Native defects in the Co$_2$Ti$Z$ ($Z=$ Si, Ge, Sn) full Heusler alloys: formation and influence on the thermoelectric properties

TL;DR

This study uses first-principles calculations to analyze native defects in Co$_2$Ti$Z$ full Heusler alloys and their impact on thermoelectric properties, explaining the experimentally observed negative Seebeck coefficient.

Contribution

It provides a detailed theoretical analysis of native defect formation energies and their effects on transport properties in Co$_2$Ti$Z$ alloys, linking defects to thermoelectric behavior.

Findings

Co vacancies and Ti$_ ext{Sn}$ anti-site have negative formation energies.

Defects significantly alter the Seebeck coefficient, sometimes making it negative.

Native defects can explain the experimentally observed negative Seebeck coefficient.

Abstract

We have performed first-principles investigations on the native defects in the full Heusler alloys Co$_2$Ti$Z$ ($Z$ one of the group IV elements Si, Ge, Sn), determining their formation energies and how they influence the transport properties. We find that Co vacancies (Vc) in all compounds and the Ti$_\text{Sn}$ anti-site exhibit negative formation energies. The smallest positive values occur for Co in excess on anti-sites (Co$_Z$ or Co$_\text{Ti}$) and for Ti$_Z$. The most abundant native defects were modeled as dilute alloys, treated with the coherent potential approximation in combination with the multiple-scattering theory Green function approach. The self-consistent potentials determined this way were used to calculate the residual resistivity via the Kubo-Greenwood formula and, based on its energy dependence, the Seebeck coefficient of the systems. The latter is shown to depend significantly on the type of defect, leading to variations that are related to subtle, spin-orbit coupling induced, changes in the electronic structure above the half-metallic gap. Two of the systems, Vc$_\text{Co}$ and Co$_Z$, are found to exhibit a negative Seebeck coefficient. This observation, together with their low formation energy, offers an explanation for the experimentally observed negative Seebeck coefficient of the Co$_2$Ti$Z$ compounds as being due to unintentionally created native defects.

Tables

Table 1. Table 1: The calculated formation enthalpy Δ ​ H Δ 𝐻 \Delta H (in eV/formula unit) for the Co 2 Ti Z 𝑍 Z ( Z = 𝑍 absent Z= Si, Ge,Sn) full Heusler alloys as well as of various Co-Ti, Co- Z 𝑍 Z , Ti- Z 𝑍 Z , and Co-Ti- Z 𝑍 Z binary and ternary compounds which compete with the Co 2 Ti Z 𝑍 Z formation. For the latter we also include the equilibrium lattice constants used in this work as obtained either through the plane-wave pseudopotential method in the SRA (results labeled as QE) or the full potential SPR-KKR Green function approach (KKR). The calculated values are compared with the experimental data. Webster and Ziebeck ( 1973 ); Carbonari et al. ( 1996 )

Compound	$\Delta H$	Lattice constant (Å)
(structure)	(eV/f.u.)	QE	KKR	Experiment
Co2TiSi ($L{2}_1$)	-2.62	5.756	5.780	5.74(0)
Co2TiGe ($L{2}_1$)	-1.95	5.848	5.874	5.83(1)
Co2TiSn ($L{2}_1$)	-1.44	6.092	6.119	6.07(3)

Table 2. Table 2: Formation energy E form subscript 𝐸 form E_{\text{form}} for the native defects in Co 2 Ti Z 𝑍 Z ( Z 𝑍 Z = Si, Ge, Sn), calculated using 128-atom supercells within the pseudopotential plane wave method. The formation energy entries, derived from Eq. ( 4 ), are given as an interval ( E form min , E form max ) superscript subscript 𝐸 form min superscript subscript 𝐸 form max (E_{\text{form}}^{\text{min}},E_{\text{form}}^{\text{max}}) corresponding to the lower and upper bounds of the chemical potentials Δ ​ μ i Δ subscript 𝜇 𝑖 \Delta\mu_{i} and/or their combinations. Also listed are the reference values Δ ​ E ​ ( Co 2 ​ Ti ​ Z , D ) Δ 𝐸 subscript Co 2 Ti 𝑍 𝐷 \Delta E(\text{Co}_{2}\text{Ti}Z,D) defined by Eq. ( 5 ). For the stoichiometric defects ’A ↔ ↔ \leftrightarrow B’ all these quantities are equal.

	$\Delta E({\text{Co}}_2\text{Ti}Z,D)\to (E_{\text{form}}^{\min },E_{\text{form}}^{\max })$ [eV]
Defect $D$	Co2TiSi		Co2TiGe		Co2TiSn
Vc${}_{\text{Co}}$	0.29 $\to$	( 0.09, 0.24)	0.01 $\to$	(-0.17, 0.01)	0.28 $\to$	(-0.08, 0.28)
Vc${}_{\text{Ti}}$	2.20 $\to$	(0.98, 2.15)	2.16 $\to$	(0.56, 1.55)	2.37 $\to$	(1.10, 1.91)
VcZ	3.21 $\to$	(1.63, 2.21)	2.87 $\to$	(1.83, 2.52)	2.96 $\to$	(2.43, 2.96)
Co${}_{\text{Ti}}$	1.90 $\to$	(0.88, 1.31)	1.96 $\to$	(0.38, 1.54)	2.16 $\to$	(0.89, 2.03)
CoZ	1.90 $\to$	(0.37, 1.10)	1.37 $\to$	(0.44, 1.12)	1.20 $\to$	(0.78, 1.56)
Ti${}_{\text{Co}}$	1.21 $\to$	(1.80, 2.23)	0.85 $\to$	(1.27, 2.43)	1.00 $\to$	(1.13, 2.27)
TiZ	1.50 $\to$	(0.86, 1.72)	0.66 $\to$	(0.29, 1.91)	-0.30 $\to$	(-0.16, 0.80)
$Z$${}_{\text{Co}}$	2.16 $\to$	(2.96, 3.69)	2.31 $\to$	(2.56, 3.24)	3.18 $\to$	(2.82, 3.60)
$Z$${}_{\text{Ti}}$	0.93 $\to$	(0.71, 1.57)	1.91 $\to$	(0.66, 2.28)	2.39 $\to$	(1.29, 2.25)
Ti $\leftrightarrow$ Co	2.57		2.28		2.63
Ti $\leftrightarrow$ $Z$	2.18		2.04		1.90
Co $\leftrightarrow$ $Z$	2.97		2.70		3.21

Equations

$${\mu }_{\alpha }=E_{\alpha }(\text{bulk})+\Delta {\mu }_{\alpha },$$

$$\Delta {\mu }_{\alpha }\le 0\forall \alpha .$$

$$\begin{array}{rl}{E}_{\text{form}}[D]=& E_{\text{tot}}[H:D]-E_{\text{tot}}[H]\\ & -\sum _{\alpha }{N}_{\alpha }{E}_{\alpha }(\text{bulk})-\sum _{\alpha }{N}_{\alpha }\Delta {\mu }_{\alpha }\end{array},$$

$$E_{\text{form}}[D]=\Delta E(H,D)-\sum _{\alpha }{N}_{\alpha }\Delta {\mu }_{\alpha },$$

$$\begin{array}{rl}\Delta E(H,D)=& E_{\text{tot}}[H:D]-E_{\text{tot}}[H]\\ & -\sum _{\alpha }{N}_{\alpha }{E}_{\alpha }(\text{bulk})\end{array}.$$

$$2\Delta {\mu }_{\text{Co}}+\Delta {\mu }_{\text{Ti}}+\Delta {\mu }_Z=\Delta H({\text{Co}}_2\text{Ti}Z),$$

$$a\Delta {\mu }_{\text{Co}}+b\Delta {\mu }_{\text{Ti}}+c\Delta {\mu }_Z\le \Delta H({\text{Co}}_a{\text{Ti}}_b{Z}_c).$$

$$L_{ij}^{(m)}=-\frac{1}{e}\int \left[\frac{\partial }{\partial E}{f}_0(E,\mu ,T)\right]{\left(E-\mu \right)}^m{\sigma }_{ij}(E)dE,$$

$$S_{ii}(T)=-\frac{1}{eT}\frac{L_{ii}^{(1)}}{L_{ii}^{(0)}}$$

$$\Delta {\mu }_{\text{Ti}}\le \Delta H({\text{Co}}_a\text{Ti})-a\Delta {\mu }_{\text{Co}}(a=1,2,3),$$

$$\Delta {\mu }_{\text{Ti}}\ge \Delta H({\text{Co}}_2\text{Ti}Z)-\Delta H(\text{Co}Z)-\Delta {\mu }_{\text{Co}}.$$

$$A(\vec{k},E)=$$

$$\int d^{3}r⟨G(\vec{r}+{\vec{R}}_m,\vec{r}+{\vec{R}}_n;E)⟩,$$

Full text

I Introduction

Intermetallic ternary compounds often possess special electronic and magnetic properties that make them interesting for technological applications. They manifest, however, a natural tendency towards off-stoichiometry, predictable from thermodynamic theory, which creates natural (intrinsic) doping via vacancies, anti-sites, or swaps.Yu et al. (2017) Therefore, thorough studies of native defects in these materials are required prior to their incorporation in actual devices. This holds particularly true for ferromagnetic half-metals in view of possible defect states being introduced in the half-metallic gap. The present work addresses this issue for Ti-derived half-metallic Heusler alloys. To motivate this research, we briefly review the use of half-metals in spintronics.

One class of such materials, under intense scrutiny in the last decades, are the Co-based Heusler alloysKübler et al. (1983); Kandpal et al. (2007a); Graf et al. (2011) crystallizing in the cubic $L2_{1}$ structure. Many of them are predicted to be half-metallic, a property describing the particular arrangement of the electronic states in which one spin channel is characterized by metallic conductance while the other one is semiconducting. While this makes them highly attractive for spintronics and spin-caloric transport, technological achievements are still rare, owing to the intrinsic complexity of these compounds. For example, first-principles electronic structure calculationsGalanakis et al. (2002, 2006) predicted 100 % spin polarization near the Fermi energy in Co2MnSi, but it took several more years of research and advanced sample preparation to provide experimental evidence in either bulk or thin film phases.Yang et al. (2012); Jourdan et al. (2014) The theoretically predicted stability of the Co2MnSi/MgO interfaceHülsen et al. (2009a) soon found experimental validation.Li et al. (2011) As a result, several successful implementations of Heusler-based magnetic tunnel junctions (MTJ), with potential use as magnetic random access memories (MRAM), have been reported.Ishikawa et al. (2009); Yang et al. (2012); Liu et al. (2012); Sterwerf et al. (2013) Consisting of Co2MnSi, Co2FeSi, or Fe2CoSi with either MgO or Al-O barriers, they are all characterized by extremely large, several hundred percent, magneto-resistance ratios. Other envisioned functionalities of half-metallic Heusler compounds include the electricallyChadov et al. (2011) and thermally drivenComtesse et al. (2014); Geisler et al. (2014) spin injection, or the magneto-caloric memory device,Geisler and Kratzer (2015) which are yet to be accomplished experimentally.

Apart from Mn and Fe, early transition metals such as titanium offer further possibilities to tune the half-metallic properties and thus to open up novel applications. Thin films of Co2TiSi were grown on MgOMeinert et al. (2011) as well as on GaAs substrates.Dau and Herfort (2015) In a previous study, we have proposed a thermal spin injector based on a thin, lattice-matched barrier layer of either Co2TiSi or Co2TiGe between metallic leads.Geisler et al. (2014) Moreover, we could show that, similar to the insulator case, the loss of half-metallicity is localized within few atomic layers in the proximity of the interface. Thus, its role in the filtering of transmission channels notwithstanding, we find the understanding of the bulk transport properties of particular importance for the design of Heusler-based heterostructures.

This aspect appears more critical when considering the example of the half-metallic Co2Ti$Z$ compounds ($Z=$ Si, Ge, and Sn), which made the subject of an extensive and exhaustive study.Barth et al. (2010) Ground-state properties, including stable crystal structure, lattice constants, or spin magnetic moments, derived from first-principles calculations agree with the experimental findings.Lee et al. (2005); Kandpal et al. (2007a); Sharma et al. (2010); Barth et al. (2010) On the other hand, theory has failed so far to satisfactorily explain the transport properties of these systems, in particular the Seebeck coefficient, whose spin dependence is at the core of many spin-caloric proposed applications. While from the calculated electronic structure at and around the Fermi energy one would expect a positive Seebeck coefficient in all three Co2Ti$Z$ compounds,Barth et al. (2010); Geisler et al. (2014) measurements indicate negative values throughout the whole temperature range, reaching $-50$ $\mu$V/K in the case of Co2TiSn close to its Curie temperature.Barth et al. (2010); Balke et al. (2010)

This proved to be a rather puzzling finding in view of the useful and robust information that can be extracted from the Seebeck coefficient in semiconductors. There, a positive (negative) Seebeck coefficient can be unambiguously associated with a $p$-($n$-)type doping of the semiconductor, a property that led to a rather common designation of conduction being either hole- or electron-like, depending on the sign of $S(T)$.

Unlike in semiconductors, where its size can be in the order of $10^{2}-10^{3}$ $\mu$V/K, the Seebeck coefficient of metallic and half-metallic systems is much smaller and thus more sensitive even to small variations in the chemical composition of the sample. This appears indeed to be the case also for the Co2Ti$Z$ Heusler alloys, for which small deviations from the 2:1:1 stoichiometry, deemed to qualify as ”within the experimental error bars”, have been reported.Barth et al. (2010)

To what extent the native defects occurring in the Co2Ti$Z$ systems may influence the thermoelectric properties is illustrated by the calculated Seebeck coefficient displayed in Fig. 1. This figure, which summarizes the main results we report in this paper, shows the $S(T)$ curves corresponding to the Co2TiSi Heusler compound including selected intrinsic defects at $3$ % atomic concentration. As can be seen, the calculations reveal the occurrence of a broad range of values, apparently varying between hole- and electron-like conduction, depending on the various off-stoichiometric deviations considered.

We investigate here native point defects in the full Heusler Co2Ti$Z$ alloys, including vacancies (Vc) on different sublattices, anti-sites, and various sublattice swaps, with emphasis on their formation energy and their influence on both the electronic structure and transport properties of the host material. Owing to the close-packed character of the $L2_{1}$ structure we have deemed more complex defects, e.g., interstitial site occupation, dumb-bells, or clustering effects, as being less likely to occur. Furthermore, we did not consider the formation and interaction of defect pairs. The main focus of our investigations is to establish a connection between the different features of the Seebeck coefficient seen in Fig. 1 with the defect-induced modifications in the electronic structure. In particular, we show that the negative $S(T)$ obtained for CoZ and Vc${}_{\text{Co}}$ defects is due to subtle changes of the electronic structure that are mediated by spin-orbit coupling. Not only are these results in qualitative agreement with experiment, but they also correspond to defects which show a small formation energy.

After briefly providing the computational details in Section II, the manuscript comprises two parts that, although methodologically independent, represent necessary logical steps in our quest. First, we employ pseudopotential-based, large supercell total energy calculations to determine the formation energyVan de Walle and Neugebauer (2004); Zhang et al. (1997); Stern et al. (2016) for all possible intrinsic defects. The results, which are discussed in Section III, also imply the evaluation of the formation energy by taking into account the limits of the chemical potentials as obtained from separate calculations of competing binary and ternary compounds. For the second part, we select six different defects, found to exhibit formation energies under $\simeq 1$ eV, and model them as dilute alloys within the framework of the coherent potential approximationSoven (1967); Taylor (1967); Soven (1970) based on multiple-scattering theory as implemented in the full-potential spin-polarized relativistic Korringa-Kohn-Rostoker method.Ebert et al. (2011, 2016); SPR A detailed analysis of the various influences these defects have on the electronic structure of the parent compound is provided in Section IV. This analysis, based on the Bloch spectral function, reveals signatures of several impurity bands appearing in the minority-spin channel, in the proximity of the half-metallic gap. With spin-orbit coupling causing majority-minority-band anti-crossings right above the Fermi energy, we further evidence the effect of the different defects on these features, that may lead, in some cases, to an enhanced conductivity and a significant reduction of the spin polarization. The last section presents the results obtained for the electronic conductivity and the Seebeck coefficient in the dilute alloys from calculations performed employing the Kubo-Greenwood formula.Butler (1985); Banhart et al. (1987) This approach has been already successfully applied to the study of various transport properties in Heusler alloys, ranging from longitudinal resistivity in Co2Mn(Al,Si)Kota et al. (2010) to the anomalous Hall effect in Co2(Mn,Cr,Co)Al including native defects.Kudrnovský et al. (2013) Applied here in the limit of dilute impurities, we include in the calculations the so-called vertex corrections,Butler (1985) that are extremely important in this regime.

II Details of the calculations

While our investigations consist of two separate steps, they do share the use of self-consistent field (SCF) calculations. These provide the crystal potential, determined within density functional theory (DFT), employing the generalized gradient approximation (GGA) parameterization for the exchange-correlation functional.Perdew et al. (1996) Specific details of the two adopted approaches are given in the following.

II.1 Formation energies from large supercells

The determination of the defect formation energies based on DFT is nowadays a standard approach, thoroughly described in the literature.Van de Walle and Neugebauer (2004); Zhang et al. (1997); Stern et al. (2016) It involves calculations of the total energies, $E_{\text{tot}}[H]$ and $E_{\text{tot}}[H:D]$, in two systems: The clean host crystal $H$ and the perturbed one $H:D$. These systems are described by large supercells, constructed by repeatedly stacking the crystal unit cell along the spatial directions. The process of replacing host atoms by impurities $D$ is regarded as a particle exchange, mediated by external reservoirs associated to each atomic species $\alpha$. The requirement to describe the equilibrium between the perturbed host and the reservoirs calls for thermodynamic concepts. In the following, the thermodynamic equilibrium conditions are worked out at zero temperature and pressure. Denoting by $E_{\alpha}(\text{bulk})$ the calculated total energy of the most stable condensed phase (solid or molecule) of element $\alpha$, the chemical potential $\mu_{\alpha}$ of each $\alpha$-reservoir can be expressed as

[TABLE]

with $\Delta\mu_{\alpha}$ the reduced chemical potential. In the present situation of the perturbed host in equilibrium with the reservoirs, it is required that

[TABLE]

In other words, the $E_{\alpha}(\text{bulk})$ provides an upper bound to the chemical potential, corresponding to the physical condition of atom $\alpha$ not precipitating into its stable phase.

With the notations introduced above, the formation energy $E_{\text{form}}[D]$ of a neutral defect $D$ is obtained from

[TABLE]

where $N_{\alpha}$ is the number of atoms, either host or impurity, that were added to ($N_{\alpha}>0$) or removed from ($N_{\alpha}<0$) the supercell upon creating the perturbed system.Van de Walle and Neugebauer (2004) Since the $N_{\alpha}$’s depend on the chosen configuration and the total energies are calculated directly, it follows that, amongst the terms on the rhs of Eq. (3), the reduced chemical potentials $\Delta\mu_{\alpha}$ are the only independent variables, leading to the compact form

[TABLE]

with

[TABLE]

The values of the reduced chemical potentials are subject to various constraints, one of which is given in Eq. (2) as an upper bound to $\Delta\mu_{\alpha}$. Additional bounds arise from the following requirements: First, thermodynamic equilibrium of the constituents with the host crystal must hold. Specific to the full Heusler alloys Co2Ti$Z$ investigated here, this condition reads

[TABLE]

where $\Delta H(\text{Co}_{2}\text{Ti}Z)$ is the formation enthalpy of the Co2Ti$Z$ compound at zero temperature and pressure. Second, the reduced chemical potentials of either host or impurity atoms have to be in ranges where competing binary or ternary systems may not form. For a generic compound with chemical formula CoaTi${}_{b}Z_{c}$ and formation enthalpy $\Delta H(\text{Co}_{a}\text{Ti}_{b}Z_{c})$ one obtains

[TABLE]

Combining all the constraints derived from the relations (2), (6), and (7) allows one to determine a stability range $\{\Delta\mu_{\text{Co}},\Delta\mu_{\text{Ti}},\Delta\mu_{Z}\}$ for which the investigated compound (here the Co2Ti$Z$ Heusler alloy) may form under equilibrium growth conditions. For ternary compounds it is common, using Eq. (6), to eliminate one of the variables (in the following $\Delta\mu_{Z}$), leaving for both the stability range and the defect formation energy $E_{\text{form}}[X]$ only a two-dimensional explicit dependence (in the following $\{\Delta\mu_{\text{Co}},\Delta\mu_{\text{Ti}}\}$).

II.2 Total energy calculations

The Co2Ti$Z$ ($Z=$ Si, Ge, Sn) compounds investigated here belong to the class of full Heusler alloysGraf et al. (2011) of prototype Cu2MnAl, crystallizing in the cubic $L2_{1}$ structure. This crystal structure, shown in Fig. 2(a), has a face-centered-cubic (fcc) primitive cell with four inequivalent atomic sites. It can be viewed either as four inter-penetrating fcc sublattices, respectively occupied by the Co, Ti, Co, and $Z$ atoms shifted against each other by $(1/4,1/4,1/4)$ lattice constants, or as two inter-penetrating CoTi and Co$Z$ zinc-blende structures shifted by $(0,0,1/2)$.

We have considered several off-stoichiometric native defects: (a) An additional Co, Ti, or $Z$ atom occupying a different sublattice (in the following denoted as Co${}_{\text{Ti}}$, Ti${}_{\text{Co}}$, $Z_{\text{Co}}$ and so on); and (b) Vacancies on one of the Co, Ti, and $Z$ sites (labeled respectively as Vc${}_{\text{Co}}$, Vc${}_{\text{Ti}}$, VcZ). Stoichiometry-preserving configurations, in which two neighboring atoms from different sublattices are switching places ($\text{Co}\leftrightarrow\text{Ti}$, $Z\leftrightarrow\text{Ti}$, and $\text{Co}\leftrightarrow{Z}$), were also investigated. Illustrative examples of the various systems are depicted in Fig. 2(a). Note that a 128 atom supercell (that is, larger than shown in the figure) was used in all calculations of the defect formation energy. On the other hand, the determination of the competing compounds formation enthalpy needed in Eq. (7) relied on their respective primitive cells.

In all cases, the total energies were calculated for spin-polarized systems in the scalar relativistic approximation (SRA) employing the plane-wave pseudopotential method as implemented in the Quantum Espresso code.Giannozzi et al. (2009) Wave functions and density have been expanded into plane waves up to cut-off energies of $40$ Ry and $400$ Ry, respectively. The neighborhood of atomic centers has been approximated by self-created ultrasoft pseudopotentials (USPPs),Vanderbilt (1990) as described previously in our study of the Al/Co2Ti$Z$ heterostructures.Geisler et al. (2014) A Methfessel-Paxton smearingMethfessel and Paxton (1989) of 10 mRy has been applied to the Brillouin zone (BZ) sampling performed with different Monkhorst-Pack $k$-point grids.Monkhorst and Pack (1976) These were chosen in such a way that they did not include the $\Gamma$ point and delivered well-converged total energies and potentials. In all systems, the lattice constants and the internal atomic positions have been accurately optimized using Hellmann-Feynman forces to reduce the force components below $1$ mRy/$a_{0}$ and the energy changes below $0.1$ mRy.

II.3 Alloy modeling for the calculation of

transport properties

In a second step, we have determined the transport properties of the Co2Ti$Z$ Heusler compounds in the presence of the intrinsic defects, which were modeled as off-stoichiometric dilute alloys within the CPA.Soven (1967); Taylor (1967); Soven (1970) As shown schematically in Fig. 2(b), each defect was assigned a mixed occupation of a single sublattice in the $L2_{1}$ structure. We illustrate here examples of Co(Co1-x$D_{x}$)Ti$Z$ and Co2Ti($Z_{1-x}$Cox) chemical formulas, respectively describing the Co2Ti$Z$:$D_{\text{Co}}$ ($D=$ Vc, Ti, $Z$) and Co2Ti$Z$:CoZ systems. The effect of a varying defect concentration was accounted for by considering different $x$ values ($x=0.01$, $0.03$, and $0.05$).

The calculations were based on a full potential, spin-polarized relativistic implementation of the Korringa-Kohn-Rostoker Green function method (FP-SPR-KKR).Ebert et al. (2011, 2016); SPR We obtained the SCF charge density and potentials by integrating the Green function in the complex energy plane over a contour consisting of 36 points and applying an angular momentum cut-off of $l_{max}=3$ for the Green function expansion. We need to emphasize here on the importance of going beyond the atomic sphere approximation (ASA) when methods employing atoms-centered basis functions are applied to Heusler alloys. Even in the case of the closed-packed $L2_{1}$ structure, ASA-based electronic structure calculations may fail to reproduce the FP results.Picozzi et al. (2002); Kandpal et al. (2007b) As pointed out by Picozzi et al.Picozzi et al. (2002) such discrepancies originate from the appreciable asphericities that may be present in the charge density.FPv

For all three parent compounds Co2Ti$Z$ we calculated the equilibrium lattice constants by minimizing the total energy. The differences between the FP-SPR-KKR determined lattice constants and those obtained by the scalar-relativistic plane-wave approach were found to be in the range of $\simeq 0.025$ Å, with an overall agreement within $1$ % of the experimental results.Webster and Ziebeck (1973); Carbonari et al. (1996); Barth et al. (2010) Including spin-orbit coupling (SOC) has therefore a very small influence on the computed equilibrium lattice parameters. We will show, however, that its role in the thermoelectric properties can not be neglected.

The transport properties of the Co2Ti$Z$:$D$ systems were determined subsequently to the SCF calculations. More specific, the temperature-dependent longitudinal Seebeck coefficient $S_{ii}(T)$ (with $i=x$, $y$, or $z$ the Cartesian coordinate) can be obtained from the diagonal elements of the energy dependent conductivity tensor $\sigma_{ii}(E)$.Wimmer et al. (2014) Introducing the transport coefficients

[TABLE]

where $f_{0}(E,\mu,T)$ is the Fermi-Dirac distribution function with chemical potential $\mu$ at energy $E$ for the temperature $T$, the Seebeck coefficient is given by

[TABLE]

with $e$ the elementary charge.Seb In our calculations, the central quantity is represented by the electronic conductivity, obtained on the basis of the Kubo-Greenwood formula,Butler (1985); Banhart et al. (1987) appropriately extended for non-spherical potentials, and including the important contributions stemming from the so called vertex corrections.Butler (1985) At each energy argument, the diagonal elements of $\underline{\sigma}(E)$ are obtained through a BZ integral evaluated over a number of $2\cdot 10^{6}$ $k$-points. For the energy integrals appearing in Eq. (8) we started by explicitly calculating $\underline{\sigma}(E)$ on an equidistant mesh of $1$ mRy separation, then refining it to a $0.01$ mRy resolution by linear interpolation. The integration boundaries around the chemical potential were set by the cut-off criterion $\partial f_{0}(E,\mu,T)/\partial E\geq 10^{-3}$ for a fixed electronic temperature $T$.

III Defect formation energy results

We present in this section the results obtained by applying the ab initio thermodynamics concepts introduced above. After establishing the stability ranges by analyzing the competing binary and ternary compounds, we derive upper and lower boundaries for the defect formation energies $E_{\text{form}}$. We show that the smallest values of $E_{\text{form}}$ occur for vacancies on Co sublattice (Vc${}_{\text{Co}}$) followed by other defects, such as the Co anti-sites Co${}_{\text{Ti}}$ and CoZ.

III.1 Boundaries of the reduced chemical potentials

The calculated stability domains for each of the Co2Ti$Z$ full Heusler alloys are displayed in Fig. 3 as two-dimensional representations in the ($\Delta\mu_{\text{C}o}$,$\Delta\mu_{\text{T}i}$)-plane. Their construction is based on the numerical evaluation of Eqs. (6) and (7) with the corresponding (zero temperature and pressure) formation enthalpies provided in Table 1. For consistency reasons, these were obtained using the same set of self-constructed pseudopotentials as employed in the supercell-related defect calculations, and were found in very good agreement with equivalent data available in various on-line repositories.Mat The list of possible competing binary phases in Table 1 is obviously not exhaustive. On the one hand side, we have only considered compounds known to exist. On the other hand, rather than exploring all possibilities, we only focused on those systems that pose restrictions for the reduced chemical potentials. In other words, from literature data we could conclude that certain stoichiometries will not give stricter boundaries than the ones already identified. This procedure appeared to be justified by the overall good agreement of our data with the previously published results.

The discussion of the results shown in Fig. 3 starts with some common features of the three panels. The stability triangles represent $(\Delta\mu_{\text{Co}},\Delta\mu_{\text{Ti}},\Delta\mu_{Z})$ triplets for which the Co2Ti$Z$ compound may form. They are defined by the upper bounds of $\Delta\mu_{i}$ given in Eq. (2). As already mentioned, the $\Delta\mu_{Z}$ variable can be eliminated using the equilibrium condition (6), such that the $\Delta\mu_{Z}\leq 0$ constraint reduces to $2\Delta\mu_{\text{Co}}+\Delta\mu_{\text{Ti}}\geq\Delta H(\text{Co}_{2}\text{Ti}Z)$. The equality sign is the equivalent of the thick red line in each panel of Fig. 3. Accounting for the formation of competing binary and ternary systems, according to Eq. (7), further reduces the allowed $\Delta\mu_{i}$ values, leading to the arbitrarily shaped polygons highlighted by the shaded areas. Their boundaries, displayed as dashed and solid lines and appropriately labeled by the respective chemical formula in the figure, are briefly discussed below.

As can be seen in Fig. 3, for the Co2TiGe and Co2TiSn compounds $\Delta\mu_{\text{Co}}=0$ remains as upper bound, whereas $\Delta\mu_{\text{Ti}}$ is limited above by either CoTi, Co2Ti, or Co3Ti. Allowed $\Delta\mu_{\text{Ti}}$ values must fulfill the conditions

[TABLE]

over the various $\Delta\mu_{\text{Co}}$ intervals. Comparing our results with those of Chepulskii and CurtaroloChepulskii and Curtarolo (2009) who calculated the full series of Co-Ti alloys, we find a very good agreement for all CoaTi ($a=1\ldots 3$) systems.Cur For CoTi2, on the other hand, these authors report a formation enthalpy of $-0.873$ eV/formula unit. It is easy to check, via Eq. (7), that the stable CoTi2 bulk phase lies inside the area already covered by one of the CoaTi compounds.

The Co2TiSi exhibits the peculiar situation of the stability range being completely determined by the formation of ternary compounds: Co3Ti2Si and Co16Ti6Si7 set the upper and lower boundary of $\Delta\mu_{\text{Ti}}$, while the crossing of their equilibrium lines fix the maximum $\Delta\mu_{\text{Co}}$ value at $-0.05$ eV. On the Co-poor (small $\Delta\mu_{\text{Co}}$) side, the lower boundary is set by the CoTiSi compound crystallizing in the orthorhombic $C23$ structure. Analogously, the ternary CoTiGe compound in the hexagonal $C22$ structure sets the lower boundary of $\Delta\mu_{\text{Co}}$ in the Co2TiGe system. In contrast, CoTiSn, with its ground state the $C1_{b}$ (half-Heusler) structure, forms outside the Co2TiSn stability range, with the Co-poor boundary determined by the Ti-Sn binaries. As seen in Table 1, TiSi and TiGe both crystallize in the $B27$ structure. A stable phase of TiSn, on the other hand, is not known to exist, although investigations performed by Colinet et al.Colinet et al. (2009) on a series of Ti-Sn binaries predict a negative formation enthalpy for TiSn in various structures. While our calculations, performed for Ti2Sn, Ti3Sn, and Ti6Sn5, delivered formation enthalpies in close agreement with their results, we chose to set the stability boundary to that provided by the Ti6Sn5 compound.

In the Co-rich, Ti-poor range (bottom-right corner), the formation of the Co2TiGe and Co2TiSn competes with that of the corresponding Co-$Z$ binary compounds. In both cases the lower boundary of $\Delta\mu_{\text{Ti}}$ is set by CoGe or CoSn, leading to the constraint

[TABLE]

Having completely defined the boundaries for $\Delta\mu_{\text{Co}}$ and $\Delta\mu_{\text{Ti}}$ we note that these can also be transferred back to $\Delta\mu_{Z}$ via Eq. (6). In combination with the observed increase of $\Delta H(\text{Co}_{2}\text{Ti}Z)$ within the $Z=$ Si, Ge, Sn series, it becomes obvious, from Fig. 3, that $Z$-rich conditions can only be attained in the Co2TiSn system. For $Z=$ Si and Ge, either CoTiSi, CoSi or Co7Ti4Ge7 set an upper bound for $\Delta\mu_{Z}$.

We close the discussion on chemical potential boundaries by analyzing the formation enthalpy of the Ti2Co$Z$ systems, as competing inverse full Heusler alloys. These compounds are stable through the whole series of group IV elements $Z$, with the formation being subject to the condition $\Delta\mu_{\text{Ti}}\geq\Delta\mu_{\text{Co}}+[\Delta H(\text{Ti}_{2}\text{Co}Z)-\Delta H(\text{Co}_{2}\text{Ti}Z)]$. According to our results shown in Table 1, this condition [labeled (B) in Fig. 3] falls inside the Co2Ti$Z$ stability triangle for all $Z$ atoms. In all cases, however, it remains outside the shaded areas indicating that, while competing precipitation of Ti2Co$Z$ may occur, it requires significantly different growth conditions than the respective stable full Heusler alloy.

III.2 Formation energies of intrinsic defects

in Co_2TiZ

We give in the following a survey of the calculated formation energies of all the defects considered. The results listed in Table 2 give the lower and upper bounds of the formation energy $E_{\text{form}}[D]$ and the reference energies $\Delta E(\text{Co}_{2}\text{Ti}Z,D)$. The latter, defined by Eq. (5), implicitly contains system-specific information related to intrinsic mechanisms concerning the defect formation, such as equilibrium bond length and electronegativity. The former, obtained from Eq. (4) by inserting the extremum values of the $\sum_{\alpha}N_{\alpha}\Delta\mu_{\alpha}$ term, can be seen as the energy required to exchange particles with the reservoirs.Stern et al. (2016) Our results can be summarized as follows:

i) Vacancies in various sublattices (Vc${}_{\text{Co}}$, Vc${}_{\text{Ti}}$, and VcZ): Those appearing in the Co sublattice are found to have the smallest formation energy, its values even turning negative for Co-poor conditions in Co2TiGe and Co2TiSn. The Vc${}_{\text{Ti}}$ and VcZ point defects have a significantly larger formation energy, whereby, in each system, the most stable sublattice is the one consisting of $Z$-atoms. A similar trend was also observed in Co2MnSi by Hülsen et al.Hülsen et al. (2009b), albeit with a somehow larger value, $\simeq 1$ eV, for $\Delta E[\text{Co}_{2}\text{Ti}Z,\text{Vc}_{\text{Co}}]$, which represents the upper bound ($\Delta\mu_{\text{Co}}=0$) of $E_{\text{form}}[\text{Vc}_{\text{Co}}]$. Experiments on Co2Mn$Z$, on the other hand,Kogachia et al. (2009) while confirming vacancy concentrations as high as $2$ %, suggest a rather random distribution over the lattice sites. An ever increasing Vc${}_{\text{Co}}$ concentration was found to lead to an L2_{1}$$\to$$C1_{b} ordering transition in Ni2-xMnSb Heusler alloys.Nagasako et al. (2015) A similar scenario may also occur in Co2TiSi$\to$CoTiSi($C23$) and Co2TiGe$\to$CoTiGe($C22$), where the compounds in the 1:1:1 stoichiometry set the lower bound for $\Delta\mu_{\text{Co}}$ and thus correspond to small (negative for $Z=$Ge) defect formation energies for the Vc${}_{\text{Co}}$ vacancies. An apparently different situation is found in Co2TiSn, for which we found the CoTiSn ground state to be the $C1_{b}$ structure but lying outside the Co2TiSn stability range. Investigations performed by Nobata et al.Nobata et al. (2002) found, on the other hand, that certain growth conditions favor the formation of the the Co1.50TiSn with a half-filled vacancy sublattice instead of the stoichiometric compound CoTiSn. This way, the obtained negative formation energy of Co2TiSn:Vc${}_{\text{Co}}$ is consistent with the experimental findings. We do emphasize, however, that all the calculated defect formation energies correspond to the limiting case of zero pressure and temperature.

ii) Co anti-sites (Co${}_{\text{Ti}}$ and CoZ): Following the sequence Si$\to$Ge$\to$Sn, opposite trends can be recognized in the two reference energies $\Delta E(\text{Co}_{2}\text{Ti}Z,\text{Co}_{\text{Ti}})$ and $\Delta E(\text{Co}_{2}\text{Ti}Z,\text{Co}_{Z})$ that correspond to the two Co anti-sites. While the former increases with the atomic number of the $Z$-atom, and thus with the lattice constant of the Co2Ti$Z$ compound, the latter decreases. Accounting, however, for the allowed variations in the reduced chemical potentials, which enter as $(-\Delta\mu_{\text{Co}}+\Delta\mu_{\text{Ti}/Z})$, the two formation energies $E_{\text{form}}[\text{Co}_{\text{Ti}}]$ and $E_{\text{form}}[\text{Co}_{Z}]$ are found to be of comparable size for a given system Co2Ti$Z$. This indicates that both anti-sites may appear with roughly equal probability during the sample preparation.

iii) Ti anti-sites (Ti${}_{\text{Co}}$ and TiZ): Similar to the previous case, the reference energy of the TiZ anti-site exhibits a strong dependence on the $Z$-atom, but with an even a more significant drop for the heavier elements. As seen in Table 2, $\Delta E(\text{Co}_{2}\text{TiSn},\text{Ti}_{\text{Sn}})$ even becomes negative One can conclude that, when Ti is in excess, a preference for the TiZ anti-site should be observed, at the expense of Ti${}_{\text{Co}}$. This is quite an opposite trend as the one seen in Co2MnSi, where the Mn${}_{\text{Co}}$ anti-site was found to have a much smaller reference energy than Mn${}_{\text{Si}}$.Hülsen et al. (2009b) As a general observation, we emphasize here the fact that the lower/upper boundary for an anti-site $A_{B}$ becomes the upper/lower boundary with opposite sign for its $B_{A}$ counterpart. Comparing now the two ’paired’ anti-sites Ti${}_{\text{Co}}$ and Co${}_{\text{Ti}}$, we note again that the allowed $\Delta\mu_{\alpha}$ intervals significantly influence the formation energy results. Indeed, whereas $\Delta E(\text{Co}_{2}\text{Ti}Z,\text{Ti}_{\text{Co}})$ is about half the size of $\Delta E(\text{Co}_{2}\text{Ti}Z,\text{Co}_{\text{Ti}})$, the corresponding $E_{\text{form}}$ values have a rather broad overlapping interval, with a lower boundary for the Co${}_{\text{Ti}}$ anti-site.

iv) $Z$ anti-sites ($Z_{\text{Co}}$ and $Z_{\text{Ti}}$): As could be anticipated from the VcZ case, the sublattice formed by the group IV elements appears to be the most stable. The $Z_{\text{Co}}$ anti-sites exhibit by far the largest formation energies and are therefore the least likely to occur. The formation energies obtained for the $Z_{\text{Ti}}$ defects are somewhat smaller, but still larger than their $\text{Ti}_{Z}$ correspondents. We further note that the $\Delta E(\text{Co}_{2}\text{Ti}Z,Z_{\text{Ti}})$ values increase along the Si$\to$Ge$\to$Sn series, in contrast to the trend observed for the $\text{Ti}_{Z}$ anti-sites.

v) swaps (Ti $\leftrightarrow$ Co,Ti $\leftrightarrow$ $Z$, and Co $\leftrightarrow$ $Z$): The calculations performed by Picozzi et al.Picozzi et al. (2004) for the Co2MnSi full Heusler alloy found an interesting feature in the formation energy of the Mn$\leftrightarrow$Co swap: its value closely matches the sum of the reference energies of the Mn${}_{\text{Co}}$ and Co${}_{\text{Mn}}$ anti-sites. Our results do not reproduce this behavior in any of the Co2Ti$Z$ compounds, regardless of the considered swaps. A possible reason may be the large differences between the corresponding reference energies of the paired anti-sites, which could efficiently curtail the swapping tendencies. As Table 2 shows, we find very large formation energies for all investigated swaps. These findings appear to be in contradiction with the experimental results obtained for the Co2TiSn Heusler alloy,Kandpal et al. (2007b) for which the authors report NMR data indicative of the crystal structure being partially $D0_{3}$-like, consistent with a Ti$\leftrightarrow$Co swap.

To conclude, we found several defects in Co2Ti$Z$ to have quite a low formation energy, whereas others, in particular the swaps and those related to the $Z$-atom sublattice, should occur with reduced probability. Some of the more likely defects have been selected for subsequent investigations concerning the effect they have on the electronic structure and the implications for the transport properties. These results form the subject of the next sections.

IV Defects-triggered changes in the

electronic structure

We present in this section the results obtained employing the FP-SPR-KKR Green function method, modeling the intrinsic defects as dilute alloys. We focus on those systems which were found to have a small formation energy and follow the changes in the electronic structure brought about by the defects, emphasizing on specific aspects relevant for the transport properties. In doing so, we use the Bloch spectral function (BSF),Faulkner and Stocks (1980) expressed as the Fourier transform of the configurationally averaged Green function

[TABLE]

performed over $N$ atomic sites with position vectors ${\vec{R}}_{m}$ and ${\vec{R}}_{n}$ in the crystal lattice. This quantity, which can be seen as a $\vec{k}$-resolved density of states (DOS), is well-defined also for ordered systems. In this case the configurational average, symbolized by $\langle\ldots\rangle$, drops out and the evaluation takes place for a complex energy argument $E+i\varepsilon$ with $\varepsilon\to 0$, providing the ordinary dispersion relation $E_{\vec{k}}$.

Taking the Co2TiSi full Heusler alloy as an illustrative example, we present the BSFs for a series of dilute alloys modeling the Co2TiSi:$D$ defects. Calculated on a fully relativistic level, the BSFs are subsequently projected on their spin components and compared with the standard band structure of the parent compound. This allows us to identify signatures of the defect-induced minority-spin impurity bands in the proximity of the half-metallic gap. We will also show that, owing to the SOC, a majority-spin d-band that crosses the Fermi energy $E_{F}$ gains minority-spin character. This rather ubiquitous effect is larger for the Co-related defects Vc${}_{\text{Co}}$ and Co${}_{\text{Si}}$, significantly reducing the spin polarization near $E_{F}$.

IV.1 Results for Co_2TiSi bulk

The spin-polarized relativistic electronic structure for the ordered compound Co2TiSi is displayed in Fig. 4, with panel (a) depicting the calculated dispersion relation along several high symmetry directions in the fcc BZ and panel (b) the corresponding spin-resolved BSF obtained for an imaginary part of the energy $\varepsilon=0.005$ Ry along the $K-\Gamma-X$ path of the BZ. The spin-resolved results of Fig. 4(b), consistent with previously reported band structure calculations on a scalar relativistic level,Barth et al. (2010); Sharma et al. (2010) demonstrate the half-metallic character of this system. The Fermi energy $E_{F}$, taken here as the energy reference, is seen to lie close to the upper edge of the band gap appearing in the minority-spin channel.

Of particular interest for the transport properties, in general, and for the Seebeck coefficient in particular, are the states located in the proximity of $E_{F}$, according to Eq. (8). As seen in Fig. 4(a), there are only two bands that cross the Fermi energy. Starting from $1.4$ eV at the $\Gamma$-point, one band goes down in energy and gets below $E_{F}$ at the $X$- and $W$-points. It thus leads to a Fermi surface enclosing the $\Gamma$-point and forming pockets at the BZ edge. The second band rises in energy from $-1.0$ eV at the $\Gamma$-point and forms a Fermi surface connecting to adjacent BZs. Stemming from the Co d-orbitals, these two bands possess a dominant majority spin character, as revealed by the BSF shown in Fig. 4(b), and are expected to dominate the transport properties in these compounds. A much flatter Co d-band is found slightly above the Fermi energy, forming the minority-spin conduction band. Its minimum lies at the $X$-point of the BZ, such that the half-metallic gap, as seen on the right side of Fig. 4(b), is indirect. We note here that, although the isoelectronic compounds Co2TiGe and Co2TiSn exhibit qualitatively similar results, one does find quantitative variations in both the width of the half-metallic gap and the position of the Fermi energy inside it. For the former, the FP-SPR-KKR calculations give $0.76$, $0.57$, and $0.48$ eV, respectively, for $Z=$ Si, Ge, and Sn. Measured relative to $E_{F}$, the minority-spin conduction band minimum, on the other hand, was found at $0.18$, $0.21$, and $0.26$ eV.

Highlighted in Fig. 4 are several areas along the $W-\Gamma$, $K-\Gamma$, and $\Gamma-X$ directions where this minority-spin d-band intersects the two majority-spin d-bands, in a range of $0.2-0.4$ eV above $E_{F}$. Qualitative differences in these intersections are easily recognizable: The falling majority-spin band has the same parity (and symmetry) as the minority-spin band, which leads to an anti-crossing between the two bands. In turn, the other majority-spin band rising from the $\Gamma$-point crosses the minority-spin band without coupling to it, as seen along the $W-\Gamma$ and $K-\Gamma$ directions.

The repelling of the two bands of different spin character is caused by SOC. By virtue of a mechanism suggested by Mavropoulos et al.Mavropoulos et al. (2004) when discussing the effect of spin-orbit coupling on the band gap of half metals, the coupling of the two bands causes a strong spin mixing. Regarding the SOC as a perturbation to the spin-dependent crystal Hamiltonian with eigenvalues $E^{\uparrow}(\vec{k})$ and $E^{\downarrow}(\vec{k})$, the potential terms causing the spins to flip are proportional, in leading order, to $1/[E^{\uparrow}(\vec{k})-E^{\downarrow}(\vec{k})]$. It becomes apparent that the spin mixing will exhibit a maximum where the unperturbed spin-up and spin-down bands would cross. Indeed, as evidenced in Fig. 4(b), the BSF of a given spin character (here obtained within a Dirac formalism) is an ”image” of the opposite spin over a broad range around the anti-crossing points. This effectively leads to a reduced spin polarization especially at the top edge of the half-metallic gap,Mavropoulos et al. (2004) since the states in its vicinity no longer posses a unique spin character. We note that a similar imaging effect occurs below the Fermi energy around the $\Gamma$-point. As can be seen in Fig. 4(b), one finds a set of valence bands that exhibit finite amplitudes of the BSF in both spin channels. The reason for spin mixing in this case is related to the intra-atomic SOC, with the states originating from d-orbitals with large magnetic quantum numbers, $\pm 5/2$ and $\pm 3/2$.

IV.2 Spectral function of dilute alloys

We have selected, from the list of all possible intrinsic defects investigated above, a total of six systems, exhibiting low formation energies. According to the results presented in Table 2, these are: Vacancies on Co sublattice Vc${}_{\text{Co}}$; the Co anti-sites CoZ and Co${}_{\text{Ti}}$; the Ti anti-sites Ti${}_{\text{Co}}$ and TiZ; and the Si anti-site $Z_{\text{Ti}}$. We employed a dilute alloy modeling for all these systems, as described in Section II.3, by considering a defect of type $D_{A}$ as a single-site effective medium $(A_{1-x}D_{x})$ while keeping the other sublattices unchanged. Without loss of generality, the discussion below is restricted to the Co2TiSi:$D$ systems ($Z=$ Si) with $x=0.03$, corresponding to a single defect in the 128-atom supercell used in the formation energy calculations.

Our primary interest is to investigate the changes in the electronic structure occurring in the proximity of the Fermi energy. As shown above, only two bands do cross $E_{F}$, both having a dominant majority-spin character. For this spin channel we found that, while the BSFs associated with these bands may differ quantitatively from one system to another, no qualitative deviations occur, irrespective of the group IV element $Z$ and the type of defect. We shall therefore focus in the following on the minority-spin BSFs in a $\simeq 2$ eV energy window around $E_{F}$, with emphasis on two aspects: i) the defect-induced states evolving in impurity bands (IB) near the half-metallic gap; and ii) the so far unexplored effect of the defects on the SOC-induced band anti-crossings observed for the ordered compound.

The minority-spin BSFs for the off-stoichiometric Co2TiSi:$D$ systems are shown in Fig. 5. Note that, while choosing the same $K-\Gamma-X$ direction in the BZ as for the Co2TiSi BSFs in Fig. 4(b), for clarity purposes the energy interval around the Fermi energy $E_{F}$ has been reduced. Moreover, since we are dealing now with an alloy system, Eq. (12) is evaluated for real energy arguments and the broadening of the alloy bands is due to the intrinsic disorder of the CPA effective medium. We first analyze, on the basis of Fig. 5, the defect-related states appearing in the investigated systems, appropriately labeled from (1) to (5) in each panel.

(1) Vacancies in the Co-sublattice, Fig. 5(a), induce minority-spin states at the edges of the half-metallic gap, thus reducing its width and, consequently, the size of the spin gap. The latter is defined as the minimum energy required to flip the electron spin. The side-by-side comparison of the different panels of Fig. 5 reveals that the highest minority-spin valence band maximum is obtained for Co2TiSi:Vc${}_{\text{Co}}$. Actually, within a broad range around the $\Gamma$-point the minority-spin BSF of this system is characterized by a much stronger intensity as compared to the other alloys. These results are consistent with those obtained by Özdoğan et al.Özdoğan et al. (2007) for various Co-based Heusler alloys from the series Co2(Mn,Cr)(Al,Si). Similar to these systems, we find that, although the presence of Co-vacancies reduces the total magnetization (e.g., from $2.00$ to $1.90$ $\mu_{B}$ per formula unit in the case of $3$ % vacancy concentration in Co2TiSi), the half-metallicity is not destroyed. Because of the large value of the spin gap, we expect the half-metallicity to be quite robust against Co-vacancy formation, at reasonably low concentrations, in all Co2Ti$Z$ compounds.

(2) The Co${}_{\text{Si}}$ anti-site, Fig. 5(b), creates two minority-spin IBs near the Fermi energy. These are distributed rather symmetrically below and above $E_{F}$, centered around $-0.3$ and $0.2$ eV, respectively. Thus, the lower band falls inside the half-metallic gap, whereas the higher one overlaps with the edge of the minority-spin conduction band. In spite of the IB present in the gap, the Co2TiSi:Co${}_{\text{Si}}$ also remains half-metallic. As further indicated in Fig. 5(b), the BSF has been multiplied by a factor $10$ in the energy interval $[-0.4,-0.2]$ eV for all $\vec{k}$-vectors, a value that represents the relative weight of the DOS associated to the high and low IBs. Although running rather parallel to each other, these two bands differ qualitatively in their dependence on $\vec{k}$. While the upper IB has a large weight near the $\Gamma$-point, enhancing and significantly broadening the BSF, the low one exhibits a maximum near the $X$-point, at the edges of the BZ.

(3) The Co${}_{\text{Ti}}$ anti-site, Fig. 5(c), creates a single IB in the energy region of interest. Located inside the half-metallic gap, this band is nearly touching the Fermi level, such that the Co2TiSi:Co${}_{\text{Ti}}$ system is predicted to loose its half-metallic character. A comparison with the low Co${}_{\text{Si}}$-IB reveals qualitatively similar $\vec{k}$-distributions in the BSF, with the amplitude increased far away from the $\Gamma$-point. The two cases are, however, quantitatively substantially different. As indicated by the different multiplicative factors applied, the Co${}_{\text{Ti}}$-BSF amplitude is about twice as large as compared to the Co${}_{\text{Si}}$ one. One further notes, in addition, the much weaker broadening of the BSFs in this system, with the alloy bands largely preserving their bulk character.

The features related to the Co anti-sites discussed here are in very good agreement, both in their energy position and in their amplitude, to those reported in the literature for other Co-based full Heusler alloys such as Co2MnSi,Picozzi et al. (2004); Hülsen et al. (2009b) Co2MnGe,Picozzi et al. (2004) Co2CrAl,Alhaj and Hamad (2013) and Co2VSn.Mahmoud et al. (2013)

(4) Another IB falling inside the half-metallic gap is the one related to the Ti${}_{\text{Co}}$ anti-site, shown in Fig. 5(d). One notes that, compared to the previously surveyed IBs, this has the highest intensity, no up-scaling having been applied in this panel. It is located $\simeq 0.2$ eV below $E_{F}$, thus bearing no influence on the half-metallicity of the system and, in contrast to the previous ones, is strongly peaked around the BZ center.

(5) Finally, the signature of a Ti${}_{\text{Si}}$-related IB can be seen in Fig. 5(e) in the vicinity of the $X$-point, at $\simeq 0.3$ eV above the Fermi energy. It is close to and eventually merges into a band complex stemming from the d-orbitals of the Ti sublattice. Owing to its location, this IB has no direct influence neither on the half-metallic nor on the transport properties of the system.

Highlighted by red ellipses in Fig. 5 are the spin-mixing d-d band anti-crossings which lie in the vicinity of the Fermi energy. We focus on the minority-spin image of the falling majority-spin band. Its spectral function shows variations from system to system, with the SOC-mediated spin-mixing effect strongly depending on the defect type. The Vc${}_{\text{Co}}$ and Co${}_{\text{Si}}$ defects are characterized by a continuous BSF with large values extending deep into the half-metallic gap. For the Ti-sublattice-related defects Co${}_{\text{Ti}}$ and Si${}_{\text{Ti}}$, the amplitude of spin mixing is smaller and the BSF exhibits merely a discrete rather than a continuous character. In the case of the Ti${}_{\text{Co}}$ and Ti${}_{\text{Si}}$ anti-sites, the flat Co band preserves almost completely its minority-spin character. The band repulsion is very weak and large BSF values are found only in a small energy interval around the anti-crossing point.

The most important observation to be made at this point concerns the possible changes in conductivity that may result from these anti-crossings. This is because the two alloy bands involved differ not only in their dominant spin character, but also in their associated Fermi velocity. One can think of a defect creating additional minority-spin states in the appropriate energy range. As a result of the spin mixing, these states will not only gain majority-spin character; but they will also borrow the dispersion of the band with which they mix. A direct consequence, shown below to occur in the two systems containing Vc${}_{\text{Co}}$ and CoZ, is an increased conductivity in the energy range near the band mixing.

We close this section with a brief comment on the dependence of the discussed effects on the group IV element $Z$ in the Co2Ti$Z$ series. Taking as an example the CoZ anti-site system modeled as Co2Ti($Z_{0.97}$Co0.03), we show in Fig. 6 the BSFs of the three different alloys for both spin components.

The majority-spin states (top panel) close to the Fermi energy do not vary too much when changing the $Z$-atom. A slight upward shift of the valence bands can be noted upon increasing the atomic number from Si to Sn. It originates from hybridization between the Co-d and the $Z$-p states which occurs already in the ordered compounds,Sharma et al. (2010); Picozzi et al. (2002) and is manifesting itself in both spin channels. Indeed, as seen in the bottom panel of Fig. 6, also the minority-spin d-bands are found at higher energies in Co2TiSn:Co${}_{\text{Sn}}$ as compared to Co2TiSi:Co${}_{\text{Si}}$. The effect is more pronounced in the valence band, leading to a monotonous reduction of the half-metallic gap in the series Si$\to$Ge$\to$Sn. The two IBs related to the CoZ anti-site follow a similar upwards shift while maintaining a nearly constant separation. The striking resemblance of the minority-spin BSFs in the Si- and Ge-based systems should be noted here, while Co2TiSn:Co${}_{\text{Sn}}$ exhibits qualitative differences. On the one hand side, the low IB approaches the Fermi energy, thus strongly reducing its half-metallic character. The upper IB, on the other hand, moves above the Co-d band at $0.3$ eV, with a subsequent strong reduction of the BSF around the $\Gamma$-point.

V Influence of defects on the transport

properties

This section presents the calculated transport properties of the Co2Ti$Z$:$D$ systems, obtained by employing the CPA-based dilute alloy modeling. We start with a detailed analysis of the series of Co2TiSi:$D$ systems with $3$ % defect concentration surveyed in the previous section. The results obtained for the energy-dependent electronic conductivity are used to interpret the calculated Seebeck coefficient, shown in Fig. 1. The connection of both quantities with the particular features of the underlying electronic structure will be appropriately evidenced. Our discussion will then be extended by covering different defect concentrations and all the group IV elements $Z$ in Co2Ti$Z$:$D$. We show that the conclusions derived from the Co2TiSi:$D$ systems are of general validity in the dilute limit, with the Seebeck coefficient exhibiting only a weak dependence on both factors.

V.1 Electronic conductivity in Co_2TiSi:D

The calculated energy dependent electronic conductivity of the Co2TiSi:$D$ dilute alloys is shown in Fig. 7. We only plot the longitudinal $\sigma_{xx}(E)$, given relative to the Fermi energy $E_{F}$ of each system (dashed vertical line). Indeed, we found that, over a broad energy range, $E_{F}\pm 0.1$ eV, the difference $\sigma_{zz}(E)-\sigma_{xx}(E)$, a measure for the anisotropic magnetoresistance (AMR), does not exceed $1$ % for all the investigated systems.

The results of our calculations indicate a strong variation of the residual conductivity $\sigma_{xx}(E=E_{F})$ with the type of defect, with values ranging from $1.75$ $\mu\Omega^{-1}\text{m}^{-1}$ for Ti${}_{\text{Si}}$ to $5.20$ $\mu\Omega^{-1}\text{m}^{-1}$ for Vc${}_{\text{Co}}$ at $3$ % defect concentration. All these values are higher than the $0.45$ $\mu\Omega^{-1}\text{m}^{-1}$ deduced from the experimentally reported residual resistivity,Barth et al. (2010) indicating additional sources of carrier scattering in the samples. Consistent with the decrease of $\sigma_{xx}(E=E_{F})$ with defect composition observed in our calculations, these could be point defects in high concentration and/or other crystal imperfections. Analogous results were obtained also for Co2TiGe:$D$ and Co2TiSn$:D$. These defect-dependent quantitative differences directly reflect the changes occurring in the majority-spin bands that cross the Fermi energy. Qualitatively, while a smooth dependence of all $\sigma_{xx}(E)$-curves in the $E_{F}\pm 0.15$ eV energy interval is obvious, several peculiarities can be noted: (i) in four cases the conductivity dependence on its energy argument exhibits a negative slope around $E_{F}$; (ii) a steep increase of $\sigma_{xx}(E)$ occurs at energies above $\simeq 0.18$ eV; (iii) in two of the investigated systems, Co2TiSi:Vc${}_{\text{Co}}$ and Co2TiSi:Co${}_{\text{Si}}$, the conductivity is characterized by a positive slope around the Fermi energy. The reasons for these qualitative differences are discussed in detail in the following.

(i) The negative slope in the energy dependence of $\sigma_{ii}(E)$ around $E_{F}$ obtained in the case of Co${}_{\text{Ti}}$, Si${}_{\text{Ti}}$, Ti${}_{\text{Co}}$ and Ti${}_{\text{Si}}$ defects is similar to that found in the bulk Co2Ti$Z$ systems.Geisler et al. (2014); Barth et al. (2010) Indeed, for all these materials the DOS decreases above $E_{F}$ such that, in the approximation of an energy-independent scattering, the conductivity is expected to decrease with increasing energy $E$. One can then conclude that, at least in a range extending not far away from the Fermi energy, the presence of these four defects leads to a weak, nearly energy-independent scattering of the electrons. This generic behavior can be modified, however, if IBs are present. This is the case for the Ti${}_{\text{Co}}$ anti-site, where the IB found below $E_{F}$ [labeled (4) in Fig. 5(d)] adds a hole-like contribution to the conductivity. The $\sigma_{xx}(E)$-curve for this system (right panel of Fig. 7) exhibits a peak below the Fermi energy, not seen for example, in the Ti${}_{\text{Si}}$ case. Let us note that no evidence was found for contributions to $\sigma_{ii}(E)$ in the case of Co${}_{\text{Si}}$- or Co${}_{\text{Ti}}$-IBs lying below $E_{F}$.

(ii) The strong increase in $\sigma_{ii}(E)$ at higher energies, $\simeq 0.18$ above $E_{F}$, is caused by the onset of the minority-spin conduction band stemming from the Co-d orbitals. The rise to a plateau of $12-14$ $\mu\Omega^{-1}\text{m}^{-1}$ appears as an abrupt jump due to the chosen scale of the figure. One has to note here that we have purposefully chosen a narrow interval for displaying $\sigma_{xx}(E)$ in order to emphasize on its $E$-dependence around $E_{F}$. We have thoroughly checked the results against numerical in the $E$-range around the raising point by refining the BZ-integration for a $\vec{k}$-mesh as high as $6\cdot 10^{6}$ points and found no smoothening in $\sigma_{xx}(E)$. Moreover, the calculated $\sigma_{xx}(E)$ values correspond to $T=0$ K, such that, at any finite temperature, the sharp steps will be washed out. For two of the systems, containing the Vc${}_{\text{Co}}$- and Ti${}_{\text{Co}}$-defects, the $\sigma_{xx}(E)$ increase is less abrupt. In these cases the minority-spin Co band, stemming from the perturbed sublattice, is effectively broadened as a result of alloying. Moreover, the presence of the impurity on a single Co sublattice [see Fig. 2(b)] destroys the inversion symmetry of the host crystal. Under this condition, SOC leads to scattering between states that are no longer degenerate, increasing the magnitude of the energy-dependent scattering.

(iii) In contrast to the situation above, the positive slope of $\sigma_{xx}(E)$ near $E_{F}$ observed for the Co2TiSi:Vc${}_{\text{Co}}$ and Co2TiSi:Co${}_{\text{Si}}$ indicates a strong energy-dependent scattering occurring in these systems. Following the inversion symmetry suppression in the former and the presence of a strong impurity band in the latter, the electronic conductivity increases right above $E_{F}$. In both cases we suggest that a SOC-mediated broadening and redistribution of states taking place around the anti-crossing points is responsible for the observed effect. As pointed out above in the BSF analysis, additionally created minority-spin states may borrow mobility, via spin-mixing, from the majority-spin bands.

V.2 Seebeck coefficient in Co_2TiSi:D

By virtue of Eqs. (8) and (9), the Seebeck coefficient $S(T)$ is expressed as the quotient of the first and zeroth moments of the conductivity. The findings related to the changes in the energy dependence of $\sigma_{ii}(E)$ can then be directly transferred to the corresponding results obtained for $S(T)$ presented above in Fig. 1. One can easily understand the broad range of values, varying both in sign and magnitude, obtained for the different kinds of defects as arising from variations in either ${L}^{(0)}_{ii}$ (denominator) or ${L}^{(1)}_{ii}$ (numerator).

Assuming an energy-independent scattering, our theoretical results predict a positive Seebeck coefficient for the bulk Co2Ti$Z$ materials,Geisler et al. (2014); Barth et al. (2010) a finding which is in contrast to the large, negative values reported experimentally, reaching as much as $-30$ $\mu$V/K in Co2TiSi/Ge and $-50$ $\mu$V/K in Co2TiSn.Barth et al. (2010); Balke et al. (2010)

As seen in Fig. 1, this tendency of a positive Seebeck coefficient is retained for the systems that have a weakly energy-dependent scattering, such as Co${}_{\text{Ti}}$, Ti${}_{\text{Co}}$, Si${}_{\text{Ti}}$, and Ti${}_{\text{Si}}$. As the temperature increases, the contribution of the minority-spin electrons becomes more important. This is because the minority-spin bands above the half-metallic gap become populated at higher temperatures. As shown above, in the case of defects with a weakly energy-dependent scattering the onset of conductivity in the minority-spin bands leads to an abrupt rise in the conductivity and thus to a negative, electron-like contribution to the Seebeck coefficient. As a result the Seebeck drops sharply and becomes negative at elevated temperatures for the systems containing Co${}_{\text{Ti}}$, Si${}_{\text{Ti}}$, and Ti${}_{\text{Si}}$. One notes here the direct relation between the change of slope for $S(T)$ and the ever increasing contribution coming from the flat minority-spin band. The exception to this behavior is represented by the Ti${}_{\text{Co}}$ anti-site where the presence of the IB below $E_{F}$ was shown to add a hole-like contribution to the conductivity, a contribution which is large enough such as to preserve a positive Seebeck coefficient throughout the whole investigated temperature range.

In the systems with a strong energy-dependent scattering, the conductivity was found to be larger above $E_{F}$ than below. This leads to a negative Seebeck coefficient, as obtained for the Co2TiSi:Vc${}_{\text{Co}}$ and Co2TiSi:Co${}_{\text{Si}}$. We note that only in these cases the results qualitatively reproduce the experimentally observed behavior. For the Co${}_{\text{Si}}$ anti-site the calculated $S(T)$ deviates within $30$ % from the experimental data for $T\leq 300$ K. As we will show below, only varying the defect concentration is not sufficient to further improve the agreement between our calculations and experiment, which is particularly deteriorating at higher temperatures. Indeed, it is expected that spin fluctuations (not accounted for here) become important when the Curie temperature is approached, and that magnon drag effects will lead to an enhanced thermopower. We note that, within a hydrodynamic theory of such effects, the sign of the Seebeck coefficient obtained from an independent-electron calculation is preserved.Watzman et al. (2016)

V.3 Effect of defect concentration and of group

IV element

We have established so far the needed link between the electronic structure and, through the energy dependence of the electronic conductivity, the main features of the Seebeck coefficient in the Co2TiSi:$D$ systems. We now extend our analysis to a broader quantitative level by investigating two effects: (i) a varying defect concentration, and (ii) different group IV element in the Co2Ti$Z$:$D$ series.

Figure 8 shows the calculated temperature-dependent Seebeck coefficient $S(T)$ for three different off-stoichiometric compositions $x=0.01$, $x=0.03$, and $x=0.05$. Due to their fairly similar behavior to that of the Co${}_{\text{Ti}}$-system, the corresponding Ti anti-sites related curves Ti${}_{\text{Co}}$ and Ti${}_{\text{Si}}$ have been omitted in this figure.

We start by noting that a monotonous decrease of $\sigma_{ii}(E)$ as a function of defect composition was observed for all investigated systems. Accordingly reflected in the integrated quantities defined by Eq. (8), $L_{ii}^{(0)}$ decreases with $x$ irrespective of the defect considered. The differences in the $S(T)$ concentration dependence noticed in Fig. 8, weak for Vc${}_{\text{Co}}$ and Si${}_{\text{Ti}}$ and rather pronounced for Co${}_{\text{Ti}}$ and Co${}_{\text{Si}}$, can be related to the behavior of the $L_{ii}^{(1)}$ term, measuring the asymmetry of $\sigma_{ii}(E)$ about the Fermi energy.

Indeed, if this term remained constant with $x$, one would obtain a monotonous increase in the absolute value of the Seebeck coefficient, $|S(T)|$, which is obviously the case for the Co${}_{\text{Ti}}$ defect, panel (c) of Fig. 8. For Vc${}_{\text{Co}}$ and Si${}_{\text{Ti}}$ [panels (a) and (d)] the nearly concentration-independent $S(T)$ indicates an asymmetry term that decreases with $x$, without changing its sign, thus compensating the decrease of $L_{ii}^{(0)}$. Note that, in the case of Si${}_{\text{Ti}}$, the shift of the $S(T)$ peak is caused by a slight upward shift in the Fermi energy taking place with increasing Si content. An even faster decrease of $L_{ii}^{(1)}$ was found to occur for the Co${}_{\text{Si}}$ system [panel (c)]. All these cases can be understood as a consequence of a broadening and smoothening, with increasing concentration $x$, of the additional contributions to $\sigma_{ii}(E)$ from above $E_{F}$.

Analogous considerations can be applied when analyzing the dependence of the Seebeck coefficient on the group IV element $Z$ in the isoelectronic systems Co2Ti$Z$:$D$. The corresponding results for all investigated defects at a net composition of $x=0.03$ are shown in Fig. 9.

Although the conductivity is mainly dominated by majority-spin carriers, one can ascribe the various differences in the $Z$-atom dependence of $S(T)$ as being mostly related to changes occurring in the minority-spin channel. Indeed, the BSFs displayed in Fig. 6 for the Co2Ti$Z$:Co${}_{\text{Si}}$ system hint to a nearly unchanged majority-spin DOS around $E_{F}$ for all three systems with $Z=$ Si, Ge, and Sn. Yet there is no simple rule, applicable to all defects, that would explain how $\sigma_{ii}(E)$ depends on the atom $Z$ of the host material. The lack of such a rule is reflected in the Seebeck coefficient being the largest in Co2TiSi:Ti${}_{\text{Si}}$ for the TiZ anti-site [panel (e) of Fig. 9] and the smallest in Co2TiSi:Si${}_{\text{Ti}}$ for $Z_{\text{Ti}}$ [panel (f)].

We have previously identified three systems, Co${}_{\text{Ti}}$, TiZ, and $Z_{\text{Ti}}$, which are characterized by a sudden change of slope in $S(T)$ at elevated temperatures. This behavior was associated with the minority-spin bands above the half-metallic gap becoming populated with increasing temperature. As the offset of these bands relative to $E_{F}$ increases with $Z$ (see Fig. 6), the peak in $S(T)$ also shifts towards higher $T$-values when $Z$ changes from Si to Ge and then Sn.

The absence of such a peak in Co2TiSi:Ti${}_{\text{Co}}$ was shown to be caused by the IB present $0.1$ eV below $E_{F}$, states introducing a rather large hole-like contribution to the conduction. A similar IB is present in all Co2Ti$Z$:Ti${}_{\text{Co}}$ systems, with its relative position inside the half-metallic gap remaining fairly the same.

For the two defects for which the Seebeck coefficient had a negative sign, Co2TiSi:Vc${}_{\text{Co}}$ and Co2TiSi:Co${}_{\text{Si}}$, we find this sign to persist for Co2TiGe and Co2TiSn. Moreover, we note that for both defects $|S(T)|$ increases with increasing atomic number of the group IV element $Z$, a rather non-intuitive behavior. Indeed, the calculated $\sigma_{ii}(E)$, which is governing the size of $L_{ii}^{(0)}$, also increases monotonously along the series Si$\to$Ge$\to$Sn. It follows that the increase in $|S(T)|$ can only be accounted for by a much larger increase in $L_{ii}^{(1)}$, the asymmetry term. Noting that the SOC strength also increases as moving down in the group IV, we conclude that the obtained results are consistent with the interpretation given about the role of the SOC in causing a largely asymmetric conductivity. It consists of a redistribution of states in the systems lacking inversion symmetry (as in Co2Ti$Z$:Vc${}_{\text{Co}}$) and an increased mobility of the minority-spin IB by mixing it with the majority-spin band crossing the Fermi energy, in Co2Ti$Z$:CoZ.

VI Conclusions

The Ti-based Heusler alloys Co2Ti$Z$ ($Z=$ Si, Ge, Sn) have been studied by means of density-functional calculations with the goal to explore their suitability for spintronics and spincaloric applications. Since the ideal Heusler alloys of this composition are ferromagnetic half metals, it is of particular interest to learn how the half-metallicity is modified in realistic materials, including off-stoichiometry and atomic-scale disorder. To this end, the formation energies $E_{\text{form}}$ of intrinsic point defects have been calculated, and their consequences for electronic transport and thermopower have been explored.

While the formation of anti-structure defect pairs by atomic swaps is generally found to be energetically costly, our calculations indicate a very high sensitivity of the ternary Heusler alloys to deviations from their ideal stoichiometry. Once formed in appreciable numbers, point defects caused by off-stoichiometry significantly influence the electrical conductivity. Moreover, the Seebeck coefficient turns out to be a sensitive probe for the presence of defects, even in the simple situation treated here, of single defect formation, rather than interacting pairs.

The lowest values of Eform are found for Vc${}_{\text{Co}}$ and TiZ, followed by the Co-related anti-sites Co${}_{\text{Ti}}$ and CoZ. The formation energy of TiZ may become negative in the case of $Z=$ Sn, and is still small in the Si and Ge compounds ($0.48$ eV and $0.30$ eV, respectively). Under Co-deficient conditions, the formation energy of Vc${}_{\text{Co}}$ may become negative for all three alloys studied, indicating that Co vacancies may occur spontaneously. Intrinsic point defects are to be expected also under Co-rich conditions, since E${}_{\text{form}}$ of CoZ and of Co${}_{\text{Ti}}$ are fairly low (in the $0.3$ to $0.6$ eV range) for Heusler alloys containing Si or Ge, while being somewhat higher in Co2TiSn. Hence, anything that comes close to ideal Heusler alloys can be synthesized only under very special conditions when the stoichiometry is met exactly, since slight deviations from stoichiometry, both to the Co-rich or the Co-poor side, are likely to lead to point defect formation. Further support for the abundance of defects comes from measurements of the electrical conductivity, where a high residual resistivity, and only a moderate increase (in the range of $30$ to $50$ %) from low- to room-temperature, have been observed experimentally.Barth et al. (2010) This points to a relatively high concentration of native defects in the material, formed already during sample preparation.

While one would expect from the density of states at the Fermi energy a positive Seebeck coefficient in all three alloys, experiments reported negative values throughout the whole temperature range.Barth et al. (2010); Balke et al. (2010) In this context, it is important that our calculations of electronic transport demonstrate a strong effect of point defects on the thermopower of the Co2Ti$Z$ Heusler alloys. At defect concentrations of a few percent, the results obtained here indicate that the magnetic order and the half-metallicity in these materials is largely retained. Hence, the majority-spin carriers dominate the electronic transport, ruling out earlier explanations of the negative Seebeck coefficient being due to a breakdown of magnetism and a non-magnetic electronic structure. Instead, we find that the negative Seebeck coefficient is due to subtle defect-induced changes of the electronic structure that occur already for small concentrations of CoZ and, to a lesser extent, of Vc${}_{\text{Co}}$ defects. Both defect types are likely to occur in Co-rich and Co-poor samples, respectively: While Eform of Vc${}_{\text{Co}}$ may even be negative, CoZ has a formation energy of only $0.30$ eV in Co2TiSi and $0.45$ eV in Co2TiGe, being second lowest after VcCo. Hence, both for surplus or a deficiency of Co, the defects with the lowest formation energy induce a negative thermovoltage. Thus, the results of our calculations help to rationalize the experimental findings at low temperatures. In addition, we point out that, both in ideal Ti-derived Heusler alloys as well as in many defected materials (Co${}_{\text{Ti}}$, TiZ, $Z_{\text{Ti}}$), the minority-spin electrons start to contribute to the conductivity, and hence also to the Seebeck coefficient. This is another factor responsible for a persistently negative Seebeck at higher temperatures.

With the Curie temperature of the Co2Ti$Z$ lying in the range of $380$-$400$ K,Barth et al. (2010) spin fluctuations are expected to start to play a role at room temperature and above. Via the pertinent scattering mechanisms for the charge carriers, they might also have an impact on the Seebeck coefficient in this temperature range. Investigations of such complex effects, also including the temperature-induced lattice vibrations as well as a detailed analysis on the spin-dependence of the Seebeck coefficient will make the subject of future work.

In summary, the presented computational results highlight the decisive role played by off-stoichiometry in Heusler compounds, which must be taken into account when one wishes to select materials from this class for specific applications. The thermopower is found to be a sensitive probe for the existence of defects, but remains difficult to interpret even with computational results at hand, in particular in cases where several contributing factors — different types of point defects, phonon and magnon scattering — are superimposed.

Acknowledgements.

This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft – DFG) within the Priority Program 1538 ”Spin Caloric Transport (SpinCaT)”. The authors acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Jülich Supercomputing Center (JSC). Additional access to the Cray-XT6m supercomputer facility has been offered by the Center for Computational Sciences and Simulation (CCSS) at the University Duisburg-Essen.