EMBRAPA- INTA: Nº 4

Currently, there is a large variety of interference experiments that are based on the superposition of electron waves in a system that preserves a static phase relation between the interfering electron waves. The electron phase is sensitive to an external applied magnetic field ~B. The reason for this is the vector potential ~A whose curl is

equal to ~B. The vector potential experienced a transition from being a mathemat-

ical tool to simplify Maxwell’s relations to an important physical property by the work of Aharonov and Bohm in 1959 [6]. The two scientists predicted that electrons accumulate a phase along a path when exposed to an external magnetic field that does not necessarily has to cross this path and proposed an interference experiment to verify their prediction. In this experiment an electron beam is split into two paths which rejoin after having traveled a length smaller than the phase coherence length of the electrons. The prediction concerns the interference pattern that oscillates between constructive and destructive as a perpendicular magnetic field is sweeped. The period of this oscillation corresponds to the enclosed magnetic flux 2Φ0 = h/e.

Indeed, this so-called Aharonov-Bohm (AB) effect was later proven to exist in vac- uum [170], gold films [171], parallel 2DEGs of a AlxGa1-xAs/GaAs heterostructure

[172], a AlxGa1-xAs/GaAs quantum ring [173], carbon nanotubes [174], graphene

[175] and Bi2Se3 nanoribbons [176]. Further quantum interference experiments with

electrons followed and were partly inspired by optical experiments.

Thomas Young’s interference experiments with light [1] were based on the usage of a double-slit whose distance from an observing screen determines the interfer-

ence pattern. The double-slit experiment was later realized for a beam of electrons [177] that is diffracted by a copper film with slits of 0.3 µm width. Subsequently, Feynman’s assumption that this interference pattern should also be visible for single electrons [178] was confirmed experimentally [179]. The double-slit experiment was followed by several other interference experiments with light that were accompanied by their electronic counterparts. One example is the Mach-Zehnder interferometer [180, 181], that is an advanced version of the Jamin interferometer [182] to detect phase shifts of light as well as the refraction index of an optical medium. In the original version of the Mach-Zehnder interferometer a beam of light is split into two paths and recombined by a set of mirrors. In the electronic counterpart edge states are created in the quantum Hall regime and are split and recombined by quantum point contacts [183]. The optical version of a Fabry-Pérot interferometer [184] makes use of an optical cavity to filter a single wavelength from light with a wide spectrum. The idea of such a resonator can be applied to carbon nanotubes that define a cavity between two nanotube-electrode interfaces [185].

Quantum interference also becomes very important in disordered metallic sys- tems where the phase evolution - and thus the interference - of the electron waves is determined by the setup of the disorder. An applied electric field E changes the potential landscape and therefore the interference paths. This leads to a fluctu- ation of the interference and therefore to a fluctuation of the resulting electrical conductance when sweeping E. This effect is referred to as universal conductance fluctuation [186, 37]. Disorder may also cause electron waves to propagate along closed circles rather than propagating along a potential gradient. The interference of a time-reversed pair of electron waves causes localization [65] which increases the electrical resistance in the absence of a magnetic field. Strong spin-orbit interaction of the localized electrons causes the opposite effect and leads to an increase of the conductance in the absence of a magnetic field [9]. The presence of a magnetic field leads to a suppression of both effects.

The sensitivity of the above mentioned quantum interference experiments to scattering and an applied magnetic field enables to probe the phase evolution in a device, as done with quantum dots [187, 188, 189] whose influence on the electron phase is studied by means of the AB effect. A typical feature of two-terminal AB interferometer devices is the restriction of the phase evolution to 0 or π jumps, be it in the presence [187, 190] or in the absence [191, 192] of quantum dots. This effect is referred to as ’phase rigidity’ and can be understood in terms of time- reversal symmetry and current conservation in a closed system [35]. This leads to the symmetry of the transmission probabilities Tij(B) = Tji(−B) of the electrons

to propagate from channel i to j and vice versa and consequently leads to the Onsager reciprocity Rij(B) = Rji(−B) [193]. The origin of phase rigidity can also

be expressed in terms of multiple reflections of the electron waves at the crossing points in the interferometer device [194]. In order to break the phase rigidity, the reduction of the device symmetry is required, e.g. by attaching additional leads to the ring [195]. This was demonstrated for four-terminal ring devices [188]. Note that even in four-terminal devices the measurement setup may still lead to a symmetry in the system causing phase rigidity [148, 194].

Depending on the observed system, electrons may change or even lose their phase relation. Elastic scattering, i.e. scattering of electrons with static potentials, such as potential barriers in an 1DEG, changes an electron’s phase. However, these changes do not vary with time and the phase shift is fix for all electrons. Therefore, elastic

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scattering preserves the phase coherence of a given system. In contrast, inelastic scattering randomizes the electron’s phase and breaks the phase coherence, i.e. it acts ’dephasing’ with a dephasing rate of τ_ϕ−1. Such a randomization also occurs due to thermal averaging of the electron phase, which describes the smearing of the Fermi wave vector and the subsequent loss of the initial phase information after traveling the thermal length. This can be expressed with the thermal dephasing rate τT. Hence, the total dephasing rate is τdeph−1 = τϕ−1+ τ

−1

T . At low temperatures

electron-electron interactions are found to dominate the dephasing scattering in 2DEGs [158], i.e. τϕ → τee, and the according relaxation time is predicted [196, 197]

to be τee∝ (Tbath2 ln Tbath)−1.

In ballistic 1DEGs the dephasing processes are not fully understood. The am- plitude of the characteristic Aharonov-Bohm oscillations is a measure of the phase relaxation time in 1D quantum rings. In recent experiments [146, 147, 148] the am- plitude of the oscillations is found to be RAB ∝ Tbath−1 . A similar observation is made

for an increased excitation current IOsc [198]. Another method to analyze the phase

coherence ring in a quantum ring is linked to the harmonics of the base oscillation frequency. Beyond the base frequency, each further visible harmonic in the Fourier spectrum corresponds to electrons propagating an additional time around the ring [199, 200, 146], allowing to estimate the phase coherence length in units of the ring circumference. The second harmonic is often associated with the Al’tshuler-Aronov- Spivak (AAS) effect [201]. The AAS oscillations were first observed in cylindrical metal films [202], subsequently in networks of thin metallic wires in the diffusive regime [203], in antidot lattices [204, 205] and in quantum rings [206, 207]. In contrast to the AB oscillations, the AAS oscillations can still occur in disordered systems [208] and occur if both interfering electrons have traveled the complete cir- cumference of the device [209]. Depending on the experimental conditions either one or both effects can be present at the same time.