arXiv subjects

Branislav K. Nikolic

Publications and source records attributed to Branislav K. Nikolic.

At least 55 records · Page 3Linked to original sources

arXiv2018

Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation self-consistently coupled to electronic time-dependent nonequilibrium Green functions

The conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dynamics of local magnetic moments, viewed as classical vectors of fixed length, with their change assumed to take place simultaneously with the cause. Here we demonstrate that recently developed [M. D. Petrović {\em et al.}, {\tt arXiv:1802.05682}] self-consistent coupling of the LLG equation to time-dependent quantum-mechanical description of electrons microscopically generates time-retarded damping in the LLG equation described by a memory kernel which is also spatially dependent. For sufficiently slow dynamics of local magnetic moments, the memory kernel can be expanded to extract the Gilbert damping (proportional to first time derivative of magnetization) and magnetic inertia (proportional to second time derivative of magnetization) terms whose parameters, however, are time-dependent in contrast to time-independent parameters used in the conventional LLG equation. We use examples of single or multiple magnetic moments precessing in an external magnetic field, as well as field-driven motion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed from conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster DW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stemming from finite amount of time which it takes for conduction electron spin to react to the motion of classical local magnetic moments, are missing from conventional classical micromagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed numerically exact and, therefore, nonperturbative result for charge current pumped by a moving DW and the same quantity computed by perturbative spin motive force formula combined with the conventional LLG equation.

cond-mat.mes-hall↗

arXiv2018

Quantum spin-transfer torque induced nonclassical magnetization dynamics and electron-magnetization entanglement

The standard spin-transfer torque (STT)---where spin-polarized current drives dynamics of magnetization viewed as a classical vector---requires noncollinearity between electron spins carried by the current and magnetization of a ferromagnetic layer. However, recent experiments [A. Zholud et al., Phys. Rev. Lett. 119, 257201 (2017)] observing magnetization dynamics in spin valves at cryogenic temperatures, even when electron spin is collinear to magnetization, point at overlooked quantum effects in STT which can lead to highly nonclassical magnetization states. Using fully quantum many-body treatment, where an electron injected as spin-polarized wave packet interacts with local spins comprising the anisotropic quantum Heisenberg ferromagnetic chain, we define quantum STT as any time evolution of local spins due to initial many-body state not being an eigenstate of electron+local-spins system. For time evolution caused by injected spin-down electron scattering off local up-spins, entanglement between electron subsystem and local spins subsystem takes place leading to decoherence and, therefore, shrinking of the total magnetization but without rotation from its initial orientation which explains the experiments. Furthermore, the same processes---entanglement and thereby induced decoherence---are present also in standard noncollinear geometry, together with the usual magnetization rotation. This is because STT in quantum many-body picture is caused only by electron spin-down factor state, and the only difference between collinear and noncollinear geometries is in relative size of the contribution of initial many-body state containing such factor state to superpositions of separable many-body quantum states generated during time evolution.

cond-mat.str-el↗

arXiv2018

Spin and charge pumping by steady or pulse current-driven magnetic domain wall: A self-consistent multiscale time-dependent-quantum/time-dependent-classical approach

We introduce a multiscale framework which combines time-dependent nonequilibrium Green function (TD-NEGF) algorithms, scaling linearly in the number of time steps and describing quantum-mechanically conduction electrons in the presence of time-dependent fields of arbitrary strength or frequency, with classical time evolution of localized magnetic moments described by the Landau-Lifshitz-Gilbert (LLG) equation. The TD-NEGF+LLG framework can be applied to a variety of problems where current-driven spin torque induces dynamics of magnetic moments as the key resource for next generation spintronics. Using magnetic domain wall (DW) as an example, we predict that its motion will pump time-dependent spin and charge currents (on the top of unpolarized DC charge current injected through normal metal leads to drive the DW motion). The conversion of AC components of spin current, whose amplitude increases (decreases) as the DW approaches (distances from) the normal metal lead, into AC voltage via the inverse spin Hall effect offers a tool to precisely track the DW position along magnetic nanowire. We also quantify the DW transient inertial displacement due to its acceleration and deceleration by pulse current and the entailed spin and charge pumping. Finally, TD-NEGF+LLG as a nonperturbative (i.e., numerically exact) framework allows us to establish the limits of validity of the so-called spin-motive force (SMF) theory for pumped charge current by time-dependent magnetic textures---the perturbative analytical formula of SMF theory becomes inapplicable for large frequencies (but unrealistic in magnetic system) and, more importantly, for increasing noncollinearity when the angles between neighboring magnetic moments exceed $\simeq 10^\circ$.

cond-mat.mes-hall↗

arXiv2018

First-principles quantum transport modeling of spin-transfer and spin-orbit torques in magnetic multilayers

We review a unified approach for computing: (i) spin-transfer torque in magnetic trilayers like spin-valves and magnetic tunnel junction, where injected charge current flows perpendicularly to interfaces; and (ii) spin-orbit torque in magnetic bilayers of the type ferromagnet/spin-orbit-coupled-material, where injected charge current flows parallel to the interface. Our approach requires to construct the torque operator for a given Hamiltonian of the device and the steady-state nonequilibrium density matrix, where the latter is expressed in terms of the nonequilibrium Green's functions and split into three contributions. Tracing these contributions with the torque operator automatically yields field-like and damping-like components of spin-transfer torque or spin-orbit torque vector, which is particularly advantageous for spin-orbit torque where the direction of these components depends on the unknown-in-advance orientation of the current-driven nonequilibrium spin density in the presence of spin-orbit coupling. We provide illustrative examples by computing spin-transfer torque in a one-dimensional toy model of a magnetic tunnel junction and realistic Co/Cu/Co spin-valve, both of which are described by first-principles Hamiltonians obtained from noncollinear density functional theory calculations; as well as spin-orbit torque in a ferromagnetic layer described by a tight-binding Hamiltonian which includes spin-orbit proximity effect within ferromagnetic monolayers assumed to be generated by the adjacent monolayer transition metal dichalcogenide.

cond-mat.mes-hall↗

arXiv2018

Fast Summation of Divergent Series and Resurgent Transseries in Quantum Field Theories from Meijer-G Approximants

We demonstrate that a Meijer-G-function-based resummation approach can be successfully applied to approximate the Borel sum of divergent series, and thus to approximate the Borel-Écalle summation of resurgent transseries in quantum field theory (QFT). The proposed method is shown to vastly outperform the conventional Borel-Padé and Borel-Padé-Écalle summation methods. The resulting Meijer-G approximants are easily parameterized by means of a hypergeometric ansatz and can be thought of as a generalization to arbitrary order of the Borel-Hypergeometric method [Mera {\it et al.} Phys. Rev. Lett. {\bf 115}, 143001 (2015)]. Here we illustrate the ability of this technique in various examples from QFT, traditionally employed as benchmark models for resummation, such as: 0-dimensional $ϕ^4$ theory, $ϕ^4$ with degenerate minima, self-interacting QFT in 0-dimensions, and the computation of one- and two-instanton contributions in the quantum-mechanical double-well problem.

hep-th↗

arXiv2017

Spin-memory loss due to spin-orbit coupling at ferromagnet/heavy-metal interfaces: Ab initio spin-density matrix approach

Spin-memory loss (SML) of electrons traversing ferromagnetic-metal/heavy-metal (FM/HM), FM/normal-metal (FM/NM) and HM/NM interfaces is a fundamental phenomenon that must be invoked to explain consistently large number of spintronic experiments. However, its strength extracted by fitting experimental data to phenomenological semiclassical theory, which replaces each interface by a fictitious bulk diffusive layer, is poorly understood from a microscopic quantum framework and/or materials properties. Here we describe an ensemble of flowing spin quantum states using spin-density matrix, so that SML is measured like any decoherence process by the decay of its off-diagonal elements or, equivalently, by the reduction of the magnitude of polarization vector. By combining this framework with density functional theory (DFT), we examine how all three components of the polarization vector change at Co/Ta, Co/Pt, Co/Cu, Pt/Cu and Pt/Au interfaces embedded within Cu/FM/HM/Cu vertical heterostructures. In addition, we use ab initio Green's functions to compute spectral functions and spin textures over FM, HM and NM monolayers around these interfaces which quantify interfacial spin-orbit coupling and explain the microscopic origin of SML in long-standing puzzles, such as why it is nonzero at Co/Cu interface; why it is very large at Pt/Cu interface; and why it occurs even in the absence of disorder, intermixing and magnons at the interface.

cond-mat.mes-hall↗

arXiv2017

Spin Hall Effect and Origins of Nonlocal Resistance in Adatom-Decorated Graphene

Recent experiments reporting unexpectedly large spin Hall effect (SHE) in graphene decorated with adatoms have raised a fierce controversy. We apply numerically exact Kubo and Landauer- Buttiker formulas to realistic models of gold-decorated disordered graphene (including adatom clustering) to obtain the spin Hall conductivity and spin Hall angle, as well as the nonlocal resistance as a quantity accessible to experiments. Large spin Hall angles of 0.1 are obtained at zero-temperature, but their dependence on adatom clustering differs from the predictions of semiclassical transport theories. Furthermore, we find multiple background contributions to the nonlocal resistance, some of which are unrelated to SHE or any other spin-dependent origin, as well as a strong suppression of SHE at room temperature. This motivates us to design a multiterminal graphene geometry which suppresses these background contributions and could, therefore, quantify the upper limit for spin current generation in two-dimensional materials.

cond-mat.mes-hall↗

arXiv2016

Correlation-driven topological phase transition from in-plane magnetized quantum anomalous Hall to Mott insulating phase in monolayer transition metal trichlorides

Based on density functional theory (DFT) calculations, we predict that a monolayer of OsCl$_3$---a layered material whose interlayer coupling is weaker than in graphite---possesses a quantum anomalous Hall (QAH) insulating phase generated by the combination of honeycomb lattice of osmium atoms, their strong spin-orbit coupling (SOC) and ferromagnetic ground state with {\em in-plane} easy-axis. The band gap opened by SOC is \mbox{$E_g \simeq 67$ meV} (or \mbox{$\simeq 191$ meV} if the easy-axis can be tilted out of the plane by an external electric field), and the estimated Curie temperature of such {\em anisotropic planar rotator} ferromagnet is $T_\mathrm{C} \lesssim 350$ K. The Chern number $\mathcal{C}=-1$, generated by the manifold of Os $t_{2g}$ bands crossing the Fermi energy, signifies the presence of a single chiral edge state in nanoribbons of finite width, where we further show that edge states are spatially narrower for zigzag than armchair edges and investigate edge-state transport in the presence of vacancies at Os sites. Since $5d$ electrons of Os exhibit {\em both} strong SOC and moderate correlation effects, we employ DFT+U calculations to show how increasing on-site Coulomb repulsion $U$: gradually reduces $E_g$ while maintaining $\mathcal{C} = -1$ for $0 < U < U_c$; leads to metallic phase with $E_g = 0$ at $U_c$; and opens the gap of topologically trivial Mott insulating phase with $\mathcal{C}=0$ for $U > U_c$.

cond-mat.str-el↗

arXiv2015

Hypergeometric resummation of self-consistent sunset diagrams for electron-boson quantum many-body systems out of equilibrium

A newly developed hypergeometric resummation technique [H. Mera et al., Phys. Rev. Lett. 115, 143001 (2015)] provides an easy-to-use recipe to obtain conserving approximations within the self-consistent nonequilibrium many-body perturbation theory. We demonstrate the usefulness of this technique by calculating the phonon-limited electronic current in a model of a single-molecule junction within the self-consistent Born approximation for the electron-phonon interacting system, where the perturbation expansion for the nonequilibrium Green function in powers of the free bosonic propagator typically consists of a series of non-crossing \sunset" diagrams. Hypergeometric resummation preserves conservation laws and it is shown to provide substantial convergence acceleration relative to more standard approaches to self-consistency. This result strongly suggests that the convergence of the self-consistent \sunset" series is limited by a branch-cut singularity, which is accurately described by Gauss hypergeometric functions. Our results showcase an alternative approach to conservation laws and self-consistency where expectation values obtained from conserving perturbation expansions are \summed" to their self-consistent value by analytic continuation functions able to mimic the convergence-limiting singularity structure.

cond-mat.mes-hall↗

arXiv2014

Nonperturbative Quantum Physics from Low-Order Perturbation Theory

The Stark effect in hydrogen and the cubic anharmonic oscillator furnish examples of quantum systems where the perturbation results in a certain ionization probability by tunneling processes. Accordingly, the perturbed ground-state energy is shifted and broadened, thus acquiring an imaginary part which is considered to be a paradigm of nonperturbative behavior. Here we demonstrate how the low order coefficients of a divergent perturbation series can be used to obtain excellent approximations to both real and imaginary parts of the perturbed ground state eigenenergy. The key is to use analytic continuation functions with a built in analytic structure within the complex plane of the coupling constant, which is tailored by means of Bender-Wu dispersion relations. In the examples discussed the analytic continuation functions are Gauss hypergeometric functions, which take as input fourth order perturbation theory and return excellent approximations to the complex perturbed eigenvalue. These functions are Borel-consistent and dramatically outperform widely used Padé and Borel-Padé approaches, even for rather large values of the coupling constant.

cond-mat.stat-mech↗

arXiv2014

First-principles vs. semi-empirical modeling of global and local electronic transport properties of graphene nanopore-based sensors for DNA sequencing

Using first-principles quantum transport simulations, based on the nonequilibrium Green function formalism combined with density functional theory (NEGF+DFT), we examine changes in the total and local electronic currents within the plane of graphene nanoribbon with zigzag edges (ZGNR) hosting a nanopore which are induced by inserting a DNA nucleobase into the pore. We find a sizable change of the zero-bias conductance of two-terminal ZGNR + nanopore device after the nucleobase is placed into the most probable position (according to molecular dynamics trajectories) inside the nanopore of a small diameter \mbox{$D=1.2$ nm}. Although such effect decreases as the nanopore size is increased to \mbox{$D=1.7$ nm}, the contrast between currents in ZGNR + nanopore and ZGNR + nanopore + nucleobase systems can be enhanced by applying a small bias voltage $V_b \lesssim 0.1$ V. This is explained microscopically as being due to DNA nucleobase-induced modification of spatial profile of local current density around the edges of ZGNR. We repeat the same analysis using NEGF combined with self-consistent charge density functional tight-binding (NEGF+SCC-DFTB) or self-consistent extended Hückel (NEGF+SC-EH) semi-empirical methodologies. The large discrepancy we find between the results obtained from NEGF+DFT vs. those obtained from NEGF+SCC-DFTB or NEGF+SC-EH approaches could be of great importance when selecting proper computational algorithms for {\em in silico} design of optimal nanoelectronic sensors for rapid DNA sequencing.

cond-mat.mes-hall↗

arXiv2014

Spin-to-charge conversion in lateral and vertical topological-insulator/ferromagnet heterostructures with microwave-driven precessing magnetization

Using the charge-conserving Floquet-Green function approach to open quantum systems driven by external time periodic potential, we analyze how spin current pumped (in the absence of any dc bias voltage) by the precessing magnetization of a ferromagnetic (F) layer is injected {\em laterally} into the interface with strong spin-orbit coupling (SOC) and converted into charge current flowing in the same direction. In the case of metallic interface with the Rashba SOC used in experiments [Nature Comm. {\bf 4}, 2944 (2013)], both spin $I^{S_α}$ and charge $I$ current flow within it where $I/I^{S_α} \simeq$ 2--8\% (depending on the precession cone angle), while for F/topological-insulator (F/TI) interface employed in related experiments (arXiv:1312.7091) the conversion efficiency is greatly enhanced $I/I^{S_α} \simeq$ 40--60\% due to perfect spin-momentum locking on the surface of TI. The spin-to-charge conversion occurs also when spin current is pumped {\em vertically} through the F/TI interface with smaller efficiency $I/I^{S_α} \sim 0.001\%$, but with charge current signal being sensitive to whether the Dirac fermions at the interface are massive or massless.

cond-mat.mes-hall↗

arXiv2014

Spin-Seebeck effect on the surface of topological insulator due to nonequilibrium spin-polarization parallel to the direction of thermally driven electronic transport

We study the transverse spin-Seebeck effect (SSE) on the surface of a three-dimensional topological insulator (TI) thin film, such as Bi$_2$Se$_3$, which is sandwiched between two normal metal leads. The temperature bias $ΔT$ applied between the leads generates surface charge current which becomes spin-polarized due to strong spin-orbit coupling on the TI surface, with polarization vector acquiring a component $P_x \simeq 60%$ {\em parallel to the direction of transport}. When the third nonmagnetic voltage probe is attached to the portion of the TI surface across its width $L_y$, pure spin current will be injected into the probe where the inverse spin Hall effect (ISHE) converts it into a voltage signal \mbox{$|V_\mathrm{ISHE}|^\mathrm{max}/ΔT \simeq 2.5$ $μ$V/K} (assuming the SH angle of Pt voltage probe and $L_y=1$ mm). The existence of predicted nonequilibrium spin-polarization parallel to the direction of electronic transport and the corresponding electron-driven SSE crucially relies on orienting quintuple layers (QLs) of Bi$_2$Se$_3$ {\em orthogonal} to the TI surface and {\em tilted} by $45^\circ$ with respect to the direction of transport. Our analysis is based on the Landauer-Büttiker-type formula for spin currents in the leads of a multi-terminal quantum-coherent junction, which is constructed using nonequilibrium Green function formalism within which we show how to take into account arbitrary orientation of QLs via the self-energy describing coupling between semi-infinite normal metal leads and TI.

cond-mat.mes-hall↗

arXiv2014

Signatures of electron-magnon interaction in charge and spin currents in magnetic tunnel junctions: A nonequilibrium many-body perturbation theory approach

We develop a numerically exact scheme for resumming certain classes of Feynman diagrams in the self-consistent perturbation expansion for the electron and magnon self-energies in the nonequilibrium Green function formalism applied to a coupled electron-magnon (\mbox{e-m}) system which is driven out of equilibrium by the applied finite bias voltage. Our scheme operates with the electronic and magnonic GFs and the corresponding self-energies viewed as matrices in the Keldysh space, rather than conventionally extracting their retarded and lesser components. This is employed to understand the effect of inelastic \mbox{e-m} scattering on charge and spin current vs. bias voltage $V_b$ in F/I/F magnetic tunnel junctions (MTJs), which are modeled on a one-dimensional (1D) tight-binding lattice for the electronic subsystem and 1D Heisenberg model for the magnonic subsystem. For this purpose, we evaluate Fock diagram for the electronic self-energy and the electron-hole polarization bubble diagram for the magnonic self-energy. The respective electronic and magnonic GF lines within these diagrams are the fully interacting ones, thereby requiring to solve the ensuing coupled system of nonlinear integral equations self-consistently. Despite using the 1D model and treating \mbox{e-m} interaction in many-body fashion only within a small active region consisting of few lattice sites around the F/I interface, our analysis captures essential features of the so-called zero-bias anomaly observed in both MgO- and AlO$_x$-based realistic 3D MTJs where the second derivative $d^2 I/dV_b^2$ (i.e., inelastic electron tunneling spectrum) of charge current exhibits sharp peaks of opposite sign on either side of the zero bias voltage.

cond-mat.mes-hall↗

arXiv2014

Giant thermoelectric effect in graphene-based topological insulators with nanopores

Designing thermoelectric materials with high figure of merit $ZT=S^2 G T/κ$ requires fulfilling three often irreconcilable conditions, i.e., the high electrical conductance $G$, small thermal conductance $κ$ and high Seebeck coefficient $S$. Nanostructuring is one of the promising ways to achieve this goal as it can substantially suppress lattice contribution to $κ$. However, it may also unfavorably influence the electronic transport in an uncontrollable way. Here we theoretically demonstrate that this issue can be ideally solved by fabricating graphene nanoribbons with heavy adatoms and nanopores. These systems, acting as a two-dimensional topological insulator with robust helical edge states carrying electrical current, yield a highly optimized power factor $S^2G$ per helical conducting channel. Concurrently, their array of nanopores impedes the lattice thermal conduction through the bulk. Using quantum transport simulations coupled with first-principles electronic and phononic band structure calculations, the thermoelectric figure of merit is found to reach its maximum $ZT \simeq 3$ at $T \simeq 40$ K. This paves a way to design high-$ZT$ materials by exploiting the nontrivial topology of electronic states through nanostructuring.

cond-mat.mes-hall↗

arXiv2013

Negative differential resistance in graphene-nanoribbon/carbon-nanotube crossbars: A first-principles multiterminal quantum transport study

We simulate quantum transport between a graphene nanoribbon (GNR) and a single-walled carbon nanotube (CNT) where electrons traverse vacuum gap between them. The GNR covers CNT over a nanoscale region while their relative rotation is 90 degrees, thereby forming a four-terminal crossbar where the bias voltage is applied between CNT and GNR terminals. The CNT and GNR are chosen as either semiconducting (s) or metallic (m) based on whether their two-terminal conductance exhibits a gap as a function of the Fermi energy or not, respectively. We find nonlinear current-voltage (I-V) characteristics in all three investigated devices---mGNR-sCNT, sGNR-sCNT and mGNR-mCNT crossbars---which are asymmetric with respect to changing the bias voltage from positive to negative. Furthermore, the I-V characteristics of mGNR-sCNT crossbar exhibits negative differential resistance (NDR) with low onset voltage $V_\mathrm{NDR} \simeq 0.25$ V and peak-to-valley current ratio $\simeq 2.0$. The overlap region of the crossbars contains only $\simeq 460$ carbon and hydrogen atoms which paves the way for nanoelectronic devices ultrascaled well below the smallest horizontal length scale envisioned by the international technology roadmap for semiconductors. Our analysis is based on the nonequilibrium Green function formalism combined with density functional theory (NEGF-DFT), where we also provide an overview of recent extensions of NEGF-DFT framework (originally developed for two-terminal devices) to multiterminal devices.

cond-mat.mes-hall↗

arXiv2013

How to construct the proper gauge-invariant density matrix in steady-state nonequilibrium: Applications to spin-transfer and spin-orbit torques

Experiments observing spin density and spin currents (responsible for, e.g., spin-transfer torque) in spintronic devices measure only the nonequilibrium contributions to these quantities, typically driven by injecting unpolarized charge current or by applying external time-dependent fields. On the other hand, theoretical approaches to calculate them operate with both the nonequilibrium (carried by electrons around the Fermi surface) and the equilibrium (carried by the Fermi sea electrons) contributions. Thus, an unambiguous procedure should remove the equilibrium contributions, thereby rendering the nonequilibrium ones which are measurable and satisfy the gauge-invariant condition according to which expectation values of physical quantities should not change when electric potential everywhere is shifted by a constant amount. Using the framework of nonequilibrium Green functions, we delineate such procedure which yields the proper gauge-invariant nonequilibrium density matrix in the linear-response and elastic transport regime for current-carrying steady state of an open quantum system connected to two macroscopic reservoirs. Its usage is illustrated by computing: (i) conventional spin-transfer torque (STT) in asymmetric F/I/F magnetic tunnel junctions (MTJs); (ii) unconventional STT in asymmetric N/I/F semi-MTJs with the strong Rashba spin-orbit coupling (SOC) at the I/F interface and injected current perpendicular to that plane; and (iii) current-driven spin density within a clean ferromagnetic Rashba spin-split two-dimensional electron gas (2DEG) which generates SO torque in laterally patterned N/F/I heterostructures when such 2DEG is located at the N/F interface and injected charge current flows parallel to the plane.

cond-mat.mes-hall↗

arXiv2012

Magnetoresistance and negative differential resistance in Ni/Graphene/Ni vertical heterostructures driven by finite bias voltage: A first-principles study

Using the nonequilibrium Green function formalism combined with density functional theory, we study finite-bias quantum transport in Ni/Gr_n/Ni vertical heterostructures where $n$ graphene layers are sandwiched between two semi-infinite Ni(111) electrodes. We find that recently predicted "pessimistic" magnetoresistance of 100% for $n \ge 5$ junctions at zero bias voltage $V_b \rightarrow 0$, persists up to $V_b \simeq 0.4$ V, which makes such devices promising for spin-torque-based device applications. In addition, for parallel orientations of the Ni magnetizations, the $n=5$ junction exhibits a pronounced negative differential resistance as the bias voltage is increased from $V_b=0$ V to $V_b \simeq 0.5$ V. We confirm that both of these nonequilibrium effects hold for different types of bonding of Gr on the Ni(111) surface while maintaining Bernal stacking between individual Gr layers.

cond-mat.mes-hall↗