3	On the Possibility of Observing Tunable Laser-Induced Bandgaps in Graphene Hernán L. Calvo, Horacio M. Pastawski, Stephan Roche and Luis E.F. Foa Torres

CONTENTS

3.1    Introduction

3.2    Dirac Fermions: k.p Approach

3.2.1    Dynamical Gaps Unveiled Through the k.p Model

3.2.2    Energy Gap Tuning Around Engineered Low-Energy Dirac Cones

3.3    Tight-Binding Model

3.3.1    Electronic Transport Through Irradiated Graphene

3.4    Conclusions

Acknowledgments

References

3.1    INTRODUCTION

Graphene offers a wealth of fascinating opportunities for the study of truly two-dimensional physics [1]. The last years have seen an unprecedented pace of development regarding the study of its electronic, mechanical, and optical properties, among many others. More recently, the interplay between these properties is making its way to the forefront of graphene research [2,3,4,5], opening up many promising paths for technology. Here, we focus on the interplay between optical and electronic properties, and we address the issue of tunability of the latter using a laser field. In the following we give a brief overview to this topic, with emphasis on our recent work [11], which we extend in the forthcoming sections.

In 2008 Syzranov and collaborators [6] predicted the emergence of dynamical gaps in graphene when illuminated by a laser of frequency Ω. The authors [6,7] studied the effects of these gaps, located at ±ℏΩ/2 above/below the charge neutrality point on the electronic properties of n-p junctions illuminated by a linearly polarized laser. Later on, Oka and Aoki [8] predicted the emergence of another gap at the charge neutrality point when changing the polarization from linear to circular; this was confirmed in [9]. Similar results were also presented for bilayer graphene by Abergel and Chakraborty [10].

In what follows, we will show that these strong renormalizations of the electronic structure of graphene can be rationalized in a transparent way using the Floquet theory, which allows us to explore electronic and transport properties of materials in the presence of oscillating electromagnetic fields. The origin of the dynamical gaps will be shown to be related to an inelastic Bragg’s scattering mechanism occurring in a higher-dimensional space (Floquet space). In [11] we followed this line of research trying to answer questions such as: What is the role of the laser polarization on these dynamical gaps, and which experimental setup would be needed to observe them? Our results showed that the polarization could be used to strongly modulate the associated electrical response. Furthermore, the first atomistic simulations of the electrical response (dc conductance) of a large graphene ribbon (of about 1 micron lateral size) were presented. Our results hinted that a transport experiment carried out while illuminating with a laser in the mid-infrared could unveil these unconventional phenomena. There are also additional ingredients that add even more interest to this proposal: the current interest in finding novel ways to open bandgaps in graphene and the promising prospects for optoelectronics applications.

Other recent studies have also contributed to different aspects of this field, including the analogy between the electronic spectra of graphene in laser fields with that of static graphene superlattices [12], the possible manifestation of additional Dirac points induced by a superimposed lattice potential [13], a study of the effects of radiation on graphene ribbons showing that for long ribbons there is a ballistic regime where edge states transport dominates [14], a proposal to unveil the dynamical gaps using photon-echo experiments [15], and the effects of radiation on tunneling [16] or tunneling times [17]. The influence of mentioned laser-induced bandgaps on the polarizability was also recently discussed in [18]. On the experimental side, the only available experiments are in the low-intensity regime where renormalizations such as those mentioned before are not important [19]. Further theoretical calculations within a master equation approach in that regime were also presented in [20].

Another captivating possibility is the generation of a topological insulator through ac fields as proposed in [21,22]. The idea is that ac fields could be used to generate topologically protected states that conduct electricity along the sample edges. We expect for this to become one of the most promising research lines in the field of ac transport [23,24,25].

In the following sections we discuss the interplay of a laser field in both the electronic structure and transport properties of graphene. In Section 3.2 we introduce the so-called k.p model and describe in detail the underlying mechanisms for the opening of energy gaps both in the vicinity of the low-energy Dirac point and at the energies ε = ±ℏΩ/2, where new symmetries are superimposed by the laser field. We derive approximate formulas for the new dynamical gaps and contrast them with numerical data. In Section 3.3 we turn to a π-orbitals tight-binding model where the explicit structure of the lattice is taken into account. Under this scenario, we first incorporate the time-dependent field by using the Floquet theory, and then we calculate the local density of states for a linearly polarized laser. When studying the electronic properties of a graphene layer weakly interacting with a boron-nitride On the Possibility of Observing Tunable Laser-Induced Bandgaps in Graphene substrate, new Dirac points are found to occur at higher energies related with the wavelength of the Moiré pattern induced by the graphene/boron-nitride mismatch as reported in [13]. In our case, laser illumination induces a superimposed spatial pattern [12], which may lead to a similar phenomenon when the laser polarization is linear. Departure from linear polarization leads to a suppression of these new Dirac points and dynamical gaps develop. Furthermore, we analyze the transport properties of the model by calculating the dc conductance for laser wavelengths in the mid-infrared domain and explore the laser-induced dips for different choices of the parameters. Finally, in Section 3.4 we summarize the obtained results and discuss their impact on possible future applications.

3.2    DIRAC FERMIONS: K.P APPROACH

Because of the honeycomb symmetry dictating the arrangement of carbon atoms in graphene, the electronic states that are relevant to the transport properties are located close to the two independent Dirac points (valleys) K and K′ alternately arranged at the edges of the Brillouin zone. For a clean sample (i.e., without any impurity or distortions), and by considering that the proposed external ac field does not introduce any intervalley coupling, we can treat both degeneracy points separately.

For each one of the two valleys, the electronic states can be accurately described by the k.p method where the energy spectrum is assumed to depend linearly on the momentum k. The electronic states are then computed by an envelope wave function Ψ = (Ψ_A, Ψ_B)^T. These two components refer to the two interpenetrating sublattices A and B [26,27], and are usually called the sublattice pseudospin degrees of freedom, owing to their analogy with the actual spin of the carriers. The energies and wave functions are thus solutions of the Dirac equation:

$H\Psi =\text{ε}\Psi$

(3.1)

with H the Hamiltonian operator of the system and ε the eigenenergies. The time periodic electromagnetic field, with period T = 2π/Ω, is assumed to be created by a monochromatic plane wave traveling along the z axis, perpendicular to the plane defined by the graphene monolayer. The vector potential associated with the laser field is thus written as

$\mathbf{\text{A}}(t)=\Re \left[{\mathbf{\text{A}}}_0{e}^{-i\Omega t}\right]$

(3.2)

where A₀ = A₀ (1, e^iφ) refers to the intensity A₀ = E/Ω and polarization φ of the field. For this choice of the parameters, φ = 0 yields a linearly polarized field A(t) = A₀ cos Ωt (x + y), while φ = π/2 results in a circularly polarized field A(t) = A₀ (cos Ωt x + sin Ωt y). Notice here that the k.p approach preserves circular symmetry around the degeneracy point, and in consequence, there is no preferential choice for the in-plane axes x and y. As we shall see in Section 3.3, this is not the case in the tight-binding model for which an explicit orientation of the lattice has to be fixed. Therefore the graphene electronic states in the presence of the ac field are encoded in the Hamiltonian:

$H(t)=v_F\sigma \cdot [p-eA(t)]$

(3.3)

where v_F ≈ 10⁶ m/s is the typical value for the Fermi velocity in graphene and $\sigma =\left({\widehat{\sigma }}_x,{\widehat{\sigma }}_y\right)$ is the vector of Pauli matrices describing the pseudospin degree of freedom.

We operate in the nonadiabatic regime in which the electronic dwell time (i.e., the traverse time along the laser spot) is larger than the period T of the laser, such that the electron experiences several oscillations of the laser field. Hence the Floquet theory represents a suitable approach to describe such electron-photon scattering processes. The Hamiltonian presented in Equation (3.3) is thus expanded into a composite basis between the usual Hilbert space and the space of time periodic functions. This new basis defines the so-called Floquet pseudostates |k, n〉_±, where k is the electronic wave vector, characterized by its momentum, and n is the Floquet index associated with the n mode of a Fourier decomposition of the modulation. The subindex refers to the alignment of the pseudospin with respect to the momentum. Considering the K valley, a + (–) labeled state has its pseudospin oriented parallel (antiparallel) to the electronic momentum. In this representation of the Hamiltonian, the time dependence introduced by the external potential is thus replaced by a series of system’s replicas arising from the Fourier decomposition. In this so-called Floquet space, one has to solve a time-independent Schrödinger equation with Floquet Hamiltonian H_F = H−iℏ∂_t. Recursive Green’s function techniques [28] can be exploited to obtain both the dc component of the conductance and the density of states (DoS) from the so-called Floquet Green’s functions [29].

The matrix elements of the resulting Floquet Hamiltonian are then computed as [30]

$H_{i,j}^{(m,n)}=\frac{1}{T}{\displaystyle {\int }_0^{T}dt{H}_{i,j}(t)e^{i{(m-n)}^{\Omega t}}+m\hslash {\text{Ωδ}}_{mn}{\text{δ}}_{ij}}$

(3.4)

where δ stands for the Kronecker symbol, i and j indicate the pseudospin orientation with respect to the momentum, and m and n are Fourier indices related with the field modulation. Therefore for a given vector k = k (cos α, sin α), the m diagonal block matrix contains both the electronic kinetic terms ± ħv_F k and the m component of the modulation of the field mħΩ, i.e.,

$H^{m,m}=\hslash v_F\left(\begin{array}{cc}k+m\text{Ω}/v_F& 0\\ 0& -k+m\text{Ω}/v_F\end{array}\right)$

(3.5)

where we have transformed the standard basis (the one related with the nonequivalent sublattices) into a diagonal basis for the diagonal block. Due to the particular sinusoidal time dependence of the field, only inelastic transitions involving the absorption or emission of a single photon are allowed, i.e., Δm = ±1, and the corresponding off-diagonal block matrix connecting Floquet states with a different number of photons reads

$H^{(m,m+1)}=\left(\begin{array}{cc}{\text{γ}}_1& {\text{γ}}_2\\ -{\text{γ}}_2& -{\text{γ}}_1\end{array}\right)$

(3.6)

FIGURE 3.1 (a) Scheme of the quasi-energies as a function of the electronic momentum k. The relevant crossing regions, marked by circles with dotted and solid lines, yield the dynamical and central gap, respectively. (b) Representation of the Floquet Hamiltonian for the k.p approach: Circles correspond to Floquet states and lines denote inelastic hopping elements. (From Calvo, H.-L., et al., Applied Physics Letters, 98, 232103, 2011. Copyright © 2011 American Institute of Physics. With permission.)

and it is responsible for the photon-assisted tunneling processes. Here, the direct hopping term γ₁ = eA₀v_F/2 (cos α + e^–iφsin α), with α = tan^–1(k_y/k_x), sets the transition amplitude between Floquet states with same pseudospin character. On the contrary, the off-diagonal term γ₂ = ieA₀v_F/2 (e^–iφcos α − sin α) introduces an inelastic back-scattering process that enables simultaneously both m and pseudospin transitions. This leads to the analog of an inelastic Bragg reflection as found in other contexts for inelastic scattering in carbon nanotubes [31,32].

In Figure 3.1 we provide a sketch of the Floquet spectrum for a given orientation of the momentum. The complete spectrum involves an average over all possible orientations, and it looks like a series of superimposed Dirac cones touching at energies mħΩ. The effect that the ac field induces on the electronic structure manifests whenever states with different pseudospin cross. Because of the hopping elements γ₁ and γ₂, a new family of gaps will open in the vicinity of the crossings shown by circles in Figure 3.1a. This defines a region where no states are available and the transport becomes drastically suppressed. As we will discuss in the next lines, the width of these gaps strongly depends on the intensity, frequency, and polarization of the field. For eA₀v_F < ħΩ these effects are clear for energies close to a half-integer of ħΩ, where a dynamical gap [6] opens as φ increases. On the other hand, as pointed out by Oka and Aoki [8], around the Dirac point another gap emerges and becomes pronounced in the circularly polarized case. As we will make clear later, the crucial difference between these two is the involved number of (photon-assisted) tunneling processes it takes to backscatter the conduction electrons. For the considered frequencies and intensities of the field, processes beyond the second order can be ignored.

In the mentioned studies [6,8], the authors have considered lasers in two ranges: the far infrared with ħΩ = 29 meV or the visible range [8]. In the former, the gaps were predicted to be of about 6 meV for a photocurrent generated in a p-n junction; in the latter, the photon energy of about 2 eV is much larger than the typical optical phonon energy 170 meV, and severe corrections to the transport properties due to dissipation of the excess energy via electron–(optical) phonon interactions can be expected. Moreover, as we pointed out in [11], appreciable effects in this last case required a power above 1 W/μm², which could compromise the material stability. To overcome both limitations we quantitatively explore the interaction with a laser in the mid-infrared range λ = 2–10 μm, where photon energies can be made smaller than the typical optical phonon energy while keeping a much lower laser power.

3.2.1    DYNAMICAL GAPS UNVEILED THROUGH THE K.P MODEL

The leading order process in the interaction with the electromagnetic field occurs at ε ~ ±ħΩ/2. In this region, we can consider an effective two-level system in which the relevant states are |k, 0〉₊ and |k, 0〉₋. The difference between the eigenenergies is estimated as twice the hopping between the mentioned states, and the resulting gap is

${\text{Δ}}_{k=\text{Ω}/2v_F}\approx e{A}_0{v}_F\sqrt{1-\cos \text{φ}\sin \text{α}}$

(3.7)

The same analysis can be done for the crossing between the symmetrically disposed states |k, 0〉_– and |k, 1〉₊ retrieving the same value for the energy gap. This clearly shows a linear dependence with the field strength and, for a fixed value of A₀, is independent of the photon frequency. Furthermore, it is interesting to note that when averaging over all orientations of the wave vector, no net gap opens in the linearly polarized case (φ = 0) since for the particular value α = π/4 the hopping vanishes and the backscattering mechanism is suppressed. For the calculation of the DoS, we use the Floquet Green’s function technique, in which we define the following Green’s function in the product space as

$G(\text{ε},k)={\left(\text{ε}I-H_F\left(k\right)\right)}^{-1}$

(3.8)

where the k-vector dependence of the Floquet Hamiltonian enters in both the diagonal blocks (amplitudes) and off-diagonal ones (directions). The DoS related with the m = 0 level includes the sum over all possible values of k and writes

$v_0(\text{ε})=-\frac{1}{\text{π}}{\displaystyle {\int }_0^{\text{∞}}kdk{\displaystyle {\int }_0^{2\text{π}}\frac{d\text{α}}{2\text{π}}\Im \left[G_{+,+}^{0,0}\left(\text{ε,}k\right)+G_{-,-}^{0,0}\left(\epsilon ,k\right)\right]}}$

(3.9)

In Figure 3.2a we show an example of the calculated DoS around the dynamical gap region for different values of the polarization. As can be seen, even in the linearly polarized case there is a strong modification in the DoS that would resemble the usual Dirac point for a Fermi energy around ε ~ ±ħΩ/2.

The decreasing number of allowed states for different orientations of the electronic momentum becomes evident through the depletion region. By increasing the polarization one finds that immediately for finite values of φ a gap is opened and reaches its maximum for the circularly polarized case. Under this situation, it is clear from Equation (3.7) that there is no dependence with the orientation of the wave vector, which can be interpreted in terms of the recovered axial symmetry, characterized by the standard one-dimensional van Hove singularities emerging at the edges of the gap.

FIGURE 3.2 Density of states in graphene for several values of the laser polarization. (a) Lowest-order effect around ε ~ ħΩ/2, k ~ Ω/2v_F. (b) Second-order effects around (ε ~ 0, k ~ 0). The chosen parameters are given by eA₀v_F/ħΩ = 0.1.

3.2.2    ENERGY GAP TUNING AROUND ENGINEERED LOW-ENERGY DIRAC CONES

To describe this mechanism, we refer to the schematic representation of the Floquet Hamiltonian depicted in Figure 3.1b. Here, the leading contribution around the Dirac point (ε ≈ 0 and k ≈ 0) comes from the four paths connecting the crossing states |k, 0〉₊ and |k, 0〉₋ through the first neighboring states. For the calculation, we observe that the mixing between these states is produced by the presence of the neighboring states |k, ±1〉_±. The effective Hamiltonian can be reduced by a decimation procedure [28] in which we keep the relevant states at m = 0. The resulting correction from the diagonal terms of the self-energy cancels out exactly, such that the energy difference only arises from the effective hopping according to

${\text{Δ}}_{k=0}\approx 2\frac{{\left(e{A}_0{v}_f\right)}^2}{\hslash \text{Ω}}\sin \text{φ}$

(3.10)

The quadratic dependence with the field strength is due to the fact that the mixing between these two states involves both the absorption and emission of a single photon. Additionally, the inverse dependence with the frequency quantifies the amount of energy absorbed and emitted during the tunneling event. For the constraint eA₀v_F < ħΩ considered here, this gap is much smaller than the dynamical gap discussed before. Note also that there is no dependence of the gap with the orientation of the k-vector, since we are assuming k = 0. By inspecting Equation (3.10), it is easy to observe that the maximum value for the gap is reached in the circularly polarized case, while no net gap opens for the linearly polarized case. According to the present approximation, we notice that in this last case no ac effects should be observed in the DoS. However, as can be seen in Figure 3.2b, there is a strong modification around the Dirac point region reflected by an increased slope in comparison with the usual DoS without any laser field. To explain this, it is convenient to include the effect induced by the crossing between states |k, −1〉₊ and |k, 1〉₋ around ε = 0 and k = Ω/v_F (right circle with solid line in Figure 3.1a). Although in this case the total energy difference going from |k, 0〉₊ to |k, 0〉₋ is 2ħΩ, we are in the resonant condition where transitions via auxiliary states |k, ±1〉_± involve the same energy difference ħΩ. The presence of states |k, 2〉_± also contributes to the self-energy correction, and the effective hopping term connecting the m = 0 states originates a gap:

${\text{Δ}}_{k=\text{Ω}/v_F}\approx \frac{{\left(e{A}_0{v}_f\right)}^2}{2\hslash \text{Ω}}{\left[5-2\cos \text{φ}\sin 2\text{α}-3{\cos }^2\text{φ}{\sin }^{2}2a\right]}^{1/2}$

(3.11)

that drops to zero for φ = 0, α = π/4 (as in the dynamical gap) and reaches its maximum for φ = π/2. This contribution is of the same order as Δ_{k = 0} and responsible for the modification of the DoS close to the Dirac point even for φ = 0. It is interesting to observe that, in addition to the gap, this transition yields the emergence of two surrounding peaks that persist even in the linear case.

In addition, we observe that this effect reduces by approximately a half the predicted value of Oka and Aoki [8]. Here we neglect higher-order contributions since we consider that the strength of the laser is small compared to the driving frequency. However, as recently discussed in [33], in the opposite limit where the number of involved photons is large, additional scattering processes enable the existence of states around the studied regions and the gaps are effectively closed.

3.3    TIGHT-BINDING MODEL

In the previous sections, we have shown how the laser field modifies the electronic structure of two-dimensional graphene, at least in any experiment carried out over a time much larger than the period T. A natural question is if these effects would be observable in a transport experiment and how. To such end we turn now to the calculation of the transport response at zero temperature using Floquet theory applied to a tight-binding (TB) π-orbitals Hamiltonian. As we shall see, the correspondence between this model and the k.p approach becomes evident in the bulk limit, where the width of the ribbon is of the order of the laser’s spot.

In the real space defined by the sites of the two interpenetrating sublattices, the electromagnetic field can be accounted for through an additional phase in the hopping γ_ij connecting two adjacent sites r_i and r_j through the Peierls substitution:

${\text{γ}}_{ij}={\text{γ}}_0\exp \left(i\frac{2\text{π}}{{\text{Φ}}_0}{\displaystyle {\int }_{r_i}^{r_j}A(t)\cdot dr}\right)$

(3.12)

where γ₀ ≈ 2.7 eV is the hopping amplitude at zero field and Φ₀ is the quantum of magnetic flux. The intensity of the magnetic vector potential is assumed to be constant along the whole sample where the irradiation takes place. Therefore the computation of such a phase is simply the scalar product between the vector potential and the vector connecting the two sites.

For numerical convenience, we consider an armchair edge structure for the lattice, and the vector potential is defined as A(t) = A_x cos Ωt x + A_y sin Ωt y, where x and y coordinates are in the plane of the nanoribbon. It is important to observe that the particular choice of the edges does not restrict the outcoming results once the bulk limit is reached. For the armchair structure we can distinguish three principal orientations according to the angle between r_j – r_i and A(t). By considering the scheme in Figure 3.3b, we define the hopping elements γ_+,±(t) and γ_2+,0(t), where the subindex denotes the x and y components of two adjacent sites. In this representation, we have

${\text{γ}}_{+,\pm }(t)={\text{γ}}_0\exp \left[i\frac{a\text{π}}{{\text{Φ}}_0}\left(A_x\cos \text{Ω}t+\sqrt{3}{A}_y\sin \text{Ω}t\right)\right]$

(3.13)

FIGURE 3.3 (a) Schematic view of the Floquet space for the tight-binding model in the real space. (b) Opening, longitudinal, and closing hopping matrices according to the relative direction between the adjacent sites and the vector potential in an armchair edge structure.

for those terms with simultaneous x and y components and

${\text{γ}}_{2+,0}(t)={\text{γ}}_0\exp \left[i\frac{2a\text{π}}{{\text{Φ}}_0}{A}_x\cos \text{Ω}t\right]$

(3.14)

for the hopping element with only an x component, respectively (see Figure 3.3b). Here a ≈ 0.142 nm is the nearest carbon-carbon distance. The Fourier decomposition of these expressions is based on the Anger-Jacobi expansion that yields a m transition amplitude in terms of Bessel functions:

${\text{γ}}_{+,\pm }^m={\text{γ}}_0{\displaystyle \sum _{k=-\infty }^{\infty }{i}^k{J}_k\left(z_x\right)J_{m-k}\left(\pm z_y\right)}$

(3.15)

${\text{γ}}_{2+,0}^m={\text{γ}}_0{i}^m{J}_m\left(2z_x\right)$

(3.16)

where $z_y=\sqrt{3}\text{π}a{A}_y/{\text{Φ}}_0$ and $z_y=\surd 3\text{π}a{A}_y/{\text{Φ}}_0$. In contrast to the k.p approximation, in this situation we may have processes involving the absorption or emission of several photons at once. This, however, decays rapidly with the number of photons, and the main contribution still comes from the renormalization at m = 0 and the leading inelastic order m = ±1.

We compute the hopping matrices (opening, longitudinal, and closing) as those connecting two adjacent transverse layers such that the difference in the Floquet indexes is m (see Figure 3.3b). If N denotes the number of transverse states, i.e., the number of electronic sites in each vertical layer, the hopping matrices are N × N blocks expressed as

$V_1^m=\left(\begin{array}{ccc}{\text{γ}}_{+,+}^m& {\text{γ}}_{+,-}^m& \\ 0& {\text{γ}}_{+,+}^m& \\ & & \ddots \end{array}\right),V_2^m=\left(\begin{array}{ccc}{\text{γ}}_{2+0}^m& 0& \\ 0& {\text{γ}}_{2+,0}^n& \\ & & \ddots \end{array}\right),V_3^m=\left(\begin{array}{ccc}{\text{γ}}_{+,-}^m& 0& \\ {\text{γ}}_{+,+}^m& {\text{γ}}_{+,-}^m& \\ & & \ddots \end{array}\right)$

(3.17)

for the opening, longitudinal, and closing geometries, respectively. Therefore in the natural basis defined on the real space, the total Floquet Hamiltonian is composed by a periodic block tridiagonal structure where the off-diagonal block matrices contain the different $V_i^m$ terms and the diagonal block only accounts for the site energies E_m = mħΩ since no gate voltages are assumed. For the proposed values of the field and a typical regularization energy η ≈ 30 μeV, the bulk limit is reached when N > 10⁴. Hence it is necessary to decompose the Floquet Hamiltonian in terms of transverse modes, since otherwise one should deal with O(N ³) operations, increasing enormously the computation time. For this reason, we limit ourselves to the linearly polarized case in which A_y = 0, which is already enough to give a hint on the transport effects. Other choices of the laser orientation or polarization would be subjected to the knowledge of more tricking bases or the employment of parallel computing techniques that are beyond the scope of this work.

The total Hamiltonian is transformed according to the normal modes of the sublattices A and B. The rotation matrix results in a trivial expansion of that in the appendix of [34] in the composite Floquet basis. The spatial part of the Hilbert space is thus reduced, and the hopping matrices now have the dimension 2N + 1 of the (truncated) Fourier space. The resulting single-mode layers preserve the same Hamiltonian structure with $V_i^m\left(q\right)=2V_i^m\cos \left[\text{π}q/\left(2N+1\right)\right]$ for i = 1, 3 and $V_2^m\left(q\right)=V_2^m$, where q is the mode number.

The DoS is therefore obtained through the Floquet Green’s functions defined as

$G_F^q\left(\text{ε}\right)={\left(\text{ε}I-H_F^q-{\Sigma }_F^q\left(\text{ε}\right)\right)}^{-1}$

(3.18)

where $H_F^q$ is the q mode Hamiltonian containing the diagonal block of the Fourier components and is written as the following matrix:

$H_F^q\left(\text{ε}\right)=\left(\begin{array}{ccccc}\ddots & & & & \\ & -\hslash \text{Ω}& 0& 0& \\ & 0& 0& 0& \\ & 0& 0& +\hslash \text{Ω}& \\ & & & & \ddots \end{array}\right)$

(3.19)

The self-energy ${\Sigma }_F^q$ is a correction term arising from the presence of the neighbor sites at both sides of the chain. The effective Hamiltonian $H_{eff}^q\left(\text{ε}\right)=H_F^q+{\Sigma }_F^q\left(\text{ε}\right)$ for a transverse layer can be calculated recursively by using a decimation procedure, and the DoS is thus obtained from the resulting Green’s function as

$v_0\left(\text{ε}\right)=-\frac{1}{\text{π}}{\displaystyle \sum _q\Im {\left[G_F^q\left(\text{ε}\right)\right]}_{0,0}}$

(3.20)

where the subindex 0, 0 corresponds to the zero photon state. Note the similarity of this expression with Equation (3.9), where the amplitude and direction of the electronic momentum are discretized in the q modes.

By employing this numerical strategy, we compute the DoS for an armchair edge structure and compare it with the one obtained from the k.p approach. As we can observe in Figure 3.4, there is in general a good agreement between both results. In particular, the depletion region around the resonances ε ~ ±ħΩ/2 is also reproduced by the TB model. However, a closer look at the peaks reveals small differences between both curves. For the TB method, we see two peaks that surround the peak obtained in the k.p model (see inset of Figure 3.4). The origin of the splitting of the peaks in the DoS can be attributed to a trigonal warping effect that is inherently included in the TB Hamiltonian, since the explicit structure of the lattice is taken into account.

FIGURE 3.4 Density of states for linear polarization of the laser field calculated by both the k.p approach (solid line) and tight-binding (TB) model (dotted line). The chosen parameters of the laser are ħΩ = 0.3 γ₀ and A₀ = 0.01 Φ₀/πa. Inset: DoS in the depletion region.

This effect is also present in the region close to the Dirac point, where both approaches reproduce an increased slope, compared with the bare result without a laser. This, however, is not visible in Figure 3.4 since the intensity of the laser is too small compared with the frequency. To estimate the relevance of this effect, we explore the DoS around the Dirac point for several values of the intensity (Figure 3.5a) and frequency (Figure 3.5b) of the laser field. In the left panel we fix the frequency at ħΩ = 0.3 γ₀ and increase the intensity (A₀ = E/Ω) from 0 to 0.05 Φ₀/πa. As we can observe, the effect the intensity induces on the distance between the peaks is approximately linear, as well as the width of the depletion region. In Figure 3.5b we fix the intensity at this final value and decrease the frequency from 0.5 γ₀ to 0.2 γ₀. In this situation the distance between the two peaks remains approximately the same while changing the frequency, thus revealing a small (even negligible) dependence.

The numerical analysis of the DoS can be complemented by considering the mode decomposition shown in Figure 3.6. This representation allows an alternative inspection of the different effects we have been discussing throughout this section. Here, we plot the DoS in grey scale as a function of the energy for each transverse mode.

The sum of all these contributions gives the standard DoS. In the figure, each panel represents a fixed value of the laser intensity, starting from A₀ = 0 to 0.05 Φ₀/πa. In the absence of a laser field, we can observe (see upper left panel) the linear behavior of the single-mode bandgaps as we move away from the central mode where no gaps occur. This central mode is defined via the identity $V_1^m\left(N_C\right)=V_2^m$ in which the system would be homogeneous; thereby no gap should be opened for this mode. In the considered example this results in N_C ~ 6600. This point constitutes the symmetry axis where both contributions at left and right sides are the same. As we turn on the laser, a gap is suddenly opened at ħΩ/2 = 0.15 γ₀ in each mode (except the central one), and its width increases linearly with the intensity of the laser.

FIGURE 3.5 Density of states around the Dirac point for a linearly polarized field as a function of the laser intensity (ħΩ = 0.3 γ₀, left panel) and the frequency (A₀ = 0.05 Φ₀/πa, right panel). The white curve in both figures corresponds to the same choice of the parameters (ħΩ = 0.3 γ₀ and A₀ = 0.01 Φ₀/πa).

In addition, we observe that the emergence of the splitting of the peaks is manifested by a slight asymmetry in their position with respect to the central mode. For A₀ > 0.03 Φ₀/πa the contribution from side modes originates the increased slope around the Dirac point region. The DoS in this region also shows an asymmetry in the size of the lobes that originates the splitting of the peaks.

3.3.1    ELECTRONIC TRANSPORT THROUGH IRRADIATED GRAPHENE

In order to estimate the influence of the laser field in a transport experiment, we calculate the two-terminal dc conductance through a graphene stripe of 1 × 1 μm in the presence of a linearly polarized laser. The sample is connected to two leads at both sides, considered a prolongation of the stripe where no laser is applied.

By following the same strategy as we used in the calculation of the DoS, we obtain the dc conductance from the recursive calculation of the Floquet Green’s functions. In this sense, the mode decomposition is again a key tool in the numerical implementation of the recursive formula for the self-energy correction. In particular, the semi-infinite leads are incorporated through an initial self-energy where the opening, longitudinal, and closing hopping matrices result in a diagonal. Since in this region no ac fields are applied, the elements of the above hopping matrices are simply ${\text{γ}}_{+,\pm }^m={\text{γ}}_{2+,0}^m={\text{γ}}_0{\text{δ}}_{m,0}$. Therefore as shown in Figure 3.7, the resulting lattice structure for a single mode in the leads is a dimer with alternating hoppings γ₀ and 2γ₀ cos [πq/(2N + 1)]. In the sample region, however, the effect of the time-dependent

FIGURE 3.6 Transverse mode decomposition of the DoS (grey scale, arbitrary units) for a linearly polarized field. The frequency of the laser is fixed at ħΩ = 0.3 γ₀, while the intensity varies from 0 to 0.05 in units of Φ 0/πa. The central mode in this example is N_C ~ 6600.

FIGURE 3.7 Scheme of the transport setup of the transverse mode decomposition of the Floquet Hamiltonian. The central region (sample) contains additional hopping elements connecting different Fourier sheets because of the presence of the time-dependent field.

field is described by the hopping elements connecting nearest-transverse layers at different Fourier levels. According to Equations (3.15) and (3.16), these decay fast with the difference between the involved Fourier replicas. For the considered examples in which eA₀v_F/ħΩ ~ 0.1 it usually converges for a small (~3) number of photons. The dc conductance is thus calculated through the transmittance via the Floquet Green’s functions (see appendix in [29]):

$T_{RL}\left(\text{ε}\right)={\displaystyle \sum _{q,n}2{\text{Γ}}_{\left(R,n\right)}^q\left(\text{ε}\right)|G_{\left(R,n\right)\leftarrow \left(L,0\right)}^R\left(\text{ε}\right)|{}^{2}2{\text{Γ}}_{\left(L,0\right)}^q\left(\text{ε}\right)}$

(3.21)

where ${\text{Γ}}_{\left(\text{α},n\right)}^q\left(\text{ε}\right)=-\Im {\text{Σ}}_{\text{α}}^q\left(\text{ε}+n\hslash \text{Ω}\right)$. The dc conductance is then obtained from the Landauer formula [35], and we consider that the system preserves space inversion symmetry, i.e., T_RL(ε) = T_LR(ε) = T(ε). Therefore, the conductance reads:

$G\left({\text{ε}}_F\right)=-\frac{2e^2}{h}{\displaystyle \underset{-\infty }{\overset{{\text{ε}}_F}{\int }}d\text{ε}T\left(\text{ε}\right)\frac{d}{d\text{ε}}f\left(\text{ε}\right)}$

(3.22)

with $f\left(\text{ε}\right)=1/\left(1+e^{\left({\text{ε-ε}}_F\right)/kT}\right)$ the Fermi function. In the zero temperature limit the derivative of f(ε) results in a Dirac delta, and we recover a linear relation between these two functions:

$G\left({\text{ε}}_F\right)=\frac{2e^2}{h}T\left({\text{ε}}_F\right)$

(3.23)

In the numerical calculation of the conductance, we consider a laser field whose wavelength lies within the mid-infrared region. We show in Figure 3.8 the resulting conductance at zero temperature for different values of the laser power. We consider λ = 10 μm (ħΩ ~ 140 meV) in Figure 3.8a and λ = 2 μm (ħΩ ~ 620 meV) in Figure 3.8b, respectively. For the case where no external laser is applied (grey dashed lines in Figure 3.8a), the conductance shows a linear dependence with the Fermi energy. In this situation there is a perfect transmission and each channel contributes a unit of the conductance quantum G₀ = 2e²/h.

FIGURE 3.8 dc conductance through a graphene stripe of 1 × 1 μm in the presence of a linearly polarized laser as a function of the Fermi energy and laser power for a laser wavelength of (a) λ = 10 μm and (b) λ = 2 μm.

This effect becomes evident in the small plateaus along the whole curve, whose widths depend on the total number of transverse modes, i.e., the width of the sample. When increasing the energy of the carriers, more channels participate in the transport and the resulting conductance increases. However, when we turn on the laser, the interaction with the field induces a significant depletion around the region ε_F ~ ħΩ/2. In the depletion region we can observe that the minimum value at ε_F = ħΩ/2 decreases with the laser power. Additionally, this minimum value approaches zero by increasing the size of the sample (i.e., the irradiated area) since the larger the time spent by the carriers, the higher the backscattering probability. Notice that there are no visible effects close to the Dirac point ε_F = 0. This is due to the fact that the laser polarization is linear, and the position of the channel bands remains unaffected. By comparing the two plots, one can observe that the effect of the laser is relatively more significant in the left panel. For instance, we observe that even for P ~ 0.03 mW/μm² the depletion is still visible, whereas in the right panel for P ~ 0.1 mW/μm² this one could be hard to distinguish. This is related to the slope of the gapped modes in the DoS of Figure 3.6 (see, e.g., upper central panel), which approximately goes like

${\left(\frac{d\text{Δ}}{dN}\right)}_{\text{ε=}\hslash \text{Ω}}\propto \frac{\sqrt{P}}{\text{Ω}}$

(3.24)

and therefore for smaller frequencies the number of modes affected by the field is increased, and thus the stronger is the suppression in the conductance. We emphasize that in the above equation the dependence of the slope (and hence the width and depth of the depletion) with the laser power is given by a square root, and in consequence there are still visible effects when decreasing this value even by four orders of magnitude.

Regarding the effect of static disorder, it is usually found that the presence of local impurities or topological defects in the lattice tend to broaden energy gaps produced by confinement (as in a graphene nanoribbon) [36]. A priori, a similar effect is to be expected here, and further studies are needed. In order to estimate the robustness of the depletion regions we calculate the conductance for λ = 10 μm and explore its behavior for different values of the temperature. This is shown on Figure 3.9, where each panel corresponds to a fixed value of the laser power. As we take a small finite temperature (~5 K), the plateaus structure is no longer visible and the curve is smoothed. The depletion regions are almost the same for the upper panels (large power) but slightly reduced in the lower ones. Nevertheless, this is still visible for the case P = 0.03 mW/μm². By increasing the temperature, the width of the depletion region also increases, whereas its height decreases to the bare value for zero field.

FIGURE 3.9 dc conductance as a function of the Fermi energy and temperature for different values of the laser power. The chosen wavelength of the laser is λ = 10 μm.

We emphasize that the dip in the conductance persists for small values of the laser power. In particular, even for temperatures around 20 K, the depletion region in the case of a laser power P = 0.03 mW/μm² is of the order of several units of G₀; thereby this should be observable in a clean sample. Such features in the conductance could be also resolved in its derivative, where a peak emerges in the region of the depletion.

3.4    CONCLUSIONS

In summary, in this chapter we have shown how a laser field would modify the electronic structure and, more importantly, the transport response (dc conductance) of graphene, where thanks to its low dimensionality and peculiar electronic structure, bandgaps can be generated. For these features to be observable, we conclude that experiments with lasers in the mid-infrared would be particularly timely. The modifications are predicted to arise both around the Dirac point and at ħΩ/2, leading to results that are strongly dependent on the laser polarization, thereby enabling its use as a control parameter.

As pointed out already in the introduction, the potential technological impact, and the high level of recent activity in this field, makes this an outstanding area for further research and much-needed experiments.

ACKNOWLEDGMENTS

We acknowledge discussions and correspondence from Junichiro Kono, Gonzalo Usaj, and Frank Koppens, as well as support from CONICET, ANPCyT, and SeCyTUNC. L.E.F.F.T. acknowledges support from ICTP–Trieste and the Alexander von Humboldt Foundation.

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