The relation between critical Larmor frequency and angular quantum number

## Abstract

For a complete description of the electronic motion in a quantum dot, we need a method that can describe not only the trajectory behavior of the electron but also its probabilistic wave behavior. Quantum Hamilton mechanics, which possesses the desired ability of manifesting the wave-particle duality of electrons moving in a quantum dot, is introduced in this chapter to recover the quantum-mechanical meanings of the classical terms such as backscattering and commensurability and to give a quantum-mechanical interpretation of the observed oscillation in the magneto-resistance curve. Solutions of quantum Hamilton equations reveal the existence of electronic standing waves in a quantum dot, whose occurrence is found to be accompanied by a jump in the electronic resistance. The comparison with the experimental data shows that the predicted locations of the resistance jump match closely with the peaks of the measured magneto-resistance.

### Keywords

- quantum dots
- quantum Hamilton mechanics
- standing waves
- quantum trajectory
- magneto-resistance

## 1. Introduction

As the size of electronic devices is narrowed down to the nanoscale, quantum effects become so prominent that classical mechanics is no longer able to provide an accurate description for electrons moving in nanostructures. However, due to the lack of the sense of trajectory in quantum mechanics, classical or semi-classical mechanics so far has been the sole tool in determining ballistic orbits in quantum dots. Classical orbits satisfying commensurability conditions of geometrical resonances were derived in the literature to determine the magneto-transport behavior of periodic quantum systems. It was reported that the observed regular peaks in the magneto-resistance corresponded to backscattering of commensurate orbits [1], and the critical magnetic fields determined from the backscattering orbits showed an excellent agreement with the observed peak positions in the magneto-resistance curves [2]. A recent study showed that the ballistic motion of electrons within quantum dots can be controlled by an externally applied magnetic field so that the resulting conductance images resemble the classical transmitted and backscattered trajectories [3].

The use of an anisotropic harmonic function, instead of an abrupt hard potential, to describe the confining potential in a quantum dot was shown to be helpful to improve the accuracy of predicting magneto-resistance peaks based on backscattering orbits [4]. Nowadays, the confinement potential forming an electron billiard can be practically patterned to almost arbitrary profile, through which ballistic orbits with chaotic dynamics can be generated to characterize magneto transport [5]. However, the chaotic behavior and its change with magnetic field could not be described in the usual quantum-mechanical picture due to the lack of a trajectory interpretation. Regarding this aspect, the classical description becomes a valued tool for detailed understanding of the transition from low to high magnetic fields in quantum dot arrays [6]. On the other hand, quantum mechanical model for electron billiards was known as quantum billiards [7], in which moving point particles are replaced by waves. Quantum billiards are most convenient for illustrating the phenomenon of Fano interference [8] and its interplay with Aharonov-Bohm interference [9], which otherwise cannot be described by classical methods.

From the existing researches, we have an observation that the ballistic motion in electron billiards was solely described by classical mechanics, while the wave motion in quantum billiards could only be described by quantum mechanics. The aim of this chapter is to give a unified treatment of electron billiards and quantum billiards. We point out that quantum Hamilton mechanics [10, 11] can describe both ballistic motion and wave motion of electrons in a quantum dot to provide us with a quantum commensurability condition to determine backscattering orbits as well as with the wave behavior to characterize the magneto-resistance in a quantum dot.

Quantum Hamilton mechanics is a dynamical realization of quantum mechanics in the complex space [12], under which each quantum operator is realized as a complex function and each wavefunction is represented by a set of complex-valued Hamilton equations of motion. With quantum Hamilton mechanics, we can recover the quantum-mechanical meanings of the classical commensurability condition by showing that there are integral numbers of oscillation in the radial direction, as an electron undergoes a complete angular oscillation around a quantum dot. When the radial and angular dynamics are commensurable, the shape of electronic quantum orbits is found to be stationary like a standing wave. Furthermore, the wave number

The electronic standing-wave motions considered in this chapter will reveal that a jump of the magneto-resistance in quantum dots is accompanied by a phenomenon of magnetic stagnation, which is a quantum effect that an electron is stagnated or trapped within a quantum dot by an applied magnetic field in such a way that the electron’s cyclotron angular velocity is exactly counterbalanced by its quantum angular velocity. We point out that magnetic stagnation is a degenerate case from the electronic standing-wave motion as the wave number

In the following sections, we first introduce quantum Hamilton mechanics and apply it to derive Hamilton equations, which are then used in Section 2 to describe the electronic quantum motions in a quantum dot. By solving the Hamilton equations of motion, Section 3 demonstrates electronic standing-wave motions in various quantum states and characterizes the magnetic field leading to the phenomenon of magnetic stagnation. In Section 4, we show that the magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field by comparing the theoretical predictions obtained from Section 3 with the experimental results of the magneto-resistance curve [4, 13].

## 2. Quantum Hamilton dynamics in a 2D quantum dot

To probe the quantum to classical transition, which involves both classical and quantum features, quantum dots are the most natural systems [14]. Analyzing such systems, we need an approach that can provide both classical and quantum descriptions. Quantum Hamilton mechanics is one of the candidates satisfying this requirement. This chapter will apply quantum Hamilton mechanics to an open quantum dot with circular shape, which is connected to reservoirs with strong coupling. The electronic transport through an open quantum dot can be realized by nano-fabrication techniques as a two-dimensional electron gases system (2DES) at an AlGaAs/GaAs heterostructure, as depicted in Figure 1.

Under the framework of quantum Hamilton mechanics [10, 12], the equivalent mathematical model of a quantum dot is described as an electron moving in an electromagnetic field with scalar potential

We adopt polar coordinates

where

where

where

the quantum Hamilton-Jacobi Eq. (4) associated with the Hamiltonian in Eq. (3) turns out to be

The recognition of the complex Hamiltonian

to Eq. (6) to produce the expected Schrodinger equation:

Due to the time-independent nature of the applied potentials

where

On the other hand, Eq. (3) can be rewritten by using the substitutions Eqs. (5) and (7) as

where

Upon performing the differentiations

Apart from deriving the Schrodinger equation, the above complex Hamiltonian also gives electronic quantum motions in the state

The appearance of the imaginary number

The Hamilton equation for

which can be conceived of as a complex-valued version of Bohmian mechanics [18, 19]. The complex quantum trajectory method based on Eq. (16) has been recently developed into a potential computational tool to analyze wave-packet interference [20] and wave-packet scattering [21].

The wavefunction

where

where the dimensionless time is expressed by

## 3. Standing waves and critical magnetic field

The conductivity of a quantum dot depends on how electrons move under the confinement potential within the quantum dot. Eqs. (19) and (20) provides us with all the required information to describe the underlying electronic quantum motion. The radial motion

On the other hand, if

where

As shown in Figure 2d, the standing-wave motion degenerates into a confined motion such that the electron is trapped into a closed trajectory, in the extreme case

The pattern and the orientation of the standing waves can be controlled by the applied magnetic field

which, in turn, is solely determined by the magnetic field

According to the residue theorem, the contour integral in Eq. (23) is equal to

where

The sequence

**(A) Standing Wave with**

In case of

It appears that that the ground-state electron rotates with a constant angular velocity

where

The commensurability condition Eq. (21) with the calculated

where we note

where

Regarding excited states, there are multiple periods in the radial motion

which has four equilibrium points at

According to different encirclements of equilibrium points, four sets of complex trajectories

Corresponding to the four different ways of encirclement, the four quantization levels of

The commensurability condition for the occurrence of standing wave in the four contour sets now can be derived from Eq. (21) as

The related critical magnetic field

**(B) Standing Wave with**

In the case of

The time average

Substituting the above

Due to the constraint

where

The period

Using

The comparison between Eqs. (33) and (40) leads to the observation that the number of the allowed integer

**(C) Standing Waves with**

In this case, the cyclotron angular velocity

Range of | Critical | Range of integer |
---|---|---|

The cases of

where the admissible integer

The critical magnetic field

The critical

In a case study of

The radial trajectories

Set | Frequency range | Critical frequency | Integer |
---|---|---|---|

Typical standing waves in

In the state

Apart from the consequence of

There are infinitely many wavelengths distributed on the circumference of the quantum dot, as

## 4. Experimental verification

This section will compare the above theoretical predictions with the existing experimental data [4, 13] to confirm the fact that the effect of magnetic stagnation is the main cause to the resistance oscillation of quantum dots in low magnetic field. The experiment was performed in an AlGaAs/GaAs heterostructure with a carrier concentration

Thus far, our analysis on quantum trajectory focuses on some specific states. In order to know the influence of the applied magnetic field on the resistance, we have to consider all the possible quantum states occupied in the device. At temperature

where

0 | 1 | 2 | 3 | 4 | ||
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | |

−1 | 2/3 | 2/3, 0.205 | 2/3, 0.373, 0.119 | 2/3, 0.52, 0.19 | 2/3, 0.252, 0.124 | |

−2 | 4/5 | 4/5, 0.316 | 4/5, 0.543, 0.2 | 4/5, 0.73, 0.31, 0.146 | 0.4, 0.213, 0.115 | |

−3 | 6/7 | 6/7, 0.391 | 6/7, 0.64, 0.26 | 0.39, 0.196 | ||

−4 | 8/9 | 8/9, 0.445 | 0.7, 0.3, 8/9 | 0.454, 0.237 | ||

−5 | 0.486 | 0.75, 0.347 | ||||

−6 | 0.52 | 0.78, 0.38 | ||||

−7 | 0.546 | |||||

−8 | 0.57 |

An incident electron subjected to an applied magnetic field

where the summation is taken over all the states listed in Table 3. The expression of

which in turn is substituted into Eq. (46) to express the magneto-stagnation function

The electron’s total angular velocity via all admissible quantum states at

If magnetic stagnation takes place simultaneously in many states, its effect will be amplified. Stagnation frequencies such as

Figure 6 demonstrates the strong correspondence between the stagnation function

## 5. Conclusions

Parallel to the existing probabilistic description for a quantum dot by a probability density function