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Physical Review B

Interstate interference of electron wave packet tunneling through a quantum ring

Takuma Okunishi, Yusuke Ohtsuka, M. Muraguchi, Kyozaburo Takeda

Published Jun 8, 2007• 20 citations
Publisher

Abstract

We theoretically study the time-developed progress of resonant tunneling for an electron wave packet injected into a two-dimensional quantum ring (QR) by solving the time-dependent (TD) Schr\"odinger equation numerically. Focusing on an extraction of the angular momentum ${l}_{z}$, we examine the TD features in the resonant tunneling electron by projection analysis in which the resulting TD wave function at the QR is decomposed into the (static) resonant states by calculating the inner products among them. This analysis reveals that the two-states approach is well applicable for the QR system and the cross terms between these two states are crucial for the TD vacillation of both the expectation value of the angular momentum $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{l}}_{z}⟩$ and the electron density $\ensuremath{\rho}$. The quasidegeneracy of the resonant states causes a characteristic beating whose frequency is determined by the difference between the eigenenergies. We further study the corresponding TD phenomena under a magnetic field and find that the rotational direction in $\ensuremath{\rho}$ changes in accordance with the strength of the magnetic field. This feature seems to be very different from the classical prospect of the cyclotron motion where the application of the magnetic field determines the rotational direction uniquely.

DOI
10.1103/physrevb.75.245314

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We theoretically study the time-developed progress of resonant tunneling for an electron wave packet injected into a two-dimensional quantum ring (QR) by solving the time-dependent (TD) Schr"odinger equation numerically. Focusing on an extraction of the angular momentum lz{l}_{z}lz​, we examine the TD features in the resonant tunneling electron by projection analysis in which the resulting TD wave function at the QR is decomposed into the (static) resonant states by calculating the inner products among them. This analysis reveals that the two-states approach is well applicable for the QR system and the cross terms between these two states are crucial for the TD vacillation of both the expectation value of the angular momentum ⟨l\ifmmode^\elseˆ\fiz⟩⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{l}}_{z}⟩⟨l\ifmmode^\elseˆ\fi​z​⟩ and the electron density \ensuremathρ\ensuremath{\rho}\ensuremathρ. The quasidegeneracy of the resonant states causes a characteristic beating whose frequency is determined by the difference between the eigenenergies. We further study the corresponding TD phenomena under a magnetic field and find that the rotational direction in \ensuremathρ\ensuremath{\rho}\ensuremathρ changes in accordance with the strength of the magnetic field. This feature seems to be very different from the classical prospect of the cyclotron motion where the application of the magnetic field determines the rotational direction uniquely.
We have modeled the transmission of an electron wave packet through a resonant tunneling semiconductor nanostructure by solving the time-dependent Schr\"odinger equation using the finite-difference method. We have found in all cases that the passage of the electron wave packet through the tunneling barrier is accompanied by a propagation delay relative to the propagation of an undisturbed wave packet. Tunneling transport is shown to be causal, and no evidence of superluminal behavior is seen, either for resonant or for off-resonant tunneling. In the case of off-resonant tunneling, the peak of the transmitted wave packet is observed to exit a double resonant tunneling barrier before the peak of the incident wave packet enters the structure. However, these two peaks are not directly related, and the appearance of a well-formed peak in the transmitted intensity is shown to be a result of the transient behavior of the tunneling event.
The wave packet of electron hoping through quantum rings under the static magnetic field is solved by transformation to the canonical form of certain system of differential equations. The Lorentz force leads to electron asymmetry which enhances the electron passing through a quantum ring while the Aharanov›Bohm effect (AB effect) reduces the probability of transmis- sion by phase shifted interference. For zero or similar magnetic field of both rings, the wave packet can pass both quantum rings to the exit quantum wire while different magnetic field of both rings prevent the second ringus injection of electron.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
The Larmor time of an incident wave packet has recently been measured experimentally [R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature (London) 583, 529 (2020)]. It differs essentially from previous well-studied free-particle cases, whereas the unique properties of wave packets are unrevealed in their theoretical analysis and state-of-the-art Larmor clock studies. In this paper, we study the Larmor time of a bound two-component electron wave packet and supplement the theoretical lack of wave-packet tunneling. We find that the spin shows a pure Larmor precession in the plane perpendicular to the field without a rotation parallel to the field. The Larmor time is defined in the limit of vanishing field where the precession angle is proportional to the field. For relatively weak fields, this precession angle will respond to the field approximately linearly. For the wave-packet tunneling case, we propose an effective scheme that utilizes the fidelity of the tunneled wave packets to calibrate an approximate linear-response region. Furthermore, we show that in the approximate linear-response region the change in the fidelity can be approximately regarded as the result of the accumulated displacement in phase space. This result reveals the inner mechanism of wave-packet tunneling. Our findings may have implications for future experiments of the Larmor clock.
แบบจำลองของอิเล็กตรอนที่เคลื่อนที่ผ่านวงแหวนควอนตัมภายใต้สนามแม่เหล็กคงตัวสามารถจำลองได้โดยการสร้างตัวแปลงเพื่อแก้ชุดสมการเชิงอนุพันธ์ให้ไปอยู่ในรูปคาโนนิคอลซึ่งแรงลอเรนซ์ที่เกิดขึ้นทำให้อิเล็กตรอนที่เคลื่อนผ่านวงแหวนมีการเอนเอียงไปด้านหนึ่ง และขณะเดียวกันผลจากปรากฏการณ์ของAharonov-Bohmซึ่งเป็นการเลื่อนเฟสได้ทำให้อิเล็กตรอนที่ปลายวงแหวนเกิดการแทรกสอดทำให้ความสามารถในการส่งผ่านลดลงผลลัพธ์จากการคำนวณแสดงให้เห็นว่าสาหรับกรณีที่ไม่มีสนามแม่เหล็กหรือสนามแม่เหล็กที่เกิดขึ้นบนวงแหวนทั้งสองมีค่าใกล้เคียงกันอิเล็กตรอนจะสามารถเคลื่อนที่ผ่านไปได้แต่สำหรับกรณีที่สนามแม่เหล็กมีค่าแตกต่างกันมากอิเล็กตรอนจะไม่สามารถเคลื่อนที่ผ่านวงแหวนวงแรกได้The wave packet of electron hoping through quantumrings under the static magnetic field is solved by transformation to thecanonical form of certain system of differential equations. The Lorentz forceleads to electron asymmetry which enhances the electron passing through aquantum ring while the Aharanov-Bohm effect (AB effect) reduces the probabilityof transmission by phase shifted interference. For zero or similar magneticfield of both rings, the wave packet can pass both quantum rings to the exitquantum wire while different magnetic field of both rings prevent the secondring’s injection of electron.
We consider a charged particle which is driven by a time-dependent flux threading a circular ring system. Various approaches including classical treatment, Fourier expansion method, time-evolution method, and Lewis-Riesenfeld method are used and compared to solve the time-dependent problem. By properly managing the boundary condition of the system, a time-dependent wave function of the charged particle can be obtained by using a non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. The eigenfunction of the linear invariant can be realized as a Gaussian-type wave packet with a peak moving along the classical angular trajectory, while the distribution of the wave packet is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. In this topologically nontrivial system, we find that although the classical trajectory and angular momentum can determine the motion of the wave packet; however, the peak position is no longer an expectation value of the angle operator. Therefore, in such a system, the Ehrenfest theorem is not directly applicable.