TimeDependent 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 date  20 Apr 2021 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(645fe6e575674935927175cf9e2f8f6c).html 

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Abstract
Since graphene was first synthesized in 2004, it has aroused tremendous enthusiasm in the research community. Graphene is a single layer, 2dimensional 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 timedependent quantum transport in graphene, including radiation effect on bilayer graphene, and photonassisted tunneling in monolayer graphene of twobarrier structure, threebarrier structure and fourbarrier structure. First, we investigate the quasienergy 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 quasiDirac 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 quasiDirac 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 k_{x} axis, but when there is a phase difference between the two radiations, it can prevent the creation of Dirac points.
Then we investigate photonassisted tunneling in a twobarrier graphene structure of timeperiodic 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. OjedaCollado 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 (α_{1}=V_{1}/ℏω_{1} and α_{3}=V_{3}/ℏω_{3} respectively) is less than 1, i.e., α_{1}−α_{3}<1. Here we find that range of modulation amplitudes α_{1}−α_{3} 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 α_{1}−α_{3}<C where C can be controlled by the oscillating frequency (ω) and barrier width (L_{1}) at some critical values of phase difference. In addition, we also find that the range of α_{1}−α_{3} is identical as long as the value of L_{1}ω remains unchanged, and the range of α_{1}−α_{3} can be expressed as a rational function of L_{1}ω, i.e. C=b_{0}+b_{1}/(L_{1}ω−b_{2}). 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 timedependent quantum transport in graphene of threebarrier structure and fourbarrier structure of timeperiodic potentials. We find that there exist critical phase differences relations {δ_{3c}+a+b=π and δ_{5c}+2a+2b=2π} and {δ_{3c}+a+b=π, δ_{5c}+2a+2b=2π and δ_{7c}+3a+3b=3π} for threebarrier and fourbarrier 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 α_{1}−α_{3}+α_{5}<C_{1} (for threebarrier structure) and α_{1}−α_{3}+α_{5}−α_{7}<C_{2} (for fourbarrier structure) where C_{1} and C_{2} depend on the oscillating frequency and barrier width. Based on the numerical results and the result of the doublebarrier structure [6], we deduce that there is a hypothetical critical phase differences relation {δ_{3c}+a+b=π, δ_{5c}+2a+2b=2π, ..., δ_{(2n−1)c}+(n−1)a+(n−1)b=(n−1)π} where δ_{(2n1)c} is the critical phase difference between the oscillating frequencies at the first barrier and the n^{th} barrier, for an oscillating potential driven nbarrier graphene system such that complete cancellation of inelastic sidebands will occur whenever α_{1}−α_{3}+α_{5}−α_{7}+⋯+(−1)^{n−1}α_{2n−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.
In this thesis we report timedependent quantum transport in graphene, including radiation effect on bilayer graphene, and photonassisted tunneling in monolayer graphene of twobarrier structure, threebarrier structure and fourbarrier structure. First, we investigate the quasienergy 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 quasiDirac 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 quasiDirac 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 k_{x} axis, but when there is a phase difference between the two radiations, it can prevent the creation of Dirac points.
Then we investigate photonassisted tunneling in a twobarrier graphene structure of timeperiodic 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. OjedaCollado 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 (α_{1}=V_{1}/ℏω_{1} and α_{3}=V_{3}/ℏω_{3} respectively) is less than 1, i.e., α_{1}−α_{3}<1. Here we find that range of modulation amplitudes α_{1}−α_{3} 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 α_{1}−α_{3}<C where C can be controlled by the oscillating frequency (ω) and barrier width (L_{1}) at some critical values of phase difference. In addition, we also find that the range of α_{1}−α_{3} is identical as long as the value of L_{1}ω remains unchanged, and the range of α_{1}−α_{3} can be expressed as a rational function of L_{1}ω, i.e. C=b_{0}+b_{1}/(L_{1}ω−b_{2}). 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 timedependent quantum transport in graphene of threebarrier structure and fourbarrier structure of timeperiodic potentials. We find that there exist critical phase differences relations {δ_{3c}+a+b=π and δ_{5c}+2a+2b=2π} and {δ_{3c}+a+b=π, δ_{5c}+2a+2b=2π and δ_{7c}+3a+3b=3π} for threebarrier and fourbarrier 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 α_{1}−α_{3}+α_{5}<C_{1} (for threebarrier structure) and α_{1}−α_{3}+α_{5}−α_{7}<C_{2} (for fourbarrier structure) where C_{1} and C_{2} depend on the oscillating frequency and barrier width. Based on the numerical results and the result of the doublebarrier structure [6], we deduce that there is a hypothetical critical phase differences relation {δ_{3c}+a+b=π, δ_{5c}+2a+2b=2π, ..., δ_{(2n−1)c}+(n−1)a+(n−1)b=(n−1)π} where δ_{(2n1)c} is the critical phase difference between the oscillating frequencies at the first barrier and the n^{th} barrier, for an oscillating potential driven nbarrier graphene system such that complete cancellation of inelastic sidebands will occur whenever α_{1}−α_{3}+α_{5}−α_{7}+⋯+(−1)^{n−1}α_{2n−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.