Temperature dependent resistivity and anomalous Hall eﬀect in NiMnSb from the ﬁrst-principles

(Dated: We present implementation of the alloy analogy model within fully relativistic density functional theory with the coherent potential approximation for a treatment of nonzero temperatures. We calculate contributions of phonons and magnetic and chemical disorder to the temperature dependent resistivity, anomalous Hall conductivity (AHC), and spin-resolved conductivity in ferromagnetic half-Heusler NiMnSb. Our electrical transport calculations with combined scattering eﬀects agree well with experimental literature for Ni-rich NiMnSb with 1 to 2 % Ni-impurities on Mn-sublattice. The calculated AHC is dominated by the Fermi surface term in the Kubo-Bastin formula. Moreover, the AHC as a function of longitudinal conductivity consists of two linear parts in the Ni-rich alloy, while it is non-monotonic for Mn impurities. We obtain the spin polarization of the electrical current P > 90% at room temperature and we show that P may be tuned by a chemical composition. The presented results demonstrate the applicability of eﬃcient ﬁrst principle scheme to calculate temperature dependence of linear transport coeﬃcient in multisublattice bulk magnetic alloys.


I. INTRODUCTION
Microscopic description of finite temperature effects in magnetic materials represents a longstanding challenge for ab initio theory despite tremendous progress over past 20 years in numerically demanding calculation of small quantities such as magnetocrystalline anisotropies or anisotropic magnetoresistance 1-4 . A simulation of electrical transport coefficients at room temperature, that are important for spintronics, requires coupling of electrons to phonons or magnons.
One possibility of ab initio description of electronic coupling to magnons and phonons is based on the alloy analogy model (AAM) which was recently employed to calculate electrical conductivity and the anomalous Hall conductivity (AHC) in elemental ferromagnets and binary alloys 5,6 . The AAM simulates the effect of phonons by transforming atomic displacement from the equilibrium positions to the multicomponent alloy. Also spin fluctuations or the magnetic orientational disorder can be treated analogically in a similar way. The limiting case of full spin disorder is called the disordered local moment (DLM) state 7-10 and describes the paramagnetic state above the Curie temperature.
The AAM employing the coherent potential approximation (CPA) and Kubo-Bastin transport theory was implemented in the framework of the Korringa-Kohn-Rostoker (KKR) method 5,6 while the supercell AAM within the Landauer-Büttiker scattering formalism was employed in the tight-binding linear muffin-tin orbital and its Curie temperature is as high as 730 K 24 . The measured value of the spin polarization of the electrical current is from 45 to 58 % 25-28 at low temperatures and about 50 % at room temperature 29 ; spin polarized photoemission experiments show the spin polarization of the emitted electrons about 50 % at 300 K 30 . The polarization of the ballistic transport for correlated electrons about 50 % was calculated for Au-NiMnSb-Au heterostructures by the SMEAGOL DFT code 31 .
The TB-LMTO method (both LSDA and LSDA+U) was previously used to estimate the Curie temperature, exchange interactions, magnon spectra, and magnetic moments in Ni 2−x MnSb alloys 32,33 . A saturation magnetization of NiMnSb is changing only slightly (by 5 to 10 %) from zero to room temperature 26,34,35 and the magnetic moments were investigated by a polarized neutron diffraction 36 . Treating NiMnSb within LDA+U (for temperature T = 0) results only in a small correction to magnetic moments 33,37 .
Here we apply our CPA-AAM for simulating the temperature dependence of conductivity, AHC and spin polarized conductivity of the prototypical half-Heusler halfmetal NiMnSb. In contrast to the so far investigated materials using the AAM, NiMnSb is more complex and with a richer phenomenology due to two magnetic sublattices, a wide range of possible structure defects with similar formation energies 38 making it difficult to compare calculations and experiment, and Dresselhaus symmetry of its Wyckoff positions allowing for novel spintronics effects such as the observed room-temperature spin-orbit torque in strained NiMnSb 4 . The material has been intensively studied for over a 25 years including AHC and electric resistance 24,25,36,39 which makes it a favorbale system for testing of novel ab initio methods.

A. Structure model and electronic structure calculations
We employ ab initio relativistic TB-LMTO method in combination with the multicomponent CPA and the atomic sphere approximation (ASA) 40 . The effect of temperature on the electronic structure is neglected in the DFT self-consistent electronic structure calculations which turned out to be a good approximation for the temperature range from zero to room temperature. We simulate the effect of disorder via CPA-AAM in the transport calculations in conjuction with using the electronic structure determined at T = 0 K. Because of the displacement transformation of the TB-LMTO potential functions required by the AAM, the spdf −basis is used. We note that (standard) calculations without the displacements employ usually only the spd−basis, especially because of numerical expenses. The transformed potential functions must be expressed in a larger basis; therefore, also functions for f −electrons are included in our basis set.
NiMnSb has the cubic crystal structure C1 b and the experimental lattice constant 24 a latt. = 5.927Å is used. Without chemical disorder, NiMnSb consists of four FCC sublattices Ni-Mn-empty-Sb equidistantly shifted along [111] direction. The empty sublattice denotes interstitial sites, i.e., empty positions in the half-Heusler lattice which would be occupied in the full-Heusler structure. We investigate Mn-and Ni-rich alloys with substitutional disorder, i.e., systems with sublattices (Ni 1−y Mn y )-Mnempty-Sb and Ni-(Mn 1−y Ni y )-empty-Sb, respectively, with y ∈ [0, 0.2]. Notation Ni x Mn 2−x Sb with x from 0.8 (Mn-rich) to 1.2 (Ni-rich) is used for brevity.
These defects are consistent with literature 4 and they have low formation energies 38 : 0.49 and 0.92 eV per formula unit for Mn-and Ni-rich case, respectively. Lower formation energies were obtained for Ni-and Mn-atoms occupying the interstitial crystallographic positions (0.20 eV and 0.73 eV per formula unit, respectively) but our calculated resistivity as a function of temperature significantly underestimates experimental values for these systems.

B. Lattice vibrations
The AAM of finite temperature effects was recently implemented within the TB-LMTO approach and applied to transition metals and simple alloys [19][20][21] . The model treats the vibrational effects by introducing for each single lattice site a mean-field CPA medium constructed from the chemically equivalent atoms but shifted in different spatial directions from their equilibrium position 5 .
The displacements are chosen along high symmetry directions of the studied crystal. The shifts of atoms are realized via a linear transformation of the LMTO potential functions (with energy arguments omitted) whereP 0 is the LMTO potential function of an atom at equilibrium position and the potential function P 0 corresponds to the atom displaced by the vector u. The displacement vectors can be conveniently expressed in terms of displacement matrix D L s ,Ls (u) In Eq. (2), a restriction = + holds; D L s ,Ls (u) = 0 for > and D L s ,Ls (u) = δ L s ,Ls for = . After the transformation given by (1), the screened TB-LMTO potential functions P α are obtained by using the matrix of screening constants α: P α = P 0 (1 − αP 0 ) −1 . The increasing magnitudes of the displacements u correspond to the rising temperature according to the Debye formula. For N displaced atoms, the mean square displacement reads u 2 = 1/N N i=1 |u i | 2 and it is related to temperature T via the Debye approximation 41,42 for atoms with identical masses m and the materialspecific Debye temperature Θ D . For simplicity, we omit the zero temperature fluctuations (the second term in Eq. (3)) that are negligible at ambient temperatures. The Debye function is D n (x) = n/x n x 0 t n /(e t − 1)dt. A standard notation for the reduced Planck constant and the Boltzmann constant k B is used.

C. Magnetic disorder
We investigate the influence of magnetic disorder on the electrical transport within a model of tilted local moments. The mean-field alloy was constructed by substituting a given site occupied by a single local moment oriented along the z-direction by 4 different local moments tilted by the Euler angle θ from the z-axis symmetrically in the four directions x,y,−x, and −y and parametrized by the second Euler angle φ ∈ {0.0π, 0.5π, 1.0π, 1.5π}. Four directions are sufficient for our case.
This approach interpolates between fully-ordered spin ferromagnetic (FM) state (T = 0 K) and fully disordered spin state (DLM, T above the Curie temperature). Attempts to make descriptions of magnetic disorder more realistic were published 5,12,43 . However, a fully ab initio theoretical estimate of temperature-dependence of total magnetization M tot (T ) can be also rather inaccurate because it employs the classical Boltzmann statistics (Monte Carlo) method (see the discussion in quaternary Heusler alloys 44 ).
We aim to estimate only the strength of the magnetic disorder contribution relative to the contribution from phonons and chemical disorder. The order of magnitude is determined from the energy difference between the disordered DLM state and the FM ground state which amounts to ∆E ≈ 12 mRy (0.16 eV) per formula unit. In such approximation, room temperature disorder roughly corresponds to φ = 0.10π. A comparison to an experimentally observed change of the saturation magnetization 26,34,35 would give φ = 0.15π. The use of experimental M tot (T ), if available, may be a better choice but in general, an accurate relation of the tilting angle as a function of the temperature is missing.

D. Transport properties
The full conductivity tensor σ µν (µ and ν are Cartesian coordinates) is calculated by employing the Kubo-Bastin formula. It consists of σ (1) µν and σ (2) µν which are in Ref. 45 called the Fermi surface and the Fermi sea terms, respectively. The first one can be separated into the coherent part σ (1,coh) µν and vertex corrections σ 46. We note that the Fermi sea term contributes only to the antisymmetric part of the tensor σ µν ; the physical meaning is then related to the sum of σ (1,coh) µν and σ (2) µν , see later Fig. 5.
The TB-LMTO method neglects electron motion inside the Wigner-Seitz cells, the velocity operators describe only inter-site hopings 47 , and the resulting effective velocity operators in a random alloy are spinindependent and non-random. The polarization of the spin-resolved currents describes a quality of the spin-dependent transport for the spin index s =↑ and s =↓ 20,48 . In the relativistic treatment of the transport, strictly speaking, one cannot define precisely the spin-resolved conductivities because of nonzero spin-flip contribution to the total conductivity (spin-nonconserving term) The spin-flip contribution was found to be small compared to the total conductivity, e.g., for the Cu-Ni alloy in a wide range of alloy compositions 20 . On the other hand, the spin-flip contribution is essential, e.g., for the Nirich NiFe alloys 49 . Calculating the coherent part of the conductivity tensor projected onto the spin-up and spindown term is a sufficient approximation for half-metals. The projected conductivity in Eq. (4) is then and v µ is averaged Green function and velocity operator, respectively, expressed in auxiliary form suitable for the numerical implementation within the relativistic TB-LMTO formalism after performing the configurational averaging. A real-energy variable is denoted E and f (E) is the energy derivative of the Fermi-Dirac distribution. To simplify the notation, g ± (E) = g(E ±i0) is used. In Eq. (6), σ 0 = e 2 /(4πV 0 N 0 ) depends on the charge of the electron e, on the volume of the primitive cell V 0 , and on the number of cells N 0 in a large finite crystal with periodic boundary conditions. If there was no spin-orbit interaction, in the two-current model 50 , the sum σ (1,coh),↑ µν + σ (1,coh),↓ µν would correspond to the total coherent conductivity.
For an ideal half-metal (with exactly one of the spinchannels insulating), this projection is valid and P → 1 (equals one without the spin-orbit interaction). If both channels are identical, e.g., for nonmagnetic materials, P = 0.
The effect of finite temperature is treated within the AAM. Thus the configurationally averaged quantities g s ± (E) are calculated not only by averaging over the different alloy configurations, but also over distinctly displaced (or magnetization tilted) configurations. The contribution from the Fermi-Dirac distribution can be usually neglected as we checked for several transition metals (Pt, Pd, Fe, Ni). Thus we will use the zero-temperature limits of the conductivity formulas CPA configurationally averaged over the alloy and displacement configurations.

E. Computational details
The mesh of 150 × 150 × 150 k-points in the Brillouin zone was used for transport calculations if not specified otherwise. Smaller numbers of k-points as for, e.g., pure metals, are required because of a large self-energy term originating from chemical or temperature disorder. Increasing the mesh to 200 3 k-points leads to a correction of 0.05 % for the isotropic resistivity.
In previous reports, the Debye temperature was theoretically estimated to be between 250 and 300 K 39 , measured (312 ± 5) K 51 or 322 K 52 and calculated 327 K 53 and 270 K 54 . We used Θ D = 320 K (see later Fig. 4); the above scatter in Debye temperature values leads to approx. 10 % error in the root-mean-square displacement u 2 . The best agreement between experimental data 25,39,55 as concerns the slope of the calculated temperature dependence of the resistivity is obtained for Θ D = 350 K and 2 % Ni-rich NiMnSb.
The Debye theory was derived for systems with identical atomic masses; however, it has also been successfully used for alloys, e.g., Cu-Ni [m(Cu) : m(Ni) ≈ 1 : 0.92] 19 . NiMnSb has [m(Ni) : m(Mn) : m(Sb) ≈ 1 : 0.93 : 2.07]; therefore, a proper choice of atomic displacements was investigated for two cases: (a) the magnitudes identical for each atom or (b) scaled according to atomic masses. The TB-LMTO approach assumes empty spheres at the empty positions in the half-Heusler lattice which would be occupied in the full-Heusler lattice. The potential functions of the empty sphere may be (i) formally displaced like other nuclei or (ii) independent on atomic shifts.
We have tested all four possibilities, i.e., combinations of models (a) and (b), and (i) and (ii) above. We have found deviations in the isotropic resistivities of the order of 5% by assuming u 2 = 0.20 a B and 0.25 a B , where a B is the Bohr radius. This value should be considered as a systematic error of the AAM (later shown by error bars in Fig. 4). In the following sections, identical magnitudes of the displacements are assumed for all atoms. Each atom was assumed to have eight different directions of displacements (within the CPA) uniformly distributed around its equilibrium position.

A. Calculated magnetic moments and density of states
The magnetic moment of the stoichiometric NiMnSb is m = 4.0µ B per formula unit, which agrees well with the half-metallic character (the Fermi level in the minority gap), with its integer number of electrons per formula unit and it is in good agreement with experimental data 36 and previous calculations 4, 37 . In Fig. 1 we show the average moment, local magnetic moments, as well as local Mn-and Ni-impurity magnetic moments on Ni-and Mn-sublattices, respectively. Local moments for the sto- The spin-resolved densities of states (DOS) of the studied system are displayed in Fig. 2. The stoichiometric NiMnSb is the half-metal as it is indicated by the DOS in Fig. 2 (b). Our results are in agreement with literature 37 . The influence of atomic displacements slightly broadens peaks in the DOS (see Fig. 2 for 540 K) but the DOS around the Fermi level is almost the same. The halfmetallic character is thus preserved even at nonzero temperatures.
The behavior of Ni-rich and Mn-rich samples differs significantly. Mn atoms on Ni sublattice preserve the half-metallic character of the alloy, see Fig. 2 (a), while Ni atoms on the Mn sublattice give a nonzero DOS at the Fermi level (Fig. 2 (d) and later Fig. 3). This leads to an increase of the conductivity. Later presented electrical transport calculations are in agreement with these changes. The inset (Fig. 2 (c)) shows a minor influence of the magnetic disorder (tilting of moments with θ = 0.1π) on the DOS of stoichiometric NiMnSb at both zero and finite temperature (T ≈ 220 K).  For further investigation of the electronic structure in the terms of the Bloch spectral function see Appendix C that shows smeared bands for the Ni-rich NiMnSb.

B. Temperature dependent resistivity and anomalous Hall effect calculation
NiMnSb has almost linear dependence of the resistivity on temperature (from 100 to 300 K), which indicates that phonons are the most important scattering mechanism 24 . Calculated temperature dependence of the resistivity and the anomalous Hall effect (resistivity ρ xy ) are shown in Fig. 4. The results are in agreement with experimental data; measured resistivities are taken from Refs. 39 and 55, and experimental ρ xy was obtained by combining Refs. 24 and 55. The quadratic (nonlinear) behavior of electrical resistivities as a function of temperature is important especially for low temperatures (T 100 K) and experimental resistivities exhibit only a small deviation from the quadratic form 34 . The residual resistivity and the weak influence of magnons are in agreement with other studies 4,39,55 . It is consistent with the high Curie temperature, resulting in a weak influence of magnetic disorder and it also agrees with the DOS showing a negligible influence of the magnetic disorder on the number of carriers at the Fermi level (Fig. 2 (c)). Our results also agree with the observed sign of the anisotropic magnetoresistance 4 and its qualitatively good description is also given by the finite-relaxation time approximation, see Appendix B.
The comparison of calculated and measured ρ and ρ xy indicates that the presence of the Mn-rich phase in real samples is unlikely because an increasing presence of additional Mn atoms dramatically increases both the resistivity and ρ xy at the zero temperature and, moreover, slopes of these quantities as a function of temperature are much higher than the measured counterparts 24,36 , see Fig. 4. The calculated transport properties as a function of Ni impurity are non-monotonic, both the resistivity and ρ xy have maxima around a 10 % Ni-rich sample. The measured residual resistivity could correspond to a presence of additional Ni atoms on the empty atomic sites (unoccupied positions of the half-Heusler structure); however, the calculated results contradict the experimental data that exhibit much steeper temperature dependence of both the resistivity and the ρ xy for these defects.
Comparing our theoretical results with data from literature (especially Ref. 24 and 55), the best mutual agreement is obtained for Ni-rich sample with 1 to 2 % of Mn atoms replaced by Ni; we note that the exact composition and chemical disorder in the experimental samples is unknown. In real samples, a wide range of different defects may occur but a systematic investigation of the huge number of different combinations of such defects goes beyond the scope of this study.
In calculations including the magnetic disorder that corresponds to room temperature, transport properties differ less than by 1 % when only Mn moments are tilted or when moments of all atoms are tilted. It is caused by a dominant contribution to the total moment from Mn atoms. The influence of magnetic disorder on the electrical resistivity for the stoichiometric NiMnSb is negligible up to room temperature as can be seen in Tab. I. Experimentally documented decrease of the saturation magnetization is from 4.0µ B at zero temperature to 3.6µ B at room temperature 26,34,35 . When we assume magnetic disorder corresponding to the same change of magnetization, θ = 0.14π, we obtain electrical resistivity between ρ = 17 µΩ cm and ρ = 25 µΩ cm (see the caption of Tab. I). It is in perfect agreement with experimental values of ρ = 23 µΩ cm. The small influence of magnetic disorder on electrical transport properties agrees with literature 55 and it is supported by negligible influence on the DOS at the Fermi level, see the inset in Fig. 2 for θ = 0.1π.
The calculated weak dependence of the resistivity on magnetic disorder justifies neglecting magnetic disorder in further discussion for T 300 K. However, the larger magnetic disorder (for larger temperatures) dramatically decreases the total magnetic moment and increases the resistivity value, see Tab. I.
Chemical impurities decrease the total magnetic moment similarly to the pure magnetic disorder. If the scat- tering properties are considered as a function of the alloy magnetization, results obtained by the different scattering mechanisms (magnetic disorder and chemical impurities) quantitatively agree with each other.
In the present study we focus on the temperature regime T < T D ≈ 320 K. We note that at the elevated temperatures, T T D , the decomposition of the Hall conductivities into skew and side jump scattering mecha- nism complicates the phonon skew scattering 56,57 , which we do not consider here.

C. Anomalous Hall effect mechanism in NiMnSb
We calculated the σ (1) xy and σ (2) xy contributions to the anomalous Hall effect at zero temperature. In Fig. 5 we show the separation of the AHC into σ  tions see Appendix A. We observe a strong dependence of the AHC magnitude on the type of disorder. In general, the AHC is much larger for the Ni-rich system (σ xy ∼ 10 3 S/cm) than for the Mn-rich NiMnSb (σ xy ∼ 10 1 S/cm). Both the Mn and Ni rich cases show the same positive sign of the AHC in agreement with experimental literature 4,39,55 ; an exception of a small negative AHC is found for the 2 % Mn-rich material due to large negative vertex corrections. The vertex part of the AHC diverges in the dilute limit, approaching zero disorder, of both Ni-and Mn-rich branches. Similar behavior is obtained in binary transition-metal alloys due to the skew-scattering mechanism 58 . The small magnitude of the Fermi sea term allows us to neglect the σ (2) term in the temperature study of the AHC by the AAM which substantially speeds up our calculations.
Simulating up to 20 % of Mn or Ni-rich swapping disorder allows us to vary in our calculations the residual resistivities over a broad range from ρ ≈ 0 for stoichiometric NiMnSb to 150 µΩ cm for 20 % of Mn-rich and 11 µΩ cm for 10 % Ni-rich materials. In Fig. 6 (a) we show the dependence of the longitudinal resistivity on the disorder. While the resistivity monotonically increases for the Mn-rich system, consistent with the appearance of the virtual bound state (Fig. 2 (a)), for the Ni-rich case we observe a maximum around 10 % of Ni.
In Fig. 6 (b, d) we present the anomalous Hall versus longitudinal conductivity dependence for both the Mnrich and Ni-rich calculations. A linear fit of the dependences is shown in Fig. 6 (b, d). In the insets (Fig. 6 (c,e)) we show also the experimentally relevant anomalous Hall angle ρ xy /ρ xx obtained by the full inversion of the conductivity tensors (instead of the usually used approxi- mation ρ xy ∼ σ xy /σ 2 xx ). A part of the Ni-rich branch belongs to a rather high conductivity regime (10 5 S/cm) and follows linear dependence σ xy ∼ σ xx signaling the dominating extrinsic, skew-scattering mechanism of the AHC 59,60 . In contrast, the behavior of Mn-rich system with higher conductivities is non-monotonic but different from a power dependence reported in literature 60 . It is rather linear for larger conductivities (small Mn dis-FIG. 6. Total resistivity (a) for zero and finite (540 K) temperature is monotonic in the Mn-rich region but it has a maximum in the Ni-rich case at 10 % and 8 % of Ni impurities for T = 0 and 540 K, respectively. Zero temperature AHC plotted as a function of the total conductivity has (b) two piecewise linear parts for the Ni-rich NiMnSb, one having a negative slope (fitted from 1, 2, 4, 6, 8, and 10 % of Ni) and the second with a positive slope (10,12,14,16,18, and 20 % of Ni). The parts are distinguishable when the resistivity for the same data is plotted (c). The same dependence in the Mn-rich region (d) exhibits a linear (2, 4, and 6 % of Mn impurities) and a non-monotonic (8,10,12,14,16,18, and 20 % of Mn) behavior; a ratio of resistivities (e) show a smooth transition between both parts. order below 6 %), where the AHC is influenced by the disorder 59 , see Fig.6 (d).
Interestingly for Ni-rich branch around ∼10 %, the slope of the AHC as a function of σ xx changes sign. It signals multiband character of the transport (Fig.6 (b)), see also Appendix C. As long as the Friedel sum rule 60,61 can be applied, the change of the AHC sign can be attributed to the change of the dominating spin channel at the concentration of ∼10 % Ni-rich (Fig. 3).
We note that the half-metal and multi-band character of the transport in NiMnSb can be responsible for notably different behavior than that generally reported in metals. For metals, only one slope exists (variations of disorder are typical on the level of a few percents) and it is difficult to achieve more than one conductivity regime 59,60 .

D. Spin-resolved electrical conductivities
To obtain maximal efficiency of the spin-polarized currents, their polarization P should approach unity and both the spin-flip part (of the coherent conductivity) and the vertex part (of the total conductivity) should be negligible. Ni-rich NiMnSb has ten or more times larger conductivity of the majority channel than similar concentration of the Mn-rich material and, unlike the minority channel, it strongly depends on temperature (especially Ni-rich), see Appendix D.
The Mn impurities do not destroy the half-metallic character of the system while the Ni impurities lead to nonzero density of minority carriers at the Fermi level (Fig. 2). It leads to the spin polarization that is almost unity for the Mn-rich case (for all temperatures) and in the Ni-rich region it decreases with increasing impurity concentration or increasing temperature, see Fig. 7. However, even at room temperature and in the Ni-FIG. 7. The spin-polarization of the electrical current for the in-plane direction is almost unity for the Mn-rich NiMnSb (small total conductivity) and it is predicted to be larger than 90 % also in the Ni-rich system at room temperature. rich case, P > 0.9, which ensures highly polarized electrical current. The influence of the spin-flip term and vertex contributions on the polarization P is small, see Appendix D, which justifies employing Eq. (4).
Combined effects of magnetic and atomic displacements was investigated for stoichiometric NiMnSb. The change between T = 0 and room temperature ( u 2 = 0.21 a B , θ = 0.1π) is 0.8 % in the polarization value P .
We focused on systems similar to samples from literature (about 1 to 2 % Ni-rich, see Sec. III B) but experimental P (T ) was measured with a wide range of samples: 44 % for a free surface of a bulk material with M S = 3.6µ B 28 , 45 % for a thin film with M S = 4.0µ B 25 , 45 % for bulk NiMnSb with M S = 3.6µ B 26 , 58 % for thin films 27 , and from 20 to 50 % depending on temperature in polycrystalline samples 30 . Saturation magnetization M S < 4.0µ B indicated disordered samples but the disorder is unknown, which makes it hard to reproduce. The discrepancy is not caused by the magnetic disorder 18 . It is dominant close to the Curie temperature, where spin fluctuations lead to P = 0; the zero polarization cannot be achieved by phonons themselves. For room temperature, the decrease of the polarization caused by the magnetic disorder is negligible, i.e., P > 0.98 for θ ≈ 0.14π.
We also investigated the polarization anisotropy. Similarly as the small anisotropic magnetoresistance (difference between σ zz and σ xx = σ yy is around 0.25%), the polarization P zz is almost the same as P xx = P yy .
The polarization for Mn-and Ni-rich cases with impurities occupying the empty crystallographic position of the Heusler structure was also calculated. The Ni atoms on interstitial positions behave similarly to the Ni-rich system with Mn atoms substituted by Ni impurities; on the other hand, for the 20 % Mn-rich case with access Mn in the interstitial positions, P (0 K) ≈ 91 % and P (400 K) ≈ 87 %. This demonstrates a strong dependence of the polarization on the kind of chemical disorder.

IV. CONCLUSIONS
We have formulated the CPA-AAM approach in the framework of the fully relativistic TB-LMTO method and Kubo-Bastin formula for the calculation of the longitudinal and anomalous Hall conductivities and applied it to the half Heulser ferromagnetic NiMnSb with alloy and temperature induced disorder. The main conclusions are: (i) The calculated temperature dependence of the longitudinal conductivity is dominated by the phonon contribution and it is in agreement with experimental literature. Specifically, the Ni-rich alloys (from 1 to 2 % of Ni atoms on the Mn sublattice) fit the experimental data 24,55 . (ii) The Ni-rich samples are also consistent with the sign of the anisotropic magnetoresistance found in literature. (iii) The effect of the Fermi-sea contribution to the AHC is generally weak although it is stronger for the Mn-rich case. The anomalous Hall effect in Nirich NiMnSb is dominated by the σ (1) part ("integration over the Fermi sheets") of the conductivity, while for the Mn-rich case, the σ (2) ("complex integration over the valence spectrum") term represents a sizable contribution of the order of 20 %. Moreover, qualitatively different behavior of the AHC was observed for the Mn-and Ni-rich systems. (iv) The calculated spin-current polarization is typically greater than 0.9 for studied concentrations of the impurities and its behavior correlates with the halfmetallic-like character (small amount of states in the minority channel). Its values overestimate available experimental data. We study an influence of different contributions to the AHC, see Sec. II D. Its total value (Fig. 5 for T = 0) is given by the σ (1) xy and σ (2) xy terms. The major contribution comes from the former one which is about two orders of magnitude larger than σ (2) xy , see Fig. 8. This justifies omitting σ (2) xy in the temperature-dependent calculations. While the concentration dependence of σ (1,coh) xy consists of two linear parts (one in the Mn-rich region, the second one for the Ni-rich system), σ (1,v.c.) xy diverges for small concentrations of impurities. The finite-relaxation time (FRT) model corresponds to the spin-and orbital independent scatterings, which is technically realized by adding a finite imaginary constant (Im z) to the Fermi energy in corresponding Green functions in the Kubo-Bastin equation. The FRT model assumes zero vertex corrections and does not allow to separate out the phonon and spin-disorder contributions to the conductivity tensor. The calculated negative anisotropic magnetoresistance (AMR) sign for Hall bars oriented along the [110] directions within the FRT is consistent with previous estimates FIG. 8. σ (2) µν (left axes, red lines with triangles) and σ = −1.6% (for Im z = 10 −5 Ry corresponding to low temperatures) to −0.3 % (roughly to room temperature residual resistivity values, Im z = 3 · 10 −3 Ry). The sign of the AMR is the same as in Mn-doped GaAs and opposite to the typical transition metal ferromagnets Ni, Co, and Fe.