Wave-function Mapping Research Articles

Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling

The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry.

Flat bands in twisted bilayer transition metal dichalcogenides

The crystal structure of a material creates a periodic potential that electrons move through giving rise to the electronic band structure of the material. When two-dimensional materials are stacked, the twist angle between the layers becomes an additional degree freedom for the resulting heterostructure. As this angle changes, the electronic band structure is modified leading to the possibility of flat bands with localized states and enhanced electronic correlations. In transition metal dichalcogenides, flat bands have been theoretically predicted to occur for long moir\'e wavelengths over a range of twist angles around 0 and 60 degrees giving much wider versatility than magic angle twisted bilayer graphene. Here we show the existence of a flat band in the electronic structure of 3{\deg} and 57.5{\deg} twisted bilayer WSe2 samples using scanning tunneling spectroscopy. Direct spatial mapping of wavefunctions at the flat band energy have shown that the flat bands are localized differently for 3{\deg} and 57.5{\deg}, in excellent agreement with first-principle density functional theory calculations.

Controlling the non-linear optical properties of MgO by tailoring the electronic structure

The study of the non-linear response of matter to high electric fields has recently encompassed harmonic generation in solids at near-infrared (NIR) driving wavelengths. Interest has been driven by the prospect of ultrafast signal processing and all-optical mapping of electron wave-functions in solids. Engineering solid-state band structures to control the non-linear process has already been highlighted theoretically. Here, we show experimentally for the first time that second harmonic generation (SHG) can be enhanced by doping crystals of magnesium oxide (MgO) with chromium (Cr) atoms. We show that the degree of enhancement depends non-linearly on dopant concentration. The SHG efficiency is shown to increase when Cr dopants are introduced into pure MgO. A physical picture of the effect of Cr dopants is aided by density functional theory (DFT) calculations of the electronic structure for pure and doped samples. This work shows an unambiguous enhancement of the SHG efficiency by modifying the electronic structure. The observed effects are consistent with an electronic structure that facilitates the surface induced SHG and demonstrates a minimal angular dependence. This work highlights the potential of manipulating the electronic structure of solids to control or test theories of their non-linear optical response.

Density–wave-function mapping in degenerate current-density-functional theory

We show that the particle density $\ensuremath{\rho}(\mathbf{r})$ and the paramagnetic current density ${\mathbf{j}}^{p}(\mathbf{r})$ are not sufficient to determine the set of degenerate ground-state wave functions. This is a general feature of degenerate systems where the degenerate states have different angular momenta. We provide a general strategy for constructing Hamiltonians that share the same ground-state density pair yet differ in degree of degeneracy. We then provide a fully analytical example for a noninteracting system subject to electrostatic potentials and uniform magnetic fields. Moreover, we prove that when $(\ensuremath{\rho},{\mathbf{j}}^{p})$ is ensemble $(v,\mathbf{A})$-representable by a mixed state formed from $r$ degenerate ground states, then any Hamiltonian $H({v}^{\ensuremath{'}},{\mathbf{A}}^{\ensuremath{'}})$ that shares this ground-state density pair must have at least $r$ degenerate ground states in common with $H(v,\mathbf{A})$. Thus, any set of Hamiltonians that shares a ground-state density pair $(\ensuremath{\rho},{\mathbf{j}}^{p})$ by necessity has to have at least one joint ground state.

Experimental realization and characterization of an electronic Lieb lattice.

Geometry, whether on the atomic or nanoscale, is a key factor for the electronic band structure of materials. Some specific geometries give rise to novel and potentially useful electronic bands. For example, a honeycomb lattice leads to Dirac-type bands where the charge carriers behave as massless particles [1]. Theoretical predictions are triggering the exploration of novel 2D geometries [2–10], such as graphynes, Kagomé and the Lieb lattice. The latter is the 2D analogue of the 3D lattice exhibited by perovskites [2]; it is a square-depleted lattice, which is characterised by a band structure featuring Dirac cones intersected by a flat band. Whereas photonic and cold-atom Lieb lattices have been demonstrated [11–17], an electronic equivalent in 2D is difficult to realize in an existing material. Here, we report an electronic Lieb lattice formed by the surface state electrons of Cu(111) confined by an array of CO molecules positioned with a scanning tunneling microscope (STM). Using scanning tunneling microscopy, spectroscopy and wave-function mapping, we confirm the predicted characteristic electronic structure of the Lieb lattice. The experimental findings are corroborated by muffin-tin and tight-binding calculations. At higher energies, second-order electronic patterns are observed, which are equivalent to a super-Lieb lattice.

Graphene quantum dots: wave function mapping by scanning tunneling spectroscopy and transport spectroscopy of quantum dots prepared by local anodic oxidation

Graphene quantum dots (GQDs) are considered as promising alternatives to quantum dots in III–V semiconductors, e.g., for the use as spin qubits. However, this potential has not been realized in experiments so far, calling for a more fundamental investigation of GQDs including a full control of the wave function properties during transport spectroscopy. Here, we review the recent success in mapping wave functions of GQDs by scanning tunneling microscopy (STM). We show that GQDs on Ir(111) are softly confined rendering the wave functions more regular. The wave function mapping is disturbed by a sp-type surface state from Ir(111), which can be improved by Au intercalation or adequate Fourier analysis. Edge states are absent on Ir(111), but reappear after intercalating Au and coupling the σ-bonds of the zig-zag edges to BN. Finally, we show how the goal of probing such wave functions during transport spectroscopy might be achieved. Therefore, we present transport spectroscopy of GQDs on SiO2 which are prepared in a STM compatible fashion. 3D view of STM image of graphene supported by SiO2 , ; Insets (from left to right): STM image of graphene nano-islands on Ir(111) and scanning tunneling spectroscopy images of the three low energy wave functions of a particular nano-island . (© 2015 WILEY-VCH Verlag GmbH &Co. KGaA, Weinheim)

Molecular potentials and wave function mapping by high-resolution electron spectroscopy and ab initio calculations

The recent development of high brightness 3rd generation soft X-ray sources and high energy resolution electron spectrometers made it possible to accurately trace quantum phenomena associated to the vibrational dynamics in core-excited molecules. The present paper reviews the recent results on mapping of vibrational wave functions and molecular potentials based on electron spectroscopy. We discuss and compare the mapping phenomena in various systems, stressing the advantages of the resonant X-ray scattering for studying of the nuclear dynamics and spectroscopic constants of small molecules. The experimental results discussed in the paper are most often accompanied by state-of-the-art ab initio calculations allowing for a deeper understanding of the quantum effects. Besides its fundamental interest, the vibrational wave function mapping is shown to be useful for the analysis of core- and valence-excited molecular states based on the reflection principle.

Wave-Function Mapping of Graphene Quantum Dots with Soft Confinement

Using low-temperature scanning tunneling spectroscopy, we map the local density of states of graphene quantum dots supported on Ir(111). Because of a band gap in the projected Ir band structure around the graphene K point, the electronic properties of the QDs are dominantly graphenelike. Indeed, we compare the results favorably with tight binding calculations on the honeycomb lattice based on parameters derived from density functional theory. We find that the interaction with the substrate near the edge of the island gradually opens a gap in the Dirac cone, which implies soft-wall confinement. Interestingly, this confinement results in highly symmetric wave functions. Further influences of the substrate are given by the known moiré potential and a 10% penetration of an Ir surface resonance into the graphene layer.

“Artificial Atoms” in Magnetic Fields: Wave-Function Shaping and Phase-Sensitive Tunneling

We demonstrate the possibility to influence the shape of the wave functions in semiconductor quantum dots by the application of an external magnetic field B(z). The states of the so-called p shell, which show distinct orientations along the crystal axes for B(z) = 0, can be modified to become more and more circularly symmetric with an increasing field. Their changing probability density can be monitored using magnetotunneling wave function mapping. Calculations of the magnetotunneling signals are in good agreement with the experimental data and explain the different tunneling maps of the p(+) and p⁻ states as a consequence of the different sign of their respective phase factors.

Unraveling the Nature of Charge Excitations inLa2CuO4with Momentum-Resolved CuK-Edge Resonant Inelastic X-Ray Scattering

The results of model calculations using exact diagonalization reveal the orbital character of states associated with different Raman loss peaks in Cu K-edge resonant inelastic x-ray scattering (RIXS) from La₂CuO₄. The model includes electronic orbitals necessary to highlight the nonlocal Zhang-Rice singlet, charge transfer, and d-d excitations, as well as states with apical oxygen 2p(z) character. The dispersion of these excitations is discussed with prospects for resonant final state wave-function mapping. A good agreement with experiments emphasizes the substantial multiorbital character of RIXS profiles in the energy transfer range 1-6 eV.

Visualization of space charge field effect on excitons in a GaAs quantum dot by near-field optical wavefunction mapping

Near-field photoluminescence (PL) imaging spectroscopy was used to investigate multi-exciton and charged-exciton states confined in a single GaAs interface fluctuation quantum dot. We determined the origin of peaks in the PL spectra by employing a wavefunction mapping technique. We observed distortion of the exciton wavefunction due to the electric field produced by an excess electron at a nearby confined state. Near-field wavefunction mapping was demonstrated to be a powerful tool for visualizing the local environment, which affects the emission properties of quantum dots.

Wavefunction Mapping of Immobilized InP Semiconductor Nanocrystals

Wavefunction Mapping of Immobilized InP Semiconductor Nanocrystals

Near-field Optical Wavefunction Mapping and Its Application to Single Quantum Dot Spectroscopy

Currently, near-field scanning optical microscopy offers a spatial resolution down to 10−30 nm. High resolution imaging spectroscopy enables us to visualize the exciton wavefunctions confined in semiconductor quantum structures. In this article, the principle of near-field optical wavefunction mapping is described. Then as an application, we performed mapping out of exciton and biexciton wavefunctions in single GaAs quantum dots. Significant displacement of the center of emission profiles was found, in contrast to the usual difference in the emission profiles of an exciton and a biexciton. By conducting a numerical calculation, such a displacement could be reproduced by introducing a shallow potential dip, which causes a significant difference in the penetration of the wavefunctions into the barrier. Precise mapping of exciton and biexciton wavefunctions of quantum-confined structures will provide a good probe for weakly localized states due to local strain and disorder.

Visualization of weak confinement potentials by near-field optical imaging spectroscopy of exciton and biexciton in a single quantum dot

Imaging spectroscopy of a single GaAs interface fluctuation quantum dot (IFQD) was performed using a near-field scanning optical microscope (NSOM) with a spatial resolution of 40nm. A difference in the emission profiles of an exciton and a biexciton was found for several IFQDs. By comparing with a numerical simulation based on the finite-difference time-domain method, this difference was attributed to the existence of a shallow potential dip in the IFQD. The NSOM wavefunction mapping for excitons and biexcitons as quasiparticles with different masses is a tool for investigating weak confinement potentials to detect local strain and disorder.

Correlation Effects in Wave Function Mapping of Molecular Beam Epitaxy Grown Quantum Dots

We investigate correlation effects in the regime of a few electrons in uncapped InAs quantum dots by tunneling spectroscopy and wave function (WF) mapping at high tunneling currents where electron-electron interactions become relevant. Four clearly resolved states are found, whose approximate symmetries are roughly s and p, in order of increasing energy. Because the major axes of the p-like states coincide, the WF sequence is inconsistent with the imaging of independent-electron orbitals. The results are explained in terms of many-body tunneling theory, by comparing measured maps with those calculated by taking correlation effects into account.

Evidence for one-dimensional electron propagation onSi(111)−(2×1)from Coulomb blockade

Scanning tunneling spectroscopy is used to study the electronic properties of the $\mathrm{Si}(111)\text{\ensuremath{-}}2\ifmmode\times\else\texttimes\fi{}1$ surface at $8\phantom{\rule{0.3em}{0ex}}\mathrm{\text{K}}$. Wave-function mapping reveals highly anisotropic scattering of electrons with one-dimensional (1D) confinement on the $\ensuremath{\pi}$-bonded chains. The differential tunneling conductance is strongly reduced at the Fermi energy ${E}_{F}$ and exhibits a Coulomb gap. The gap width $W$ scales with the inverse length $L$ of the chain segments defined by the distance between two adjacent scatterers such as surface steps or dopant atoms within a given chain. A model of electronic transport through 1D chain segments of finite length describes the observed behavior.

Wave function mapping at Fano resonances and conductance switching in open quantum dots

We numerically simulate the effects of a controllable impurity induced by an atomic microscope tip scanned on an open quantum dot with highly transmitting channels. The coupled quantum dot – tip system may result in a transistor like device, for which the conductance switching is a function of the position, spatial dimensions and potential of the perturbation induced by the tip. For realistic dimensions we show that energy shifts of the resonances deliver a bona fide imaging of low energy isolated quantum dot states, as well as of higher energy states coupled to the continuum (Fano resonances). (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Near-field wave-function spectroscopy of excitons and biexcitons

Improvements in spatial resolution of near-field spectroscopy have recently allowed for wave-function imaging of single excitons and biexcitons localized in semiconductor quantum wells. Surprisingly, the apparent extent of the biexciton wave function was found to be considerably reduced compared to a single exciton. We analyze theoretically the image contrast in general near-field wave-function mapping, taking into account the finite spatial resolution of the experiment. In particular, we show that due to the nonlinear nature of biexciton spectroscopy, a smaller biexciton image size is expected even if the spatial resolution is insufficient to resolve the wave functions.

Electronic structure and transport properties of quantum dots

The subject of this paper are electronic properties of isolated quantum dots as well as transport properties of quantum dots coupled to two electronic reservoirs. Thereby special focus is put on the effects of Coulomb interaction and possible correlations in the quantum dot states. First, the regime of sequential tunneling to the reservoirs is investigated. It is shown that in case degenerate states participate in transport, the resonance positions in the differential conductance generally depend on temperature and the degree of degeneracy. This effect can be used to directly probe degeneracies in a quantum dot spectrum. A further effect, characteristic for sequential tunneling events, is the complete blocking of individual channels for transport. A generalisation of the well known spin blockade is found for correlated dot states transitions through which are not directly spin-forbidden. In the second part, the electronic structure of spherical quantum dots is calculated. In order to account for correlation effects, the few-particle Schrödinger equation is solved by an exact diagonalization procedure. The calculated electronic structure compares to experimental findings obtained on colloidal semiconductor nanocrystals by Scanning Tunneling Spectroscopy. It is found that the electric field induced by the tunneling tip is gives rise to a Stark effect which can break the spherical symmetry of the electronic ground state density which is in agreement with wave-function mapping experiments. The symmetry breaking depends on the competition between exchange energy and the Stark energy. Moreover, a systematic dependence on particle number is found for the excitation energies of optical transitions which explains recent experimental findings on self-organized quantum dots. In the last part, co-tunneling in the Coulomb blockade regime is studied. For this end the tunneling current is calculated up to the forth order perturbation theory in the tunnel coupling by a real-time Green's function approach for the non-equilibrium case. The differential conductance calculated for a quantum dot containing up to two interacting electrons shows complex signatures of the excitation spectrum which are explained by a combination of co-tunneling and sequential tunneling processes. Thereby the calculations show a peak structure within the Coulomb blockade regime which has also been observed in experiment.

Wave function mapping of self-assembled quantum dots by capacitance spectroscopy

We use frequency-dependent capacitance–voltage spectroscopy to study the dynamic charging of self-assembled InAs quantum dots. With increasing frequency, the AC charging becomes suppressed, beginning with the low-energy states. By applying an in-plane magnetic field, we generate an additional magnetic confinement that alters the tunneling barrier and hence the charging dynamics. In traveling through the potential barrier, the electrons acquire an additional momentum k 0, proportional to the magnetic field B. As the tunneling is enhanced, when k 0 matches the maximum of the electronic wave function Ψ (in momentum representation), we are able to map out the shape of Ψ by varying B.