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Klein tunneling through an oblique barrier in graphene ribbons

We study a transmission coefficient of graphene nanoribbons with a top gate which acts as an oblique barrier. Using a Green function method based on the Dirac-like equation, scattering among transverse modes due to the oblique barrier is taken into account numerically. In contrast to the 2-dimensional graphene sheet, we find that the pattern of transmission in graphene ribbons depends strongly on the electronic structure in the region of the barrier. Consequently, irregular structures in the transmission coefficient are predicted while perfect transmission is still calculated in the case of metallic graphene independently of angle and length of the oblique barrier.


I. INTRODUCTION
Recently, there has been considerable interest in transport on a graphene sheet, a single atomic layer usually pulled out of bulk graphite. [1][2][3]. Due to its unique two-dimensional closely-packed honeycomb structures, electrons in it behave like massless Dirac fermions. [4,5] One of its fascinating properties is the so-called Klein tunneling where perfect penetration occurs independently of potential barrier height, contrast to the conventional, non-relativistic tunneling. [6][7][8] This relativistic effect is basically originated from the gapless electronic dispersion, which in turn leads to the connection between electron and hole states in graphene.
A sufficiently strong potential, being repulsive for electrons, is attractive for holes and gives rise to hole states in the barrier to form channels through which electrons can penetrate the barrier. [7] The idea of the Klein tunneling realized on graphene sheets was suggested by Katsnelson et al. [9] and then several experimental attempts have been made to demonstrate the perfect transmission. [10][11][12] Up to now, the perfect transmission is addressed in terms of sudden phase shift of conductance as a function of magnetic field. [12] However, a more direct evidence for Dirac particles may be the incident angle-dependence of transmission coefficient, [9] which is not realized experimentally yet. For incident angle θ and barrier length D the transmission coefficient T of Dirac particles is given by, showing oscillating behavior as a function of incident momentum k, angle, and barrier length.
However, in the case of a 2-dimensional graphene sheet it is hard to adjust the incident angle θ because there are randomly directed particles which result in averaging Eq. (1) over the incident angle. A collimation-gate method to remove the randomness of electrons may be not appropriate because it requires proximate implementation to a potential barrier less than a mean free path.
A good candidate to resolve the problem is graphene nanoribbons (GNRs) because electrons in each transverse mode propagate along its axis (i.e., θ = 0). Then, we can adjust the incident angle θ definitely by adopting an oblique potential barrier with respect to the axis of GNRs. However, in this case the transverse momentum is quantized and correspondingly Dirac particles become massive depending on their occupation to transverse modes. So, it is interesting to ask about what is the transmission coefficient of GNRs with an oblique potential barrier and to see whether the perfect transmission occurs in ribbon structures even under the oblique potential barrier.
In this work, the Klein tunneling is investigated in graphene nanoribbons (GNRs) with oblique barriers theoretically. Based on Green functions taking account scattering among transverse modes we show that the transmission coefficient also oscillates, however, with rich structures as a function of incident energy, angle, and barrier length. Through the analysis of local density of states, it is found that rich structures in the transmission coefficients are resulted from detailed hole states formed in the region of the potential barrier and associated interference. We also examine effects of inelastic scattering for the experimental realization of the Kleining tunneling by employing the level broadening in the simplest approximation.

II. MODEL AND METHOD
To examine the Klein tunneling we assume a mono-layered graphene ribbon along ydirection with armchair boundaries as shown in Fig. 1. It is well known that a graphene ribbon can be metal or semiconductor depending on its width W . [13] We set W = Na 0 /2 where 2N is a total number of carbon atoms in a unit cell and a 0 = 2.46Å is the graphene lattice constant. Then, the system becomes a metal if N + 1 is a multiple of three, otherwise semiconductors.
The system under consideration is composed of two electrodes and a central device part.
The device part has a finite graphene ribbon with a gate electrode while the lower and upper electrodes are assumed to be semi-infinite perfect GNRs. We describe the system with the 4 × 4 Dirac-like equation. This effective-mass equation is known to give accurate low-energy properties of graphene. [13] With a potential V (x, y) induced by a gate, the Hamiltonian reads as, where σ x,y is a Pauli matrix, v F ∼ 10 6 m/sec the Fermi velocity, p x = −i ∂ x and p y = −i ∂ y , respectively. When a gate electrode is tilted with an angle θ as shown in Fig. 1, the potential barrier is generally a function of x− and y−coordinates and can be modeled as Here, we introduce a transition region of the length d to reduce numerical error of a finite difference method used in the followings as well as for the effective mass equation of Eq. (2) to be valid.
In the absence of a potential barrier V (x, y), since the system is a perfect GNR one can solve the equation of motion analytically. For instance, the appropriate armchair boundary conditions have been formulated [13] and used to examine bound states [14]. Wavefunctions are known to be plane waves along both x-and y directions as, where z nk = √ q n − ik/ √ q n + ik and γ = ±1 denotes conduction(+) and valence(−) bands of graphene, respectively. The boundary conditions yield the following quantization for a wave vector in x-direction; and electronic energy is given by E = γ v F q 2 n + k 2 for a propagating wave in y-direction with its wave vector k.
In general an incident wave with a certain transverse mode from the lower to upper leads is scattered to other modes due to a perturbing potential in the device region. Consequently, one should take into account multiple transverse modes to resolve a scattering problem. In our case, we choose the following basis function to describe scattered waves and adopt Green function approach to calculate the transmission coefficient. Actually, Eq.
(6) denotes eigenfunctions of an infinite GNR, namely Eq. (4) at k = 0. So, since they are orthonormal to each other, i.e., φ γ n | φ γ ′ n ′ = δ nn ′ δ γγ ′ and satisfy the armchair boundary conditions, we can expand the wavefunction of Eq. (2) as ψ(x, y) = nγ χ nγ (y)φ γ n (x). Resulting equation of motion for χ nγ (y) is summarized as, in a matrix form, where χ(y) is a column matrix with its component χ nγ (y) and matrices D andb are given by, Next, we implement a lattice version of Eq. (7) by replacing d/dy with a finite difference on a uniform grid. Detailed results are shown as, especially in a block tridiagonal form, where a m = E1 + D(y m ) and b =b/2∆ with a grid spacing ∆. Following a standard Green function technique, we define a retarded Green function of the system as inverse of the matrix in Eq. (9) with slightly shifted energy E → E + iη. Especially, we are interested in the Green function of the device part which is represented by grid points m = 0, . . . , M − 1.
For the Green function of the device we truncate the lower-and upper-lead parts from the original matrix. [15] Then effects of the truncation manifest itself to self-energies as, Here, Σ L,U is called surface self-energy from the lower and upper leads, respectively, and Then, using Eq. (8) we obtain, From the calculated Green function, the local density of states (LDOS) at the index m can be found by where G(m, m) is a block matrix of the diagonal Green function at the index m. And the lowfield conductance σ of the graphene ribbon can be calculated using the Landauer-Büttiker can be expressed as, where

III. NUMERICAL RESULTS
In this section, we numerically illustrate solutions of the Green function representing scattered waves from an oblique potential barrier and related transport properties. We consider a typical size of the device structure which may be realized experimentally; for instance, W = 99a 0 , D = 60a 0 , and d = 30a 0 . To include a pattern of an oblique potential barrier, large length of the device part is chosen. Actually since the system is in equilibrium, the transmission coefficients are independent of the total simulation length as long as the scattering potential is described properly in it. We use 100 transverse modes for the accurate description of scattering waves and a grid spacing of ∆ = 2Å. So, for the total simulation length of 260a 0 a dimension of the matrix Eq. in (10)  backwardly (−k) within the same mode. However, in the case of a metallic graphene ribbon the lowest mode has a momentum of q n = 0 and thus z kn in Eq. (4) has exactly the out-of phase for the backward-scattering relative to the incident one. Consequently, there is no the backward scattering in the metallic GNRs independently of the oblique angle. This is not the case for semiconducting GNRs because the lowest mode has q n = 0 and the out-of phase cannot be achieved for backward-scattered waves.
The barrier-length dependence of the transmission coefficients is examined in Fig. 3-(a). scattering is added to the Hamiltonian independently of position and energy. This effect is equivalent to the level broadening by setting a finite value η in the retarded Green function.
The value η is related to the scattering rate γ via the relation of η = γ. [17,18]  In Fig. 5 we show effects of the level broadening on the transmission coefficients for metallic graphene ribbons at θ = π/4. It is found that the pattern of transmission becomes smeared with increasing the broadening together with accompanying many new dips in the region of the perfect transmission, and eventually has no structures up to η = 10meV. We also find similar behavior for different size of GNRs as shown in the second and third rows of According to the Fermi golden rule, the level broadening gives rise to scattering to more diverse states by releasing the energy conservation from the delta function to the Lorentzian one. So, the scattering matrix is averaged over the energy interval of ∆E ∼ min{η, v F /D, v F /W tan θ}. For this reason, the out-of phase of the back-scattered waves is no longer achieved and thereby the transmission coefficient becomes suppressed. However, we find that the newly developed dips reflecting more strong suppression cannot be explained by the Fermi golden rule and may be attributed to higher-order perturbations representing multiple scattering and associated interference. As a results, since the pattern of transmission is washed out for the level broadening about η = 10meV, it is necessary to reduce scattering rates for the experimental realization of Dirac particles, for instance, by lowering temperature and preparing clean GNRs.

IV. SUMMARY
In this work, we studied the transmission coefficient of graphene ribbons with an oblique barrier based on the Dirac-like equation. In contrast to the 2-dimensional graphene sheet, the transmission in graphene ribbons is found to depend strongly on the electronic structure in the region of barriers. Consequently, irregular structures in the transmission coefficient are predicted, however the perfect transmission is shown in the case of metallic graphene independently of angle, width, length of oblique barriers. This behavior still demonstrates the nature of Dirac particles in graphene ribbons because very large potential barrier is assumed. We also examine effects of scattering by employing the simplest approximation.