A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

The pursuit of high charge carrier mobility in nanostructures is crucial for advancing both device development and exploring fundamental physics. This study focuses on minimizing scattering to achieve high mobility, employing a polytype type-II interface formed between wurtzite and zincblende InP. On the zincblende side, electrons accumulate, while holes gather on the wurtzite side, resulting in the creation of spatially separated indirect excitons with aligned dipoles. The perfectly flat polytype type-II interface serves to minimize scattering. Furthermore, modulation doping is implemented to reduce Coulomb scattering, achieved by doping the entire wurtzite segment n-type, excluding the vicinity of the interface. The investigation particularly focuses on understanding the impact of doping on the spatial distribution of indirect excitons at polytype type-II heterointerfaces. Spatially and temporally resolved photoluminescence spectroscopy at low temperatures is utilized to explore the spatial distribution and recombination dynamics of indirect excitons. The study reveals a significantly reduced spatial distribution and enhanced recombination rate of indirect excitons in the presence of an electron background for the n-type doped platelets compared to the undoped platelets. This aligns with expectations due to enhanced recombination rates provided by numerous electrons. Additionally, we were able to estimate a lower limit for the expansion speed of the indirect excitons at high excitation power density in undoped platelets.

Indirect excitonsare the focus of intense research due to theopportunity of studying degenerate quantum gases and liquids in anexcitonic system. To realize such systems, it is highly advantageousto have as little scattering as possible. A polytype type-II interfaceis formed between wurtzite and zincblende InP due to the band alignment.Electrons gather on the zincblende and holes on the wurtzite sideof the interface. Therefore, electrons and holes that are spatiallyseparated by the interface form indirect excitons with aligned dipoles.This polytype type-II interface is perfectly flat, which limits scattering.Here we report that repulsive interaction between the indirect excitonsis the driving force behind the long-range transport of indirect excitonsalong the interface at high exciton densities. This is indicativeof less scattering than in conventional type-II heterostructures.The spatial separation of the charge carriers across the interfaceleads to a low recombination rate of the indirect excitons since theoverlap of the electron-hole wavefunction at the interfaceis small. Emission from the long-lived indirect excitons can be detectedeven after 40 & mu;s. Our studies have been performed by using spatiallyand temporally resolved photoluminescence at the low temperature.

Indirect excitons offer a platform to gain an understanding of the realization of the excitonic-based device and the physics of excitonic condensation. Exploiting transport properties of the indirect excitonic requires a large interface where electrons and holes are spatially separated like polytype type-II interface consisting of the same compound, such as zinc blende and wurtzite InP. The indirect excitons are dipoles and aligned at the type-II interface. Indirect excitons transport a long distance before recombination from the excitation spot by the presence of high-density excitons due to the strong repulsive interaction between dipolar indirect excitons.

The interface between wurtzite and zinc blende InP has been identified as type-II, where electrons gather on the zinc blende side and holes on the wurtzite side of the interface. The photoluminescence resulting from recombination across the interface is expected to be long-lived and to exhibit non-exponential decay of emission intensity after pulsed excitation. We verify this prediction using time-resolved photoluminescence spectroscopy on nanowires containing a single heterostructure between a single segment of wurtzite and zinc blende. We find that a significant intensity of type-II emission remains even more than 30 ns after excitation. The decay of the emission intensity is also non-exponential and considerably longer than the exponential decay of the wurtzite InP segment (260 ps). Our results are consistent with the expected photoluminescence characteristics of a type-II interface between the two polytypes. We also find that the lifetime becomes shorter if we create an electron gas at the interface by n-type doping the entire wurtzite segment of the nanowire. This is expected since there are many electrons that a given hole can recombine with, in contrast to the undoped case.

Tight binding models are widely used in large scale electronic structure calculations of nanostructures. Their atomistic nature makes them flexible, but also means the computational cost increases rapidly with system size. The large number of calculations required to design nanostructures makes computational efficiency desirable. We have developed a method to increase computational speed while retaining most of its accuracy. The method is based on the use of supercells and zone folding combined with a truncation of the Hamiltonians to only include states close to the band-edges. We apply the method to model the band edge energies of a GaAs/AlAs quantum well grown along the [110]-directions with 3D and 2D periodic boundary conditions as well as the density of states and dielectric function of the quantum well. We typically find a speed-up of ten times with only a small loss of accuracy of the calculation result.

Calculating the electronic structure of systems involving very different length scales presents a challenge. Empirical atomistic descriptions such as pseudopotentials or tight-binding models allow one to calculate the effects of atomic placements, but the computational burden increases rapidly with the size of the system, limiting the ability to treat weakly bound extended electronic states. Here we propose a new method to connect atomistic and quasi-continuous models, thus speeding up tight-binding calculations for large systems. We divide a structure into blocks consisting of several unit cells which we diagonalize individually. We then construct a tight-binding Hamiltonian for the full structure using a truncated basis for the blocks, ignoring states having large energy eigenvalues and retaining states with energies close to the band edge energies. A numerical test using a GaAs/AlAs quantum well shows the computation time can be decreased to less than 5% of the full calculation with errors of less than 1%. We give data for the trade-offs between computing time and loss of accuracy. We also tested calculations of the density of states for a GaAs/AlAs quantum well and find a ten times speedup without much loss in accuracy.

Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A problem studying such quantum graphs is that the spectrum is timeconsuming to compute by hand and the inverse problem of finding a quantum graph having a specified spectrum is difficult. We solve the forward problem, to find the eigenvalues, using a previously developed computer program. We obtain all eigenvalues analytically for not too big graphs that have rationally dependent edges. We solve the inverse problem using "spectrally grown graphs". The spectrally grown graphs are evolved from a starting (parent) graph such that the child graphs have eigenvalues are close to some criterion. Our experiments show that the method works and we can usually find graphs having spectra which are numerically close to a prescribed spectrum. There are naturally exceptions, such as if no graph has the prescribed spectrum. The selection criteria (goals) strongly influence the shape of the evolved graphs. Our experiments allows us to make new conjectures concerning the spectra of quantum graphs. We open-source our software at https://github.com/meapistol/Spectra-of-graphs.

Crystal phase defined heterostructures, or polytype heterostructures, are atomically sharp with no intermixing, which makes them ideal contenders for a wide range of applications. Although polytype quantum dots have shown promising results as single photon sources, a high degree of control on the dimensions and the number of polytype quantum dots is necessary before any application can be developed. In this work, we show results from optical characterization of highly controlled wurtzite (wz)–zinc blende (zb) GaAs quantum dots with sharp photoluminescence signal and a strong indication of 0D density of states. One band effective mass calculations show good agreement with the measured data. Radially confined nanowires with a single wz–zb GaAs interface also show sharp photoluminescence signal and 0D density of states. This indicates the existence of quantum dot like states in triangular wells formed at the wz–zb GaAs interface. These results show the potential of polytype quantum dots for physics and optics applications.

Inverse problems are important in quantum mechanics and involve such questions as finding which potential give a certain spectrum or which arrangement of atoms give certain properties to a molecule or solid. Inverse problems are typically very hard to solve and tend to be very compute intense. We here show that neural networks can easily solve inverse problems in quantum mechanics. It is known that a neural network can compute the spectrum of a given potential, a result which we reproduce. We find that the (much harder) inverse problem of computing the correct potential that gives a prescribed spectrum is equally easy for a neural network. We extend previous work where neural networks were used to find the electronic many-particle density given a potential by considering the inverse problem. That is, we show that neural networks can compute the potential that gives a prescribed many-electron density.

Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we do a slight excursion into graphs with Dirichlet and $\delta$-type boundary condititons towards the end of the paper. The spectrum of $\mathbf{L}$ does not determine the graph uniquely, that is, there exist non-isomorphic graphs with the same spectra. There are few known examples of pairs of non-isomorphic but isospectral quantum graphs. In this paper we start to correctify this situation by finding hundreds of isospectral sets, using computer algebra. We have found all sets of isospectral but non-isomorphic equilateral connected quantum graphs with at most nine vertices. This includes thirteen isospectral triplets and one isospectral set of four. One of the isospectral triplets involves a loop where we could prove isospectrality. We also present several different combinatorial methods to generate arbitrarily large sets of isospectral graphs, including infinite graphs in different dimensions. As part of this we have found a method to determine if two vertices have the same Titchmarsh-Weyl $M$-function. We give combinatorial methods to generate sets of graphs with arbitrarily large number of vertices with the same $M$-function. We also find several sets of graphs that are isospectral under both Neumann and Dirichlet boundary conditions as well as under more general, $\delta$-type and $\delta'_s$-type, boundary conditions. We discuss the possibilities that our program is incorrect, present our tests and open source it for inspection at http://github.com/meapistol/Spectra-of-graphs

The quality, such as long-range correlation and mobility, of a two-dimensional electron gas (2DEG) is limited by, among other factors, interface roughness, which is inherent to the use of compositional heterostructures. Polytypic heterostructures have atomically sharp interfaces and minimal strain, decreasing the interface roughness, which may increase the mobility and long-range correlation of 2DEGs. In this work, we show the formation of a 2DEG at the wurtzite–zinc blende interface in partially n-type-doped InP nanowires using power-dependent photoluminescence. We additionally determined the wurtzite–zinc blende InP valence band offset to be 35 meV <ΔEv< 70 meV. Our results may enable the study of electron gases at interfaces, which are atomically flat over large areas.

It is now possible to synthesize the wurtzite crystal phase of most III-V semiconductors in the form of nanowires. This sparks interest for fundamental research and adds extra degrees of freedom for designing novel devices. However, the understanding of many properties, such as phonon dispersion, of these wurtzite semiconductors is not yet complete, despite the extensive number of studies published. The E 2 L and E 2 H phonon modes exist in the wurtzite crystal phase only (not in zinc blende) where the E 2 H mode has been already experimentally observed in Ga and In arsenides and phosphides, while the E 2 L mode has been observed in GaP, but not in GaAs or InP. In order to determine the energy of E 2 L in wurtzite GaAs and InP, we performed Raman scattering measurements on wurtzite GaAs and InP nanowires. We found clear evidence of the E 2 L phonon mode at 64 cm−1 and 54 cm−1, respectively. Polarization-dependent experiments revealed similar selection rules for both the E 2 L and the E 2 H phonon modes (as expected) where the intensity peaked with excitation and detection polarization being perpendicular to the [0001] crystallographic direction. We further find that the splitting between the E1(TO) and A1(TO) modes is around 2 cm−1 in wurtzite GaAs and below 1 cm−1 in wurtzite InP. We believe these results will be useful for a better understanding of phonons in wurtzite crystal phase of III-V semiconductors as well as for testing and improving phonon dispersion calculations.

III-nitrides are considered the material of choice for light-emitting diodes (LEDs) and lasers in the visible to ultraviolet spectral range. The development is hampered by lattice and thermal mismatch between the nitride layers and the growth substrate leading to high dislocation densities. In order to overcome the issue, efforts have gone into selected area growth of nanowires (NWs), using their small footprint in the substrate to grow virtually dislocation-free material. Their geometry is defined by six tall side-facets and a pointed tip which limits the design of optoelectronic devices. Growth of dislocation-free and atomically smooth 3D hexagonal GaN micro-prisms with a flat, micrometer-sized top-surface is presented. These self-forming structures are suitable for optical devices such as low-loss optical cavities for high-efficiency LEDs. The structures are made by annealing GaN NWs with a thick radial shell, reforming them into hexagonal flat-top prisms with six equivalents either m- or s-facets depending on the initial heights of the top pyramid and m-facets of the NWs. This shape is kinetically controlled and the reformation can be explained with a phenomenological model based on Wulff construction that have been developed. It is expected that the results will inspire further research into micron-sized III-nitride-based devices.

One advantage of nanowires is the possibility to create heterostructures of the same material, but different crystal structures. These polytype-interfaces are atomically sharp and, in many cases, have a type-II band-alignment. We show, using optical methods, that n-type doped polytype InP nanowires form a 2D electron gas at the polytype-interfaces. Using the photoluminescence from doped and undoped polytype nanowires, we can estimate a lower bound for the band-offset of the type-II band-alignment.

Semiconductor nanowire arrays are a promising candidate for next generation solar cells due to enhanced absorption and reduced material consumption. However, to optimize their performance, time consuming three-dimensional (3D) opto-electronics modeling is usually performed. Here, we develop an accurate one-dimensional (1D) modeling method for the analysis. The 1D modeling is about 400 times faster than 3D modeling and allows direct application of concepts from planar pn-junctions on the analysis of nanowire solar cells. We show that the superposition principle can break down in InP nanowires due to strong surface recombination in the depletion region, giving rise to an IV-behavior similar to that with low shunt resistance. Importantly, we find that the open-circuit voltage of nanowire solar cells is typically limited by contact leakage. Therefore, to increase the efficiency, we have investigated the effect of high-bandgap GaP carrier-selective contact segments at the top and bottom of the InP nanowire and we find that GaP contact segments improve the solar cell efficiency. Next, we discuss the merit of p-i-n and p-n junction concepts in nanowire solar cells. With GaP carrier selective top and bottom contact segments in the InP nanowire array, we find that a p-n junction design is superior to a p-i-n junction design. We predict a best efficiency of 25% for a surface recombination velocity of 4500 cm s(-1), corresponding to a non-radiative lifetime of 1 ns in p-n junction cells. The developed 1D model can be used for general modeling of axial p-n and p-i-n junctions in semiconductor nanowires. This includes also LED applications and we expect faster progress in device modeling using our method.

The band offset between wurtzite (wz) and zinc-blende (zb) GaAs is an important fundamental parameter in polytype heterostructure engineering. Since the interface has a type-II band alignment, it is reasonably straightforward to measure the band offset using photoluminescence (PL) e.g. on nanowires containing heterostructures between wz and zb GaAs. It has, however, been found that the transition energy in such heterostructures depends on the diameter of the nanowires which introduces uncertainties in the determined value of the band offset. In order to extract a more accurate value and to further elucidate the diameter-dependent behavior of the transition energy we have performed PL studies on a large set of GaAs nanowires. Those nanowires have different diameters and contain one wz–GaAs segment embedded in otherwise zb–GaAs. We have also studied the effect of a passivating capping layer of AlAs on the determined band offset. We find that our data is well explained by a diameter-dependent radial band bending in the nanowires. Combining modeling of the band bending with the experimental data we extract a band offset of about 125 meV and a p-type doping concentration of 1016 cm−3. Our results will improve the accuracy of future modeling of the electronic properties of wz–zb GaAs heterostructures and other engineered polytypic materials.

A tandem solar cell consisting of a III-V nanowire subcell on top of a planar Si subcell is a promising candidate for next generation photovoltaics due to the potential for high efficiency. However, for success with such applications, the geometry of the system must be optimized for absorption of sunlight. Here, we consider this absorption through optics modeling. Similarly, as for a bulk dual-junction tandem system on a silicon bottom cell, a bandgap of approximately 1.7 eV is optimum for the nanowire top cell. First, we consider a simplified system of bare, uncoated III-V nanowires on the silicon substrate and optimize the absorption in the nanowires. We find that an optimum absorption in 2000 nm long nanowires is reached for a dense array of approximately 15 nanowires per square micrometer. However, when we coat such an array with a conformal indium tin oxide (ITO) top contact layer, a substantial absorption loss occurs in the ITO. This ITO could absorb 37% of the low energy photons intended for the silicon subcell. By moving to a design with a 50 nm thick, planarized ITO top layer, we can reduce this ITO absorption to 5%. However, such a planarized design introduces additional reflection losses. We show that these reflection losses can be reduced with a 100 nm thick SiO2 anti-reflection coating on top of the ITO layer. When we at the same time include a Si3N4 layer with a thickness of 90 nm on the silicon surface between the nanowires, we can reduce the average reflection loss of the silicon cell from 17% to 4%. Finally, we show that different approximate models for the absorption in the silicon substrate can lead to a 15% variation in the estimated photocurrent density in the silicon subcell.

We demonstrate one-dimensional (1D) electrical modeling of InP nanowire array solar cells. This 1D modeling gives accurate description of the current voltage response even at high surface recombination velocity. The 1D electrical model decreases the simulation time by 3 orders of magnitude compared to a full three-dimensional (3D) model.

Geometrically designed III-V nanowire arrays are promising candidates for optoelectronics due to their possibility to excite nanophotonic resonances in absorption spectra. Strong absorption resonances can be obtained by proper tailoring of nanowire diameter, length and pitch. Such enhancement of the light absorption is, however, accompanied by undesired resonance dips at specific wavelengths. In this work, we theoretically show that tilting of the nanowires mitigates the absorption dips by exciting strong Mie resonances. In particular, we derive a theoretical optimum inclination angle of about 30 degrees at which the inclined nanowires gain 8% in absorption efficiency compared to vertically standing nanowires in a spectral region matching the intensity distribution of the sun. The enhancement is due to engineering the excited modes inside the nanowires regarding the symmetry properties of the nanowire/light system without increasing the absorbing material. We expect our results to be important for nanowire-based photovoltaic applications.