Sixfold Degeneracy Research Articles

Insights into electron and phonon topology: multi-fold fermions and Dirac nodal-line net bosons in SiX2 (X=P and As)

There has been a notable increase in enthusiasm for delving into new (quasi) particles within the domains of condensed matter and materials. Particularly, a distinctive excitement has arisen over a recently theorized exotic excitation found in a three-dimensional material, showcasing linear dispersion and the possibility of multi-fold degeneracy. In this article, we embark on an extensive investigation into the multi-fold degenerate exotic fermions beyond Dirac and Weyl in the pyrite-structured SiX2 compounds (with X as P and As). We delve into the topological attributes of band-touching nodes in fermions, featuring four- and six-fold degeneracy positioned at the Γ and R points in the first Brillouin zone, analyzing them through symmetry analysis. Regarding bosons, employing symmetry analysis, this study identifies distinctive properties where three-nodal surfaces (NSs) and phononic Dirac nodal-line (DNL) nets coexist. These phononic DNL nets comprise three phononic DNLs that share a common nodal point in momentum space, positioned at the intersection of two out of the three k i planes (i = x, y, z). The presence of observable phononic surface states in these materials may facilitate their experimental detection, suggesting these materials’ potential as prime candidates for realizing such phononic states in solid-state materials. Our investigation highlights the pyrite-structured SiX2 as a promising platform for studying multi-fold fermions and delving into their potential applications.

Phononic nodal points with quadratic dispersion and multifold degeneracy in the cubic compound Ta3Sn

Topological states with quadratic dispersion and multiple-fold band degeneracy have not only fundamentally updated our knowledge of the phases of matter but also become a major cutting-edge research direction in condensed matter physics. Note that the sixfold band degeneracy corresponds to the maximum degeneracy in phononic systems. Remarkably, in this work, based on first-principles calculations, we proposed an authentic material, ${\mathrm{Ta}}_{3}\mathrm{Sn}$, which hosts phononic nodal points with both quadratic dispersion and maximum band degeneracy, i.e., quadratic contact triple point, quadratic contact Dirac point and sixfold-degenerate point. Furthermore, the corresponding symmetry analysis with the help of the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ model further deepens the understanding of the relative physics. Evident arc-shaped surface states, originating from the projected phononic nodal points, can strongly benefit the experimental detection. It is hoped that the rich types of phonon points with quadratic dispersion and multiple-fold band degeneracy as well as the surface states can be confirmed in experiments soon.

Cryogen-free scanning gate microscope for the characterization of Si/Si0.7Ge0.3 quantum devices at milli-Kelvin temperatures

Silicon can be isotopically enriched, allowing for the fabrication of highly coherent semiconductor spin qubits. However, the conduction band of bulk Si exhibits a six-fold valley degeneracy which may adversely impact the performance of silicon quantum devices. To date, the spatial characterization of valley states in Si has remained limited. Moreover, techniques for probing valley states in functional electronic devices are needed. Here, we describe a cryogen-free scanning gate microscope for the characterization of Si/Si0.7Ge0.3 quantum devices at mK temperatures. The newly built instrument is the first cryogen-free scanning gate microscope capable of forming and measuring a quantum dot on a Si/SiGe device with an overlapping gate structure without compromising the ability to host multiple DC and microwave lines for quantum control experiments. The microscope is based on the Pan-walker design, with coarse positioning piezostacks and a fine scanning piezotube. A tungsten microscope tip is attached to a tuning fork for active control of the tip-to-sample distance. To reduce vibration noise from the pulse tube cooler, we utilize both active and passive vibration isolation mechanisms and achieve a root-mean-square noise in z of ∼2 nm. Our microscope is designed to characterize fully functioning Si/Si0.7Ge0.3 quantum devices. As a proof of concept, we use the microscope to manipulate the charge occupation of a Si quantum dot, opening up a range of possibilities for the exploration of quantum devices and materials.

Interfacial symmetry breaking induced spin-orbit coupling in wurtzite GaN nanowires

The interfacial asymmetry-induced spin–orbit coupling (SOC) has a distinct effect on the spin relaxation and gate manipulation of spintronic devices. However, it is hard to separate the contributions from the interface and bulk by commonly used techniques, such as weak antilocalization and time-resolved Kerr rotation. Owing to the different distribution of the effective magnetic field in the momentum space, the SOC originated from the interfacial effect and the wurtzite structure inversion asymmetry (WSIA) could be distinguished by circular photo-galvanic effect (CPGE) measurements. In this work, the angle and external gate-voltage dependences of the interfacial SOC in GaN nanowires (NWs) were investigated using the CPGE. The interfacial SOC-induced CPGE shows a twofold degeneracy, while the removal of the sixfold degeneracy is attributed to the asymmetrical optical absorption. Furthermore, the interfacial SOC-induced CPGE shows a weak dependence on the external gate voltage, which is totally different from the WSIA-related SOC. This result is consistent with the clarification that the interfacial asymmetry originates from the Fermi level pinning at the GaN NW sidewall surfaces, thus clearly illustrating the special distribution and gate dependences of the interfacial SOC, which is critical for the design of spintronic devices based on GaN NWs.

Phase transitions in the classical exchange-anisotropic Kitaev-Heisenberg model.

The Kitaev model on the honeycomb lattice has been receiving substantial attention due to the discovery of quantum spin liquid state associated with this model. Consequently, its classical partners such as the Kitaev-Heisenberg (KH) model and associated phase transitions become concerned. Specifically, an intermediate Kosterlitz-Thouless (KT) phase engaged in the transition from the high-temperature (T) disordered state to the low-T sixfold degenerate state is predicted in the isotropic KH model [Phys. Rev. Lett. 109, 187201 (2012)10.1103/PhysRevLett.109.187201], but so far no sufficient experimental proof has been reported. In this work, we consider an essential extension of this KH model on the honeycomb lattice by including the Kitaev exchange anisotropy that is non-negligible in realistic materials. The associated phase transitions are thus investigated using the Monte Carlo simulations. It is found that such an anisotropy will result in a degradation of the sixfold degeneracy of the ground state in the isotropic KH model down to the fourfold or twofold degenerate ground state, and the finite-T phase transitions will also be modified remarkably. Interestingly, the intermediate KT phase can be suppressed by this Kitaev exchange anisotropy. This work thus provides a more realistic description of the physics ingredient with the KH model and presents a possible explanation on absence of the intermediate phase in real materials where the Kitaev exchange anisotropy can be more or less available.

Effect of group-3 elements doping on promotion of in-plane Seebeck coefficient of n-type Mg3Sb2

Mg3Sb2-based alloys are promising thermoelectric materials with a reasonably low thermal conductivity. However, their electrical transport property is usually limited by the low carrier concentration. Mg3Sb2 has a multi-valley conduction band with a six-fold degeneracy, benefiting n-type thermoelectric performance. Recently, n-type Y-doped Mg3Sb1.5Bi0.5 and Sc-doped Mg3Sb2Mg3Bi2 alloys show a large figure of merit (ZT). In this paper, the doping effect of group-3 and chalcogen elements on the electronic structures and electrical transport properties of Mg3Sb2 was investigated via the first-principles calculations. Chalcogen elements have a slight effect on the electronic structure, and Te-doped Mg3Sb2 shows better normalized power factors in both the out-of-plane and in-plane directions, compared to the S-doped and Se-doped systems. Distinctly different doping effects appear in Mg3Sb2 doped with group-3 elements. A increased density of states near the bottom of the conduction band can be induced by Sc or Y. Sc-doped and Y-doped Mg3Sb2 show higher normalized power factors along the in-plane direction than those doped with chalcogens.

A thorough study of Si nanowire FETs with 3D Multi-Subband Ensemble Monte Carlo simulations

We thoroughly compare the DC electrical behavior of n-MOS transistors based on Si nanowires with 〈100〉 and 〈110〉 channel orientations by means of Multi-Subband Ensemble Monte Carlo simulations. We find that the drain current depends on the nanowire diameter and it is slightly, but consistently, larger for 〈100〉 than for 〈110〉 nanowires. The observed differences in mobility, velocity and spatial charge distribution are interpreted in terms of the effective masses and populations of the different Si conduction band valleys, whose sixfold degeneracy is lifted by quantum confinement in narrow nanowires. Finally, we study the scaling behavior for channel lengths down to 8 nm, concluding that the differences observed between orientations are minimal.

Electronic band structure and proximity to magnetic ordering in the chiral cubic compound CrGe

CrGe belongs to the family of cubic B20 intermetallics. From experimental investigations by susceptibility and de Haas-van Alphen (dHvA) measurements and from calculations of its electronic band structure by density-functional theory (DFT), CrGe is found to form a metallic paramagnetic ground state. Combining dHvA and DFT data, a detailed picture of the Fermi surface of CrGe is provided. The proximity to a magnetic long-range ordering in CrGe is suggested from a prominent thermal magnetic susceptibility. The possibility to induce magnetic long-range order in CrGe is discussed based on calculated properties for CrGe substituting Ge by As or Sn, and from a comparison with MnGe and the alloy series ${\mathrm{Cr}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{Ge}$. Owing to the noncentrosymmetric and nonsymmorphic crystal structure of CrGe, in absence of broken time reversal symmetry, its band structure is marked by forced nodal lines at the Fermi edge. Moreover, this material hosts degenerate unconventional electronic quasiparticles. In particular, CrGe exhibits a sixfold degeneracy of fermions crossing within about 5 meV of the Fermi energy at the $R$ point of the Brillouin zone.

Chiral and helical p -wave superconductivity in doped bilayer BiH

We investigate the superconductivity (SC) driven by correlation effects in electron-doped bilayer BiH near a type-II van Hove singularity (vHS). By functional renormalization group, we find triplet $p$-wave pairing prevails in the interaction parameter space, except for spin density wave (SDW) closer to the vHS or when the interaction is too strong. Because of the large atomic spin-orbital coupling (SOC), the $p$-wave pairing occurs between equal-spin electrons, and is chiral and two-fold degenerate. The chiral state supports in-gap edge states, even though the low energy bands in the SC state are topologically trivial. The absence of mirror symmetry allows Rashba SOC that couples unequal spins, but we find its effect is of very high order, and can only drive the chiral $p$-wave into helical $p$-wave deep in the SC state. Interestingly, there is a six-fold degeneracy in the helical states, reflected by the relative phase angle $\theta=n\pi/3$ (for integer $n$) between the spin components of the helical pairing function. The phase angle is shown to be stable in the vortex state.

Continuous degeneracy of the fcc kagome lattice with magnetic dipolar interactions

Results are presented on analytic and computational analyses of the spin states associated with a 3D fcc lattice composed of ABC stacked kagome planes of magnetic ions with only long-range dipole-dipole interactions. Extending previous work on the 2D kagome system, where discrete six-fold discrete degeneracy of the ground state was revealed [Holden et al. Phys. Rev. B 91, 224425 (2015)], we show that the 3D lattice exhibits a continuous degeneracy characterized by just two spherical angles involving six sublattice spin vectors. Application of a Heat Bath Monte Carlo algorithm shows that thermal fluctuations reduce this degeneracy at very low temperature in an order-by-disorder process. A magnetic field applied along directions of high symmetry also results in lifting the continuous degeneracy to a subset of states from the original set of ground states. Metropolis Monte Carlo simulation results are also presented on the temperature and system size dependence of the energy, specific heat, and magnetization, providing evidence for a phase transition at T $\simeq$ 0.38 (in units of the dipole strength). The results can be relevant to a class of magnetic compounds having the AuCu$_3$ crystal structure.

On the possibility of magnetic Weyl fermions in non-symmorphic compound PtFeSb

Weyl fermions are expected to exhibit exotic physical properties such as the chiral anomaly, large negative magnetoresistance or Fermi arcs. Recently a new platform to realize these fermions has been introduced based on the appearance of a three-fold band crossing at high symmetry points of certain space groups. These band crossings are composed of two linearly dispersed bands that are topologically protected by a Chern number, and a at band with no topological charge. In this paper we present a new way of inducing two kinds of Weyl fermions, based on two- and three-fold band crossings, in the non-symmorphic magnetic material PtFeSb. By means of density functional theory calculations and group theory analysis we show that magnetic order can split a six-fold degeneracy enforced by non-symmoprhic symmetry to create three-fold or two-fold degenerate Weyl nodes. We also report on the synthesis of a related phase potentially containing two-fold degenerate magnetic Weyl points and extend our group theory analysis to that phase. This is the first study showing that magnetic ordering has the potential to generate new threefold degenerate Weyl nodes, advancing the understanding of magnetic interactions in topological materials.

Finite-temperature behavior of a classical spin-orbit-coupled model for YbMgGaO4 with and without bond disorder

We present the results of finite-temperature classical Monte Carlo simulations of a strongly spin-orbit-coupled nearest-neighbor triangular-lattice model for the candidate $\mathrm{U}(1)$ quantum spin liquid $\mathrm{YbMgGaO}_4$ at large system sizes. We find a single continuous finite-temperature stripe-ordering transition with slowly diverging heat capacity that completely breaks the sixfold ground-state degeneracy, despite the absence of a known conformal field theory describing such a transition. We also simulate the effect of random-bond disorder in the model, and find that even weak bond disorder destroys the transition by fragmenting the system into very large domains -- possibly explaining the lack of observed ordering in the real material. The Imry-Ma argument only partially explains this fragility to disorder, and we extend the argument with a physical explanation for the preservation of our system's time-reversal symmetry even under a disorder model that preserves the same symmetry.

Topological dynamics of vortex-line networks in hexagonal manganites

Hexagonal manganites REMnO3 (RE, rare earths) have attracted significant attention due to their potential applications as multiferroic materials and the intriguing physics associated with the topological defects. The two-dimensional (2D) and 3D domain and vortex structure evolution of REMnO3 is predicted using the phase-field method based on a thermodynamic potential constructed from first-principles calculations. In 3D spaces, vortex lines show three types of topological changes, i.e. shrinking, coalescence, and splitting, with the latter two caused by the interaction and exchange of vortex loops. Compared to the coarsening rate of the isotropic XY model, the six-fold degeneracy gives rise to negligible differences with the vortex-antivortex annihilation controlling the scaling dynamics, whereas the anisotropy of interfacial energy results in a deviation. The temporal evolution of domain and vortex structures serves as a platform to fully explore the mesoscale mechanisms for the 0-D and 1-D topological defects.

Weyl points and Dirac lines protected by multiple screw rotations

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron’s energy bands to have Weyl points at which two bands touch. This is illustrated for space groups No. 19 (P212121) and No. 198 (P213), which have three orthogonal screw rotation axes. In the case of space groups No. 61 (Pbca) and No. 205 (Pa-3) that have extra inversion symmetry, Weyl points are promoted to four-fold degenerate line nodes in glide-invariant planes. The three-fold rotation symmetry present in the space groups No. 198 and No. 205 allows Weyl and Dirac points, respectively, to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the Γ point and R point, respectively.

Three-dimensional higher-spin Dirac and Weyl dispersions in the strongly isotropicK4crystal

We analyze the electronic structure in the three-dimensional (3D) crystal formed by the $sp^2$ hybridized orbitals ($K_4$ crystal), by the tight-binding approach based on the first-principles calculation. We discover that the bulk Dirac-cone dispersions are realized in the $K_4$ crystal. In contrast to the graphene, the energy dispersions of the Dirac cones are isotropic in 3D and the pseudospin $S=1$ Dirac cones emerge at the $\Gamma$ and $H$ points of the bcc Brillouin zone, where three bands become degenerate and merge at a single point belonging to the $T_2$ irreducible representation. In addition, the usual $S=1/2$ Dirac cones emerge at the $P$ point. By focusing the hoppings between the nearest-neighbor sites, we show an analytic form of the tight-binding Hamiltonian with a $4\times 4 $ matrix, and we give an explicit derivation of the $S=1$ and $S=1/2$ Dirac-cone dispersions. We also analyze the effect of the spin-orbit coupling to examine how the degeneracies at Dirac points are lifted. At the $S=1$ Dirac points, the spin-orbit coupling lifts the energy level with sixfold degeneracy into two energy levels with two-dimensional $\bar E_2$ and four-dimensional $\bar F$ representations. Remarkably, all the dispersions near the $\bar F$ point show the linear dependence in the momentum with different velocities. We derive the effective Hamiltonian near the $\bar F$ point and find that the band contact point is described by the $S=3/2$ Weyl point.

Dispersive readout of valley splittings in cavity-coupled silicon quantum dots

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 "fingerprint" of the microscopic energy level structure of a semiconductor double quantum dot, providing useful information on valley splittings and intervalley coupling rates.

Charge frustration in a triangular triple quantum dot.

We experimentally investigate the charge (isospin) frustration induced by a geometrical symmetry in a triangular triple quantum dot. We observe the ground-state charge configurations of sixfold degeneracy, the manifestation of the frustration. The frustration results in omnidirectional charge transport, and it is accompanied by nearby nontrivial triple degenerate states in the charge stability diagram. The findings agree with a capacitive interaction model. We also observe unusual transport by the frustration, which might be related to elastic cotunneling and the interference of trajectories through the dot. This work demonstrates a unique way of studying geometrical frustration in a controllable way.

Charge stability of a triple quantum dot with a finite tunnel coupling

We study charge stability of a symmetric triple quantum dot by using the Hund-Milliken method. As the interdot tunnel coupling increases from zero, the stability deviates continuously from that of the capacitance model. We demonstrate how the stability diagram is modified by the tunneling coupling, especially, around the sextuple point. It is shown that the characteristic feature of the sextuple point, omnidirectional charge transport, is maintained even though the sixfold degeneracy is lifted by the tunnel coupling. This is possible because the ground state near the broken sextuple point shows election or hole delocalization over the three quantum dots.

Theory of transport properties of the sixfold-degenerate two-dimensional electron gas at the H-Si(111) surface

We consider the mobility of the interacting two-dimensional electron gas as realized at the hydrogen terminated Si(111) surface with sixfold valley degeneracy. For zero temperature we calculate the mobility as function of the electron density for charged-impurity scattering. We take into account many-body effects. Multiple-scattering effects, leading to a metal-insulator transition at low electron density, are also taken into account. Our calculation is in agreement with recent experimental results. We also present analytical results for the mobility and we study the importance of interface-roughness scattering. For the spin-polarized electron gas, created by applying a parallel magnetic field, we predict the mobility and the phase diagram of the metal-insulator transition.

Electronic structure and effective masses in strained silicon

AbstractThe structural and electronic properties of strained silicon are investigated quantitatively with ab initio computational methods. For this purpose we combine densityfunctional theory within the local‐density approximation and the GW approximation for the electronic self‐energy. From the variation of the total energy as a function of applied strain we obtain the elastic constants, Poisson ratios and related structural parameters, taking a possible internal relaxation fully into account. For biaxial tensile strain in the (001) and (111) planes we then investigate the effects on the electronic band structure. These strain configurations occur in epitaxial silicon films grown on SiGe templates along different crystallographic directions.The tetragonal deformation resulting from (001) strain induces a valley splitting that removes the sixfold degeneracy of the conduction‐band minimum. Furthermore, strain in any direction causes the band structure to warp. We present quantitative results for the electron effective mass, derived from the curvature of the conduction band, as a function of strain and discuss the implications for the mobility of the charge carriers. The inclusion of proper self‐energy corrections within the GW approximation in our work not only yields band gaps in much better agreement with experimental measurements than the localdensity approximation, but also predicts slightly larger electron effective masses (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)