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Anomalous Photon-Assisted Tunneling in Graphene

We investigated the Dirac electrons transmission through a potential barrier in the presence of circularly polarized light. An anomalous photon-assisted enhanced transmission is predicted and explained in a comparison with the well-known Klein paradox. It is demonstrated that the perfect transmission for nearly-head-on collision in an infinite graphene is suppressed in gapped dressed states of electrons, which is further accompanied by shift of peaks as a function of the incident angle away from the head-on collision. In addition, the perfect transmission in the absence of potential barrier is partially suppressed by a photon-induced gap in illuminated graphene. After the effect of rough edges of the potential barrier or impurity scattering is included, the perfect transmission with no potential barrier becomes completely suppressed and the energy range for the photon-assisted perfect transmission is reduced at the same time.


I. INTRODUCTION
Research activity on the properties of graphene has been growing rapidly ever since its experimental discovery and demonstration of its unusual properties arising from its energy band structure [1,2]. The novel properties of graphene may be attributed to its massless Dirac fermions at the Fermi energy [3]. An interesting consequence of the Dirac electron is the Klein paradox [4] in which an electron in graphene undergoes unimpeded tunneling through potential barriers of arbitrary height and thickness. This property of Dirac electrons is due to their linear energy dispersion relation or helicity. Electrons are said to be chiral if their wave functions are eigenstates of the chirality operatorĥ = σ · p/(2p) where σ = {σ x , σ y } is the Pauli vector consisting of Pauli matrices and p = {p x , p y } is the electron momentum in graphene layers. Electrons in graphene near the K points (around the corners of the hexagonal Brillouin zone) are chiral due to the fact that the chirality operator is proportional to the Dirac Hamiltonian which automatically makes chirality a good quantum number.
The fact that the Klein paradox is also obtained in bilayer graphene makes this effect even more sophisticated. This leads us to realize that the Klein paradox is not simply due to linear electron dispersion but may be observed for both massless and massive quasiparticles [4]. In this paper, we consider a sharp p − n − p junction or potential barrier profile. This type of potential can be constructed by underlying metal contact or insulating strip [5] and was employed to demonstrate unimpeded transmission [3].
Both the tight-binding model and k · p approximations for infinite graphene sheet accurately show that electrons and holes have linear energy dispersions ε(k) = s v F k = s v F k 2 x + k 2 y with no gap, where s is the electron-hole parity with s = 1 for electrons and s = −1 for holes. For the potential barrier described above, there is a translational symmetry in the y direction parallel to the boundaries of the potential so that k y is conserved.
In contrast, the longitudinal component k x is modified by the potential so that when the particle has energy E, we have k xi = (E − V i ) 2 /( v F ) 2 − k 2 y , where V i is the potential in the region i.
The electron effective mass and its properties will play a crucial role in our analysis. In infinite intrinsic graphene, the Dirac electron is massless although the mass can be introduced [6] or implemented experimentally. For bilayer graphene, the particle effective mass exists in any possible approximation [7], leading to the existence of evanescent terms in the wave function [4] although this does not violate chirality symmetry and the Klein paradox.
The outline of the rest of this paper is as follows. In Sec. II, we derive the eigenstates and transmission coefficient for Dirac electrons dressed with photons from a circularly polarized light. Section III is devoted to a formalism in which the roughness of the boundaries for the potential barrier is included phenomenologically in calculating the transmission coefficient and numerical results are presented. A brief summary is given in Sec. IV.

II. TRANSMISSION COEFFICIENT FOR DRESSED ELECTRON STATES
It was shown recently [8,9] that when Dirac electrons in a single graphene layer are interacting with an intense circularly polarized light, electron states will be dressed by photons. The main idea of the present study is to investigate the transmission properties of such dressed electrons for the case of single layer graphene. We go beyond the approximations used in Ref. [8] by retaining the results up to the order of O(∆ 4 ) so that we are able to investigate the difference between the dressed states and massive Dirac electrons described below by the Hamiltonian in (24). Here, ∆ is a quantity measuring the induced gap between the valence and conduction bands of dressed electrons.
We begin with the electron-photon interaction Hamiltonian where v F is the Fermi velocity and the vector potential for circularly polarized light of frequency ω 0 can be expressed as in terms of photon creation and destruction operatorsâ † andâ † , respectively. Here, V is the mode volume of an optical field. In order to study the complete electron-photon interacting system, we must add the field energy term ω 0â †â to the Hamiltonian (1). As usual, we seek the wave function in the form of a plane wave Ψ(r) = e ik·r ψ(k), which results in the following reduced Hamiltonian where and the Jaynes-Cummings Hamiltonian which corresponds to a two-level quantum optical system and, most importantly, can be solved analytically. Here, we have defined k ± = k x ± ik y and N represents the number of radiation quanta (intensity) for the incident optical field. We only consider the situation such that the electron-photon interaction amplitude w is much less than either the photon or Dirac electron energy, i.e., where α ≡ w/ ω 0 ≪ 1. The two eigenstates of the Hamiltonian (5) could be obtained with an expansion over the basis of just two functions |1 N ≡ | ↑, N and |2 N ≡ | ↓, N + 1 for each N value, that is, space. This expansion yields the eigenvalue equation and its solution below where ǫ ↑ and ǫ ↓ correspond to the lower + and upper − signs in the solution. For simplicity, we assume here the radiation is classically strong with N ≫ 1. However, we note that we may set N + 1 ⋍ N only in the terms O(α 2 ) but not in the terms with N ω 0 . Furthermore, it is a simple matter to obtain the expansion coefficients µ N and ν N as One can easily verify that the wave functions (7) and (8) are the eigenstates of the Jaynes-Cummings Hamiltonian (5) with energies (11), to any order of α. Here, for all the above derivations, we have employed the standard relations, i.e., We now look for the eigenstates of the Hamiltonian (3) as expansions over the set of Jaynes-Cummings Hamiltonian eigenfunctions (7) and (8). We will confine our attention to the field source with only three nearest photon occupation numbers, i.e., N = N 0 − 1, N 0 and N 0 + 1, leading to We know that the eigenfunctions (7), (8) corresponding to different numbers N are orthogonal to each other. First acting the Hamiltonian (3) on (14), and then, multiplying both sides of the expansion (14) by Ψ ↑,N 0 |, Ψ ↓,N 0 −1 |, Ψ ↑,N 0 +1 | and Ψ ↓,N 0 |, this result in the following four equations For chosen number N 0 , each pair of these equations describe two energy subbands that are separated by ⋍ α 2 ω 0 /2 at k = 0. Here, we include only four nearest energy subbands, corresponding to the dressed states with different photon occupation numbers N and electron states with subband indices ↑ (lower energy) and ↓ (higher energy).
Taking into account the following relations for the Dirac Hamiltonian, and with the simplifications described above, we can explicitly write out the Dirac Hamiltonian terms in equations (15) and (16). The presence of photon occupation numbers does not follow from the dressed states Hamiltonian and cannot be determined only from the ratio between the photon filed energy N ω 0 and the electron-photon interaction amplitude w. It should results from the model of the circularly polarized light source. If we consider electrons near the K point such that the kinetic energy v F k ⋍ α 2 ω 0 /2, we can only retain only three photon occupation numbers N 0 − 1, N 0 and N 0 + 1, which is consistent with the approximation described above. Consequently, we arrive at the system which consists of weakly coupled equations to determine the two nearest subbands, i.e., This leads to the following energy dispersions with µ ≈ 1: where ξ = (2v F /ω 0 ) 2 and η = (2∆/ ω 0 ) 2 ≈ α 4 /4. Another approximation which we may use here is to consider only small wave vectors k so that we get two independent pairs of subbands separated by energy ω 0 . This gives a two-component spinor wave function where each component consists of two independent terms. If we consider the transmission of such states through a potential barrier whose height is equal to ω 0 , one of the terms in region "1" will exactly match the other in the region "2" (the potential region) so that a part of the wave function will be completely transmitted. This will lead to a substantial increase in the total transmission. On the other hand, if we consider a larger number of such subband pairs, this effect will not be significant since this complete transmission occurs only for one of the wave function terms. This is the reminiscent of the Klein paradox since it does not depend on the barrier width as long as the barrier height is exactly equal to ω 0 . Revealing such an anomalous increase for the dressed states tunneling is an important discovery of the present paper. The corresponding eigenvalue equation for the case of two independent pairs of subbands yields, as expected The simplest possible approximation is to keep only two terms with the coefficients leading to the two nearest energy subbands separated by a gap 2∆ ⋍ α 2 ω 0 /2 at k = 0 due to electron-photon interaction. This approximation results in a simplified algebraic system determining the pair of coefficients C 1 (k) and C 2 (k), i.e., The non-trivial solution of these equations gives the energy dispersion which was previously obtained in Ref. [8] as If the electron-photon interaction is removed, then α ≡ w/2 → 0, and the energy dispersion relations (23) demonstrate a non-interacting system consisting of a Dirac electron ε β (k) = β v F |k| and photons N 0 ω 0 . All the other energy subbands are separated at least by ω 0 ≫ w and could be neglected, which justifies the above two-subband approximation. In this notation, β = ±1 is the dressed conduction/valence band index corresponding to bare electron/hole bands for infinite graphene when ∆ → 0.
The system (22) is formally similar to the eigenvalue equations for the case of the effectivemass Dirac Hamiltonian where σ 3 is a Pauli matrix and V(x) is a one-dimensional potential. The electron dispersion and transmission properties for both a single as well as multiple square potential barriers have been studied [6,10] for monolayer and bilayer graphene [11]. It was also shown that a one-dimensional periodic array of potential barriers leads to multiple Dirac points [12].
Several papers have introduced an effective mass term into the Dirac Hamiltonian for infinite graphene which may be justified based on different physical reasons [13]. For example, it has been shown [14] that an energy bandgap in graphene can be created by boron nitride substrate resulting in a finite electron effective mass. However, we emphasize that the analogy between the Hamiltonian (24) and that for irradiated graphene is not complete since that would correspond to ∆ < 0. Although this difference does not result in any modification of the energy dispersion term containing ∆ 2 , it certainly modifies the corresponding wave function.
The interaction between Dirac electrons in graphene and a circularly polarized light has been considered in the classical limit in Ref. [15]. In this limit, a gap in the Dirac cone opens up due to nonlinear effects. The dressed state wave function has the form with C 1 (k) = C 2 (k) given by corresponding to the energy subbands ε(k) = N 0 ω 0 ± ∆ 2 + 2 v 2 F k 2 . In this notation, γ = ∆/( v F k) and φ is the angle which k makes with the longitudinal x axis. Without an optical field, i.e. ∆ = 0, we obtain C ± 1 = C ± 2 = 1/ √ 2. In the limit (∆ ≪ k), the coefficients exhibit peculiar symmetry with We note that this expansion is not valid too close to the Dirac point and should not be used for arbitrary wave vector to calculate, for example, the polarization function. Additionally, one may verify that C 1 (k) = C 2 (k) for any chosen ∆ in the range of validity. Consequently, the chiral symmetry is broken for electron dressed stateŝ Clearly, it follows from (29) that the non-chirality of the dressed electron states becomes significant if the electron-photon interaction (the leading γ term) is increased. This affects the electron tunneling and transport properties. We now turn to an investigation of the transmission of electron states through a potential barrier when graphene is irradiated with a circularly polarized light.
For simplicity, we consider a square potential barrier of height V 0 given by where W 0 is the barrier width and θ(x) is the Heaviside step function. Since the wave number k x is the same in region 1 (x < 0) and region 3 (x > W 0 ) and the current component is j x = Φ † σ x Φ, we only need the wave-function continuity at the potential boundaries for the system considered. From this continuity condition, the transmission probability T can be determined from T = |t| 2 where t is the transmission coefficient or the amplitude of the wave propagating forward in region 3. Here, we only show an analytical expression for the transmission coefficient of the dressed states in the limit of ε ≪ V 0 corresponding to k x1 ≪ k x2 : Here, θ = tan −1 (k y /k x2 ) → 0, φ = tan −1 (k y /k x1 ), the second term includes the effect of electron-photon interaction (∝ ∆ 2 ) to the leading order. In addition, we only show those terms of the lowest order in ε/V 0 . There is another relevant study [16], investigating the tunneling of Dirac electrons with a finite effective mass, a parabolic dispersion in the presence of a energy gap, and a certain chirality, through a potential region. In that study, the particle tunneling through a square potential barrier differs from both Dirac electrons and the dressed states of electrons under a circularly polarized light illumination.
For nearly-head-on collision with k y ≪ k x1 ≪ k x2 for high potential as well as for infinite graphene (∆ → 0), transmission coefficient has the following simplified form where we assume V 0 ≫ ε, θ ≪ φ ≪ 1 and β = ±1.

III. NUMERICAL RESULTS AND DISCUSSION
In our numerical calculations, energies will be measured in units of (3k F at/2) with the carbon-carbon distance a ≈ 1.42Å and the hopping parameter t = 2.7 √ 3/2 eV . We measure the wave vector in units of the Fermi wave number k F and write its components as k x = cos φ and k y = sin φ in terms of the angle of incidence φ.
In Fig. 1, we present the transmission for dressed electron states with arbitrary energy and angle of incidence. We clearly see that dressing ruins the Klein paradox in (  φ dependence is greatly suppressed when the dressed-state gap is increased in (c), leaving us a set of equally-distant branches due to photo-assisted electron tunneling.
As mentioned above, by including more than the two nearest subbands, as shown in in Fig. 3(a), the electron dispersion, wave function and transmission amplitude will be modified. According to the approximation adopted by Kibis [8], the two subband pairs may be considered as independent. Under the condition of equal transverse momentum k y for both terms of the particle wave function, the first term of the wave function in region "2" with a potential barrier is similar to the second term in the regions without potential. Therefore, the states exactly match across the potential boundary, which should definitely increase the total transmission amplitude. The other possibility is that incoming angle and momentum of the second term is totally independent of the first one and leads to resonant transmission regardless of the transmission amplitude of the first term. Based on our derived results, we find that the transmission should increase even for non-split energy subband pairs and corresponding wave function. Since the existence of certain photon occupation numbers N is determined by the laser source, we can consider only one pair of the dressed states subbands is occupied while the other subband pairs are unpopulated. As illustrated in Fig. 3(b), in the potential region "2", a particle may populate another subband corresponding to a different number of photons instead of changing its longitudinal momentum for the barrier heights exactly equal to the multiple of ω 0 . The opposite transition will occur at the boundary between regions "2" and "3". This will result in unimpeded tunneling T = 1 independent of the barrier width, as seen from Fig. 3(c), which is expected to be a major contribution to the current.
We now investigate the effect of disorder on the transmission probability through a potential barrier in graphene. This can appear as short-range disorder, inter-valley scattering and trigonal distortion. In a single layer graphene, disorder also induces a metal-insulator transition by creating a dynamical gap [17]. Consequently, this effect can modify the gap created by the electron-photon interaction. In bilayer grpahene, disorder directly leads to energy dispersion with a gap as well as modify the energy dispersion close to the band edges. Additionally, disorder may also lead to localized states inside a gap in bilayer graphene [18,19].
In this paper, we introduce disorder phenomenologically through the non-conservation of the transverse electron momentum k y . As mentioned above, k y is conserved for both Dirac electrons in infinite graphene and dressed states by photons. Introducing the quantity Γ as a measure for the disorder [20,21], we model its stochastic distribution as a Lorentzian, where t 0 (k x , q y ) denotes the transmission coefficient in the absence of any imperfections.
As long as the disorder is weak with Γ ≪ 1, different distributions, which give δ-function in the limit of Γ → 0, will result in almost equal transmission coefficients. For nearlyhead-on collision, k y ≪ k x2 , the transmission coefficient may be obtained analytically using a Gaussian distribution. The imperfect boundary of the potential region can also be the result of some stochasticity of k x2 to make the effect stronger. We neglect this effect since our goal is to investigate the role played by disorder using a simple approximation.
Our numerical results showing the effect due to disorder are presented in Fig. 4.

IV. CONCLUDING REMARKS
According to recently published results [8], the interaction between Dirac particles in graphene and circularly polarized light leads to the formation of quantum electron dressed states. These states are appreciably different from conventional Dirac electrons in ordinary infinite graphene. From Ref. [8], laser power 100 mW leads to a gap on the order of ∆ ⋍ 100 meV, which is required to make the effect significant for infrared light frequencies and room temperature. This enables possible experimental demonstrations of the described effects. We have shown that electron-photon interaction gives rise to states of broken chiral symmetry. The non-symmetrical properties of the states become more significant when the electron-photon interaction is increased. In addition, there are no dressed states with chirality symmetry. In general, incoming electrons or holes passing unimpeded through a square potential barrier require chiral symmetry. Under the illumination from a circularlypolarized light, we can control the degree of partially-broken chiral symmetry in dressed states or the degree of partially-perfect transmission through a potential barrier in a graphene layer.
In the simplest approximation when only the two nearest subbands are included, the model is formally similar to the so-called σ 3 Hamiltonian used to describe the particles in a single layer graphene with parabolic energy dispersion, giving non-zero electron effective mass. We discussed the similarities as well as the differences affecting the wave functions but not the energy dispersion. By including more the next-nearest subbands, the tunneling amplitude is modified in a significant way. In the approximation when two independent pairs of subbands are included, we obtain an enhanced transmission probability when the barrier height is close to ω 0 . This is due to the fact that one of the terms in the corresponding wave function is perfectly transmitted. By including more than two independent pairs, this effect will decrease since perfect transmission will occur only for the two wave function terms. The effect is not sensitive to the barrier width, and therefore, can be considered as a reminiscent of the Klein paradox.
We have introduced disorder phenomenologically in this paper through non-conservation of the transverse electron momentum component, which is shown to suppress the perfect transmission along the diagonal k x1 = k x2 . From a physical point of view, this disorder model could be interpreted as arising from surface roughness of the potential barrier. The same type of statistical distribution can be applied to fluctuations in the the barrier width which will result in modification of the intensity and location of the transmission peaks.
Consequently, the transmission maxima observed experimentally will not exactly match to those theoretically predicted for clean samples.