Wave propagation through an assembly of spheres: II. The density of single-particle eigenstates

Observation of time-reversal symmetry breaking in the band structure of altermagnetic RuO2

Altermagnets are an emerging elementary class of collinear magnets. Unlike ferromagnets, their distinct crystal symmetries inhibit magnetization while, unlike antiferromagnets, they promote strong spin polarization in the band structure. The corresponding unconventional mechanism of time-reversal symmetry breaking without magnetization in the electronic spectra has been regarded as a primary signature of altermagnetism but has not been experimentally visualized to date. We directly observe strong time-reversal symmetry breaking in the band structure of altermagnetic RuO2 by detecting magnetic circular dichroism in angle-resolved photoemission spectra. Our experimental results, supported by ab initio calculations, establish the microscopic electronic structure basis for a family of interesting phenomena and functionalities in fields ranging from topological matter to spintronics, which are based on the unconventional time-reversal symmetry breaking in altermagnets.

Altermagnetic lifting of Kramers spin degeneracy

Lifted Kramers spin degeneracy (LKSD) has been among the central topics of condensed-matter physics since the dawn of the band theory of solids1,2. It underpins established practical applications as well as current frontier research, ranging from magnetic-memory technology3-7 to topological quantum matter8-14. Traditionally, LKSD has been considered to originate from two possible internal symmetry-breaking mechanisms. The first refers to time-reversal symmetry breaking by magnetization of ferromagnets and tends to be strong because of the non-relativistic exchange origin15. The second applies to crystals with broken inversion symmetry and tends to be comparatively weaker, as it originates from the relativistic spin-orbit coupling (SOC)16-19. A recent theory work based on spin-symmetry classification has identified an unconventional magnetic phase, dubbed altermagnetic20,21, that allows for LKSD without net magnetization and inversion-symmetry breaking. Here we provide the confirmation using photoemission spectroscopy and ab initio calculations. We identify two distinct unconventional mechanisms of LKSD generated by the altermagnetic phase of centrosymmetric MnTe with vanishing net magnetization20-23. Our observation of the altermagnetic LKSD can have broad consequences in magnetism. It motivates exploration and exploitation of the unconventional nature of this magnetic phase in an extended family of materials, ranging from insulators and semiconductors to metals and superconductors20,21, that have been either identified recently or perceived for many decades as conventional antiferromagnets21,24,25.

Discontinuity in the Electronic Structure and Magnetic Order of β-Co1+xGa1-x

The present work reports on the calculated electronic and magnetic structure of the binary Co-Ga system at high Co content. β-CoGa adopts a simple cubic CsCl type structure. Well-ordered CoGa does not exhibit collective magnetism but is a paramagnetic, metallic compound. Neither Co nor Ga deficiency induces magnetic order; however, ferromagnetism is observed for Co-Ga anti-site disorder. The magnetic moment per cell increases by up to approximately 1.2 μB in the completely disordered body-centered cubic structure. With increasing Co content, Co1+xGa1-x maintains the CsCl type structure and becomes ferromagnetic. Most importantly, a discontinuity of the magnetic order with composition is observed at about 10% excess Co, where a change from a low magnetic moment state to a high moment state is observed. This is accompanied by a change in the electronic structure and transport properties. The discontinuity is forced by the increasing exchange splitting related to the localized moment of the additional Co atoms that replace Ga. Subsequently, the magnetic moment increases continuously up to 2.5 μB for x=0.6. For x≳0.6, the structure changes to a face-centered cubic structure with random site occupation and the magnetic moment further increases. Above the magnetic discontinuity, the Curie temperature increases linearly with the Co content from the onset of ferromagnetism, until it reaches its maximum in pure Co.

Magnetic ground state of supported monatomic Fe chains from first principles

A new computational scheme is presented based on a combination of the conjugate gradient and the Newton-Raphson method to self-consistently minimize the energy within local spin-density functional theory, thus to identify the ground state magnetic order of a finite cluster of atoms. The applicability of the newab initiooptimization method is demonstrated for Fe chains deposited on different metallic substrates. The optimized magnetic ground states of the Fe chains on Rh(111) are analyzed in details and a good comparison is found with those obtained from an extended Heisenberg model containing first principles based interaction parameters. Moreover, the effect of the different bilinear spin-spin interactions in the formation of the magnetic ground states is monitored. In case of Fe chains on Nb(110) spin-spiral configurations with opposite rotational sense are found as compared to previous spin-model results which hints on the importance of higher order chiral interactions. The wavelength of the spin-spiral states of Fe chains on Re(0001) was obtained in good agreement with scanning tunneling microscopy experiments.

Electronic and Magnetic Properties of Building Blocks of Mn and Fe Atomic Chains on Nb(110)

We present results for the electronic and magnetic structure of Mn and Fe clusters on Nb(110) surface, focusing on building blocks of atomic chains as possible realizations of topological superconductivity. The magnetic ground states of the atomic dimers and most of the monatomic chains are determined by the nearest-neighbor isotropic interaction. To gain physical insight, the dependence on the crystallographic direction as well as on the atomic coordination number is analyzed via an orbital decomposition of this isotropic interaction based on the spin-cluster expansion and the difference in the local density of states between ferromagnetic and antiferromagnetic configurations. A spin-spiral ground state is obtained for Fe chains along the [11¯0] direction as a consequence of the frustration of the isotropic interactions. Here, a flat spin-spiral dispersion relation is identified, which can stabilize spin spirals with various wave vectors together with the magnetic anisotropy. This may lead to the observation of spin spirals of different wave vectors and chiralities in longer chains instead of a unique ground state.

First-principles Dzyaloshinskii-Moriya interaction in a non-collinear framework

We have derived an expression of the Dzyaloshinskii–Moriya interaction (DMI), where all the three components of the DMI vector can be calculated independently, for a general, non-collinear magnetic configuration. The formalism is implemented in a real space—linear muffin-tin orbital—atomic sphere approximation (RS-LMTO-ASA) method. We have chosen the Cr triangular trimer on Au(111) and Mn triangular trimers on Ag(111) and Au(111) surfaces as numerical examples. The results show that the DMI (module and direction) is drastically different between collinear and non-collinear states. Based on the relation between the spin and charge currents flowing in the system and their coupling to the non-collinear magnetic configuration of the triangular trimer, we demonstrate that the DMI interaction can be significant, even in the absence of spin-orbit coupling. This is shown to emanate from the non-collinear magnetic structure, that can induce significant spin and charge currents even with spin-orbit coupling is ignored.

Role of temperature-dependent spin model parameters in ultra-fast magnetization dynamics

In the spirit of multi-scale modelling magnetization dynamics at elevated temperature is often simulated in terms of a spin model where the model parameters are derived from first principles. While these parameters are mostly assumed temperature-independent and thermal properties arise from spin fluctuations only, other scenarios are also possible. Choosing bcc Fe as an example, we investigate the influence of different kinds of model assumptions on ultra-fast spin dynamics, where following a femtosecond laser pulse, a sample is demagnetized due to a sudden rise of the electron temperature. While different model assumptions do not affect the simulational results qualitatively, their details do depend on the nature of the modelling.

The influence of Gaussian strain on sublattice selectivity of impurities in graphene

Among the different strategies used to induce the opening of a band gap in graphene, one common practice is through chemical doping. While a gap may be opened in this way, disorder-induced scattering is an unwanted side-effect that impacts the electron mobility in the conductive regime of the system. However, this undesirable side effect is known to be minimised if dopants interact asymmetrically with the two sublattices of graphene. In this work we propose that mechanical strain can be used to introduce such a sublattice asymmetry in the doping process of graphene. We argue that a localised out-of-plane deformation applied to a graphene sheet can make one of the graphene sublattices more energetically favourable for impurity adsorption than the other and that this can be controlled by varying the strain parameters. Two complementary modelling schemes are used to describe the electronic structure of the flat and deformed graphene sheets: a tight-binding model and density functional theory. Our results indicate a novel way to select the doping process of graphene through strain engineering.

Magnetic anisotropy and chirality of frustrated Cr nanostructures on Au(1 1 1)

By using a fully relativistic embedded cluster Green's function technique we investigated the magnetic anisotropy properties of four different compact Cr trimers (equilateral triangles) and Cr mono-layers deposited on the Au(1 1 1) surface in both fcc and hcp stackings. For all trimers the magnetic ground state was found to be a frustrated 120° Néel configuration. Applying global spin rotations to the magnetic ground state, predictions of an appropriate second order spin Hamiltonian were reproduced with high accuracy by first principles calculations. For the Cr trimers with adjacent Au atoms in similar geometry, we obtained similar values for the in-plane and out-of-plane anisotropy parameters, however, the Dzyaloshinskii-Moriya (DM) interactions appeared to differ remarkably. For two kinds of trimers we found an unconventional magnetic ground state showing 90° in-the-plane rotation with respect to the high symmetry directions. Due to higher symmetry, the in-plane anisotropy term was missing for the mono-layers and distinctly different DM interactions were obtained for the different stackings. The chiral degeneracy of the Néel configurations was lifted by an energy less than 2 meV for the trimers, while this value increased up to about 15 meV per 3 Cr atoms for the hcp packed mono-layer.

Langevin spin dynamics based on ab initio calculations: numerical schemes and applications

A method is proposed to study the finite-temperature behaviour of small magnetic clusters based on solving the stochastic Landau-Lifshitz-Gilbert equations, where the effective magnetic field is calculated directly during the solution of the dynamical equations from first principles instead of relying on an effective spin Hamiltonian. Different numerical solvers are discussed in the case of a one-dimensional Heisenberg chain with nearest-neighbour interactions. We performed detailed investigations for a monatomic chain of ten Co atoms on top of a Au(0 0 1) surface. We found a spiral-like ground state of the spins due to Dzyaloshinsky-Moriya interactions, while the finite-temperature magnetic behaviour of the system was well described by a nearest-neighbour Heisenberg model including easy-axis anisotropy.

Variable range of the RKKY interaction in edged graphene

The indirect exchange interaction is one of the key factors in determining the overall alignment of magnetic impurities embedded in metallic host materials. In this work we examine the range of this interaction in magnetically doped graphene systems in the presence of armchair edges using a combination of analytical and numerical Green function approaches. We consider both a semi-infinite sheet of graphene with a single armchair edge, and also quasi-one-dimensional armchair-edged graphene nanoribbons (GNRs). While we find signals of the bulk decay rate in semi-infinite graphene and signals of the expected one-dimensional decay rate in GNRs, we also find an unusually rapid decay for certain instances in both, which manifests itself whenever the impurities are located at sites which are a multiple of three atoms from the edge. This decay behavior emerges from both the analytic and numerical calculations, and the result for semi-infinite graphene can be interpreted as an intermediate case between ribbon and bulk systems.

Interatomic exchange interactions for finite-temperature magnetism and nonequilibrium spin dynamics

We derive ab inito exchange parameters for general noncollinear magnetic configurations, in terms of a multiple scattering formalism. We show that the general exchange formula has an anisotropiclike term even in the absence of spin-orbit coupling, and that this term is large, for instance, for collinear configuration in bcc Fe, whereas for fcc Ni it is quite small. We demonstrate that keeping this term leads to what one should consider a biquadratic effective spin Hamiltonian even in the case of collinear arrangement. In noncollinear systems this term results in new tensor elements that are important for exchange interactions at finite temperatures, but they have less importance at low temperature. To illustrate our results in practice, we calculate for bcc Fe magnon spectra obtained from configuration-dependent exchange parameters, where the configurations are determined by finite-temperature effects. Our theory results in the same quantitative results as the finite-temperature neutron scattering experiments.

Projection potentials and angular momentum convergence of total energies in the full-potential Korringa-Kohn-Rostoker method

Although the full-potential Korringa-Kohn-Rostoker Green function method yields accurate results for many physical properties, the convergence of calculated total energies with respect to the angular momentum cutoff is usually considered to be less satisfactory. This is surprising because accurate single-particle energies are expected if they are calculated by Lloyd's formula and because accurate densities and hence accurate double-counting energies should result from the total energy variational principle. It is shown how the concept of projection potentials can be used as a tool to analyse the convergence behaviour. The key factor blocking fast convergence is identified and it is illustrated how total energies can be improved with only a modest increase of computing time.

Nonequilibrium thermodynamics of interacting tunneling transport: variational grand potential, density functional formulation and nature of steady-state forces

The standard formulation of tunneling transport rests on an open-boundary modeling. There, conserving approximations to nonequilibrium Green function or quantum statistical mechanics provide consistent but computational costly approaches; alternatively, the use of density-dependent ballistic-transport calculations (e.g., Lang 1995 Phys. Rev. B 52 5335), here denoted 'DBT', provides computationally efficient (approximate) atomistic characterizations of the electron behavior but has until now lacked a formal justification. This paper presents an exact, variational nonequilibrium thermodynamic theory for fully interacting tunneling and provides a rigorous foundation for frozen-nuclei DBT calculations as a lowest-order approximation to an exact nonequilibrium thermodynamic density functional evaluation. The theory starts from the complete electron nonequilibrium quantum statistical mechanics and I identify the operator for the nonequilibrium Gibbs free energy which, generally, must be treated as an implicit solution of the fully interacting many-body dynamics. I demonstrate a minimal property of a functional for the nonequilibrium thermodynamic grand potential which thus uniquely identifies the solution as the exact nonequilibrium density matrix. I also show that the uniqueness-of-density proof from a closely related Lippmann-Schwinger collision density functional theory (Hyldgaard 2008 Phys. Rev. B 78 165109) makes it possible to express the variational nonequilibrium thermodynamic description as a single-particle formulation based on universal electron-density functionals; the full nonequilibrium single-particle formulation improves the DBT method, for example, by a more refined account of Gibbs free energy effects. I illustrate a formal evaluation of the zero-temperature thermodynamic grand potential value which I find is closely related to the variation in the scattering phase shifts and hence to Friedel density oscillations. This paper also discusses the difference between the here-presented exact thermodynamic forces and the often-used electrostatic forces. Finally the paper documents an inherent adiabatic nature of the thermodynamic forces and observes that these are suited for a nonequilibrium implementation of the Born-Oppenheimer approximation.

Understanding the resistivity and absolute thermoelectric power of disordered metals and alloys

We recall definitions of the electronic transport properties, direct coefficients like electrical and thermal transport conductivities and crossed thermoelectric coefficients like the Seebeck, Peltier and Thomson coefficients. We discuss the links between the different electronic transport coefficients and the experimental problems in measuring these properties in liquid metals. The electronic transport properties are interpreted in terms of the scattering of electrons by 'pseudo-atoms'. The absolute thermoelectric power (ATP), thermopower or Seebeck coefficient is known as the derivative of the electrical resistivity versus energy. The key is to understand the concept of resistivity versus energy. We show that the resistivity follows approximately a 1/E curve. The structure factor modulates this curve and, for a Fermi energy corresponding to noble and divalent metals, induces a positive thermopower when the free electron theory predicts a negative one. A second modulation is introduced by the pseudopotential squared form factor or equivalently by the squared t matrix of the scattering potential. This term sometimes introduces an anti-resonance (divalent metals) which lowers the resistivity, and sometimes a resonance having an important effect on the transition metals. Following the position of the Fermi energy, the thermopower can be positive or negative. For heavy semi-metals, the density of states splits into an s and a p band, themselves different from a free electron E(0.5) curve. The electrons available to be scattered enter the Ziman formula. Thus if the density of states is not a free electron one, a third modulation of the [Formula: see text] curve is needed, which also can change the sign of the thermopower. For alloys, different contributions weighted by the concentrations are needed to explain the concentration dependent resistivity or thermopower. The formalism is the same for amorphous metals. It is possible that this mechanism can be extended to high-temperature crystalline alloys or even disordered semiconductors since we can separate the transport coefficients between the effect of the number of charge carriers and a scattering term linked to carrier mobility.