Time-Dependent Quantum Transport in Graphene

石墨烯時間依賴的量子運輸

Student thesis: Doctoral Thesis

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Award date20 Apr 2021

Abstract

Since graphene was first synthesized in 2004, it has aroused tremendous enthusiasm in the research community. Graphene is a single layer, 2-dimensional carbon lattice that has no bandgap between the valence and the conduction bands. Together with its linear energy dispersion relation in low energy, we can observe some very unusual electronic properties including Klein tunneling, quantum Hall effect, and finite minimal conductivity.

In this thesis we report time-dependent quantum transport in graphene, including radiation effect on bilayer graphene, and photon-assisted tunneling in monolayer graphene of two-barrier structure, three-barrier structure and four-barrier structure. First, we investigate the quasi-energy band structure of bilayer graphene irradiated by two laser radiations of frequencies ω and 2ω by using Floquet formalism. Our objective is to examine the creation of Dirac points on the dynamic gap between the conduction band of Floquet branch index −1 and the valence band of Floquet branch index +1. We find that in the momentum space, the dynamical gap between the two bands does not follow rotational symmetry. Thus the dynamical gap becomes zero along some radial directions and creates some quasi-Dirac points, in which the conduction band almost meets the valence band, when some particular parameters are applied. By using Lowdin perturbation theory, we can express analytically the directional dependence of the dynamical gaps. When two circularly polarized radiations are applied and under the first order and second order perturbations, the dynamical gap will be reduced to zero along some directions. In higher order perturbations, a small gap will be induced. When two linearly polarized radiations are applied and under the fourth order perturbation, the dynamical gap will be reduced to zero and more than one quasi-Dirac point will be created.  Then we examine the velocity of electrons that are close to a dynamical gap. We find that magnitudes of electron velocities are reduced to almost zero when the k vector is above the dynamical gap. In addition, the direction of the electron velocity changes when it crosses the gap. When the gap is large, the change in the velocity direction takes place slowly. We also find that trigonal warping has no effect on the creation of a Dirac point along the kx axis, but when there is a phase difference between the two radiations, it can prevent the creation of Dirac points.

Then we investigate photon-assisted tunneling in a two-barrier graphene structure of time-periodic potential. We use Floquet theory and the transfer matrix method to calculate the total transmission probabilities and the central band transmission probabilities for different values of oscillating frequency, barrier width, quantum well width, incident angle, and modulation amplitude. Ojeda-Collado et al have pointed out that at a critical phase difference between the oscillating frequencies at the barriers, all inelastic sidebands will be cancelled due to destructive interference whenever the difference of the two modulation amplitudes (α1=V1/ℏω1 and α3=V3/ℏω3 respectively) is less than 1, i.e., |α1−α3|<1. Here we find that range of modulation amplitudes |α1−α3| can be controlled by barrier width and oscillating frequency but not the width of the quantum well, i.e. transmission is perfect and attributed solely to the central band within |α1−α3|<C where C can be controlled by the oscillating frequency (ω) and barrier width (L1) at some critical values of phase difference. In addition, we also find that the range of |α1−α3| is identical as long as the value of L1ω remains unchanged, and the range of |α1−α3| can be expressed as a rational function of L1ω, i.e. C=b0+b1/(L1ω−b2). We also study the effect of quantum interference on transmission probabilities in subtle changes of oblique angles. We note that there are critical phase differences in which all sidebands are cancelled completely and resonant peaks are restored to almost unity for different values of incident angle and modulation amplitude, except when the modulation amplitudes are extremely high. We also find that suppression of resonant peaks increases as incident angles and decreases as modulation amplitudes.

Next, we examine time-dependent quantum transport in graphene of three-barrier structure and four-barrier structure of time-periodic potentials. We find that there exist critical phase differences relations {δ3c+a+b=π and δ5c+2a+2b=2π} and {δ3c+a+b=π, δ5c+2a+2b=2π and δ7c+3a+3b=3π} for three-barrier and four-barrier structures respectively where a is the barrier width and b is the quantum well width, such that all inelastic sidebands completely cancel each other even at large values of modulation amplitudes in which |α1−α35|<C1 (for three-barrier structure) and |α1−α35−α7|<C2 (for four-barrier structure) where C1 and C2 depend on the oscillating frequency and barrier width. Based on the numerical results and the result of the double-barrier structure [6], we deduce that there is a hypothetical critical phase differences relation {δ3c+a+b=π, δ5c+2a+2b=2π, ..., δ(2n−1)c+(n−1)a+(n−1)b=(n−1)π} where δ(2n-1)c is the critical phase difference between the oscillating frequencies at the first barrier and the nth barrier, for an oscillating potential driven n-barrier graphene system such that complete cancellation of inelastic sidebands will occur whenever |α1−α35−α7+⋯+(−1)n−1α2n−1|<C where C is a constant that depends on the oscillating frequency and barrier width. We also study the transmission probability as functions of barrier height as well as incident energy at different oblique angles and modulation amplitudes; it is found that the resonant transmission peaks can be restored to almost unity when the conditions of the critical phase differences relations are met.