Comment on "Photon-assisted electron transport in graphene: Scattering theory analysis"
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Comment on ”Photon-assisted electron transport in graphene:Scattering theory analysis”
M. Ahsan Zeb ∗ Department of Earth Sciences, University of Cambridge, UK.
Abstract
It is argued that Trauzettel et al . [Phys. Rev. B 75, 035305 (2007)] made some mistakes in theircalculations regarding the photon-assisted transport in graphene that lead to uncoupled sidebands and emergence of step-like features in dG/dV (G is differential conductance and V is the biasvoltage). We discuss the relevant corrections and explain in detail how the correct results areexpected to be quite different than the incorrect ones. ∗ Electronic address: [email protected] ε weightedby the Bessel functions, which apparently seems correct. But, it can be easily seen that noneof these ”components” satisfies the wave equation. Nevertheless, since we can consider theincident particles at a single energy, this issue can be resolved simply by restating theproblem. However, the following mistake makes all their calculations incorrect so that theirresults become useless. The reflected and transmitted wavefunctions need to account for thepossible transitions to lower and higher energies on emission or absorption of modulationquantum/quanta of the ac signal. The reflected wavefunction Ψ ( ac ) r ( −→ x , t ) they consider isΨ ( ac ) r ( −→ x , t ) = ∞ X m = −∞ r m J m ( eV ac ~ ω )Ψ ( ac )0 , − e − i ( ε + ~ mω ) t/ ~ = Ψ ( ac )0 , − e − iεt/ ~ ∞ X m = −∞ r m J m ( eV ac ~ ω ) e − i ~ mωt which even does not satisfy the wave equation unless the factor r m J m ( eV ac ~ ω ) equals Bessel2unction of order m with argument ( eV ac ~ ω ), i.e., if r m = 1 for all m ( in such a case itwill represent particles only at energy ε ). The correct form of the reflected wavefunctionwould be a linear combination of components at all sideband energies with a phase factor e − i ( eVac ~ ω ) sin( ωt ) (= ∞ X m = −∞ J m ( eV ac ~ ω ) e − imωt ) due to the ac signal, i.e.,Ψ ( ac ) r ( −→ x , t ) = ∞ X n = −∞ r n Ψ ( ac ) − ,n e − i ( ε + ~ nω ) t/ ~ × ∞ X m = −∞ J m ( eV ac ~ ω ) e − imωt where Ψ ( ac ) − ,n is the solution of the wave equation without the ac potential representing parti-cles at energy ε + ~ nω moving along negative x-direction and r n will give the amplitude for thetransitions to the n th sideband in reflected states. The transmitted wavefunction Ψ ( in ) tr ( −→ x , t )they considered is given by Ψ ( in ) tr ( −→ x , t ) = ∞ X m = −∞ t m J m ( eV ac ~ ω )Ψ ( in )+ ,m e − i ( ε + ~ mω ) t/ ~ , which is correctbut contains unnecessary Bessel functions J m ( eV ac ~ ω ) that also means that the amplitude forthe transmission in m th sideband would be t m J m ( eV ac ~ ω ) instead of t m as considered by theauthors. Correcting for all above mistakes and omitting the Bessel functions in transmit-ted wavefunction, boundary conditions lead to a set of coupled equations that cannot bedecoupled so analytical solutions for t m cannot be found. Further, the relation X n | r n | + X n | t n | = 1, where the sums are only over the bands with the propagating modes, would hold in thiscase instead of | r n | + | t n | = 1 for obvious reasons. In section-II of ref[5], expressionsfor the current and conductance are given. Authors missed the factors Ψ † ( in )+ ,m σ x Ψ ( in )+ ,m ′ inthe summations in the expression for the current given in equation(15) that also affectsthe expression for the differential conductance given in equation(16). Finally, consider thestep-like features in dG/dV presented in figure(2). Authors attribute them to the vanishingdensity of states at the Dirac point, which shows that these features may still persist in thecorrect results. In the following we will explain how these steps arise in their calculationsand why they are not expected in the correct results.The first point to note is the fact that all ”sidebands” in their calculations are indepen-dently contributing to the transmission, each like as if it were a dc potential problem. Thereis no photon-assisted transport at all. At values of eV that are integer multiple of ~ ω , contri-bution of a ”sideband” is included to/excluded from G. This amounts to ”adding/removing” sharp increase in the number of transmitted particles on emergenceof a new contributing sideband (Note that the sharp rise in current in the problem con-sidered by Dayem and Martin[6] has two different reasons. First, the absorption of energyquantum/quanta from the microwave field, help electrons cross the energy gap and maketransition to empty conduction band from the filled valence band. Second, the density ofstates is very large at the gap edges. ). Since the correct value of transmission probabilityfor any sideband is likely to have strong and complex dependence on the energy and thepropagation angle of incident particles (Chiral Dirac fermions), the precise dependence ofG or any other quantity like dG/dV on V is hard to predict. However, due to the reasonsdiscussed above, one thing is obvious: the correct results are expected to be significantlydifferent than the incorrect ones presented in figure(2) in ref[5]. [1] P. K. Tien and J. P. Gordon, Phys. Rev. 129, 647 (1963).[2] Mathias Wagner, Phys. Rev. B 49, 16544 (1994)[3] M. Ahsan Zeb, K. Sabeeh and M. Tahir, Phys. Rev. B 78, 165420 (2008).[4] M. Ahsan Zeb, K. Sabeeh and M. Tahir, arxive 0804.2061v1[5] B. Trauzettel, Ya. M. Blanter, and A. F. Morpurgo, Phys. Rev. B 75, 035305 (2007).[6] A. H. Dayem and R. J. Martin, Phys. Rev. Lett. 8, 246 (1962).[1] P. K. Tien and J. P. Gordon, Phys. Rev. 129, 647 (1963).[2] Mathias Wagner, Phys. Rev. B 49, 16544 (1994)[3] M. Ahsan Zeb, K. Sabeeh and M. Tahir, Phys. Rev. B 78, 165420 (2008).[4] M. Ahsan Zeb, K. Sabeeh and M. Tahir, arxive 0804.2061v1[5] B. Trauzettel, Ya. M. Blanter, and A. F. Morpurgo, Phys. Rev. B 75, 035305 (2007).[6] A. H. Dayem and R. J. Martin, Phys. Rev. Lett. 8, 246 (1962).