Investigation of the coupling asymmetries at double-slit interference experiments
QQuantum phases: 50 years of the Aharonov-Bohm Effect and 25years of the Berry phase
Investigation of the coupling asymmetries atdouble-slit interference experiments
A. I. Mese , A. Bilekkaya , S. Arslan , S. Aktas andA. Siddiki , Trakya University, Department of Physics, 22030 Edirne, Turkey Department of Physics, Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilans-Universitat, Theresienstrasse 37, 80333 Munich, Germany Istanbul University, Faculty of Sciences, Physics Department, Vezneciler-Istanbul34134, Turkey
Abstract.
Double-slit experiments inferring the phase and the amplitude of thetransmission coefficient performed at quantum dots (QD), in the Coulomb blockaderegime, present anomalies at the phase changes depending on the number of electronsconfined. This phase change cannot be explained if one neglects the electron-electroninteractions. Here, we present our numerical results, which simulate the real samplegeometry by solving the Poisson equation in 3D. The screened potential profile is usedto obtain energy eigenstates and eigenvalues of the QD. We find that, certain energylevels are coupled to the leads stronger compared to others. Our results give strongsupport to the phenomenological models in the literature describing the charging of aQD and the abrupt phase changes.
1. Introduction
One of the most interesting experiments in the history of physics is the double-slitexperiments, which infers to the quantum mechanical nature of the particles. Thetechnological developments in producing low dimensional high mobility charge carriersystems, enabled experimentalists to re-do the double-slit experiments consideringnanostructures. In the experiments performed at cryogenic temperatures andconsidering a two dimensional electron system (2DES), the phase and the transmissionamplitude were measured simultaneously [1, 2]. The findings of these and consequentexperiments activated a huge number of theoreticians to understand the physicsunderlying the abrupt phase changes [3, 4, 5, 6, 7, 8], for a comprehensive review wesuggest the reader to check especially Ref. [5] and the references given thereby. Inparticular, G. Hackenbroich et. al investigated the effect of shape deformation of aparabolic quantum dot (QD), in the absence of Coulomb interaction, and showed thatthe degeneracy due to the symmetry of the QD is lifted, however, for the deformed QD a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b oupling asymmetries δ . The interplay between the level widthΓ and δ is used to give an explanation to the observed phase anomalies [6, 8]. On theother hand, the effect of interactions was included by M. Stopa by solving the relatedPoisson and Schr¨odinger equations self-consistently within a Hartree-Fock type meanfield approximation, however, its influence on the phase was left unresolved. We shouldalso note that, in these calculations a rather simplified QD geometry was investigatedcompared to the experiments.This work aims to provide numerical support to the theories which rely on theformation of a wide state at certain QD geometries. We obtain the potential profile ofthe real samples by solving the Poisson equation in 3D using fast Fourier transformation,iteratively [9, 10]. In our calculations, we consider the sample geometry presented inRef. [2]. The next step is to obtain the energy eigenstates and values for the calculatedeffective potential. We solve the Schr¨odinger equation by diagonalizing the single-particle Hamiltonian implementing the finite difference techniques.
2. Theory
Here, we investigate the single particle eigen-energies and eigenfunctions of the reduced2D Hamiltonian H = p m ∗ + V ( x, y ) , (1)where p is the momentum operator in 2D, m ∗ is the effective mass (= 0 . m e inGaAs) and V ( x, y ) = V G ( x, y ) + V D ( x, y ) + V l ( x, y ) + V H ( x, y ) is the mean field potentialcomposed of gates, leads, donors and Hartree terms, respectively.In physics, WKB approximation is one of the most frequently applied approximation tosolve Schrdinger’s equation. The transmission amplitude W n ( a, b ) is calculated at thebarrier along the classical turning points (a,b), via W n ( a, b )( E ) = e ξ ( E ) e ξ ( E ) ,ξ ( E ) = − (cid:90) ba dx (cid:114) m (cid:126) ( V ( x ) − E ) . (2)It is known that the transmission amplitude depend almost linearly to the energy of theincoming state, assuming plane waves and within the WKB approximation [11], whichwe utilize likewise in the following to calculate transport through the QD.We proceed our work by considering the real geometry and the potential profilecalculated within the self-consistent Thomas-Fermi-Poisson (TFP) theory. In thefollowing section we first discuss the limitations of such a mean field approximationand compare our method with the existing calculation schemes in the high electronoccupation regime, i.e. N (cid:38) oupling asymmetries Figure 1.
The self-consistent potential plotted in 3D as a function of the lateralcoordinates. The lengths are in units of effective Bohr radius and energy is normalizedwith the effective Rydberg energy. The inset depicts the sample geometry, where Sstands for source lead and D stands for the drain lead. Coupling of the QD to theleads is manipulated by changing the applied potential V .
3. Results and Discussion
The calculation of the electrostatic potential considering real sample geometries togetherwith the electron-electron (e-e) interaction is a challenging issue. Since such a calculationcannot be done analytically for almost all the cases, usually numerical techniques aredeployed. It is clear that for ”more than a few” electron regime (
N >
10) exactdiagonalization methods are either impossible or very costly in terms of computationaleffort. It is favorable to use a mean field approximation to describe the (e-e) interactions,which is questionable in the ”less than a few” electron regime. The commonly usedapproach to determine the bare confinement potential generated by the gates is the”frozen charge” approximation [12], which takes into account properly the gate patternand the effect of the spacer between the gates and the 2DES. Since, it is not self-consistent this approximation cannot account for the induced charges on the metallicgates defining the QD. The effects resulting from the induced charges and donor layercan be handled by solving the 3D Poisson equation self-consistently. Almost a decadeago M. Stopa introduced a very effective numerical scheme to describe the electrostaticsof such samples [11], including the e-e interactions either using a full Hartree, i.e. solvingthe Poisson and Schr¨odinger equations self-consistently, or considering Thomas-Fermiapproximation (TFA). The exchange-correlation interaction was accounted by a localdensity approximation (LDA) using the density functional theory (DFT). It was shownthat the TFA is powerful enough to describe the electrostatic potential even if the oupling asymmetries Figure 2. (a-g) Selected eigenfunctions residing at the E = 1 .
96 plateau (indicatedby the horizontal line in i) calculated for the color plotted potential (h), together withthe energy spectrum near E F . The state = 160 present a slight asymmetry in couplingto source lead compared to the drain. The asymmetric potential distribution with inthe dot is visible. electrons are fully depleted in some regions of the sample [11].Here, we stay in the TFA to calculate the electrostatic properties of the real samplegeometry using the algorithm developed by A. Weichselbaum et. al [13, 14], whichimplements an efficient grid relaxation technique to solve the 3D Poisson equation.This approach was shown to be reliable to obtain the potential profiles in the ”morethan a few” electron regime considering QDs and quantum point contacts [10]. Thenext step in our calculation scheme is to obtain the single particle energies and states,which we do same as described in the previous section.Figure. 1 presents the calculated potential profile for the sample geometry measuredin Ref. [2]. We apply negative voltages to the gates shown in the inset. The upper andlower two gates (denoted by red areas) are kept at the same potential V , whereas thecenter gate (left black) and the plunger gate (right black) are biased with a fixed voltage V . Here, we consider a unit cell of 440x440 nm spanned by 128x128 mesh matrix tocalculate the self-consistent potential. The surface potential is fixed to -0.75 V pinningthe Fermi energy at the mid gap. The 2DES is some 100 nm below the surface followedby a thick GaAs layer. To achieve numerical convergence and satisfy the open boundaryconditions 3 mesh points of dielectric material is assumed at all boundaries. In Figure. 1fixed voltages of V = − . V = − . × cm − corresponding to E F ≈ .
75 meV, with the given density,the number of electrons in the dot N is similar to 200. Figure. 2 presents the calculatedsingle particle wave functions as a function of spatial coordinates, together with the oupling asymmetries Figure 3.
The transmission coefficients calculated at the barrier. The turning pointsare obtained from the self-consistent potential at the center of the barrier, where theenergy of the incoming wave cuts along. potential counter plot and the corresponding eigen energies versus the state number.We show the states residing at the energy plateau , which lay in the close vicinity of E F (depicted by the horizontal solid line in Figure. 2i). The states shown at the upperpanel present the chaotic behavior, whereas the first two states of the mid panel arethe non-propagating states. At n = 165 a resonant channel is observed, meanwhile thehighest state shown presents the chaotic behavior. These results show that, qualitatively,transport through state 165 is much probable compared to the others sitting at the sameplateau. Although the single particle energy eigenvalues are close to each other a singlechannel is in charge of transport. At these gate voltages, the QD is loosely defined asone can see that it is possible to find an electron also at the left side of the actual QD.This situation is changed by applying a higher negative potential to the central and theplunger gates, V = − . W n ( a, b )( E ), in Fig. 3 we show the this quantity as a function of energy of the incomingwave calculated within the WKB at various gate voltages, V , V . We see that, when theupper and the lower gates are biased with small potentials, the transmission increaseslinearly. This linearity changes if one applies higher voltages to the barriers, however,for higher energies the linearity is recovered. Such an observation leads us to concludethat, essentially the probability distribution determines the level widths, which may oupling asymmetries Acknowledgments
We would like to thank Jan von Delft for introducing us the “phase lapses” problem andmotivating us to perform numerical calculations. Moty Heiblum is also acknowledgedfor his enlightening discussions. This work is partially supported by TUBiTAK, undergrant no:109T083.
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