation <cit.>

    χ”(ϕ)+F(ϕ)χ(ϕ)=0,

where F(ϕ) is

    F(ϕ)=i(ζ_x/ħΩσ̂_xsinϕ-ζ_y/ħΩσ̂_ycosϕ)
    
    +(1/ħΩ)^2[ϵ^2-2κ_v·S
    +1/2(ζ^2_x+ζ^2_y).
    .+1/2(ζ^2_x-ζ^2_y)cos(2ϕ)],

and

    S   =   (ζ_xcosϕ,ζ_xsinϕ),
    ζ_x   =   eE_xv_x/Ω,
    ζ_y   =   eE_yv_y/Ω.

The unitless parameter ϵ/ħΩ
is the ratio of the electron energy to the photon energy.
Similarly, the parameter ζ_x/ħΩ 
(ζ_y/ħΩ) is the ratio of the work done by the
electromagnetic wave along the x (y) direction
to the photon energy.   

The determination of the stability regions of
the differential Eq. (<ref>)
is quite challenging mainly due to the
imaginary part in the first term of the
right-hand side of Eq. (<ref>).
While the real part gives rise to the Whittaker-Hill equation<cit.>, the imaginary term yields a Mathieu-like equation
with complex characteristic values,
rarely discussed in literature <cit.>.






Fortunately, in the intense electric field
or long wavelength regimes the imaginary part is negligible.
Other limits are treatable by
perturbation theory <cit.>.

Here we focus on the intense electric field regime.
We thus assume that ζ_i/ħΩ≫1 with i=x, y,
which is equivalent to ecE_x/ħΩ^2≫ 349 and
ecE_y/ħΩ^2≫ 435. This corresponds to
electric fields E_x≫1.91 V/m and E_y≫2.39 V/m.
Thereby, we can neglect the linear terms of ζ_i/ħΩ
in Eq. (<ref>) that yield the
imaginary terms.
The obtained expression,
best-known for describing the dynamics of the
parametric pendulum<cit.>,
is the Mathieu differential equation 

    χ”(ϕ)+[a - 2q cos(2 ϕ)]χ(ϕ)=0.


The purely real parameters q and a are given by

    q    =   ζ^2_y-ζ^2_x/(2ħΩ)^2
      =(e/2ħΩ^2)^2
      (v^2_yE^2_y-v^2_xE^2_x)
      ,
    
    a   =   ϵ^ 2+ζ^2_y/(ħΩ)^2-2q
      =2ϵ^ 2+ζ^2_x+ζ^2_y/2(ħΩ)^2
               = ϵ^2/(ħΩ)^2+
         (e/ħΩ^2)^2
          (v^2_xE^2_x+v^2_yE^2_y).

The characteristic value of the Mehtieu equation

    √(a)=Ω_0/Ω,

with Ω_0=√(2ϵ^2+ζ^2_x+ζ^2_y)/√(2)ħ
is the ratio between the fundamental frequency Ω_0 and the 
frequency of the electromagnetic wave Ω.
The characteristic value a parametrizes
the family of ellipses in the k_x-k_y plane
that are characterized by the 
eccentricity [1-(v_y^2/v_x^2)]^1/2.
Stated differently, each value of the parameter a
corresponds to a particular elliptical section of the Dirac cone.
However,
not all the ordered pairs in the q-a plane produce stable solutions
of the Mathieu equation.
Consequently, in the presence of an intense electromagnetic
radiation not all the elliptical sections of the Dirac cones
correspond to stable solutions. In fact, the interaction with
light induces elliptical sections of the
Dirac cone that alternate between forbidden (unstable)
and allowed (stable) solutions.
As can be seen in Fig. <ref>,
the stability chart in the q-a plane
consists of tongue-like stable regions (light blue)
that neighbor  with unstable
regions (white). The Mathieu equation might have either even or
odd stable solutions.
Even stable solutions of (<ref>) have the form

    χ(ϕ)=𝒞(a,q,ϕ)
     =exp[-ir(a,q)ϕ]f_C(ϕ),

where 𝒞(a,q,ϕ) is the even Mathieu function,
f_C(ϕ) is an even function with period π
and a=a_r(q) is the Mathieu even characteristic value.
Conversely, odd stable solutions have the form

    χ(ϕ)=𝒮(a,q,ϕ)
     =exp[-ir(b,q)ϕ]f_S(ϕ),

where 𝒮(a,q,ϕ) is the odd Mathieu function,
f_S(ϕ) is an odd function with period π
and b=b_r(q) is the Mathieu odd characteristic value.
When r is a non-integer rational number, inside the stable regions,
the even and odd characteristic values are identical, namely 
a_r(q)=b_r(q).
The rational function r(a,q) depends on the
Mathieu characteristic value a and the parameter q.
On the boundaries between the stable and unstable regions
(solid and dashed blue lines Fig. <ref>)
r takes an integer value and inside the stable regions
r is a non-integer rational number.
Thus, inside the stability regions the even and odd
Mathieu functions have the same characteristic value.
For the particular situation in which q=0, (<ref>)
reduces to the differential equation
of an harmonic oscillator whose solutions are  
cos(√(a)ϕ) and sin(√(a)ϕ) <cit.>.
Evidently, in this case r=√(a).
Moreover, in the special case where r=√(a)∈Z,
resonant states are generated for which

    Ω_0/Ω=1, 2, 3,...,

and therefore when q → 0 two contiguous
stability zones are connected.







 §.§ Wave function and stability spectrum


The general solution of Eq. (<ref>) is
the superposition of the even and odd Mathieu functions 
𝒞(a,q,ϕ) and 𝒮(a,q,ϕ).
The wave function is then given by

    Ψ(r,t)
      = 𝒩exp[
        i(k·r
         -v_t k_y t 
         - v_tζ_y/v_yħΩcos(Ω t)
         )] 
    ×[ 𝒞(a,q,Ω t)
        ± iη𝒮(a,q,Ω t) ]
        ([        1; ηexp(iΘ) ]),

where 𝒩 is a normalization constant,
Θ= tan^-1(v_y k_y/v_x k_x),
and η=± 1 denotes the conduction and valence bands, respectively.
The wave function (<ref>) reduces
to the free-particle wave function (<ref>)
when the electric field vanishes.

Since the time-dependent wave function is expressed
in terms of the Mathieu functions, its stability
is governed by the stability chart in <ref>
that we discussed previously.
Indeed, the structure of the dynamical gaps of Weyl electrons,
generated in the presence of an intense electromagnetic radiation,
is inherited from the properties of the characteristic
values of the Mathieu functions.

The chart can be divided into two key regions according to the
shape of the electromagnetic wave: q<0 (ζ_x>ζ_y) and
q>0 (ζ_y>ζ_x) and q=0 (ζ_x =ζ_x).
If ζ_x>ζ_y (E_x v_x>E_y v_y) the work done by the
electromagnetic
wave on the electrons is higher along the x axis. Conversely, if  
ζ_x<ζ_y (E_x v_x<E_y v_y) the work is higher along y.
Finally, if ζ_x=ζ_y (E_x v_x=E_y v_y) the electromagnetic
wave contributes with equal amounts of work in each direction.
Nevertheless, the electron state can not access any point (q,a)
in the stability chart shown in Fig. <ref>;
Eq. (<ref>) imposes an extra constraint.
Defining 
Γ=ϵ/ħΩ=√(v_x^2k_x^2+v_y^2k_y^2)/Ω,
Eq. (<ref>) takes the form
of a straight line a=Γ^2+(ζ_y/ħΩ)^2-2q in the q-a
plane. 
Hence, for any state to be accessible to the electron,
the ordered pair (q,a) must
satisfy the inequality

    a ≥(ζ_y/ħΩ)^2-2q.

In Fig. <ref> the solid purple and
solid red lines illustrate the limiting case

    a=(ζ_y/ħΩ)^2-2q,

for ζ_x>ζ_y and ζ_y>ζ_x respectively. 
Naturally, any of these points should also fall on the stable regions
allowed by the Mathieu equation in order to produce a stable
solution of the wave function.

For fixed E_x, E_y and Ω, q is constant,
and therefore the allowed states should be located on
the vertical line q=const (see for example
the green dotted line or the gray dotted line
in Figs. <ref> and <ref>).
Along this lines, the ranges of stable and unstable states alternate
producing the appearance
of bands separated by dynamical energy gaps.
The opening of these gaps is due to the space-time diffraction
of electrons in phase with the electromagnetic field,
and effect akin to the magnetoacoustic diffraction of electrons
in phase with acoustic waves <cit.>.

To further comprehend the connection
between the Mathieu stability chart
and the consequent wave function gap structure,
it is illustrating to project
the stable and unstable regions
of Fig. <ref>
on the surface of the tilted Dirac cones
that arise from the free particle Weyl equation.
To this end, we explicitly express the normalized energy dispersion
E=(1/ħΩ)E^+_± 1, k from
Eq. (<ref>)
in terms of the normalized wave vector components 
(k_x,k_y)
and the parameter a in the Eq. (<ref>)
obtaining

    (v_x/v_F)^2k^2_x
    +(v_y/v_F)^2k^2_y
    =a+2q-(ζ_y/ħΩ)^2,

where k_x= (v_F/ħΩ) k_x and
k_y= (v_F/ħΩ) k_y.
Elliptical rings of allowed and forbidden states form in the
(k_x,k_y) plane or on the
surface of the Dirac cone
for fixed values of E_x and E_y
(or fixed values of q and ζ_y/ħΩ).
The dressed Dirac cones, the Dirac cones over whose
surfaces the allowed and forbidden states have been projected,
are shown in Fig. <ref>.
The light blue portion of the surface represents the allowed states
and the white rings are the forbidden ones. The first
correspond to the stable regions and the latter to the unstable
regions
of Fig. <ref>.

In Figs. <ref> (a) and (b) we plot
the vertical line q= - 1.77 (green dotted line) and
the line (<ref>)
(solid purple line) superimposed to a zoom
of the stability chart
for typical electromagnetic
field values E_x=5.5 V/m, E_y=2.6 V/m and
Ω=50 GHz.
The crossing between these two lines,
seen in Fig. <ref> (a), is the starting point
for the search of stable solutions.
However, in the immediate region above the crossing we observe
a gap of unstable solutions, that projected on to the
Dirac cone produces the appearance of forbidden
states at the tip, forming a gap.
At higher energies, we observe the rings corresponding to
the third order gap as can be appreciated in
Figs. <ref> and <ref> (b).
This gap yields
and energy range Δ^L_3=13.22 μeV
(see Appendix <ref>) of forbidden states.
It should be noted that the origins of the first gap at the
Dirac point and the following ring-like forbidden regions
are essentially the same.
Both of them are generated in points that comply with
the inequality (<ref>),
and as a result of
inherent instabilities of the Mathieu solutions.

When the parameters are chosen to
fall on the opposite side of the stability chart (ζ_x<ζ_y)
the arrangement of the gaps is quite different.
In Fig. <ref> (c) we observe the
crossing of the vertical line q=0.87
and the limiting line (<ref>)
for electromagnetic field values E_x=2.6 V/m, E_y=5.5 V/m and
Ω=50 GHz.
In contrast to the previous case, above the crossing
we find ourselves well inside a stability region.
Hence, the tip of the Dirac cone
is dressed entirely with allowed states and the forbidden rings
appear well above it as can be seen
in Fig. <ref>.
At high energies the line q=0.87 crosses the second order gap
as can be seen in Fig. <ref> (d).
A ring of unstable states with an energy gap
of Δ^R_2=19.45 μeV is projected on to the Dirac cone
until the line reaches the next stability zone
[see Fig. <ref> (b)].





To systematize the search of gaps in the K point
of the Dirac cone we define
the indicator

    Δ E(q)/2=ħΩ√(c_r(q)
      -[(ζ_y/ħΩ)^2-2 q]),

where r∈Z and c_r(q) is either a_r(q)
or b_r(q), depending on which one is at the bottom
of the allowed band.
This indicator corresponds to the energy difference between
the lower allowed band edge and the limiting case of
the inequality (<ref>) given by (<ref>).
The integer r is chosen so that the
purple line in Fig. <ref>
is situated directly below
the top band edge associated with the stable region.
Therefore, if for a given value of q
the purple line falls on a forbidden region
of states then Δ E(q)>0. If, on the other hand,
the purple line falls on an allowed band Δ E(q) is
a pure imaginary number. The domain where Δ E(q) is a
pure real number corresponds, thus, to a gap of forbidden states.
Hence, the function E(q) provides with a clear-cut
criterion to detect the formation of gaps in
the surroundings of the Dirac point:
Δ E(q)∈R.

In Fig. <ref> we analyze
the gap formation through the behaviour of Δ E(q)
in the case ζ_x>ζ_y for fixed E_y=2.6V/m.
The domains of q where Δ E(q)>0 are shown as solid brown lines
in Fig. <ref> (a).
The point given by E_x=5.5 V/m, q=-1.77,
examined previously in Fig. <ref> (a),
is shown in Fig. <ref> (a).
We notice that this value of q falls inside one of the
domains where Δ E(q)>0,
therefore indicating the presence of a gap opening at the tip
of the cone. Moreover, another useful property of Δ E(q)
is that it gives the energy of the gap.
In this example Δ E(-1.77)/2=17μeV
which corresponds to the gap shown in Fig. <ref>(a).
In contrast, for E_x=2.6 V/m, E_y=5.5 (ζ_x<ζ_y)
and q=0.89, Fig. <ref> (b) shows
that there is no gap formation as expected from
Fig. <ref>.




These results clearly show that if E_x>E_y, the function
Δ E(q)/2 presents more domains that yield forbidden gaps
close to the Dirac points than E_y>E_x.
This strongly suggests that
the shape and position of the gaps is strongly influenced
by the relative orientation of the minor and major axes of
the elliptical profiles of the radiation and the
free-Weyl particle Dirac cone. If the major and minor
axes of the ellipses arising from the Dirac cone are perpendicular
to the major and minor axes of the ellipses of
the electromagnetic profile, a gap
at the Dirac point is more likely to form.
Otherwise, if the ellipses are oriented
in the same direction, gaps are more improbable.
It is important to emphasize that this does not
imply that the two ellipses necessarily have to have the
the same proportions.
Nevertheless, the only way to correctly predict the formation
of a gap in the Dirac point is to determine
if the value of q falls inside one of the domains
were Δ E>0, as we discussed above.



 §.§ Quasi-energy spectrum


The Hamiltonian in Schrödinger equation (<ref>)
is a periodic function of time,
therefore the Floquet theorem must hold
and, consequently, the wave function must be of the form

    Ψ(r,t)
        =𝒩exp(-iℰt/ħ)
        f(Ω t).

The phase ℰ is usually termed the quasi-energy and
f(Ω t)=f(Ω t+2π)
is a periodic function of time with the same period
T=2π/Ω as the Hamiltonian.
Using Eqs. (<ref>) and (<ref>)
we can rearrange
the Mathieu functions as

    𝒞(a,q,Ω t)-iη𝒮(a,q,Ω t)=
    exp[-i η r(a,q) Ω t]f(Ω t),

where r(a,q) is purely rational and therefore a_r=b_r as we
discussed before.
Substituting Eq. (<ref>) into
(<ref>) and
comparing with (<ref>)
we find the explicit expression for the
quasi-energy

    ℰ/ħΩ= v_tk_y/Ω+η r(a,q).


To better visualize the shape of the spectrum
we may use r≈√(a) for r≫ 1
to approximate the quasi-energy. Using
the definition of a (<ref>) and taking the limit
for large quasi-momenta we get

    lim_k_x,k_y→∞ℰ=
        ħ v_tk_y+ħ√(v_x^2 k_x^2+v_y^2k_y^2).

Hence, the quasi-energy spectrum asymptotically approaches
the Dirac cone for large quasi-momentum values.
This feature is clearly seen
in Fig. <ref>. The light blue
straight lines are the section of the Dirac cone
cut by the k_x=0 plane and the solid gray lines are
the quasi-energy spectrum in the approximation r=√(a).
The blue, orange, green and red lines correspond to the
different bands arising from the exact expression
of the quasi-energy (<ref>).
We readily confirm that
both, the exact and the approximated quasi-energy spectrum lines,
asymptotically come close to the Dirac cone.
This is also seen in Fig. <ref>, where
the full quasi-energy surface is shown and
the Dirac cone is depicted as a light blue semi-transparent surface.
The most striking characteristic of
these plots is the formation of gaps.
In Fig. <ref> (a) a gap
appears close to the Dirac point for E_x=5.5 V/m, E_y=2.6 V/m
and q=-1.77.
It is located inside the same region
as the forbidden states in the dressed Dirac cone 
shown in Fig. <ref> (a).
In the full spectrum of Fig. <ref> (a)
this gap translates into a disc-shaped vacuum of states in the tip
of the quasi-energy spectrum.
Instead, for E_x=2.6 V/m, E_y=5.5 V/m and q=0.87
there is no gap formation close to the Dirac point,
though,  a ring-shaped gap appears around the tip of the quasi-energy
spectrum as it is shown in
Figs. <ref> (b) and <ref> (b).
These results are consistent with the ones found for the dressed
Dirac cones. Newly, the relative orientation of the minor and
major axes of the radiation and the Dirac cone determine
the location and structure of the gaps.





§ CONCLUSIONS


We have systematically investigated the wave function stability
of charge carries in a Weyl Hamiltonian under an intense
elliptically polarized electromagnetic radiation.
To this end, we have worked out the time-dependent
wave function from the Schrödinger equation in the
limit of strong electric field (or long wavelength).
We have proven that the stability properties of the wave
functions are inherited from the Mathieu functions,
in terms of which they are expressed.
The analysis of the stability chart of the Mathieu functions
projected onto the Dirac cones shows the formation of gaps
of unstable states for certain domain regions of the quasi-momentum
space. We have shown that the structure of the gaps
strongly depends on the alignment between the minor and major axes
of the elliptical profiles of the radiation and the Dirac cone
of the free-Weyl particle spectrum. In summary, the formation
of a gap at the Dirac point is more likely if the radiation
and Dirac cone axes do not match, or are perpendicular.
Otherwise, if the
axes are aligned, ring-shaped gaps form
for higher energies.
The quasi-energy spectrum extracted from the phase of the
wave function consistently reproduces the position and shapes
of these gaps.
Magnitude estimations of the electromagnetic fields
and gap values were presented for the 8-Pmmn borophene phase.




§ ACKNOWLEDGEMENTS


This work was supported by DCB UAM-A grant numbers
2232214 and 2232215, and UNAM DGAPA PAPIIT IN102717.
 V.G.I.S and J.C.S.S. acknowledge the total support from 
DGAPA-UNAM fellowship. 





§ 

In this Appendix, we derive Eqs. (<ref>) and
(<ref>) from the ansatz (<ref>) 
and (<ref>). We start from  the Dirac equation 
(<ref>) and  applying the solution 
(<ref>), we obtain Eq. (<ref>), 
where

    H(t)= 
    [   α_𝐤 β^*_𝐤;   β_𝐤   α_𝐤 ],

whose matrix elements are given by

    α_𝐤=ħ v_t k_y- v_t e A_y,
          β_𝐤=ħ(v_x k_x+i v_y k_y)-e[v_x A_x+i v_y A_y].

In the previous equatins,
the vector components A_x and A_y only depend on
the time variable. 

Now, we substitute Eq. (<ref>) in Eq. 
(<ref>), to get

    H(t)χ(t)
    =iħd/dtχ(t),

where

    H(t)= 
    [     0 β^*_𝐤;   β_𝐤     0 ].

From Eq. (<ref>) we obtain

    d^2/dt^2χ(t)
    +i/ħ[d/dtH(t)]χ(t)
    +1/ħ^2[H(t)]^2χ(t)=0,

where

    d/dtH(t)
    =-σ̂·dS/dt.

In the equations above we used the vectors 
σ̂=(σ̂_x, σ̂_y) and
S=e(v_x A_x, v_y A_y). Therefore, Eq. 
(<ref>) can be reduced into

    d^2/dt^2χ(t)+F(t)χ(t)=0,

with

    F(t)=-i/ħσ̂·dS/dt+1/ħ^2[H(t)]^2,

and

    [H(t)]^2
    =|κ_v-S|^2,

where κ_v=ħ(v_x k_x, v_y k_y).



§ 

To estimate the gap size, first,
we calculate the value of the parameter q from the Eq. 
(<ref>), and subsequently use the expression 
<cit.>

    Δ_r=ħΩ√(|b_r(q)-a_r(q)|),

where a_r(q) and b_r(q) for r∈Z
are the boundaries of a forbidden region as it is shown
in Fig. <ref>.  

For a microwave with frequency Ω= 50 GHz,
we estimate the gap in the following cases:
a) For E_x=5.5 V/m and E_y=2.6 V/m ( ζ_x>ζ_y),
we find q=-1.77 and Δ^L_3 = 13.22 μ eV. 
b) For E_x=2.6 V/m and  E_y=5.5 V/m (ζ_x<ζ_y),
we find q=0.87 and Δ^R_2 = 19.45 μ eV. 



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