Correlated electronic structure with uncorrelated disorder

Correlated electronic structure with uncorrelated disorder

Abstract

We introduce a computational scheme for calculating the electronic structure of random alloys that includes electronic correlations within the framework of the combined density functional and dynamical mean-field theory. By making use of the particularly simple parameterization of the electron Green’s function within the linearized muffin-tin orbitals method, we show that it is possible to greatly simplify the embedding of the self-energy. This in turn facilitates the implementation of the coherent potential approximation, which is used to model the substitutional disorder. The computational technique is tested on the Cu-Pd binary alloy system, and for disordered Mn-Ni interchange in the half-metallic NiMnSb.

I Introduction

Disordered metallic alloys find applications in a large number of areas of materials science. Designing alloys with specific thermal, electrical, and mechanical properties such as conductivity, ductility, and strength, nowadays commonly starts at the microscopic level Vitos et al. (2002). First-principles calculations of the electronic structure offer a parameter-free framework to meet specific engineering demands for materials prediction. Advances in the atomistic simulation of physical properties should to a large extent be attributed to the development of density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) within the local density approximation (LDA) or beyond-LDA schemes.

In solids exhibiting disorder the calculation of any physical property involves configurational averaging over all realizations of the random variables characterizing the disorder. In the case of substitutional disorder the symmetry of the lattice is kept, but the type of atoms in the basis is randomly distributed. This causes the crystal to lose translational symmetry, making the Bloch theorem inapplicable. Perhaps the most successful approach to solve the problems associated with substitutional disorder is the coherent potential approximation Soven (1967) (CPA). The presence of random atomic substitution generates a fluctuating external potential which within the CPA is substituted by an effective medium. This effective medium is energy dependent and is determined self-consistently through the condition that the impurity scattering should vanish on average. The CPA is a single-site approximation, which becomes exact in certain limiting cases Velický et al. (1968). The explanation for the good accuracy of the CPA can be traced back to the fact that it becomes exact in the limit of infinite lattice coordination number,  Vlaming and Vollhardt (1992).

The CPA was initially applied to tight-binding Hamiltonians Velický et al. (1968), where for binary alloys the CPA equations take a complex polynomial form that can be solved directly. Later Győrffy Gyorffy (1972) formulated the CPA equations for the muffin-tin potentials within the multiple-scattering Korringa-Kohn-Rostocker (KKR) method Korringa (1947); Kohn and Rostoker (1954). Consequently, the configurational average could be performed over the scattering path operator, instead of the Green’s function, simplifying the implementation of the CPA for materials calculations Stocks et al. (1971). Later the CPA was also implemented within the linearized muffin-tin orbitals Andersen (1970) (LMTO) basis set Kudrnovský et al. (1987); Kudrnovský and Drchal (1990); Abrikosov et al. (1991); Abrikosov and Skriver (1993). With the advent of the third-generation exact muffin-tin orbitals Andersen et al. (1994); Vitos (2001, 2010) (EMTO) method, and the full-charge density Vitos et al. (1994) (FCD) technique, it was possible to go beyond the atomic-sphere approximation (ASA) with CPA calculations Vitos et al. (2001), and investigate the energetics of anisotropic lattice distortions. For interacting model Hamiltonians, dynamical mean-field theory Metzner and Vollhardt (1989); Georges et al. (1996); Kotliar and Vollhardt (2004) (DMFT) was also combined with the CPA Janis and Vollhardt (1992); Ulmke et al. (1995); Kakehashi (2002). Later on the methodology was extended to study realistic materials containing significant electronic correlations within the framework of a combined DFT+DMFT method Kotliar et al. (2006); Held (2007). To treat weak disorder within the framework of charge self-consistent LDA+DMFT, the CPA has been implemented within the KKR method Minár et al. (2005). In an alternative approach, the band structures computed from DFT were mapped to tight-binding model Hamiltonians were the disorder was treated within the CPA Wissgott et al. (2011); Korotin et al. (2014); Belozerov et al. (2015); Belozerov and Anisimov (2016).

In this paper, we introduce a method that can treat substitutional disorder effects through the CPA, and electronic correlation effects through DMFT, on an equal footing for real materials. The method is based on the LDA+DMFT method, MTO+DMFT, which was recently introduced by us Östlin et al. (2017). By making use of the particularly simple parameterized form of the electron Green’s function in a linearized basis set, we show that the self-energy can be incorporated naturally within the LMTO formalism as a modification of a single, self-consistently determined parameter. Due to this straightforward inclusion of the many-body effects, the CPA within the LMTO method retains its general form, and can be used almost unaltered for the LDA+DMFT approach.

The paper is organized as follows: Section II briefly reviews the muffin-tin basis sets used in this work. A short overview of the LMTO method is given in Sec. II.1, with additional formulas given in Appendix A, in order to introduce the most important quantities needed for the CPA+DMFT implementation. A review of the EMTO method, which is the second basis set used in this paper, is given in Appendix B. In Sec. III we present the most important development in this paper, a combination of the CPA and the LDA+DMFT method. First, in Sec. III.1 we discuss the principles behind the configuration average used for the CPA. Sec. III.2 shows how the electronic self-energy can be incorporated into the LMTO potential functions, while Sec. III.3 demonstrates how the potential functions are used as an effective medium within the CPA. The section ends with an outline of the full computational scheme. In Sec. IV we present the results of the method applied to the binary copper-palladium system, treating the -electrons of palladium as correlated. The half-metallic semi-Heusler compound NiMnSb with disorder is also investigated. Section V provides a conclusion of our paper.

Ii Electronic structure with muffin-tin orbitals

The standard approach adopted in first-principles electronic structure calculations is the mapping to an effective single-particle equation. The Hohenberg-Kohn-Sham density functional formalism Hohenberg and Kohn (1964); Kohn and Sham (1965) provides a self-consistent description for the effective one-particle potential . Within the muffin-tin orbital methods, the effective potential in the Kohn-Sham equations,

 [∇2−Veff(r)]Ψj(r)=ϵjΨj(r), (1)

is approximated by spherical potential wells centered on lattice sites R, and a constant interstitial potential , viz.,

 Veff(r)≈Vmt(rR)≡V0+∑R[VR(rR)−V0], (2)

where we introduced the notation . We will in the following omit the vector notation for simplicity. This form of the potential makes it possible to divide the Kohn-Sham equation (1) into radial Schrödinger-like equations within the muffin-tin spheres, and wave equations in the interstitial region, which can be solved separately. The computational scheme that we present in this paper is based on the muffin-tin theories developed by Andersen and coworkers, namely the LMTO Andersen (1970); Andersen and Jepsen (1984); Andersen et al. (1986a); Skriver (1984), and the EMTO Andersen et al. (1994); Andersen and Saha-Dasgupta (2000); Vitos (2001); Vitos et al. (2000); Vitos (2010), methods. In the following, we briefly review the main ideas and notations behind the LMTO-ASA method.

ii.1 Linear Muffin-Tin Orbitals

The energy-independent linearized muffin-tin orbitals , centered at the lattice site , are given as:

 χαRL(rR) = ϕRL(rR)+∑R′L′˙ϕαR′L′(rR)hαRL,R′L′, (3) ˙ϕαRL(rR) = ˙ϕRL(rR)+ϕRL(rR)oαRL, (4) ϕRL(rR) = ϕRL(rR)YL(^rR). (5)

is the solution of the radial Schrödinger equation at an arbitrary energy , usually taken to be the center of gravity of the occupied part of the band, and is an overlap integral. denotes the orbital and azimuthal quantum numbers, respectively. The superscript denotes the screening representation used in the tight-binding LMTO theory. The expansion coefficients are determined from the condition that the wave function is continuous and differentiable at the sphere boundary at each sphere: . Note that we are now assuming a translationally invariant lattice system, so that the Bloch wave vector k is well defined. From now on, will denote the atoms in the unit cell only. The coefficient is parametrized by the center of the band, , and the band width parameter , both expressed in terms of the potential function and its energy derivative evaluated at . The potential function and the structure constant are expressed using the conventional potential function and the conventional structure constant matrix  Andersen et al. (1986a):

 PαRL = [P0(1−αP0)−1]RL, (6) SαRLR′L′ = [S0(1−αS0)−1]RLR′L′. (7)

is proportional to the cotangent of the phase shift created by the potential centered at the sphere at . Thus, the potential parameters characterize the scattering properties of the atoms placed at the lattice sites. The geometry of the lattice enters through the structure constants , which is independent of the type of atoms occupying the sites. has a long range behavior in real space, but is decaying nearly exponentially in the tight-binding -representation Andersen and Jepsen (1984). The so-called band distortion parameter , which is also used to denote the screening representation, gives a relation between the -representation and the unscreened representation. In the nearly-orthogonal -representation, , and , hence the Hamiltonian is given by:

 HγRLR′L′(k)=CRl+√ΔRlSγRLR′L′(k)√ΔR′l′, (8)

where , and are the representation-independent band center-, width- and distortion potential parameters, respectively Andersen et al. (1986a); Skriver (1984). The corresponding Green’s function in the -representation is given by:

 GγRLR′L′(k,z)=[(z−Hγ(k))]−1RLR′L′, (9)

where is an arbitrary complex energy. Further important relations among the LMTO representations can be found in Appendix A.

Iii Electronic correlations and disorder: the single-site approximation

In this section we present an LMTO-CPA scheme, that allows to include local self-energies, on the level of DMFT, for the alloy components. The scheme is implemented within the Matsubara representation, and is combined with the MTO+DMFT method Östlin et al. (2017). Section III.1 briefly discusses the configurational averaging and the CPA, while in Sec. III.2 we show how DMFT through the Dyson equation leads to a renormalization of the parameters of the LMTO-ASA formalism. Section III.3 combines the ideas of the previous two sections, and proposes a combined CPA and DMFT loop.

iii.1 Configuration averaging and the CPA

In disordered systems the configurational degrees of freedom characterizing the composition are described by a random variable. Consequently, the potentials at sites are random in space as in quenched disordered solids. A particular realization of the random variable constitutes a configuration of the system in discussion. According to Anderson Anderson (1958) only physically measurable quantities such as diffusion probabilities, response functions, and densities of states should be configurationally averaged. As these quantities are themselves Green’s functions, electronic structure methods using Green’s functions are favored for the study of disordered systems.

A major development of the theory of disordered electronic systems was achieved using the CPA. The CPA belongs to the class of mean-field theories according to which the properties of the entire material are determined from the average behavior at a subsystem, usually taken to be a single site (cell) in the material. In the multiple scattering description of a disordered system, one considers the propagation of an electron through a disordered medium as a succession of elementary scatterings at the random atomic point scatterers. In the single-site approximation one considers only the independent scattering off different sites and finally takes the average over all configurations of the disordered system consisting of these scatterers. One may then consider any single site in a specific configuration and replace the surrounding material by a translationally invariant medium, constructed to reflect the ensemble average over all configurations. In the CPA this medium is chosen in a self-consistent way. One assumes that averages over the occupation of a site embedded in the effective medium yield quantities indistinguishable from those associated with a site of the medium itself.

In view of the great progress achieved through the previous implementations of the CPA within muffin-tin orbitals methods Stocks et al. (1971); Gyorffy (1972); Kudrnovský et al. (1987); Kudrnovský and Drchal (1990); Abrikosov et al. (1991); Abrikosov and Skriver (1993) we present here in detail a novel combination of CPA+DMFT in the recently developed MTO+DMFT method Östlin et al. (2017).

iii.2 Self-energy-modified effective potential parameters

To deal with the important question concerning the effect of interaction, we start by observing that within the DMFT the self-energy is local and primarily modifies the local parameters of the model that describes disorder. In this section, we show that the presence of a local self-energy, , modifies the potential function entering in the expression of the Green’s function in the LMTO-ASA formalism.

We start from the Dyson equation used to construct the LDA+DMFT Green’s function:

 [GDMFTRLR′L′(k,z)]−1=[GLDARLR′L′(k,z)]−1−ΣRLRL′(z), (10)

where denotes the LDA+DMFT/LDA-level Green’s function and the self-energy. It is useful to define an auxiliary Green’s function, the path operator , as

 gαRLR′L′(k,z)=[PαRl(z)−SαRLR′L′(k)]−1, (11)

which is valid for a general representation . In Appendix A we present explicit expressions for the potential functions and auxiliary Green’s functions in different representations, as well as their connection to the physical Green’s function. As is apparent in Eq. (11), the full energy- and k-dependence of the path operator is contained in the potential function and the structure constants, respectively. Furthermore, the potential function is fully local, i.e., it is diagonal in site index. In the following, it will prove convenient to first work in the -representation, since here the Green’s function takes a particularly simple form (see Eq. (24)):

 Gγ,LDARLR′L′(k,z)=Δ−1/2RlgγRLR′L′(k,z)Δ−1/2R′l′, (12)

i.e., it is the path operator normalized by the potential parameter . Using Eqs. (12) and (23), the LDA Green’s function can be evaluated as follows:

 [Gγ,LDARLR′L′(k,z)]−1 =Δ1/2Rl[Pγ,LDARl(z)−SγRLR′L′(k)]Δ1/2R′l′ =z−CRl−Δ1/2RlSγRLR′L′(k)Δ1/2R′l′. (13)

Hence, the LDA+DMFT Green’s function (10) can be written as

 [Gγ,DMFTRLR′L′(k,z)]−1=z−CRl−Δ1/2RlSγRLR′L′(k)Δ1/2R′l′−ΣRLRL′(z), (14)

i.e., as the resolvent of the Hamiltonian (8) with an embedded self-energy. From Eq. (14), it is obvious that the same result will follow if the potential parameter is replaced by an effective parameter, in which the self-energy is embedded, viz.,

 CDMFTRLRL′(z)≡CRl+ΣRLRL′(z). (15)

Hence the effective potential parameter will now in general be complex, energy-dependent, and have off-diagonal elements. However, is still local. With this effective potential parameter, the LDA+DMFT level Green’s function can be expressed in a similar form as the LDA-level Green’s function, viz.,

 [Gγ,DMFTRLR′L′(k,z)]−1=Δ1/2Rl[Pγ,DMFTRLRL′(z)−SγRLR′L′(k)]Δ1/2R′l′, (16)

where

 Pγ,DMFTRLRL′(z) ≡Pγ,LDARl(z)−Δ−1/2RlΣRLRL′(z)Δ−1/2R′l′ =Δ−1/2Rl(z−CRl−ΣRLRL′(z))Δ−1/2R′l′ =Δ−1/2Rl(z−CDMFTRLRL′(z))Δ−1/2R′l′. (17)

Note that due to the self-energy, the effective potential function now has off-diagonal elements.

iii.3 The combined CPA and DMFT loop

For disorder calculations using the CPA, it is more convenient to use the tight-binding -representation, since in this case only the potential functions (and not the structure constants) are random Gunnarsson et al. (1983); Kudrnovský and Drchal (1990); Abrikosov et al. (1991); Abrikosov and Skriver (1993). In this case, the representation-dependent potential parameter (Eq. (26)) will be modified accordingly:

 Vβ,DMFTRLRL′(z)≡CRl+ΣRLRL′(z)−ΔRlγRl−βl. (18)

For LDA+DMFT calculations, this form of should be used for the potential function (25), the path operator , and the Green’s function (28), in order for the Dyson equation (10) to be fulfilled. Note that the transformations in Eqs. (27) and (28) are now no longer simply scaled as in the LDA case, but are matrix multiplications due to the presence of off-diagonal terms in the self-energy.

In the following, all quantities will be on the dynamical mean-field level, and we suppress the common superscript “DMFT” for the coherent medium path operators , the alloy component path operators , the coherent potential functions , and the alloy component potential functions , which will be defined below. The representation-superscript is kept. An additional superscript appears to represent the iterative nature of the equations. The superscript refers to the index enumerating the alloy components at a certain site. We also introduce the concentration of the respective alloy components , at a site , as ( 1, ). The DMFT impurity problem is solved within the imaginary-axis Matsubara frequency representation, where the Matsubara frequencies are defined as , where , and is the temperature. In the following, we use the shorthand to represent the set of all Matsubara frequencies.

The CPA self-consistency condition requires that the sequential substitution by an impurity atom into an effective, translationally-invariant, coherent medium should produce no further electron scattering, on average. This can be realized by averaging the Green’s function, or, following Győrffy Gyorffy (1972), the path operator:

 ~gβRLRL′(iω)=∑iciRgi,βRLRL′(iω), (19)

where the coherent path operator in the -representation,

 ~gβRLRL′(iω)=∫[~Pβ(iω)−Sβ(k)]−1RLRL′dk, (20)

has been integrated over the Brillouin zone (BZ). In Eq. (20), the coherent potential function has been introduced. The alloy component path operators in Eq. (19) are found through a Dyson equation,

 gi,βRLRL′(iω)=~gβRLRL′(iω)+∑L′′L′′′~gβRLRL′′(iω)×[Pi,βRL′′RL′′′(iω)−~PβRL′′RL′′′(iω)]gi,βRL′′′RL′(iω). (21)

Here the potential functions are computed according to Eq. (25), for each type respectively. In order to close the CPA equations self-consistently, a new coherent potential function has to be determined at each iteration. This is done by taking the difference between the inverses of the coherent path operators from the present iteration and the previous iteration , as follows:

 ~Pβ,n+1(iω)=~Pβ,n(iω)−[~gβ,n+1(iω)]−1+[~gβ,n(iω)]−1. (22)

The new coherent potential function can be inserted into Eq. (20), and the cycle can be repeated until self-consistency has been reached. This is performed for each Matsubara frequency . Once self-consistency in the CPA equations has been achieved, the Green’s functions for each alloy component can be obtained by normalizing the alloy component path operators in Eq. (21), using the transformation in Eq. (28). These (local) Green’s functions are then used as input for the DMFT impurity problem, with a separate impurity problem for each alloy component.

The scheme presented above can easily be incorporated within the formalism of the MTO+DMFT method Östlin et al. (2017). Here, the EMTO method (see Appendix B for a brief review) is employed to solve the Kohn-Sham equations for random alloys self-consistently, within the CPA, on the level of the LDA. The DMFT impurity problem is then solved in the Matsubara representation, using linearization techniques to evaluate the alloy components Green’s functions, as presented above. This can be done both on the LDA-level (setting ) and on the DMFT-level (using the self-energy from the DMFT impurity problem). The charge self-consistency is achieved similarly as in Ref. [Östlin et al., 2017], by computing moments of the alloy component Green’s function at LDA and DMFT level. The difference between the charge densities computed in this way can then be added as a correction on the LDA-level charge computed within the EMTO method. In Figure 1, we present a schematic picture of the self-consistent loops.

Iv Results

In order to demonstrate the feasibility of our proposed method, we apply it to investigate the electronic structure of the binary CuPd alloy and the semi-Heusler compound NiMnSb, with partially exchanged Ni and Mn components.

iv.1 Computational details

In all calculations, the kink cancellation condition was set up for 16 energy points distributed around a semi-circular contour with a diameter of 1 Ry, enclosing the valence band. The BZ integrations were carried out on an equidistant k-point mesh in the fcc BZ. For the exchange-correlation potential the local spin density approximation with the Perdew-Wang parameterization Perdew and Wang (1992) was used. To solve the DMFT equations, we used the spin-polarized -matrix fluctuation-exchange (SPTFLEX) solver Bickers and Scalapino (1989); Lichtenstein and Katsnelson (1998); Katsnelson and Lichtenstein (1999); Pourovskii et al. (2005); Grånäs et al. (2012). In this solver, the electron-electron interaction term can be considered in a full spin and orbital rotationally invariant form. Since specific correlation effects are already included in the exchange-correlation functional, so-called “double counted” terms must be subtracted. To achieve this, we replace with Lichtenstein et al. (2001) in all equations of the DMFT procedure Kotliar et al. (2006). Physically, this is related to the fact that DMFT only adds dynamical correlations to the DFT result Petukhov et al. (2003). The Matsubara frequencies were truncated after 1024 frequencies, and the temperature was set to K. The values for the Coulomb and the exchange parameters are discussed in connection with the presentation of the results in each case. The densities of state were computed along a horizontal contour shifted away from the real energy axis. At the end of the self-consistent calculations, to obtain the self-energy on the horizontal contour, was analytically continued by a Padé approximant Vidberg and Serene (1977); Östlin et al. (2012). For the studied alloys a -basis was used. For the Cu-Pd system, the Cu and states, and for Pd the and states, were treated as valence. For the case of NiMnSb, the Ni and Mn and states, and for Sb the and states, were treated as valence. The core electron levels were computed within the frozen-core approximation, and were treated fully-relativistically. The valence electrons were treated within the scalar-relativistic approximation. After self-consistency was achieved for NiMnSb, the density of states (DOS) was evaluated with a k-point mesh, in order to get an accurate band gap.

iv.2 Spectral functions and the Fermi surface of Cu1−xPdx random alloys

Discrepancies between the measured photoemission spectra Weightman and Cole (2010) and the KKR-CPA spectral functions Wright et al. (1987); Winter et al. (1986) for various Cu-Pd alloys were often discussed in the literature. In particular LDA-CPA results for the Pd partial DOS of the CuPd alloy reveal a three-peak structure (black line, Figure 2, peaks marked by A, B, and C), similar to the DOS of pure fcc-Pd Östlin et al. (2016). Experimental data Wright et al. (1987) on the other hand, see also inset of Figure 2, show a contracted band width for the partial DOS and do not resolve the peak at the bottom of the band (marked by C). A detailed discussion concerning these discrepancies can be found in Ref. [Weightman and Cole, 2010]. We note that the frequently discussed reasons for these discrepancies are connected to matrix element effects Nahm et al. (1993), broadening by electronic self-energy Nahm et al. (1993), and local lattice distortions Weightman et al. (1987); Thornton et al. (1994); Kucherenko et al. (1998), that go beyond the capabilities of standard CPA. Although it is not our intention to address all of the above inconsistencies, our current implementation allows us to address the possible source of discrepancy in connection to the combined disorder and correlation effects.

We have previously investigated the electronic structure of fcc-Pd Östlin et al. (2016), within the framework of the LDA+DMFT method using the perturbative FLEX impurity solver Katsnelson and Lichtenstein (2002). Recently, the properties of fcc-Pd were revisited using a lattice (non-local) FLEX solver Savrasov et al. (2018). These recent calculations Savrasov et al. (2018) support our results using the local approximation of the self-energy. Consequently, we study the electronic correlations in the CuPd alloys using the same local DMFT technique as we used before. In particular we consider modeling correlations only for the Pd alloy component.

In Figure 2 we present the spectral function (DOS) for the CuPd alloy, as a function of the Coulomb parameter . All curves were evaluated at the lattice constant given by a linear interpolation between that of pure Cu and pure Pd (Vegard’s law), which in this case corresponds to Å. Vegard’s law has previously been shown to hold in a larger range of concentrations for Cu-Pd within KKR-CPA Wilkinson et al. (2001). As the Coulomb interaction is increased, the peak close to the bottom of the band (C) shifts towards the Fermi energy, while the major peak close to (A) remains unchanged. The high binding energy peak (C) loses intensity with increasing , and the spectral weight is shifted to higher binding energy, where it builds up a satellite structure (not shown). A similar behavior in the spectral weight shift was also found for pure Pd Östlin et al. (2016). The results of the calculations including self-energy effects shown in Figure 2, bring the spectral function more in line with experimental photoemission data Wright et al. (1987) (see also inset). Since we neglect matrix element effects due to the photoemission process, as well as local lattice relaxations, we do not make a quantitative statement concerning the differences between theory and experiment. However, our calculation shows that the proposed method which combines correlation and alloy disorder effects, provides the correct trend in the spectral function.

In the following we comment upon the disorder and correlation induced modifications in the shape of the Fermi surface of CuPd alloys. On the basis of KKR-CPA calculations Győrffy and Stocks Gyorffy and Stocks (1983) proposed an electronic mechanism which determines short-range order effects experimentally seen in CuPd alloys. The experimental observation, namely the dependence of the scattering intensities on concentration in these alloys, was traced back to the flattening of the Fermi surface sheets with increasing Pd concentration. According to their results Gyorffy and Stocks (1983) the Fermi surface must change from a convex shape in the Cu-rich alloy to a concave one for the Pd-rich limit, in a continuous fashion. Consequently the Fermi surface is forced to be almost flat for some concentration, giving rise to nesting phenomena. This was later confirmed by further experiments and CPA calculations Wilkinson et al. (2001); Bruno et al. (2001).

According to previous calculations Gyorffy and Stocks (1983) a flattened Fermi surface in the plane was obtained for the CuPd alloy. In Figure 3(a) we plot our results for the Fermi surface of the same alloy. In our calculations we used for the lattice constant the value  Å (from Vegard’s law), and electronic interactions on the Pd alloy component were parameterized by  eV, and  eV. The Fermi surface is represented in the (010) and (110) planes of the fcc BZ. The major part of the Fermi surface consists of the electron sheet centered at the -point. This sheet goes from convex to concave with Pd-alloying, forcing parts of the sheet to be nearly flat at Pd. Our result is in good agreement with previous KKR-CPA calculations Wilkinson et al. (2001); Bruno et al. (2001). To quantify the effect of correlation, we plot in Figure 3(b) the difference between the correlated ( eV) and the non-correlated () case. Note the relatively small scale, which shows that the Fermi surface is insensitive to correlation effects.

iv.3 Interplay of correlation and disorder in Mn-Ni partially interchanged NiMnSb

Half-metallic ferromagnets (HMF) Katsnelson et al. (2008) are ferromagnetic systems which are metallic in one spin channel, while for the opposite spin direction the Fermi level is situated in a gap. Such systems would therefore present a full spin-polarization at the Fermi level, and have consequently drawn considerable interest due to their potential application in spintronics. One of the first systems to be characterized as a HMF is the semi-Heusler NiMnSb de Groot et al. (1983). The crystal structure of the NiMnSb compound is cubic with the space group (No. 216). It consists of four interpenetrating fcc sublattices equally spaced along the [111] direction. The Ni lattice sites are situated at , Mn sites are at , and Sb is situated at . The position at is unoccupied in the ordered alloy. In experiment, contrary to the DFT prediction, the measured spin-polarization of NiMnSb is only  Soulen et al. (1998). Several suggestions have been given to explain this large reduction in spin-polarization. Among them we mention electronic correlation effects Chioncel et al. (2003); Katsnelson et al. (2008) and disorder Orgassa et al. (1999); Attema et al. (2004); Ekholm et al. (2010).

Within the current implementation we have the opportunity to study the combined effect at equal footing. In the present calculations we use the experimental lattice constant,  Å. To parametrize the Coulomb interaction the values eV and eV was used, which are in the range of previous studies Chioncel et al. (2003); Morari et al. (2017). Only the Mn states were treated as correlated. Because the Ni bands in NiMnSb are almost filled, these are subject to minor correlation effects, as shown previously Morari et al. (2017). The effect of electronic correlations is the appearance of nonquasiparticle (NQP) states in the minority spin gap (spin down channel) just above the Fermi level. The origin of these many-body NQP states is connected with “spin-polaron” processes: the spin-down low-energy electron excitations, which are forbidden for the HMF in the one-particle picture, turn out to be allowed as superpositions of spin-up electron excitations and virtual magnons Edwards and Hertz (1973); Katsnelson et al. (2008). By direct computation, spin-orbit effects were found to be negligible Mavropoulos et al. (2004) in NiMnSb. A partially filled minority spin gap was obtained but the material remains essentially half-metallic with a polarization of the DOS of about  Mavropoulos et al. (2004). The interplay of spin-orbit induced states and NQP states have been also discussed Chioncel et al. (2006). In contrast with the spin-orbit coupling, correlation induced NQP states have a large asymmetric spectral weight in the minority-spin channel, leading to a peculiar finite-temperature spin depolarization effect. It has been shown that also disorder induces minority-spin states in the energy gap of the ordered material Orgassa et al. (1999). These states widen with increasing disorder. This behavior leads to a reduced minority-spin band gap and a displacement of the Fermi energy within the original band gap.

The current implementation give us the opportunity to investigate the possible interplay of disorder-induced and NQP states. To this end we perform electronic structure calculations considering the partial interchange of Ni and Mn, (NiMn)(MnNi)Sb, which leaves the overall stoichiometry and number of electrons constant. In Figure 4 we show the total DOS around the Fermi level for a number of different disorder levels. In the Figure 4(a)/(b) the LDA+DMFT DOS for smaller degrees of disorder , and respectively for larger disorder , is seen. The results for the clean, (ideal) case, NiMnSb, are presented with read lines (non-interacting, ), and light blue (DMFT). The minority spin gap is about 0.5 eV wide and is formed between the occupied Ni , Sb and Mn states. In the majority spin channel (spin up), Mn states dominates at . Already at disorder (dark blue dashed line) minority states appear below . These states are generated by the presence of Ni impurities at the Mn site, as previously shown by Orgassa et al. Orgassa et al. (1999). Furthermore, the upper band edge is shifted to higher energy. As the disorder is increased, the width of the Ni impurity states are increased. With correlation, minority spin states appear just above the Fermi level. These NQP states arise from many-body electron-magnon interactions Katsnelson et al. (2008). At larger degrees of disorder, see Figure 4 (b), the impurity states and the NQP states overlap in energy, removing the spin-down gap. Hence, the combination of exchange disorder and correlation effects remove the half-metallic state in NiMnSb.

Figure 5(a)/(b) displays the self-energy along the real energy axis for the Mn / states, respectively, for a Mn-Ni interchange of . The blue lines correspond to the Mn at the Mn-site, and is similar to the self-energy for the pure NiMnSb (not shown). The self-energy behaves as in previous calculations Chioncel et al. (2003), namely: The spin-down channel (blue down-triangles) has a self-energy that is fairly small below , but starts to increase above . At around eV above , the self-energy shows a hump, which gives rise to the NQP peak in the spectral function. The spin-up channel self-energy (blue up-triangles) behaves differently, it is relatively large below , while being small in magnitude above . The self-energy for the impurity Mn, situated at the Ni-site, is marked by the red lines in Figure 5(a)/(b). It is interesting to note that the trend seen for the spin channels has the reversed behavior compared to the self-energy on the Mn-site. For the spin-down channel (red down-triangle), the self-energy is large below , while it is small above . The trend is opposite for the spin-up channel (red up-triangle). This reversal of behavior between the Mn-site self-energy and the Ni-site self-energy can be understood by looking at the magnetic moments in the system. In pure NiMnSb, the total magnetic moment is (integer) 4 , with the main contribution stemming from the Mn-site ( ). As the pure system starts to be disordered, a Mn-moment of opposite sign ( , depending on the interchange concentration) develops at the Ni-site. Hence, the moment of the impurity Mn on the Ni-site aligns antiferromagnetically with the Mn moment on the Mn-site.

It is of interest to investigate how the effect of disorder, i.e. the degree of Mn-Ni interchange, influence the formation of NQP states. For this reason, in the inset of Figure 5(b), we plot the Mn-site self-energies of the dominant orbitals for different disorder concentrations. For minor degrees of Mn-Ni interchange (up to ), the sudden increase in just above (dashed blue lines of Figure 5), signaling the departure from Fermi liquid behavior, remains unaffected. It should also be noted that the Ni-site self-energy (Figure 5, red lines), follows the Fermi liquid (quasiparticle) behavior . The Ni band in NiMnSb is almost fully occupied, leaving little possibility for magnons to be excited, therefore weak electron-magnon interaction exists in the Ni-sublattice and no NQP states are visible in the density of states.

V Conclusion and Outlook

In this paper we developed a calculation scheme within the framework of the density functional theory, which allows one to study properties of disordered alloys including electronic correlation effects. We model disorder using the coherent potential approximation and include local but dynamic correlations through dynamical mean field theory. Similar to our previous implementation Östlin et al. (2017), the DFT-LDA Green’s function is computed directly on the Matsubara contour. Simultaneously the CPA is implemented within the LMTO formalism also in the Matsubara representation. Within the LMTO formalism the CPA effective medium is naturally encoded in the potential function, which alone contains the necessary information about the atomic configuration (assuming that a suitable screening representation is chosen). As shown in this paper, the simple parameterization of the potential function allows us to easily embed the self-energy into the standard LMTO potential parameters. Accordingly, the previously developed CPA schemes within the various muffin-tin approximations can then be used with only minor changes.

We presented results of the electronic structure calculation for two disordered alloys: the CuPd system, and the half-metallic NiMnSb semi-Heusler, in which correlations were considered for Pd and Mn alloy components respectively. For the case of the binary CuPd system, we see that the inclusion of electronic correlation improves the agreement with the experimental spectral functions for . For a Pd concentration of the Fermi surface, which is well captured already on the level of the LDA, remains more or less unchanged as correlation effects are turned on. In the second example, the partial exchange of Mn and Ni in NiMnSb was investigated, simultaneously with correlation effects. Already for low levels of disorder, impurity states appear below the Fermi level, while many-body induced nonquasiparticle states appear just above the Fermi level. Both these states contribute to the closing of the minority-spin gap.

In the future, the present method will be extended to compute total energies within the full-charge density technique Vitos et al. (1994), making it possible to study the energetics of anisotropic lattice distortions Vitos et al. (2001) in alloys. Another interesting venue is to change the arithmetic configuration average used in the CPA to the geometric average used in typical medium methods Terletska et al. (2017). This will allow one to investigate the effects of Anderson localization Anderson (1958) in realistic materials.

Acknowledgments

We greatly benefited from discussions with O. K. Andersen and D. Vollhardt, whose advices are gratefully acknowledged. AÖ thanks W. H. Appelt, M. Sekania, and M. Dutschke for help with the graphics. Financial support of the Deutsche Forschungsgemeinschaft through the Research Unit FOR 1346 and TRR80/F6 is gratefully acknowledged. LV acknowledges financial support from the Swedish Research Council, the Swedish Foundation for Strategic Research, the Swedish Foundation for International Cooperation in Research and Higher Education, and the Hungarian Scientific Research Fund (OTKA 84078). We acknowledge computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) in Linköping.

Appendix A Relations and formulas within the LMTO method

Within the nearly-orthogonal -representation, the potential function takes the simple form

 PγRl(z)=z−CRlΔRl. (23)

An insertion of this form into Eq. (11), and comparing with Eq. (9), one sees that

 gγRL′RL′(k,z)=√ΔRlGγRLR′L′(k,z)√ΔR′l′, (24)

i.e., the Green’s function is the normalized path operator.

In the case of a random alloy, the potential parameters , and will be site-dependent random parameters. Hence, both the potential function , Eq. (23), and the structure constants , will be random within the -representation. To avoid this, it is useful to switch to the tight-binding -representation Andersen and Jepsen (1984), as has been pointed out previously Gunnarsson et al. (1983); Kudrnovský and Drchal (1990); Abrikosov et al. (1991); Abrikosov and Skriver (1993). Within the tight-binding -representation, the potential function takes the form

 PβRl(z)=ΓβRlVβRl−z+1γRl−βl, (25)

where here the representation-dependent potential parameters and are given by

 VβRl=CRl−ΔRlγRl−βl,ΓβRl=ΔRl(γRl−βl)2. (26)

The -parameters can be found tabulated in several sources Andersen and Jepsen (1984); Skriver and Rosengaard (1991). The structure constants depend only on the geometry of the underlying lattice, and only the potential function is random. The path operator in the -representation, , is given similarly as in Eq. (11). The following relation allows to transform path operators between different representations Andersen and Jepsen (1984); Andersen et al. (1986b):

 gβ(z)=(β−γ)Pγ(z)Pβ(z)+Pγ(z)Pβ(z)gγ(z)Pγ(z)Pβ(z), (27)

where we have omitted the indices for simplicity. Using this transformation, and Eqs. (23), (25), and (24), the Green’s function can be obtained from as Skriver and Rosengaard (1991)

 GγRLR′L′(k,z)=1z−VβRl+√ΓRlz−VβRlgβRLR′L′(k,z)√ΓR′l′z−VβR′l′. (28)

Note that the transformations in Eqs. (27) and (28) are simply energy-dependent scalings of the path operator, since the potential parameters and the potential functions are diagonal matrices.

We here briefly mention the accuracy of the presented expressions. The formulas as written above give correct energies up to second order in . A way to improve on this is by a variational procedure Andersen et al. (1999), which produces a new Hamiltonian, giving eigenvalues correct to third order. Correspondingly, the substitution in Eq. (28) gives a third-order expression for the potential function Andersen et al. (1986b, a). Here, is a (relatively small) potential parameter. In order to compare the spectra arising from the different orders of LMTO’s, we investigated the DOS for various systems using either second or third order potential functions, and comparing the result with the DOS computed from the Hamiltonian through the spectral representation. We found that while at second order there was no difference between the DOS, for third order there were clear differences between the spectra. This can be attributed to the false poles present in the third-order potential function Andersen et al. (1999), since the energy dependence is now not linear, but cubic. In practice, we found that this lead to a loss of spectral weight in the Green’s function of Eq. (28), compared to the spectral representation. Hence, we in this paper only consider second order potential functions in Eq. (28).

Appendix B Exact Muffin-Tin Orbitals method

One choice of basis for the solution of the Kohn-Sham equation (1) is the energy-dependent exact muffin-tin orbitals Andersen et al. (1994); Vitos (2001, 2010), . They are constructed as a sum of the so called partial waves, the solutions of the radial equations within the spherical muffin-tins, and of the solutions in the interstitial region. Using this basis, the Kohn-Sham eigenfunctions can be expressed as

 Ψj(r)=∑RL¯ψaRL(ϵj,rR)vaRL,j, (29)

where the superscript denotes the screening representation used in the EMTO theory Andersen et al. (1994); Vitos (2001).

The expansion coefficients, , are determined so the is a continuous and differentiable solution of Eq. (1) in all space. This leads to an energy-dependent secular equation, , where is the so called kink matrix, viz.

 KaRLR′L′(k,z)≡aδRR′δLL′DaRL(z)−aSaRLR′L′(k,z). (30)

denotes the EMTO logarithmic derivative function Vitos et al. (2000); Vitos (2001), and is the slope matrix Andersen and Saha-Dasgupta (2000). The energy dependence of the kink matrix and the secular equation poses no difficulties, since the DFT problem can be solved by Green’s function techniques (see, for example, Ref. Zeller et al., 1982). By defining the path operator as the inverse of the kink matrix,

 ∑R′′L′′KaR′L′R′′L′′(k,z)gaR′′L′′RL(k,z)=δR′RδL′L, (31)

the poles of the path operator in the complex energy plane will correspond to the eigenvalues of the system. The energy derivative of the kink matrix, , gives the overlap matrix for the EMTO basis set Andersen and Saha-Dasgupta (2000), and hence it can be used to normalize the path operator , which gives the EMTO Green’s function Vitos et al. (2000); Vitos (2001)

 GRLR′L′(k,z) = ∑R′′L′′gRLR′′L′′(k,z)˙KR′′L′′R′L′(k,z) (32) −δRR′δLL′IRL(z),

where accounts for the unphysical poles of  Vitos (2001, 2010). The use of Green’s functions also facilitates the implementation of the CPA, the reader is referred to Refs. Vitos, 2001; Vitos et al., 2001; Vitos, 2010 for more detailed discussions.

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