Berry Phase Contribution Research Articles

Hall Effect Anisotropy in the Paramagnetic Phase of Ho0.8Lu0.2B12 Induced by Dynamic Charge Stripes.

A detailed study of charge transport in the paramagnetic phase of the cage-cluster dodecaboride Ho0.8Lu0.2B12 with an instability both of the fcc lattice (cooperative Jahn-Teller effect) and the electronic structure (dynamic charge stripes) was carried out at temperatures 1.9-300 K in magnetic fields up to 80 kOe. Four mono-domain single crystals of Ho0.8Lu0.2B12 samples with different crystal axis orientation were investigated in order to establish the singularities of Hall effect, which develop due to (i) the electronic phase separation (stripes) and (ii) formation of the disordered cage-glass state below T*~60 K. It was demonstrated that a considerable intrinsic anisotropic positive component ρanxy appears at low temperatures in addition to the ordinary negative Hall resistivity contribution in magnetic fields above 40 kOe applied along the [001] and [110] axes. A relation between anomalous components of the resistivity tensor ρanxy~ρanxx1.7 was found for H||[001] below T*~60 K, and a power law ρanxy~ρanxx0.83 for the orientation H||[110] at temperatures T < TS~15 K. It is argued that below characteristic temperature TS~15 K the anomalous odd ρanxy(T) and even ρanxx(T) parts of the resistivity tensor may be interpreted in terms of formation of long chains in the filamentary structure of fluctuating charges (stripes). We assume that these ρanxy(H||[001]) and ρanxy(H||[110]) components represent the intrinsic (Berry phase contribution) and extrinsic (skew scattering) mechanism, respectively. Apart from them, an additional ferromagnetic contribution to both isotropic and anisotropic components in the Hall signal was registered and attributed to the effect of magnetic polarization of 5d states (ferromagnetic nano-domains) in the conduction band of Ho0.8Lu0.2B12.

Room-temperature weak collinear ferrimagnet with symmetry-driven large intrinsic magneto-optic signatures

Here we present a magnetic thin film with a weak ferrimagnetic (FIM) phase above the N\'eel temperature (${T}_{N}=240\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) and a noncollinear antiferromagnetic (AFM) phase below, exhibiting a small net magnetization due to strain-associated canting of the magnetic moments. A long-range ordered FIM phase has been predicted in related materials, but without symmetry analysis. We now perform this analysis and use it to calculate the magneto-optical Kerr effect (MOKE) spectra in the AFM and FIM phases. From the good agreement between the form of the measured and predicted MOKE spectra, we propose the AFM and FIM phases share the magnetic space group $C{2}^{\ensuremath{'}}/{m}^{\ensuremath{'}}$ and that the symmetry-driven magneto-optic and magneto-transport properties are maximized at room temperature in the FIM phase due to the nonzero intrinsic Berry phase contribution present in these materials. A room temperature FIM with large optical and transport signatures, as well as sensitivity to lattice strain and magnetic field, has useful prospects for high-speed spintronic applications.

Dirac materials under linear polarized light: quantum wave function time evolution and topological Berry phases as classical charged particles trajectories under electromagnetic fields

The response of electrons under linearly polarized light in Dirac materials as borophene or graphene is analyzed in a continuous wave regime for an arbitrary intense field. Using a rotation and a time-dependent phase transformation, the wave function evolution is shown to be governed by a spinor-component decoupled Whittaker–Hill equation. The numerical solution of these equations enables to find the quasienergy spectrum. For borophene it reveals a strong anisotropic response. By applying an extra unitary transformation, the wave functions are proven to follow an Ince equation. The evolution of the real and imaginary parts of the wave function is interpreted as the trajectory of a classical charged particle under oscillating electric and magnetic field. The topological properties of this forced quantum system are studied using this analogy. In particular, in the adiabatic driving regime, the system is described with an effective Matthieu equation while in the non-adiabatic regime the full Whittaker–Hill equation is needed. From there, it is possible to separate the dynamical and Berry phase contributions to obtain the topological phase diagram due to the driving. Therefore, a different path to perturbation theory is developed to obtain time-driven topological phases.

Strain dependence of Berry-phase-induced anomalous Hall effect in the non-collinear antiferromagnet Mn3NiN

The anomalous Hall effect (AHE) has been shown to be present in certain non-collinear antiferromagnets due to their symmetry-breaking magnetic structure, and its magnitude is dependent primarily on the non-zero components of the Berry curvature. In the non-collinear antiferromagnet Mn3NiN, the Berry phase contribution has been predicted to have strong strain dependence, although in practice, direct observation may be obscured by other strain-related influences—for instance, magnetic phase transitions mediated by strain. To unravel the various contributions, we examine the thickness and temperature dependence of the AHE for films grown on the piezoelectric substrate BaTiO3. We observe a systematic reduction in TN due to increased compressive strain as film thickness is reduced and a linear decrease in the AHE magnitude as the films are cooled from their ferrimagnetic phase above TN to their antiferromagnetic phase below. At 190 K, we applied an electric field across a 0.5 mm thick BaTiO3 substrate with a 50 nm thick Mn3NiN film grown on top and we demonstrate that at the coercive field of the piezoelectric substrate, the tensile in-plane strain is estimated to be of the order of 0.15%, producing a 20% change in AHE. Furthermore, we show that this change is, indeed, dominated by the intrinsic strain dependence of the Berry curvature.

Anomalous Hall effect in MnAs: Intrinsic contribution due to Berry curvature

We present an experimental and theoretical study of the anomalous Hall effect (AHE) in MnAs epilayers grown over GaAs, with the aim to identify the intrinsic contribution to the AHE, which can be accurately evaluated using ab initio electronic structure calculations. Our magnetotransport measurements show a quadratic behavior of the Hall resistivity with longitudinal resistivity, characteristic of scattering-independent processes, thus enabling the comparison with our ab initio calculations. The calculated Berry phase contribution to the AHE is in quantitative agreement with the measured AHE in these epilayers. Moreover, the predicted anisotropic dependence of the experimental AHE on the magnetization is well reproduced.

Berry phase manipulation in ultrathin SrRuO3 films

A notion of the Berry phase is a powerful means to unravel the non-trivial role of topology in various novel phenomena observed in chiral magnetic materials and structures. A celebrated example is the intrinsic anomalous Hall effect (AHE) driven by the non-vanishing Berry phase in the momentum space. As the AHE is highly dependent on details of the band structure near the Fermi edge, the Berry phase and AHE can be altered in thin films whose chemical potential is tunable by dimensionality and disorder. Here, we demonstrate that in ultrathin SrRuO$_3$ films the Berry phase can be effectively manipulated by the effects of disorder on the intrinsic Berry phase contribution to the AHE, which is corroborated by our numerically exact calculations. In addition, our findings provide ample experimental evidence for the superficial nature of the topological Hall effect attribution to the protected spin texture and instead lend strong support to the multi-channel AHE scenario in ultrathin SrRuO$_3$.

Anomalous Hall effect in the kagome ferrimagnet GdMn6Sn6

We present magnetotransport data on the ferrimagnet GdMn$_6$Sn$_6$. From the temperature dependent data we are able to extract a large instrinsic contribution to the anomalous Hall effect $\sigma_{xz}^{int} \sim$ 32 $\Omega^{-1}cm^{-1}$ and $\sigma_{xy}^{int} \sim$ 223 $\Omega^{-1}cm^{-1}$, which is comparable to values found in other systems also containing kagome nets of transition metals. From our transport anisotropy, as well as our density functional theory calculations, we argue that the system is electronically best described as a three dimensional system. Thus, we show that reduced dimensionality is not a strong requirement for obtaining large Berry phase contributions to transport properties. In addition, the coexistence of rare-earth and transition metal magnetism makes the hexagonal MgFe$_6$Ge$_6$ structure type a promising system to tune the electronic and magnetic properties in future studies.

Intrinsic spin-orbit torque in an antiferromagnet with a weakly noncollinear spin configuration

An antiferromagnet is a promising material for spin-orbit torque generation. Earlier studies of the spin-orbit torque in an antiferromagnet are limited to collinear spin configurations. We calculate the spin-orbit torque in an antiferromagnet whose spin ordering is weakly noncollinear. Such noncollinearity may be induced spontaneously during the magnetization dynamics even when the equilibrium spin configuration is perfectly collinear. It is shown that deviation from perfect collinearity can modify properties of the spin-orbit torque since noncollinearity generates extra Berry phase contributions to the spin-orbit torque, which are forbidden for collinear spin configurations. In sufficiently clean antiferromagnets, this modification can be significant. We estimate this effect to be of relevance for fast antiferromagnetic domain wall motion.

Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects

In solid state conductors, linear response to a steady electric field is normally dominated by Bloch state occupation number changes that are correlated with group velocity and lead to a steady state current. However, for a number of important physical observables, the most important response even in conductors can be electric-field induced coherence between Bloch states in different bands, such as that responsible for screening in dielectrics. Examples include the anomalous and spin-Hall effects, spin torques, minimum conductivities and the chiral anomaly. In this paper we present a general quantum kinetic theory of linear response to an electric field which can be applied to solids with arbitrarily complicated band structures and includes the inter-band coherence response and the Bloch-state repopulation responses on an equal footing. We aim to enable extensive transport theory applications using computational packages based on maximally localized Wannier functions. The formalism is based on density-matrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. The quasiparticle bands are treated in a completely general manner that allows for arbitrary forms of the spin-orbit interaction and for the broken time reversal symmetry of magnetic conductors. We demonstrate that the inter-band response in conductors consists primarily of two terms: an intrinsic contribution due to the entire Fermi sea that captures the Berry curvature contribution to wave-packet dynamics, and an anomalous contribution caused by scattering that is sensitive to the presence of the Fermi surface. We solve for some electric-field response properties of simple model systems that are known to be dominated by interband coherence contributions and discuss an extensive list of complicated problems that cannot be solved analytically.

Thermoelectric Transport Signatures of Dirac Composite Fermions in the Half-Filled Landau Level

The half filled Landau level is expected to be approximately particle-hole symmetric, which requires an extension of the Halperin-Lee-Read (HLR) theory of the compressible state observed at this filling. Recent work indicates that, when particle-hole symmetry is preserved, the composite Fermions experience a quantized $\pi$-Berry phase upon winding around the composite Fermi-surface, analogous to Dirac fermions at the surface of a 3D topological insulator. In contrast, the effective low energy theory of the composite fermion liquid originally proposed by HLR lacks particle-hole symmetry and has vanishing Berry phase. In this paper, we explain how thermoelectric transport measurements can be used to test the Dirac nature of the composite Fermions by quantitatively extracting this Berry phase. First we point out that longitudinal thermopower (Seebeck effect) is non-vanishing due to the unusual nature of particle hole symmetry in this context and is not sensitive to the Berry phase. In contrast, we find that off-diagonal thermopower (Nernst effect) is directly related to the topological structure of the composite Fermi surface, vanishing for zero Berry phase and taking its maximal value for $\pi$ Berry phase. In contrast, in purely electrical transport signatures the Berry phase contributions appear as small corrections to a large background signal, making the Nernst effect a promising diagnostic of the Dirac nature of composite fermions.

Optical gyrotropy from axion electrodynamics in momentum space.

Several emergent phenomena and phases in solids arise from configurations of the electronic Berry phase in momentum space that are similar to gauge field configurations in real space such as magnetic monopoles. We show that the momentum-space analogue of the "axion electrodynamics" term E·B plays a fundamental role in a unified theory of Berry-phase contributions to optical gyrotropy in time-reversal invariant materials and the chiral magnetic effect. The Berry-phase mechanism predicts that the rotatory power along the optic axes of a crystal must sum to zero, a constraint beyond that stipulated by point-group symmetry, but observed to high accuracy in classic experimental observations on alpha quartz. Furthermore, the Berry mechanism provides a microscopic basis for the surface conductance at the interface between gyrotropic and nongyrotropic media.

Berry phase, Lorentz covariance, and anomalous velocity for Dirac and Weyl particles

We consider the relation between spin and the Berry-phase contribution to the anomalous velocity of massive and massless Dirac particles. We extend the Berry connection that depends only on the spatial components of the particle momentum to one that depends on the the space and time components in a covariant manner. We show that this covariant Berry connection captures the Thomas-precession part of the Bargmann-Michel-Telegdi spin evolution, and contrast it with the traditional (unitary, but not naturally covariant) Berry connection that describes spin-orbit coupling. We then consider how the covariant connection enters the classical relativistic dynamics of spinning particles due to Mathisson, Papapetrou and Dixon. We discuss the problems that arise with Lorentz covariance in the massless case, and trace them mathematically to a failure of the Wigner-translation part of the massless-particle little group to be an exact gauge symmetry in the presence of interactions, and physically to the fact that the measured position of a massless spinning particle is necessarily observer dependent.

How disorder affects the Berry-phase anomalous Hall conductivity: A reciprocal-space analysis

The anomalous Hall conductivity of "dirty" ferromagnetic metals is dominated by a Berry-phase contribution which is usually interpreted as an intrinsic property of the Bloch electrons in the pristine crystal. In this work we evaluate the geometric Hall current directly from the electronic ground state with disorder, and then recast it as an integral over the crystalline Brillouin zone. The integrand is a generalized k-space Berry curvature, obtained by unfolding the Berry curvature from the small Brillouin zone of a large supercell. Therein, disorder yields a net extrinsic Hall contribution, which we argue is related to the elusive side-jump effect. As an example, we unfold the first-principles Berry curvature of an Fe3Co ordered alloy from the original fcc-lattice Brillouin zone onto a bcc-lattice zone with four times the volume. Comparison with the virtual-crystal Berry curvature clearly reveals the symmetry-breaking effects of the substitutional Co atoms.

Topological Response Theory of Doped Topological Insulators

We generalize the topological response theory of three-dimensional topological insulators (TI) to metallic systems-specifically, doped TI with finite bulk carrier density and a time-reversal symmetry breaking field near the surface. We show that there is an inhomogeneity-induced Berry phase contribution to the surface Hall conductivity that is completely determined by the occupied states and is independent of other details such as band dispersion and impurities. In the limit of zero bulk carrier density, this intrinsic surface Hall conductivity reduces to the half-integer quantized surface Hall conductivity of TI. Based on our theory we predict the behavior of the surface Hall conductivity for a doped topological insulator with a top gate, which can be directly compared with experiments.

Quasiparticle scattering from vortices ind-wave superconductors. II. Berry phase contribution

In the mixed state of a d-wave superconductor, Bogoliubov quasiparticles are scattered from magnetic vortices via a combination of two effects: Aharonov-Bohm scattering due to the Berry phase acquired by a quasiparticle upon circling a vortex, and effective potential scattering due to the superflow swirling about the vortices. In this paper, we consider the Berry phase contribution in the absence of superflow, which results in branch cuts between neighboring vortices across which the quasiparticle wave function changes sign. Here, the simplest problem that captures the physics is that of scattering from a single finite branch cut that stretches between two vortices. Elliptical coordinates are natural for this two-center problem, and we proceed by separating the massless Dirac equation in elliptical coordinates. The separated equations take the form of the Whittaker-Hill equations, which we solve to obtain radial and angular eigenfunctions. With these eigenfunctions in hand, we construct the scattering cross section via partial wave analysis. We discuss the scattering effect of Berry phase in the absence of superflow, having considered the superflow effect in the absence of Berry phase in a separate paper. We also provide qualitative comparison of transport cross sections for the Berry phase and the Superflow effects. The important issue of interference between the two effects is left to future work.

Quasiparticle scattering from vortices ind-wave superconductors. I. Superflow contribution

In the vortex state of a $d$-wave superconductor, massless Dirac quasiparticles are scattered from magnetic vortices via a combination of two basic mechanisms: effective potential scattering due to the superflow swirling about the vortices, and Aharonov-Bohm scattering due to the Berry phase acquired by a quasiparticle upon circling a vortex. In this paper, we study the superflow contribution by calculating the differential cross section for a quasiparticle scattering from the effective noncentral potential of a single vortex. We solve the massless Dirac equation in polar coordinates and obtain the cross section via a partial wave analysis. We also present a more transparent Born-limit calculation, and in this approximation we provide an analytic expression for the differential cross section. The Berry phase contribution to the quasiparticle scattering is considered in a separate paper.

Confinement-Induced Berry Phase and Helicity-Dependent Photocurrents

The photocurrent in an optically active metal is known to contain a component that switches sign with the helicity of the incident radiation. At low frequencies, this current depends on the orbital Berry phase of the Bloch electrons via the "anomalous velocity" of Karplus and Luttinger. We consider quantum wells in which the parent material, such as GaAs, is not optically active and the relevant Berry phase only arises as a result of quantum confinement. Using an envelope approximation that is supported by numerical tight-binding results, it is shown that the Berry-phase contribution is determined for realistic wells by a cubic Berry phase intrinsic to the bulk material, the well width, and the well direction. These results for the Berry-phase effect suggest that it may already have been observed in quantum well experiments.

Quantum transport theory of anomalous electric, thermoelectric, and thermal Hall effects in ferromagnets

In ferromagnets, the spontaneous magnetization bears the Hall effect through the relativistic spin-orbit interaction. Similar effects also occur in thermoelectric and thermal transport phenomena. Their mechanism, if it is of the intrinsic or extrinsic origin, has been controversial for many decades. We present a unified theory of these Hall transport phenomena in ferromagnetic metals with dilute impurities at the zero temperature, in terms of a fully quantum-mechanical transport theory for multiband systems with the self-consistent $T$-matrix approximation. This theory becomes exact with a single impurity and is appropriate for treating the dilute limit of the impurity concentration ${n}_{\mathrm{imp}}$. With the Fermi energy ${E}_{F}$ and the spin-orbit interaction energy ${E}_{\mathrm{SO}}$ being fixed $({E}_{F}g{E}_{\mathrm{SO}})$, three regimes and the associated two crossovers are found in the anomalous Hall conductivity ${\ensuremath{\sigma}}_{xy}$ as a function of ${n}_{\mathrm{imp}}$ that controls the longitudinal conductivity ${\ensuremath{\sigma}}_{xx}$. (i) In the superclean case with the relaxation rate $\ensuremath{\hbar}∕\ensuremath{\tau}\ensuremath{\lesssim}{u}_{\mathrm{imp}}{E}_{\mathrm{SO}}D$, the skew scattering arising from the vertex correction yields a dominant contribution that is inversely proportional to ${n}_{\mathrm{imp}}$, where ${u}_{\mathrm{imp}}$ is the impurity potential strength and $D$ is the density of states. With increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, this extrinsic skew-scattering contribution rapidly decays. (ii) In the moderately dirty regime ${u}_{\mathrm{imp}}{E}_{\mathrm{SO}}D\ensuremath{\lesssim}\ensuremath{\hbar}∕\ensuremath{\tau}\ensuremath{\lesssim}{E}_{F}$, ${\ensuremath{\sigma}}_{xy}$ becomes insensitive to the scattering strength because of the intrinsic dissipationless topological Berry-phase contribution. It is resonantly enhanced to the order of the quantization unit of conductance when an accidental degeneracy of band dispersions around the Fermi level is lifted by the spin-orbit interaction. Further increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, another crossover occurs to (iii) the scaling regime of ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}{\ensuremath{\sigma}}_{xx}^{\ensuremath{\varphi}}$ with $\ensuremath{\varphi}\ensuremath{\sim}1.6$, which has recently been verified by experiments on a wide class of ferromagnets. Similar behaviors also appear in the temperature-linear coefficient of the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$. The thermoelectric Hall conductivity ${\ensuremath{\alpha}}_{xy}$ strongly diverges in the clean limit when the Fermi level crosses edges of the avoided crossing, which may be observed by careful experiments. With increasing $\ensuremath{\hbar}∕\ensuremath{\tau}$, there occurs an interference between positive and negative contributions to ${\ensuremath{\alpha}}_{xy}$, which often leads to a sign change and obscures similar crossovers in the anomalous Nernst effect.

Competition between superconductivity and charge density waves

We derive an effective field theory for the competition between superconductivity (SC) and charge density waves (CDWs) by employing the SO(3) pseudospin representation of the SC and CDW order parameters. One important feature in the effective nonlinear $\ensuremath{\sigma}$ model is the emergence of a Berry phase even at half filling, originating from the competition between SC and CDWs, i.e., the pseudospin symmetry. A-well known conflict between the previous studies of Oshikawa [Phys. Rev. Lett. 84, 1535 (2000)] and Lee and Shankar [Phys. Rev. Lett. 65, 1490 (1990)] is resolved by the appearance of the Berry phase. The Berry phase contribution allows a deconfined quantum critical point of fractionalized charge excitations with $e$ instead of $2e$ in the SC-CDW quantum transition at half filling. Furthermore, we investigate the stability of the deconfined quantum criticality against quenched randomness by performing a renormalization group analysis of an effective vortex action. We argue that, although randomness results in a weak disorder fixed point differing from the original deconfined quantum critical point, deconfinement of the fractionalized charge excitations still survives at the disorder fixed point owing to a nonzero fixed point value of the vortex charge.

Interplay between Carrier and Impurity Concentrations in AnnealedGa1−xMnxAs: Intrinsic Anomalous Hall Effect

Investigating the scaling behavior of annealed Ga1-xMnxAs anomalous Hall coefficients, we note a universal crossover regime where the scaling behavior changes from quadratic to linear. Furthermore, measured anomalous Hall conductivities in the quadratic regime when properly scaled by carrier concentration remain constant, spanning nearly a decade in conductivity as well as over 100 K in T_[C] and comparing favorably to theoretically predicated values for the intrinsic origins of the anomalous Hall effect. Both qualitative and quantitative agreements strongly point to the validity of new equations of motion including the Berry phase contributions as well as the tunability of the anomalous Hall effect.