We study the optical properties of hydrogen passivated silicon, germanium and mixed Ge/Si core/shell quantum dots (QDs) using high accuracy Density Functional Theory (DFT) and time-dependent DFT (TDFT). We employ the hybrid DFT functional of Becke, Lee, Yang and Parr (B3LYP) in combination with good quality basis sets. As we have shown in our previous work, this combination is an accurate and computationally efficient way for such calculations. The mixed quantum dots, as would be expected, are more versatile and offer more possibilities for band gap engineering, with gap values (electronic and optical) between those of the corresponding Si and Ge dots. Our results support the quantum confinement theory for all three types of QDs.
We study spin polarized transport in silicon germanium nanowires using a semiclassical monte carlo approach. Spin depolarization in the channel is caused due to D'yakonov-Perel (DP) relaxation associated with Rashba spin orbit coupling and due to Elliott- Yafet (EY) relaxation. We investigate the dependence of spin dephasing on germanium mole fraction in silicon germanium nanowires. The spin dephasing lengths decrease with an increase in the germanium mole fraction. We also find that the temperature has a strong influence on the dephasing rate and spin relaxation lengths increase with decrease…
We have simultaneously measured conductance and thermoelectric power (TEP) of individual silicon and germanium/silicon core/shell nanowires in the field effect transistor device configuration. As the applied gate voltage changes, the TEP shows distinctly different behaviors while the electrical conductance exhibits the turn-off, subthreshold, and saturation regimes respectively. At room temperature, peak TEP value of $\sim 300 μ$V/K is observed in the subthreshold regime of the Si devices. The temperature dependence of the saturated TEP values are used to estimate the carrier doping of Si nano…
Transport measurements at cryogenic temperatures through a few electron top gated quantum dot fabricated in a silicon/silicon-germanium heterostructure are reported. Variations in gate voltage induce a transition from an isolated dot toward a dot strongly coupled to the leads. In addition to Coulomb blockade, when the dot is strongly coupled to the leads, we observe the appearance of a zero bias conductance peak due to the Kondo effect. The Kondo peak splits in a magnetic field, and the splitting scales linearly with the applied field. We also observe a transition from pure Coulomb blockade to…
Nanocrystalline silicon and silicon-germanium alloys are promising thermoelectric materials that have achieved substantially improved figure of merits compared to their bulk counterparts. This enhancement is typically attributed to a reduction in lattice thermal conductivity by phonon scattering at grain boundaries. However, further improvements are difficult to achieve because grain boundary scattering is poorly understood, with recent experimental observations suggesting that the phonon transmissivity may depend on phonon frequency rather than being constant as in the commonly used gray mode…
The Thomas-Fermi-Dirac (TFD) approximation and an sp3d5s* tight binding method were used to calculate the electronic properties of a delta-doped phosphorus layer in silicon. This self-consistent model improves on the computational efficiency of "more rigorous" empirical tight binding and ab initio density functional theory models without sacrificing the accuracy of these methods. The computational efficiency of the TFD model provides improved scalability for large multi-atom simulations, such as of nanoelectronic devices that have experimental interest. We also present the first theoretically …
Superconductors and semiconductors are crucial platforms in the field of quantum computing. They can be combined to hybrids, bringing together physical properties that enable the discovery of new emergent phenomena and provide novel strategies for quantum control. The involved semiconductor materials, however, suffer from disorder, hyperfine interactions or lack of planar technology. Here we realise an approach that overcomes these issues altogether and integrate gate-defined quantum dots and superconductivity into a material system with strong spin-orbit coupling. In our germanium heterostruc…
The origin of visible light emission from nanostructures has been a subject of an intense debate since the early work by L. E. Brus and A. P. Alivisatos in 1980s. The intense research that followed has paved the way towards applications of quantum structures in optoelectronics and in bio-sensing and contributed to the development of nanotechnology. The major new challenge is in accessing the structural, electronic and optical properties of quantum dots on a nanoparticle scale in order to understand complex relationships between structural motifs and their contributions to the relevant physical…
The prospect of building quantum circuits using advanced semiconductor manufacturing positions quantum dots as an attractive platform for quantum information processing. Extensive studies on various materials have led to demonstrations of two-qubit logic in gallium arsenide, silicon, and germanium. However, interconnecting larger numbers of qubits in semiconductor devices has remained an outstanding challenge. Here, we demonstrate a four-qubit quantum processor based on hole spins in germanium quantum dots. Furthermore, we define the quantum dots in a two-by-two array and obtain controllable c…
The dynamics of the luminescence decay from germanium nanocrystals embedded in crystalline silicon has been studied for temperatures varied between 16 K and room temperature. At room temperature the characteristic decay time is of the order of 50 nanoseconds while it extends into the microsecond range at low temperatures. The decay is dominated by non-radiative processes, which show a typical thermal activation energy of a few meV.
Dopant-based quantum computing implementations often require the dopants to be situated close to an interface to facilitate qubit manipulation with local gates. Interfaces not only modify the energies of the bound states but also affect their symmetry. Making use of the successful effective mass theory we study the energy spectra of acceptors in Si or Ge taking into account the quantum confinement, the dielectric mismatch and the central cell effects. The presence of an interface puts constraints to the allowed symmetries and lead to the splitting of the ground state in two Kramers doublets [J…
The idea of quantum computation is the most promising recent developments in the high-tech domain, while experimental realization of a quantum computer poses a formidable challenge. Among the proposed models especially attractive are semiconductor based nuclear spin quantum computer's (S-NSQC), where nuclear spins are used as quantum bistable elements, ''qubits'', coupled to the electron spin and orbital dynamics. We propose here a scheme for implementation of basic elements for S-NSQC's which are realizable within achievements of the modern nanotechnology. These elements are expected to be ba…
Computationally efficient modeling of the thermal conductivity of materials is crucial to thorough experimental planning and theoretical understanding of thermal properties. We present a modeling approach in this work that utilizes frequency-dependent effective medium to calculate lattice thermal conductivity of nanostructured solids. The method accurately predicts a significant reduction in the thermal conductivity of nanostructured Si80Ge20 systems, along with previous reported thermal conductivities in nanowires and nanoparticles-in-matrix materials. We use our model to gain insight into th…
We present a comparative micro-photoluminescence study of the emission intensity of self-assembled germanium islands coupled to the resonator mode of two-dimensional silicon photonic crystal defect nanocavities. The emission intensity is investigated for cavity modes of L3 and Hexapole cavities with different cavity quality factors. For each of these cavities many nominally identical samples are probed to obtain reliable statistics. As the quality factor increases we observe a clear decrease in the average mode emission intensity recorded under comparable optical pumping conditions. This clear…
The thermal coefficient of resistance (TCR) for epitaxial silicon-germanium (SiGe) layers has been analyzed by experiment and simulation. Predictive simulation using drift-diffusion formalism and self-consistent quantum-mechanical solutions yielded similar results, TCR around 2%/K at 300 K. This modeling approach can be used for different, graded and constant, SiGe profiles,. It is also capable of predicting the influence of background auto-doping on the TCR of the detectors
The lifting of the two-fold degeneracy of the conduction valleys in a strained silicon quantum well is critical for spin quantum computing. Here, we obtain an accurate measurement of the splitting of the valley states in the low-field region of interest, using the microwave spectroscopy technique of electron valley resonance (EVR). We compare our results with conventional methods, observing a linear magnetic field dependence of the valley splitting, and a strong low-field suppression, consistent with recent theory. The resonance linewidth shows a marked enhancement above $T\simeq 300$ mK.
We examine a silicon-germanium heterojunction bipolar transistor (HBT) for cryogenic pre-amplification of a single electron transistor (SET). The SET current modulates the base current of the HBT directly. The HBT-SET circuit is immersed in liquid helium, and its frequency response from low frequency to several MHz is measured. The current gain and the noise spectrum with the HBT result in a signal-to-noise-ratio (SNR) that is a factor of 10-100 larger than without the HBT at lower frequencies. The transition frequency defined by SNR = 1 has been extended by as much as a factor of 10 compared …
Silicon-germanium heterostructures have successfully hosted quantum dot qubits, but the intrinsic near-degeneracy of the two lowest valley states poses an obstacle to high fidelity quantum computing. We present a modification to the Si/SiGe heterostructure by the inclusion of a spike in germanium concentration within the quantum well in order to increase the valley splitting. The heterostructure is grown by chemical vapor deposition and magnetospectroscopy is performed on gate-defined quantum dots to measure the excited state spectrum. We demonstrate a large and widely tunable valley splitting…
Strain engineering has proven to be vital for germanium-based photonics, in particular light emission. However, applying a large permanent biaxial strain to germanium has been a challenge. We present a simple, CMOS-compatible technique to conveniently induce a large, spatially homogenous strain in microdisks patterned within ultrathin germanium nanomembranes. Our technique works by concentrating and amplifying a pre-existing small strain into the microdisk region. Biaxial strains as large as 1.11% are observed by Raman spectroscopy and are further confirmed by photoluminescence measurements, w…
We investigate spin polarized electron transport in ultra-thin Si-Core/Ge-Shell and Ge-Core/Si-Shell nanowire system using semi-classical Monte Carlo simulation method. Depolarization of electron's spin occurs in nanowire mainly due to D'yakonov-Perel dephasing (DP-mechanism) and Elliott-Yafet dephasing (EY-mechanism). We studied the dependence of spin dephasing on ultra-thin silicon core diameter in Si-Core/Ge-Shell nanowire and germanium core diameter in Ge-Core/Si-Shell nanowire. Variation in spin dephasing length with varying core diameter ranging from 1 nm to 9 nm indicate that spin depha…
In this work we report new silicon and germanium tubular nanostructures with no corresponding stable carbon analogues. The electronic and mechanical properties of these new tubes were investigated through ab initio methods. Our results show that the structures have lower energy than their corresponding nanoribbon structures and are stable up to high temperatures (500 and 1000 K, for silicon and germanium tubes, respectively). Both tubes are semiconducting with small indirect band gaps, which can be significantly altered by both compressive and tensile strains. Large bandgap variations of almos…
We present a first-principles study of bare and hydrogen passivated armchair nanoribbons of the puckered single layer honeycomb structures of silicon and germanium. Our study includes optimization of atomic structure, stability analysis based on the calculation of phonon dispersions, electronic structure and the variation of band gap with the width of the ribbon. The band gaps of silicon and germanium nanoribbons exhibit family behavior similar to those of graphene nanoribbons. The edges of bare nanoribbons are sharply reconstructed, which can be eliminated by the hydrogen termination of dangl…
We theoretically investigate the properties of holes in a Si$_{x}$Ge$_{1-x}$/Ge/ Si$_{x}$Ge$_{1-x}$ quantum well in a perpendicular magnetic field that make them advantageous as qubits, including a large ($>$100~meV) intrinsic splitting between the light and heavy hole bands, a very light ($\sim$0.05$\, m_0$) in-plane effective mass, consistent with higher mobilities and tunnel rates, and larger dot sizes that could ameliorate constraints on device fabrication. Compared to electrons in quantum dots, hole qubits do not suffer from the presence of nearby quantum levels (e.g., valley states) that…
The promise of quantum computation with quantum dots has stimulated widespread research. Still, a platform that can combine excellent control with fast and high-fidelity operation is absent. Here, we show single and two-qubit operations based on holes in germanium. A high degree of control over the tunnel coupling and detuning is obtained by exploiting quantum wells with very low disorder and by working in a virtual gate space. Spin-orbit coupling obviates the need for microscopic elements and enables rapid qubit control with Rabi frequencies exceeding 100 MHz and a single-qubit fidelity of 99…
Electrons and holes confined in quantum dots define an excellent building block for quantum emergence, simulation, and computation. In order for quantum electronics to become practical, large numbers of quantum dots will be required, necessitating the fabrication of scaled structures such as linear and 2D arrays. Group IV semiconductors contain stable isotopes with zero nuclear spin and can thereby serve as excellent host for spins with long quantum coherence. Here we demonstrate group IV quantum dot arrays in silicon metal-oxide-semiconductor (SiMOS), strained silicon (Si/SiGe) and strained g…
Quantum dots fabricated using techniques and materials that are compatible with semiconductor manufacturing are promising for quantum information processing. While great progress has been made toward high-fidelity control of quantum dots positioned in a linear arrangement, scalability along two dimensions is a key step toward practical quantum information processing. Here we demonstrate a two-dimensional quantum dot array where each quantum dot is tuned to single-charge occupancy, verified by simultaneous measuring with two integrated radio frequency charge sensors. We achieve this by using pl…
We demonstrate third harmonic generation in plasmonic antennas made of highly doped germanium and designed to be resonant in the mid infrared. Owing to the near-field enhancement, the result is an ultrafast, sub-diffraction, coherent light source tunable between 3 and 5 micrometer wavelength on a silicon substrate. To observe nonlinearity in this challenging spectral region, a high-power femtosecond laser system equipped with parametric frequency conversion in combination with an all-reflective confocal microscope setup is employed. We show spatially resolved maps of the linear scattering cros…
In this article, the results of investigation of processes occurring during the molecular-beam deposition of germanium layers on Si$_3$N$_4$ dielectric substrates within a wide range of the Ge film growth temperatures (30 to 600°C) are presented. The intensity of the IR absorption bands related to the vibrations of the N$-$H and Si$-$N bonds are established to decrease with the increase of the Ge deposition temperature. It appears that this phenomenon cannot be explained only as a thermally activated process. Simultaneously, the peak corresponding to the Ge$-$N vibration bonds emerges in the X…
Germanium has emerged as an exceptionally promising material for spintronics and quantum information applications, with significant fundamental advantages over silicon. However, efforts to create atomic-scale devices using donor atoms as qubits have largely focussed on phosphorus in silicon. Positioning phosphorus in silicon with atomic-scale precision requires a thermal incorporation anneal, but the low success rate for this step has been shown to be a fundamental limitation prohibiting the scale-up to large-scale devices. Here, we present a comprehensive study of arsine (AsH$_3$) on the germ…
The mechanism responsible for the emission of clusters from heavy ion irradiated solids is proposed to be thermal spikes. Collision cascade-based theories describe atomic sputtering but cannot explain the consistently observed experimental evidence for significant cluster emission. Statistical thermodynamic arguments for thermal spikes are employed here for qualitative and quantitative estimation of the thermal spike-induced cluster emission from silicon, germanium and zinc oxide. The evolving cascades and spikes in elemental and molecular semiconducting solids are shown to have fractal charac…
We report electronic transport on silicon double and triple dots created with the optimized number of two gates. Using silicon nitride spacers two dots in series are created below two top gates overlapping a silicon nanowire. Coupling between dots is controlled by gate voltages. A third dot is created either by combined action of gate voltages or local doping depending on the spacers length. The main characteristics of the triple dot stability diagram are quantitatively fitted.
Spins of donor electrons and nuclei in silicon are promising quantum bit (qubit) candidates which combine long coherence times with the fabrication finesse of the silicon nanotechnology industry. We outline a potentially scalable spin qubit architecture where donor nuclear and electron spins are coupled to spins of electrons in quantum dots and discuss requirements for donor placement aligned to quantum dots by single ion implantation.
Semiconductors with strong spin-orbit interactions can exhibit a helical gap with spin-momentum locking opened by a magnetic field. Such a gap is highly spin selective as a result of a topologically protected spin-momentum locking, which can be used for spin filtering. We experimentally demonstrate such a spin filtering effect in a quasi-ballistic p-type germanium/silicon core/shell nanowire (NW), which possesses a pseudo-helical gap without the application of magnetic field. Polarized hole spin injection to the NW is achieved using cobalt ferromagnetic contacts with controlled natural surface…
We electrically measure intrinsic silicon quantum dots with electrostatically defined tunnel barriers. The presence of both p-type and n-type ohmic contacts enables the accumulation of either electrons or holes. Thus we are able to study both transport regimes within the same device. We investigate the effect of the tunnel barriers and the electrostatically defined quantum dots. There is greater localisation of charge states under the tunnel barriers in the case of hole conduction leading to higher charge noise in the p-regime.
Donor electron spins in semiconductors make exceptional quantum bits because of their long coherence times and compatibility with industrial fabrication techniques. Despite many advances in donor-based qubit technology, it remains difficult to selectively manipulate single donor electron spins. Here, we show that by replacing the prevailing semiconductor host material (silicon) with germanium, donor electron spin qubits can be electrically tuned by more than an ensemble linewidth, making them compatible with gate addressable quantum computing architectures. Using X-band pulsed electron spin re…
We report on ambipolar gate-defined quantum dots in silicon on insulator (SOI) nanowires fabricated using a customised complementary metal-oxide-semiconductor (CMOS) process. The ambipolarity was achieved by extending a gate over an intrinsic silicon channel to both highly doped n-type and p-type terminals. We utilise the ability to supply ambipolar carrier reservoirs to the silicon channel to demonstrate an ability to reconfigurably define, with the same electrodes, double quantum dots with either holes or electrons. We use gate-based reflectometry to sense the inter-dot charge transition(IDT…
Elastic strain distributions in SiGe heterostructures with quantum dots have been simulated with the use of the finite element method. The effect of a non-uniform germanium distribution in the nanoislands on the spatial dependence and the magnitude of elastic fields was studied. It is shown that quantum dots with a uniform component content are more strained in comparison with non-uniform nanoislands
We report charge sensing measurements on a silicon quantum dot (QD) with a nearby silicon single electron transistor (SET) acting as an electrometer. The devices are electrostatically formed in bulk silicon using surface gates. We show that as an additional electron is added onto the quantum dot, a charge is induced on the SET of approximately 0.2e. These measurements are performed in the many electron regime, where we can count in excess of 20 charge additions onto the quantum dot.
Implicit summation is a technique for the conversion of sums over intermediate states in multiphoton absorption and the high-order susceptibility in hydrogen into simple integrals. Here, we derive the equivalent technique for hydrogenic impurities in multi-valley semiconductors. While the absorption has useful applications, it is primarily a loss process; conversely, the non-linear susceptibility is a crucial parameter for active photonic devices. For Si:P, we predict the hyperpolarizability ranges from $χ^{(3)}/n_{\text{3D}}=2.9 $ to $580 \times 10^{-38}$ $\text{m}^5/\text{V}^2$ depending on …
We integrate ambipolar quantum dots in silicon fin field-effect transistors using exclusively standard complementary metal-oxide-semiconductor fabrication techniques. We realize ambipolarity by replacing conventional highly-doped source and drain electrodes by a metallic nickel silicide with Fermi level close to the silicon mid-gap position. Such devices operate in a dual mode, either as classical field-effect or single-electron transistor. We implement a classical logic NOT gate at low temperature by tuning two interconnected transistors into opposite polarities. In the quantum regime, we dem…
We present a novel reconfigurable metal-oxide-semiconductor multi-gate transistor that can host a quadruple quantum dot in silicon. The device consist of an industrial quadruple-gate silicon nanowire field-effect transistor. Exploiting the corner effect, we study the versatility of the structure in the single quantum dot and the serial double quantum dot regimes and extract the relevant capacitance parameters. We address the fabrication variability of the quadruple-gate approach which, paired with improved silicon fabrication techniques, makes the corner state quantum dot approach a promising …
We present data on the electrical transport properties of highly-doped silicon-on-insulator quantum dots under the effect of pulsed magnetic fields up to 48 T. At low field intensities, B<7 T, we observe a strong modification of the conductance due to the destruction of weak localization whereas at higher fields, where the magnetic field length becomes comparable to the effective Bohr radius of phosphorous in silicon, a strong decrease in conductance is demonstrated. Data in the high and low electric field bias regimes are then compared to show that close to the Coulomb blockade edge magnetica…
The thermoelectric properties in one- and two-dimensional silicon and germanium structures have been investigated using first-principle density functional techniques and linear response for the thermal and electrical transport. We have considered here the two-dimensional silicene and germanene, together with nano-ribbons of different widths. For the nano-ribbons, we have also investigated the possibility of nano-structuring these systems by mixing silicon and germanium. We found that the figure of merit at room temperature of these systems is remarkably high, up to 2.5.
We report charge detection in degenerately phosphorus-doped silicon double quantum dots (DQD) electrically connected to an electron reservoir. The sensing device is a single electron transistor (SET) patterned in close proximity to the DQD. Measurements performed at 4.2K show step-like behaviour and shifts of the Coulomb Blockade oscillations in the detector's current as the reservoir's potential is swept. By means of a classical capacitance model, we demonstrate that the observed features can be used to detect single-electron tunnelling from, to and within the DQD, as well as to reveal the DQ…
Reliable detection of single electron tunneling in quantum dots (QD) is paramount to use this category of device for quantum information processing. Here, we report charge sensing in a degenerately phosphorus-doped silicon QD by means of a capacitively coupled single-electron tunneling device made of the same material. Besides accurate counting of tunneling events in the QD, we demonstrate that this architecture can be operated to reveal asymmetries in the transport characteristic of the QD. Indeed, the observation of gate voltage shifts in the detector's response as the QD bias is changed is …
We have observed a negative differential conductance with singular gate and source-drain bias dependences in a phosphorus-doped silicon quantum dot. Its origin is discussed within the framework of weak localization. By measuring the current-voltage characteristics at different temperatures as well as simulating the tunneling rates dependences on energy, we demonstrate that the presence of shallow energy defects together with an enhancement of localization satisfactory explain our observations. Effects observed in magnetic fields are also discussed.
Deterministic control over the location and number of donors is crucial to donor spin quantum bits (qubits) in semiconductor based quantum computing. In this work, a focused ion beam is used to implant antimony donors close to quantum dots. Ion detectors are integrated next to the quantum dots to sense the implants. The numbers of donors implanted can be counted to a precision of a single ion. In low-temperature transport measurements, regular coulomb blockade is observed from the quantum dots. Charge offsets indicative of donor ionization are also observed in devices with counted donor implan…
Many electrical applications of quantum dots rely on capacitively coupled gates; therefore, to make reliable devices we need those gate capacitances to be predictable and reproducible. We demonstrate in silicon nanowire quantum dots that gate capacitances are reproducible to within 10% for nominally identical devices. We demonstrate the experimentally that gate capacitances scale with device dimensions. We also demonstrate that a capacitance simulator can be used to predict measured gate capacitances to within 20%. A simple parallel plate capacitor model can be used to predict how the capacita…
The bandstructure of bulk silicon has a six-fold valley degeneracy. Strain in the Si/SiGe quantum well system partially lifts the valley degeneracy, but the materials factors that set the splitting of the two lowest lying valleys are still under intense investigation. We propose a method for accurately determining the valley splitting in Si/SiGe double quantum dots embedded into a superconducting microwave resonator. We show that low lying valley states in the double quantum dot energy level spectrum lead to readily observable features in the cavity transmission. These features generate a "fin…
Two-dimensional electron and hole gas systems, enabled through band structure design and epitaxial growth on planar substrates, have served as key platforms for fundamental condensed matter research and high performance devices. The analogous development of one-dimensional (1D) electron or hole gas systems through controlled growth on 1D nanostructure substrates, which could open up opportunities beyond existing carbon nanotube and nanowire systems, has not been realized. Here we report the synthesis and transport studies of a 1D hole gas system based on a free-standing germanium/silicon (Ge/S…
We replace the established aluminium gates for the formation of quantum dots in silicon with gates made from palladium. We study the morphology of both aluminium and palladium gates with transmission electron microscopy. The native aluminium oxide is found to be formed all around the aluminium gates, which could lead to the formation of unintentional dots. Therefore, we report on a novel fabrication route that replaces aluminium and its native oxide by palladium with atomic-layer-deposition-grown aluminium oxide. Using this approach, we show the formation of low-disorder gate-defined quantum d…
The magnitude of spin accumulation created in semiconductors by electrical injection of spin-polarized electrons from a ferromagnetic tunnel contact is investigated, focusing on how the spin signal detected in a Hanle measurement varies with the thickness of the tunnel oxide. An extensive set of spin-transport data for Si and Ge magnetic tunnel devices reveals a scaling with the tunnel resistance that violates the core feature of available theories, namely, the linear proportionality of the spin voltage to the injected spin current density. Instead, an anomalous scaling of the spin signal with…
Semiconductor superlattices have been extensively investigated for thermoelectric applications, to explore the effects of compositions, interface structures, and lattice strain environments on the reduction of thermal conductivity, and improvement of efficiency. Most studies assumed that their electronic properties remain unaffected compared to their bulk counterparts. However, recent studies demonstrated that electronic properties of silicon (Si)/germanium (Ge) superlattices show significant variations depending on compositions and growth substrates. These studies used a constant relaxation t…
The presence of degenerate conduction band valleys and how they are mixed by interfaces play critical roles in determining electron interaction and spectrum in a silicon nanostructure. Here we investigate how the valley phases affect the exchange interaction in a symmetric two-electron silicon double quantum dot. Through a configuration interaction calculation, we find that exchange splitting is suppressed at a finite value of valley phase difference between the two dots, and reaches its minimum value ({\sim} 0) when the phase difference is π. Such a suppression can be explained using the Hubb…
