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Electron phase coherence Interference is the addition (superposition) of two or more waves that result in a new wave pattern. Simple example: Two slit experiment φ d L Electron phase coherence Q1: Why do we use a two-slit set? Q2: What to we need to observe interference from two different sources? A: We can observe interference of two sources, if they are coherent. Two signal are coherent if the phase difference, stable. Electron phase coherence , is According to quantum mechanics, electrons can behave as waves. What is the role of phase coherence? To answer this question let us discuss the effect which would not exist in the absence of interference – the Aharonov-Bohm effect. An important difference between electrons and electromagnetic waves is that electrons have a finite charge. Electron phase coherence Aharonov-Bohm effect In classical mechanics the motion of a charged particle is not affected by the presence of magnetic fields in regions from which the particle is excluded. For a quantum charged particle there can be an observable phase shift in the interference pattern recorded at the detector. The Aharonov-Bohm effect demonstrates that the electromagnetic potentials, rather than the electric and magnetic fields, are the fundamental quantities in quantum mechanics. Electron phase coherence Let us make a confined tube of magnetic field 1 2 Will the interference pattern feel this magnetic field? For a plane wave, the wave function The phase gain along some way is then As we know, in a magnetic field Additional phase difference Electron phase coherence Aharonov-Bohm Effect for Nanowires Φ Magnetic flux quantum Electron phase coherence Aharonov-Bohm oscillations, Webb 1985, Au Fourier analysis shows that there are also weak oscillations with half period Electron phase coherence High order interferences Here the clock-wise and counterclockwise paths are exactly the same: The waves “strengthen” each other, constructive interference The backscattering probability is enhanced. Sharvin, 1981 Finite magnetic field destroys the interference, the period being a half of the period in the ring. Electron phase coherence Altshuler, Aronov, Spivak (AAS) oscillations The period is 2Φ0 Experiment by Sharvin, 1981, Mg-coated human hair (different samples, 1.12 K) AB oscillations vanish in an ensemble of small ring since the phases φ0 are random. In contrast, AAS oscillations survive ensemble averaging. Electron phase coherence Test of the ensemble averaging, Umbach 1986 Ag loops, 940x940 nm2, width of the wires 80 nm Fourier series N-dependence of the AB oscillations amplitude Electron phase coherence Weak localization Let us calculate the probability for an electron to move from point 1 to point 2 during time t. in terms of the transition amplitude, Ai, along different paths. Classical probability Interference contribution Vanishes for most paths since phases are almost random Electron phase coherence Consider now a close loop with two identical paths traversed in opposite directions. Then the amplitude Aj is just a time reversal of Ai. Hence The backscattering probability is enhanced by factor 2! This is a predecessor of localization. This effect is called the weak localization since the relative number of closed loops is small. However, the effect is very important since it is sensitive to very weak magnetic fields. Electron phase coherence Let us calculate the probability for an electron to return to the “interference volume” during the time dt. One obtains: Interference volume Volume of diffusive trajectory Thus, the relative quantum correction is decoherence time Electron phase coherence D – diffusion constant, b – thickness, d dimensionality In 2D case, introducing the sheet conductance we get This contribution is suppressed by very weak magnetic fields, where bending of the trajectories by magnetic field is still not important. This value corresponds to one magnetic flux quantum trapped in a typical interfering trajectory. Electron phase coherence Experiment: Si/SiGe quantum well Weak localization is a very important phenomenon – it allows find the decoherence time, spin-orbit interaction, etc. Electron phase coherence Universal conductance fluctuations Let us look at the resistance of the wire we have already seen, in a weak magnetic field, ωc < 0.45ω0 WL CF The patterns are not random, the are fingerprints of the system Electron phase coherence How they can be explained? For a rectangular sample LxW the Drude formula can be written as Conductance per mode Electron phase coherence Number of modes How they can be explained? For a rectangular sample LxW the Drude formula can be written as Conductance per mode Number of modes From the Landauer formula, Now we are interested in Electron phase coherence For the transition amplitude, Since reflections are dominated by few events, they can be considered as uncorrelated. Then Now we have to calculate the variance of the reflections Electron phase coherence Result: Averaging and substituting we get: Universal conductance fluctuations (UCF) Thus Electron phase coherence Typical fluctuation is very large, and relative fluctuation does not decay as it would follow from statistical physics. Quantum low-dimensional systems do not possess the property of self-averaging. At relatively large temperatures this properties is restored due to decoherence. At , Electron phase coherence What happens in magnetic field? Usually it is assumed that in a magnetic field the sample properties change significantly. Thus one can think that at each magnetic field a sample represents a realization of an ensemble (so-called ergodic hypothesis) . Consequently, one can average over magnetic fields rather then over different samples. As a result, we have fluctuations as a function of magnetic field, the theory being rather complicated. Electron phase coherence Phase coherence in ballistic systems Electrostatic Aharonov-Bohm Effect Split gates Interference pattern Electron phase coherence Tunneling transport Classical motion Quantum tunneling E V0 1 2 a Electron phase coherence 3 4 unknown coefficients, and 4 boundary conditions at : continuity of wave functions and their derivatives (currents) we find transition and reflection amplitudes, t and r, respectively We show expression for the transition probability, Tunneling Quantum effects: Over-barrier reflection Electron phase coherence Resonant tunneling and S-matrices Electron phase coherence Properties: For a symmetric system they imply What will happen with transmission probability if we place two barriers in series? Electron phase coherence + Geometric series Transmission probability: Electron phase coherence +… At some values of For small transmission, Since the phase is given by the relation we have Electron phase coherence We observe that at E=Es and T1=T2 the total transparency is one though partial transparency are very small. This phenomenon is called the resonant tunneling – it is fully due to quantum interference. Often people introduce the attempt frequency as and partial escape rates, s-resonance Electron phase coherence . Then near the Thus, the double-barrier structure behaves as an optical interferometer. Will resonant tunneling survive if the barriers are far from each other? Unfortunately, NO, since then the coherence would be lost. Decoherence can be modeled by introducing a reservoir between the barriers which can absorb and re-emit the electrons whose phase memory gets lost. Electron phase coherence Current-voltage curve: Negative differential conductance! Electron phase coherence Transmission through a ballistic quantum ring Calculation can be done using Smatrix approach Novel feature – triple junctions, described by the matrix Electron phase coherence Open ring, AB oscillations Closed ring Electron phase coherence Summary In this lecture, we have discussed • The Aharonov-Bohm effect in mesoscopic conductors • Weak localization • Universal conductance fluctuations • Phase coherence in ballistic 2DEGs • Resonant tunneling Electron phase coherence