Electronic transport properties of Sierpinski lattices

Liu, Youyan; Hou, Zhipeng; Hui, P. M.; Sritrakool, W.

doi:10.1103/physrevb.60.13444

Cited by 62 publications

(42 citation statements)

References 28 publications

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“…19 In the following we rewrite separately the network equation ͑4͒ in the absence and presence of dissipation. In the former case Јϭ0, ϭ 0 , so the wave vector kϭ ͱ 0 /c 0 .…”

Section: Network Equations and Generalized Eigenfunction Methods Fmentioning

confidence: 99%

“…To numerically solve this problem we can use the generalized eigenfunction method which was developed to deal with the electronic transport problems of Sierpinski gasket lattices. 19 The trick of the GEM lies in that we treat the amplitudes r and t like the wave functions i . In this way the coupled equations ͑9͒ and ͑10͒ can be rewritten as a matrix equation of order N ϩ2, where N is the number of nodes in the Sierpinski network: If we denote the above matrix equation as…”

Section: Network Equations and Generalized Eigenfunction Methods Fmentioning

confidence: 99%

“…17,18 The effects of magnetic field on the electronic structure of SGs have also been studied. 18 Liu and co-workers 19,20 have developed a generalized eigenfunction method ͑GEM͒ to study the electronic transport properties of an open SG which is connected to electron reservoirs. The electronic localization induced by the fractal structure and by the quan-tum coherence effect has been systematically studied.…”

Section: Introductionmentioning

confidence: 99%

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Photonic band structure of Sierpinski waveguide networks

Liu

Zhang

2000

Phys. Rev. B

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The photonic band structure and transmission properties of Sierpinski fractal networks ͑SNs͒ made of one-dimensional waveguides are studied with the generalized eigenfunction method. In the absence or presence of dissipation and in different exit situations, we have numerically calculated the transmission coefficient as a function of frequency in the range 0-500 MHz for the first four generations of SNs. As the number of generations is increased, the structures in the transmission spectra show explicitly the evolution of discrete eigenmodes and the corresponding photonic band gap structures in such fractal networks. The gap structures are not altered by the presence of dissipation and are independent of the exit situation. An interesting anomaly due to dissipation is found at certain frequencies where a resonant peak is split into two peaks with zero transmission in the valley.

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“…19 In the following we rewrite separately the network equation ͑4͒ in the absence and presence of dissipation. In the former case Јϭ0, ϭ 0 , so the wave vector kϭ ͱ 0 /c 0 .…”

Section: Network Equations and Generalized Eigenfunction Methods Fmentioning

confidence: 99%

Section: Network Equations and Generalized Eigenfunction Methods Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Photonic band structure of Sierpinski waveguide networks

Liu

Zhang

2000

Phys. Rev. B

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“…2(a) and statistical Sierpiński carpets with p = 0.5. From resistor network simulations [41], we extract the classical conductance exponent in the 5 × 5 pattern with d h = log 5 (23): β ∞ ≈ −0.1055. In the p = 0.5 statistical fractal with Hausdorff dimensiond h = log 3 (8.5), after averaging conductance for an ensemble of 10 4 geometrically disordered lattices at each L, we find β ∞ ≈ −0.1063 [41].…”

Section: (C) and 2(d)]mentioning

confidence: 99%

“…The scaling theory of phase transitions on fractals holds for the Ising model [15,16], the percolation transition [17], as well as the metal-insulator transition of the Anderson model on bifractal lattices [18,19]. Tangential to our study are the investigations of shot noise in fractal resistor networks [20,21] and quantum transport on clean Sierpiński gaskets [22][23][24] and carpets [25]. …”

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confidence: 99%

Attractive critical point from weak antilocalization on fractals

Sticlet

Akhmerov

2016

Phys. Rev. B

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We report an attractive critical point occurring in the Anderson localization scaling flow of symplectic models on fractals. The scaling theory of Anderson localization predicts that in disordered symplectic two-dimensional systems weak-antilocalization effects lead to a metal-insulator transition. This transition is characterized by a repulsive critical point above which the system becomes metallic. Fractals possess a noninteger scaling of conductance in the classical limit which can be continuously tuned by changing the fractal structure. We demonstrate that in disordered symplectic Hamiltonians defined on fractals with classical conductance scaling g ∼ L −ε , for 0 < ε < β max ≈ 0.15, the metallic phase is replaced by a critical phase with a scale-invariant conductance dependent on the fractal dimensionality. Our results show that disordered fractals allow an explicit construction and verification of the ε expansion. DOI: 10.1103/PhysRevB.94.161115Introduction. The one-parameter scaling hypothesis [1] is central to the study of disordered electronic systems. The hypothesis states that in disordered noninteracting systems the beta function β = d log g/d log L determining the change of conductance g with the system size L is a universal function of g. Single-parameter scaling is known to be violated in quantum Hall systems [2] or topological insulators [3,4], where the topological invariant is the second scaling variable required to capture the scaling flow, and in systems where disorder itself is an irrelevant scaling variable [5][6][7]. Despite that, the scaling flow of Anderson localization holds in an extremely broad range of systems [8,9].The scaling flow has two universal regimes. In the insulating regime g 1 the exponential localization of the wave functions leads to a further decrease of conductance with the system size leading to β ∝ log g + constant. At high conductance, the beta function recovers the classical Ohm law, lim g→∞ β ≡ β ∞ = d − 2 with d the Euclidean dimension. A successful prediction of the theory was the occurrence of a metal-insulator transition in 3d as the flow passes between these two limits. Later studies have refined the theory in the diffusive regime by taking into account quantum corrections to the Ohmic conductance [8,9]. In time-reversal-invariant systems with spin-orbit interactions, also called symplectic, the corrections to g are positive, yielding weak-antilocalization effects [10]. Consequently, these systems exhibit a metal-insulator transition even in 2d, with logarithmic corrections to conductance g ∝ log L, and a metallic phase at large conductance (see Fig. 1).A successful approach in treating Anderson localization is the ε expansion, which treats d as a continuous variable and constructs a series expansion of the scaling flow [11][12][13][14]. The ε expansion is a mathematical construct which is not expected to have a physical meaning when d is not integer; nevertheless, fractals are examples of systems with noninteger dimensionality. This motivates the main questi...

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Theoretical Design of a Pump‐Free Ultrahigh Efficiency All‐Optical Switching Based on a Defect Ring Optical Waveguide Network

Yang

2019

Annalen der Physik

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A theoretical design of a defect ring optical waveguide network is proposed to construct a pump-free ultrahigh efficiency all-optical switch. This switch creates ultrastrong photonic localization and causes the nonlinear dielectric in the defect waveguide to intensely respond. At its ON state, this material defect without Kerr response helps to produce a pair of sharp pass bands in the transmission spectrum to form the dual channel of the all-optical switch. When it is switched to its OFF state, the strong Kerr response induced refractive index change in the high nonlinear defect waveguide strongly alters the spectrum, leading to a collapse of the dual channels. Network equation and generalized eigenfunction method are used to numerically calculate the optical properties of the switch and obtain a threshold control energy of about 2.90 zJ, which is eight orders of magnitude lower than previously reported. The switching efficiency/transmission ratio exceeds 3×10 11 , which is six orders of magnitude larger than previously reported. The state transition time is nearly 108 fs, which is approximately two orders of magnitude faster than the previously reported shortest time. Furthermore, the switch size can be much smaller than 2.6 µm and will be suitable for integration.

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Electronic transport properties of Sierpinski lattices

Cited by 62 publications

References 28 publications

Photonic band structure of Sierpinski waveguide networks

Photonic band structure of Sierpinski waveguide networks

Attractive critical point from weak antilocalization on fractals

Theoretical Design of a Pump‐Free Ultrahigh Efficiency All‐Optical Switching Based on a Defect Ring Optical Waveguide Network

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