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Conductance enhancement due to atomic potential fluctuations in graphene

We solve the Dirac equation, which describes charge massless chiral relativistic carriers in a two-dimensional graphene. We have identified and analysed a novel pseudospin-dependent scattering effect. We compute the tunneling conductance and generalize the analytical result in the presence of the tunable atomic potential of a graphene strip. The absence of back scattering in graphene is shown to be due to Berry's phase which corresponds to a sign change of the wave function under a spin rotation of a particle. We use the transfer matrix approach and find that the electric conductance of doped graphene increases due to atomic potential fluctuations.


I. INTRODUCTION
Graphene is a single layer of carbon atoms densely packed in a honeycomb lattice. This material was found in its free state only recently, when individual graphene samples of a few microns in size were isolated by micromechanical cleavage of graphite [1]. The current intense interest in graphene is driven by both the unusual physics involved and a realistic promise of device applications. Among relativistic like phenomena observed in graphene so far, there are two new types of the integer quantum Hall effect and the presence of minimal metallic conductivity of about one conductivity quantum [2]. Transport properties of graphene are interesting because of their unique topological structure [3][4][5][6]. There have been some reports on experimental study of transport in graphene [7]. Various calculations have been performed to understand energy bands of graphene [8,9]. It has been successful in the study of various properties including the Aharonov-Bohm effect [10], optical absorption spectra [11], quantum tunneling in graphene-based structures [12][13][14], ferromagnetism [15] and instabilities in the presence and absence of a magnetic field [16].
In a two-dimensional world, there are two basic motions: forward and backward. Random scattering can cause them to mix, which leads to resistance. Just as we have learned from the basic traffic control, it would be much better if we could spatially filter the counterflow directions. This effect could be used to produce a valley-polarized current out of an unpolarized stream of electrons due to line defect particularly in graphene [17][18][19][20][21][22][23][24][25][26].
The electronic transport properties of graphene with defects and impurities exhibit pronounced differences from those of conventional two dimensional electron systems investigated in the past. The basic question that we seek to answer is what happens to the tunneling as we approach the Dirac point of zero carrier concentration with the presence of electrically charged scatters. In the present work we consider the effects of atomic potential fluctions in the setup depicted in fig. 1. At low doping where u is a spinor of wave amplitudes for two nonequivalent sites of the honeycomb lattice. The Fermi energy ε and the impurity potential V (x, y) in the graphene sample (0 < x < L) are considered to be much smaller than the Fermi energy E F in the ideal metallic leads (x < 0 and x > L). For zero doping the conductance is determined by the states at the Dirac point, ε = 0. Transport properties at finite energies determine the conductance of doped graphene. Since for aspect ratios W ≫ L the boundary conditions in the y-direction are irrelevant, so we take periodic boundary conditions for simplicity. Different wave vectors q n =2πn/W (with n=0, ± 1, ± 2, . . . ) in the ydirection parallel to the interfaces(at the points x = 0 and x = L) and not coupled, so we can consider each transverse mode separately. Without impurity in the bulk at a given energy ε and transverse wave vector q we have up to two basis states with the definitions The angle α ∈ (-π/2,π/2) is the angle between the initial and final wave vectors k i = (k, q)| x=0 and k f = (k, q)| x=L . With this sign convention the state u + move in the +x, while u − move in the −x direction. The factor 1/ √ cosα ensure that two states carry the same particle current.
Considering a back scattering process k → −k in an arbitrary external potential we will confine ourselves to states in the vicinity of the K point, but the extension to states near a K ′ point is straightforward. We have α(k) = 0 and α(−k) = π. Rotation in spin space can be obtained through the replacement (s i , k i ) → (s f , −k f ) corresponding to the electron motion of a time-reversal path. When the wave vector k is rotated in the anticlockwise direction adiabatically as a function of time t around the origin for a time interval 0 < t < T , the wavefunction is changed into u s (k) exp(−iφ), where u s (k) is the "spin" part of an eigenfunction of the Eq. 1 and φ is Berry's phase. Choosing φ(k) = α(k)/2 in such a way that the wave function becomes continuous as a function of α(k)/2 and given by This shows that the rotation in the k space by 2π leads to the change in the phase by +π, i.e., a sign change. A similar phase change of back scattering process is the origin of the so-called anti-localization effect in systems with strong spin-orbit scattering [27]. This anti-localization effect was observed experimentally [28,29]. The absence of back scattering can be destroyed by various effects. When the potential range becomes shorter than the lattice constant, the back scattering appears two reasons. The first is the appearance of inter-valley matrix elements between K and K ′ points [30]. The second, which we investigate in this work, is that the effective potential of an impurity for A and B sites in honeycomb lattice can be different. This causes mixing of the spin space and the momentum space.

B. Long range impurities
There is an absence of back scattering for long range impurities. This can be seen easily: using the formalism of the previous subsection we can simply show that the matrix element of the back-scattering process is zero (here S is the unit cell area of the graphene lattice). This is rather dramatic since all real materials have impurities, and it is normally these which determine the transport properties of the material. However, in the case of graphene the impurities don't scatter, i.e. they are effectively not there, and hence one has the possibility of coherent transport in graphene.

III. TRANSFER MATRIX APPROACH
We first determine the transfer matrix M n (x, 0) of the nth mode u n (x)e iqn y through the undoped graphene ribbon. The evolution of u n (x) inside the graphene sample can be written as u n (x) = M n (x, 0)u n (0), where the transfer matrix M n (x) should satisfy a generalized unitarity condition In general Eq.6 (ignoring an irrelevant scalar phase factor) restricts M to a three-parameter form The real parameters α 1 , α 2 , α depend on the boundary at the scale of the lattice constant and they cannot be determined from the Dirac equations. In this work we consider the ideal interfaces x = 0 and x = L case, when α 1 = 0 = α 2 . From the Dirac Eq.1 we obtain the differential equation with solution for V (x, y) = 0 M n (x, 0) = cos k n x + sin k n x k n iε v σ x + q n σ z (10) where k n = ( ε v 2 − q 2 n is the longitudinal wave vector, q n is the transversal wave vector. The total transfer matrix through the impurity-free graphene ribbon (0 < x < L) is Now we construct the transfer matrix for the electron (hole) quasiparticle excitations in the graphene ribbon for disorder potential case, where V is the charged impurity potential given by a diagonal matrix, i.e., where γ A (γ B ) is the corresponding microscopic potential and smooth on atomic scales, which is localized along a line x = x 0 and placed in the zone corner K (K ′ ) of sublattice A (B). The Fermi-energy ε and impurity potential V (Eq.12) in graphene sample (0 < x < L) are considered to be much smaller than the Fermi-energy E F in the ideal metallic leads (x < 0 and x > L). For zero doping the conductance is determined by the states at the Dirac point, ε = 0. Transport properties at finite energies determine the conductance of doped graphene.
The great physical and analytical advantage of the M matrix is that it is multiplicative. In order to obtain transfer matrix for disorder region (along line Making use of well-known algebraic property for Pauli matrices we easily obtain the final form for total transfer matrix M in disorder graphene sample where andγ A(B) = −iγ A(B) / v F . The transfer matrix of the whole sample is straightforwardly related to the matrices of transmission and reflection amplitudes The conductance of the graphene strip is expressed through the transmission amplitudes by the Landauer where T = T r[t † t]. At the Dirac point V gate = 0 the transmission probability for the case N ≫ W/L reads When parameters P = 0 and α = 0 the Eq.18 reduces to equation (4), derived in the paper [31]. Parameter P =| γ A − γ B | reperesents atomic potential fluctuations, when finite P ≪ υ F .

IV. CONCLUSION
In this paper we used the transfer matrix approach to calculate the tunneling conductance. We generalize the analytical result in the presence of the tunable atomic potential of a graphene strip and find that the electric conductance of doped graphene increases due to atomic potential fluctuations. The absence of back scattering in graphene is shown to be due to Berry's phase which corresponds to a sign change of the wave function under a spin rotation of a particle. We have identified and analysed a novel pseudospin-dependent scattering effect. The low cost experimental techniques allow for easy verification of the proposed hypothesis.

V. ACKNOWLEDGEMENTS
We are indebted to Ben Still for assistance. D. B. thanks Myerscough Bequest for financial support. D. B. acknowledges Thomas Young Centre for Junior Research Fellowship and Cornell University (Neil Ashcroft and Roald Hoffmann) for hospitality.