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Semiconductor Science and Technology

Aharonov-Bohm effect in semiconductor quantum wells

Ahmet Elçi, D. A. Depatie

Published Feb 1, 1994• 5 citations
Publisher

Abstract

We present a stationary state description of the Aharonov-Bohm effect in semiconductor quantum wells in terms of Landau wavefunctions and discuss electron injection into wells. This description explicitly shows that, if the potential drops between the quantum well interfaces are negligible, a net ballistic charge current in the direction perpendicular to the magnetic field and perpendicular to the axis of the wells is produced only by electrons which are injected into the structure in Aharonov-Bohm superposition states.

DOI
10.1088/0268-1242/9/2/006

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Revista Mexicana de Física•Jan 1, 2001•1 citation
We present a stationary state description of the Aharonov-Bohm effect in semiconductor quantum wells in terms of Landau wavefunctions and discuss electron injection into wells. This description explicitly shows that, if the potential drops between the quantum well interfaces are negligible, a net ballistic charge current in the direction perpendicular to the magnetic field and perpendicular to the axis of the wells is produced only by electrons which are injected into the structure in Aharonov-Bohm superposition states.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.
The Aharonov-Bohm effect is a quantum mechanical effect, that is, has no classical counterpart. The effect was predicted in 1959 in a seminal paper of Y. Aharonov and D. Bohm [AB59] in which they demonstrated that a beam of electrons is affected by the existence of the electric/magnetic field even though electrons travel through field-free regions. Aharonov and Bohm carried out two hypothetical experiments to support their claim that potentials are more fundamental than fields and they are respon-sible of the effect. Since then, the debate has arisen around whether potentials are mathematical tools or fundamental entities in physics. Different arguments have been set to explain the results predicted by Aharonov and Bohm and experimentally confirmed. Amongst these arguments, the first argument adopted by Aharonov and Bohm was that potentials are physically significant. Many have claimed that fields do have non-local features, i.e. action at a distance. Others have claimed that topological effects may interpret the effect in which potentials are modeled as connections in higher-dimensional fiber bun-dle geometries. The most recent argument has been proposed by Vaidman [Vai12] who claimed that the the composite system is represented by one state, an entangled state, and due to the electromagnetic interactions part of this state is changed, hence, the total state. In the present essay, I will discuss the latter argument as well as reviewing some other arguments. i
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
This work presents an analysis of the Aharonov-Bohm effect taking into account both regions where the electron is present and where it is excluded, and also the time variations of the magnetic vector potential and the magnetic induction field as they change between two of their stationary values. This analysis allows the identification of the electrical force field induced in the region where the electron is present by the time variation of the magnetic flux in the region where the electron is excluded, thus providing a classical explanation of the changes in the stationary properties of the electron when the magnetic potential and field change from one set of stationary values to another. Some apparent contradictions in attempts to interpret the A-B effect are also discussed and clarified.