Kubo-Greenwood Equation Research Articles

An investigation of high entropy alloy conductivity using first-principles calculations

The Kubo–Greenwood equation, in combination with the first-principles Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) can be used to calculate the DC residual resistivity of random alloys at T = 0 K. We implemented this method in a multiple scattering theory based ab initio package, MuST, and applied it to the ab initio study of the residual resistivity of the high entropy alloy AlxCoCrFeNi as a function of x. The calculated resistivities are compared with experimental data. We also predict the residual resistivity of refractory high entropy alloy MoNbTaVxW. The calculated resistivity trends are also explained using theoretical arguments.

Electronic structure and optical properties of quantum crystals from first principles calculations in the Born-Oppenheimer approximation.

We develop a formalism to accurately account for the renormalization of the electronic structure due to quantum and thermal nuclear motions within the Born-Oppenheimer approximation. We focus on the fundamental energy gap obtained from electronic addition and removal energies from quantum Monte Carlo calculations in either the canonical or grand-canonical ensembles. The formalism applies as well to effective single electron theories such as those based on density functional theory. We show that the electronic (Bloch) crystal momentum can be restored by marginalizing the total electron-ion wave function with respect to the nuclear equilibrium distribution, and we describe an explicit procedure to establish the band structure of electronic excitations for quantum crystals within the Born-Oppenheimer approximation. Based on the Kubo-Greenwood equation, we discuss the effects of nuclear motion on optical conductivity. Our methodology applies to the low temperature regime where nuclear motion is quantized and, in general, differs from the semi-classical approximation. We apply our method to study the electronic structure of C2/c-24 crystalline hydrogen at 200 K and 250 GPa and discuss the optical absorption profile of hydrogen crystals at 200 K and carbon diamond at 297 K.

On the delta function broadening in the Kubo–Greenwood equation

Understanding DC electrical conductivity is crucial for the study of materials. Macroscopic DC conductivity can be calculated from first principles using the Kubo–Greenwood equation. The procedure involves finding the thermodynamic limit of the current response to an electric field that is slowly switched on, and then taking the limit of the switching rate to zero. We introduce a nonlinear extrapolation procedure executed in systems with periodic boundary conditions, which predicts conductivity close to the thermodynamic limit even for very small systems. The scheme also overcomes a large part of the usual ambiguities of the DC conductivity definition for finite systems. We numerically compare our method to the Landauer equation-based approach and find both techniques to be consistent with each other.

Thermally driven Fermi glass states in warm dense matter: Effects on terahertz and direct-current conductivities

There is a growing interest in the electrical conductivity of warm dense matter from terahertz-frequency alternating current to direct current. Herein, using first-principles molecular dynamics simulations, we show that ionic thermal motion in warm dense matter drives thermal fluctuations in the electronic valence band that produce localized states in Lifshitz tails on the top and bottom of the bands. We predict Fermi glass states when these localized states extend and fill the gap between valence and conduction bands. This significantly affects the ultralow-frequency and direct current conductivity because of the very small but nonzero energy gaps between these localized states. An order parameter is proposed to describe the degree of glassiness of an electron energy band using the local density-of-state distribution. To take into account thermal hopping, we introduce electron energy-level broadening as a thermal correction term in the Kubo–Greenwood equation. The calculated terahertz conductivities of warm dense helium and argon show the differences between the Fermi glass and normal metal states.

Spin projection and spin current density within relativistic electronic-transport calculations

A spin projection scheme is presented which allows the decomposition of the electric conductivity into two different spin channels within fully relativistic $ab$ $initio$ transport calculations that account for the impact of spin-orbit coupling. This is demonstrated by calculations of the spin-resolved conductivity of Fe$_{1-x}$Cr$_x$ and Co$_{1-x}$Pt$_x$ disordered alloys on the basis of the corresponding Kubo-Greenwood equation implemented using the Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA) band structure method. In addition, results for the residual resistivity of diluted Ni-based alloys are presented that are compared to theoretical and experimental ones that rely on Mott's two-current model for spin-polarized systems. The application of the scheme to deal with the spin-orbit induced spin Hall effect is discussed in addition.

Ultrafast switching of a spin valve with perpendicular anisotropy

The switching properties of a spin valve with perpendicular anisotropy is investigated in terms of the fully relativistic screened Korringa-Kohn-Rostoker method, the corresponding Kubo-Greenwood equation of electric transport, and the Landau-Lifshitz-Gilbert equation. In excellent agreement with very recent experimental data, it is found that (a) the inverse of the switching time is linear with respect to the critical current, (b) the ratio of critical currents for switching from parallel (P) to antiparallel (AP) and from AP to P is about 2, and (c) the tiny magnetoresistance found in experiment is a consequence of the perpendicular anisotropy.

Domain walls and domain wall resistivities in Co x Pd1−x (111) and Co x Pt1−x (111)

In using the relativistic screened Korringa–Kohn–Rostoker method and a multi-scale approach based on a parametrization of the Ginzburg–Landau expansion of the free energy as a functional of the magnetization density, domain wall widths in Co x Pd1−x and Co x Pt1−x are determined by considering these substitutionally disordered alloys as two-dimensionally invariant systems. It is shown that in Co x Pd1−x , domains are formed nearly over the whole concentration range, while in Co x Pt1−x , domain wall formation only occurs near 50% Co and for Co-rich alloys. Based on these results resistivities in the presence and absence of domains walls are evaluated in terms of the relativistic Kubo–Greenwood equation. In contrast to Co x Ni1−x , it turns out that both systems are unsuitable as race track materials.

Electric properties of generalized domain walls and spin spirals in permalloy

In using the fully relativistic versions of the screened Korringa-Kohn-Rostoker method and of the Kubo-Greenwood equation for permalloy, ${\text{Ni}}_{85}{\text{Fe}}_{15}$, the equilibrium width of generalized domain walls and the corresponding resistivities are calculated for the case that the orientations of the magnetization in the two domains separated by a wall form a relative angle, which is assumed to vary continuously between $0\ifmmode^\circ\else\textdegree\fi{}$ and $360\ifmmode^\circ\else\textdegree\fi{}$. It is found, e.g., that the width of a $90\ifmmode^\circ\else\textdegree\fi{}$ domain wall is shorter by 100 nm than the one for a $180\ifmmode^\circ\else\textdegree\fi{}$ domain wall yielding a reduction in the anisotropic magnetoresistance ratio (AMR), which as compared to the bulk value (no domain wall present) is even larger than that for a $180\ifmmode^\circ\else\textdegree\fi{}$ domain wall. In particular it is demonstrated that the reduction in the AMR of generalized domain walls can safely be correlated with the anisotropy contribution to their formation energies.

Anisotropy effects in switching the direction of the magnetization in a spin-valve system with exchange-bias pinning

The electronic, magnetic, and electric transport properties of spin valves of the type Cu111 /C oO /C o/C u/C o/C u111 are investigated in terms of the local density functional theory by means of the fully relativistic screened Korringa-Kohn-Rostoker and the Kubo-Greenwood equation. This investigation concentrates on anisotropy effects for twisting energies and electric transport perpendicular to the planes of atoms. It is found that by rotating uniformly the orientation of the magnetization in the free layer with respect to the one in the pinned layer, for most of the spacer thicknesses considered 14–48 monolayers the twisting energy shows strong deviations from a 1 � cos behavior, while the corresponding sheet resistances and the magnetoresistance do follow this simple law. Furthermore, it is shown that interdiffusion in the two atomic layers forming the Co/ Cu interfaces is only of maginal importance for anisotropy effects in the twisting energies; however, it reduces—as to be expected—the magnetoresistance.

Electric transport perpendicular to the planes

AbstractSince the discovery of the giant magnetoresistance (GMR) in magnetic multilayers, several theoretical descriptions have been used to determine the resistivity of such layered structures. The resistance for the current in direction of the planes of layers can easily be measured, and has been intensively studied theoretically. However, the investigation of the GMR for the current perpendicular to the planes (CPP) is slightly more difficult. Here, a microscopic formalism for the study of the CPP GMR is reported by making use of the Kubo–Greenwood equation. Within this method perturbations of the interfaces like interdiffusion, alloy formation, or impurities can easily be included, which is of importance, because the discussion of the GMR is always related to the structure of the interfaces. The presentation of the Kubo–Greenwood formalism for CPP transport is complemented by a brief discussion of some exemplary results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reorientation transition inFen∕Au(100)

Experimental investigations and theoretical magnetic anisotropy energy calculations show a reorientation transition of the magnetization in ${\mathrm{Fe}}_{n}∕\mathrm{Au}(100)$ from a normal-to-plane to an in-plane direction at about three monolayers of Fe. In the present paper the magneto-optical properties of this system are investigated theoretically by using the spin-polarized relativistic screened Korringa-Kohn-Rostoker method, the Kubo-Greenwood equation for finite photon frequencies, and a classical optical approach that takes into account all reflections and interferences. By varying the thickness of the Fe film, the reorientation of the ground-state magnetization is clearly traced as a strong decrease in the calculated Kerr rotation angles for oblique incidence of light. For all film thicknesses under consideration, it is found that by continuously varying the angle of the incident light the Kerr rotation angle reaches a maximum at an incidence of about 70\ifmmode^\circ\else\textdegree\fi{}. In the case of normal incidence a direct proportionality of the Kerr angles to the normal component of the magnetization is demonstrated by changing the orientation of the magnetization. When relating Kerr angles as calculated for a set of angles between the surface normal and the orientation of the magnetization to the corresponding magnetic anisotropy energy a very compact description of the occurring reorientation transition can be given. Moreover, based on these data and using a simple phenomenological picture a qualitative description of the Kerr angles with respect to applied external fields is provided.

Ab initiostudy of the electric transport in gold nanocontacts containing single impurities

By employing a real-space formulation of the Kubo-Greenwood equation based on a Green's function embedding technique combined with the fully relativistic spin-polarized Korringa-Kohn-Rostoker method a detailed investigation of the electrical transport through atomic-scaled contacts between two Au(001) semi-infinite systems is presented. Following a careful numerical test of the method the conductance of Au nanocontacts with different geometries is calculated and, in turn, correlated with the thickness of the constriction. Particular emphasis is paid to the influence of transition metal impurities (Pd, Fe, and Co) placed on various positions near the center of a particular contact. We found that the conductance is very sensitive to the position of the magnetic impurities and that the mechanism for the occurring relative changes can mainly be attributed to the impurities' minority $d$-band inducing resonant line shapes in the $s$-like density of states at the center of the contact.

CPP transport in Fe/Cr/Fe trilayers with mn impurities: an ab initio study

The influence of Mn impurities on the electric transport properties of Fe/Cr/Fe sandwiches has been investigated as a function of the thickness of the Cr layer by employing the Kubo–Greenwood equation. The electronic and magnetic properties of the Fe/Cr sandwich-structures have been evaluated within the screened Korringa–Kohn–Rostoker method for layered systems. In this article, two types of impurities are studied: atomic layers of Mn placed on top of the Cr layers, and spacers consisting of disordered CrMn alloys. In agreement with experimental findings it is found that the size of the giant magnetoresistance (GMR) is not much affected by Mn impurities. However, Mn is known to transform the spin density wave (SDW) of Cr into a commensurate antiferromagnetic structure. These kinds of changes in the magnetic coupling are reflected in the GMR. The present results show that 5% of Mn in the spacer is sufficient to destroy the SDW.

Ab initiodescription of domain walls in Permalloy: Energy of formation and resistivities

To determine the formation energy and resistivity for domain walls in permalloy (fcc-${\mathrm{Ni}}_{85}{\mathrm{Fe}}_{15})$ we use the fully relativistic spin-polarized screened Korringa-Kohn-Rostoker method for layered systems and the corresponding Kubo-Greenwood equation in the context of the (inhomogeneous) coherent potential approximation. We find that the difference in formation energy between 90\ifmmode^\circ\else\textdegree\fi{} and 180\ifmmode^\circ\else\textdegree\fi{} domains becomes very small if the domain wall width increases. Furthermore, we show that regardless of the configuration within a domain wall the in-plane components of the resistivity are larger than in a single domain and, in particular, that the anisotropic magnetoresistance ratio within the domain wall vanishes.

Electronic transport in disordered n-alkanes: From fluid methane to amorphous polyethylene

We use a fast Fourier transform block Lanczos diagonalization algorithm to study the electronic states of excess electrons in fluid alkanes (methane, ethane, and propane) and in a molecular model of amorphous polyethylene (PE) relevant to studies of space charge in insulating polymers. We obtain a new pseudopotential for electron–PE interactions by fitting to the electronic properties of fluid alkanes and use this to obtain new results for electron transport in amorphous PE. From our simulations, while the electronic states in fluid methane are extended throughout the whole sample, in amorphous PE there is a transition between localized and delocalized states slightly above the vacuum level (∼+0.06 eV). The localized states in our amorphous PE model extend to −0.33 eV below this level. Using the Kubo–Greenwood equation we compute the zero-field electron mobility in pure amorphous PE to be μ≈2×10−3 cm2/V s. Our results highlight the importance of electron transport through extended states in amorphous regions to an understanding of electron transport in PE.

Ab initiostudies of electric transport in terms of the real space Kubo-Greenwood equation

We propose a method suitable to describe electrical transport properties of nanostructures. In this approach a Green's function embedding technique based on the fully relativistic spin-polarized Korringa-Kohn-Rostoker method and the coherent potential approximation is combined with a real-space formulation of the Kubo-Greenwood equation. We present calculations for the Ag(100) surface, Ag bulk, two types of CuPt bulk alloys in the ``large cluster'' limit as well as finite Fe and Co chains embedded into the surface layer of Ag(100) in order to illustrate the reliability and applicability of this approach.

Role of interfaces in the magnetoresistance of Au/Fe/Au/Fe multilayers

By using the fully relativistic spin-polarized screened Korringa-Kohn-Rostoker method and the corresponding Kubo-Greenwood equation resistivities and the giant magnetoresistance are evaluated for Fe/Au/Fe/Au spin valves assuming both bcc-like Fe and fcc-Au leads. The theoretically obtained values for a spacer thickness of 7 monolayers of Au are in a rather good agreement with the corresponding experimental data. These data, in particular the temperature dependence of the resistivities reported in here, are analyzed with respect to the experimental setup and related to the theoretical models assumed. Since in the theoretical calculations interdiffused interfaces are taken into account by means of the inhomogeneous coherent potential approximation, the role of the interfaces in the magnetoresistance of this system can be discussed rigorously.

Ab Initio Study of Electric Transport and Interlayer Exchange Coupling in Fe-Si-Fe Systems

We present a first principles study of the magnetoresistance (MR) perpendicular to the planes of atoms and the interlayer exchange coupling (IEC) in Fe-Si-Fe trilayers. In both cases the dependence on the number of spacer layers is investigated, whereby the spacer thickness ranges between 3 and 21 Å for the IEC and extends to 33 Å for the MR in order to obtain the asymptotic behavior. Additionally, the influence of alloy formation at the interfaces on the MR and the IEC is examined. The calculations of the electronic structure are performed within the fully relativistic spin-polarized screened Korringa-Kohn-Rostoker method and the transport properties are derived from the Kubo-Greenwood equation. Our results give evidence that inter-diffusion is one of the origins of the small MR, which is observed experimentally in Fe/Si/Fe trilayers. AFM coupling occurs for spacers thicker than 4 Å which is in accordance with the experimental findings. It seems that interdiffusion stabilizes AFM coupling in Fe-Si-Fe trilayers.

Ab initiocharacterization of the giant magnetoresistance in realistic spin valves

The electric transport properties of a rather complicated spin-valve system containing NiFe permalloy, CoFe hard magnets and two types of spacers (Cu,Ru) are investigated theoretically in terms of the relativistic spin-polarized versions of the Screened Korringa-Kohn-Rostoker method and the Kubo-Greenwood equation. It is found that the regimes of antiferromagnetic coupling relevant for the giant magnetoresistance (GMR) are mostly determined by the thickness of the Cu-spacer: the nodes of oscillation of the interlayer exchange coupling (IEC) with respect to the Cu-spacer thickness are shifted only marginally by the presence of a very thin Ru cap. Also found are oscillations of the IEC and GMR with respect to the thickness of the hard magnet parts of the system (CoFe) and with respect to the Ru-spacer thickness. Viewed with respect to the thickness of the Cu-spacer the minima in and the actual value of the giant magnetoresistance as a function of the thickness of the Ru-spacer are in rather good agreement with experiment.

Interlayer exchange coupling and perpendicular electric transport in Fe/Si/Fe trilayers

The interlayer exchange coupling and the perpendicular magnetoresistance of Fe/Si/Fe systems have been investigated within the fully relativistic screened Korringa-Kohn-Rostoker method and the Kubo-Greenwood equation considering interdiffusion effects, i.e., inhomogeneous Fe-Si alloy formation at the interfaces. It is shown that the experimentally observed strong antiferromagnetic interlayer exchange coupling is caused by the formation of Fe-Si alloys at the interface. Furthermore, our calculations give evidence that the small magnetoresistance, which has been observed experimentally in Fe/Si/Fe trilayers has a similar origin. The results presented here give no evidence for a direct connection between the magnetoresistance and interlayer exchange coupling in Fe/Si/Fe systems.