Twofold and Fourfold Symmetric Anisotropic Magnetoresistance Effect in A Model with Crystal Field

We theoretically study the twofold and fourfold symmetric anisotropic magnetoresistance (AMR) effects of ferromagnets. We here use the two-current model for a system consisting of a conduction state and localized d states. The localized d states are obtained from a Hamiltonian with a spin--orbit interaction, an exchange field, and a crystal field. From the model, we first derive general expressions for the coefficient of the twofold symmetric term ($C_2$) and that of the fourfold symmetric term ($C_4$) in the AMR ratio. In the case of a strong ferromagnet, the dominant term in $C_2$ is proportional to the difference in the partial densities of states (PDOSs) at the Fermi energy ($E_{\rm F}$) between the $d\varepsilon$ and $d\gamma$ states, and that in $C_4$ is proportional to the difference in the PDOSs at $E_{\rm F}$ among the $d\varepsilon$ states. Using the dominant terms, we next analyze the experimental results for Fe$_4$N, in which $|C_2|$ and $|C_4|$ increase with decreasing temperature. The experimental results can be reproduced by assuming that the tetragonal distortion increases with decreasing temperature.


Introduction
The anisotropic magnetoresistance (AMR) effect is a phenomenon in which the electrical resistivity depends on the relative angle between the magnetization (M ) direction and the electric current (I) direction (see Fig. 1). [1][2][3][4][5][6][7] The AMR effect has been studied extensively both experimentally and theoretically since 1857, when it was discovered by W. Thomson. 1) The AMR ratio, which is the efficiency of the effect, is generally defined * E-mail address: kokado.satoshi@shizuoka.ac.jp by ∆ρ(φ) with ρ ⊥ =ρ(π/2). Here, φ is the relative angle between the thermal average of the spin S (∝−M ) and I, and ρ(φ) is the resistivity at φ.
The d states were indexed by M=0, ±1, and ±2, with M being the magnetic quantum number of the 3d states. Furthermore, the partial density of states (PDOS) of each d state at the Fermi energy (E F ) was assumed to be constant regardless of the d state.
Here, C 2 (C 4 ) is the coefficient of the twofold (fourfold) symmetric term, and C 0 is chosen to be C 2 − C 4 so as to satisfy ∆ρ(π/2)/ρ=0. For example, |C 2 | and |C 4 | for Fe 4 N increase with decreasing temperature T as shown later in Fig. 11. [22][23][24][25][26] The coefficients C 2 and C 4 were measured to be C 2 =−0.0343 and C 4 =0.00556 at T =4 K. 22) The set of C 0 , C 2 , and C 4 , however, has seldom been derived within the framework of transport theory and has often been represented by phenomenological expressions. 27-29, 32, 33) We anticipate that expressions for C 0 , C 2 , and C 4 obtained by transport theory will play an important role in the analysis and understanding of the AMR effect. We also predict that the fourfold symmetric term in Eq. (3) may appear under the crystal field of the d states, which was neglected in the previous models 19) [i.e., Eq. (2)].
In this paper, we obtained C 2 and C 4 by extending our model 19,20) to one with a crystal field. We first performed a numerical calculation of C 2 and C 4 for a strong ferromagnet using the d states, which were obtained by applying the exact diagonalization method (EDM) to a Hamiltonian of the d states with a crystal field. The result revealed that C 4 appears under a crystal field of tetragonal symmetry, whereas it vanishes under a crystal field of cubic symmetry. We next derived general expressions for the resistivity, C 2 , and C 4 for ferromagnets with the tetragonal field using the d states, which were obtained by applying first-and second-order perturbation theory (PT) to the Hamiltonian. From the expressions, we obtained expressions for C 2 and C 4 for the strong ferromagnet with the tetragonal field. The result showed that C 2 cos 2φ is related to the real part of the probability amplitudes of the specific hybridized states and C 4 cos 4φ is related to the probabilities of the specific hybridized states. In addition, we performed a simple analysis of the experimental results of C 2 and C 4 for Fe 4 N using the dominant terms in C 2 and C 4 obtained by PT. The experimental results could be reproduced by assuming that the tetragonal distortion increases with decreasing T .
The present paper is organized as follows: In Sec. 2, we obtain wave functions of the localized d states by applying first-and second-order PT to the Hamiltonian of the localized d states. Using the wave functions, we derive general expressions for the resistivity, C 2 , and C 4 for ferromagnets. In Sec. 3, we obtain expressions for C 2 and C 4 for a strong ferromagnet from the above-mentioned C 2 and C 4 . In addition, we perform the numerical calculation of C 2 and C 4 using the d states, which are obtained by applying the EDM to the Hamiltonian. We then compare C 2 and C 4 obtained by PT and the respective values obtained by the EDM. In Sec. 4, we analyze the experimental results of C 2 and C 4 for Fe 4 N. The conclusion is presented in Sec. 5. In Appendix A, we show the matrix of the Hamiltonian. In Appendix B, we give the zero-order states of the d states, which are obtained by performing the unitary transformation on the perturbation term.
In Appendix C, we describe the overlap integrals of the s-d scattering rate. In Appendix D, we give an expression for the s-d scattering rate. Section E shows that the present ∆ρ(0)/ρ (=2C 2 ) coincides with our previous model 19,20) and the CFJ model 2) under appropriate conditions.

Theory
In this section, we obtain general expressions for the resistivity, C 2 , and C 4 in a model in which I flows in the x direction and S (∝−M ) lies in the xy plane (see Fig. 1). We here use the two-current model with s-d scattering in which the conduction electron is scattered into the localized d states by nonmagnetic impurities. 2,3,5,6,19,20) The d states are obtained by applying PT to a Hamiltonian of the d states. We also explain the numerical calculation method for C 2 and C 4 , in which the d states are obtained by applying the EDM to the Hamiltonian.

Hamiltonian
We first present the Hamiltonian H of the localized d states of a single atom 19,34) in a ferromagnet with a spin-orbit interaction, an exchange field, and a crystal field of tetragonal symmetry. This crystal field represents the case that distortion in the z direction is added to the crystal field of cubic symmetry. 35) Note that C 4 appears under a crystal field of tetragonal symmetry, whereas it vanishes under a crystal field of cubic symmetry, as will be described in Sec. 3.2.
The Hamiltonian H is expressed as with and S = (S x , S y , S z ), where H > 0. Here, S is the spin angular momentum and L is the orbital angular momentum. The spin quantum number S and the azimuthal quantum number L are chosen to be S=1/2 and L=2. 19) The above terms are explained as follows: The term H cubic represents the crystal field of cubic symmetry. The term −H ·S is the Zeeman interaction due to the exchange field of the ferromagnet H, where H∝−M and H∝ S . The term V so is the spin-orbit interaction, where λ is the spin-orbit coupling constant. The term V tetra is an additional term to reproduce the crystal field of tetragonal symmetry. The state |i, χ σ (φ) is expressed by |i, χ σ (φ) =|i |χ σ (φ) . The state |i is the orbital state, defined by |xy =xyf (r), |yz =yzf (r), |xz =xzf (r), |x 2 − y 2 = 1 2 (x 2 − y 2 )f (r), and |3z 3 − r 2 = 1 2 √ 3 (3z 2 − r 2 )f (r), with f (r) being the radial part of the 3d orbital, where r= x 2 + y 2 + z 2 . The states |xy , |yz , and |xz are referred to as dε orbitals and |x 2 − y 2 and |3z 2 − r 2 are referred to as dγ orbitals. The quantity E ε is the energy level of |xy and E γ is that of |x 2 − y 2 .
Equations (23) and (24) play an important role in C 2 and C 4 as described in the φ dependence of the wave functions in this section.
Applying the usual first-and second-order PT to H in Table I, we obtain |i, χ ς (φ)), where i (ς) denotes the orbital index (spin index) of the dominant state in |i, χ ς (φ)).
sin 2φ term normalized by c i,ς , while c i,ς is the coefficient of the constant term, which does not depend on φ. Such w i,ς j,σ cos 2φ and w i,ς j,σ sin 2φ generate the twofold and fourfold symmetric terms of ∆ρ(φ)/ρ as described in Sec. 2.5. Hybridized state x y x y 2 2 cos 2 | 3 , Hybridized state  , respectively. The hybridized states are represented by expressions with a probability amplitude of cos 2φ or sin 2φ, i.e., cos 2φ|3z 2 − r 2 , χ ± (φ) and sin 2φ|3z 2 − r 2 , χ ± (φ) , where the prefactor of cos 2φ or sin 2φ is ignored. In each panel, the upper part shows the top view (looking down along the z axis) of the dominant state. In (a) and (c), the middle part shows cos 2φ|3z 2 − r 2 , χ + (φ) in the xy plane, and the lower part shows sin 2φ|3z 2 − r 2 , χ − (φ) in the xy plane. In (b), the middle part shows sin 2φ|3z 2 − r 2 , χ − (φ) in the xy plane, and the lower part shows cos 2φ|3z 2 − r 2 , χ + (φ) in the xy plane. The state |yz, χ − (φ) is shown by the pink or pinkbordered orbital, |xz, χ − (φ) is shown by the sky-blue or sky-blue-bordered orbital, and |xy, χ − (φ) is shown by the gray or gray-bordered orbital. The state |3z 2 − r 2 , χ − (φ) is represented by the yellow or yellow-bordered orbital. The color-filled orbitals (white orbitals with a colored border) express regions with a negative sign (positive sign) in the wave function including the probability amplitude. In the middle part of (a) and (c) and the lower part of (b), the blue and red curves are cos 2φ and cos 2 2φ [=(1 + cos 4φ)/2], respectively. In the lower parts of (a) and (c) and the middle part of (b), the blue and red curves are sin 2φ and sin 2 2φ [=(1 − cos 4φ)/2], respectively. is shown by the yellow or yellow-bordered orbital. The state |x 2 − y 2 , χ − (φ) is shown by the green or green-bordered orbital. The color-filled orbitals (white orbitals with a colored border) express regions with a negative sign (positive sign) in the wave function, where the probability amplitude is taken into consideration in regard to |3z 2 − r 2 , χ − (φ) and |x 2 − y 2 , χ − (φ) . In the lower parts of (a) and (b), the blue and red curves are cos 2φ and cos 2 2φ [=(1 + cos 4φ)/2], respectively.
does not depend on φ (see Table A·1).

General expression for resistivity
Using |i, χ ς (φ)) of Eqs. (25)−(34), we can obtain a general expression for ρ(φ). The resistivity ρ(φ) is first described by the two-current model, 2) i.e., The quantity ρ σ (φ) is the resistivity of the σ spin at φ with σ=+, −, where σ=+ (−) denotes the up spin (down spin) for the case in which the quantization axis is chosen along the direction of S . The resistivity ρ σ (φ) is written as where e is the electric charge and n σ (m * σ ) is the number density (effective mass) of the electrons in the conduction band of the σ spin. 40,41) The conduction band consists of the s, p, and conductive d states. 19) In addition, 1/τ σ (φ) is the scattering rate of the conduction electron of the σ spin, expressed as 1 where i=ξ + , δ ε , ξ − , x 2 − y 2 , and 3z 2 − r 2 . Here, 1/τ s,σ is the s-s scattering rate, which is proportional to the PDOS of the conduction state of the σ spin at E F , D where k σ is the Fermi wavevector of the σ spin in the x direction (i.e., the I direction) and Ω is the volume of the system. The quantitiy V imp (R n ) is the scattering potential at R n due to a single impurity, where R n is the distance between the impurity and the nearest-neighbor host atom. 19) The quantity N n is the number of nearest-neighbor host atoms around a single impurity, 19) n imp is the number density of impurities, and is the Planck constant h divided by 2π.

Calculation method of C 2 and C 4 by exact diagonalization method
As a different approach from PT, we perform a numerical calculation of C 2 and C 4 using the d states, which are obtained by applying the EDM to H in Table I.
The first purpose of this approach is to find the crystal field that leads to C 4 =0. The second purpose is to check the validity of the results obtained by PT (see Sec. 3). The calculation in the EDM is as follows: (i) We numerically obtain |i, χ ς (φ)) in Eq. (38) by applying the EDM to H in Table I. (ii) Utilizing the obtained |i, χ ς (φ)) and Table C·1, we numerically calculate (iv) When the AMR ratio is expressed as Eq. (3), we have From Eqs. (56) and (57), we obtain C 2 and C 4 as

Application to Strong Ferromagnets
On the basis of C 2 of Eq. (54) and C 4 of Eq. (55), we obtain expressions for C 2 and C 4 for a strong ferromagnet with D i,− =0. The coefficients C 2 and C 4 are compared with those obtained by the EDM. In addition, from the results of the EDM we find that C 4 appears under a crystal field of tetragonal symmetry, whereas it vanishes under a crystal field of cubic symmetry.

Expressions for C 2 and C 4
Using Eqs. (43)−(47), (54), and (55), we obtain expressions for C 2 and C 4 for a sim- where ρ 2,− is given by Eq. (45). In addition, in accordance with previous studies 42) we assume n + =n − , m * + =m * − , and v + =v − , where v + =v − is satisfied by setting k + =k − in Eqs. (53) and (C·5). The expressions for C 2 and C 4 are then written as (63) Here, we have where with i=ξ + , δ ε , ξ − , x 2 − y 2 , and 3z 2 − r 2 . The resistivity ρ s→d i ,− is given by Eqs. (49) and (52), where σ in ρ s,σ→d i ,− is unspecified because the σ dependences of n σ , m * σ , and k σ are ignored as noted above. Furthermore, we note that r i,− satisfies the relation r i,− ∝ D On the basis of (i) and (ii) of Sec. 2.5, the features of C 2 cos 2φ and C 4 cos 4φ are described as follows: (i) The term C 2 cos 2φ is related to the real part of the probability amplitudes of 2,σ cos 2φ is related to the real part of the probability amplitudes of |3z 2 − r 2 , χ σ (φ) and |x 2 − y 2 , χ σ (φ) as noted in (i) of Sec. 2.5.
4,σ cos 4φ is related to the probabilities of |3z 2 − r 2 , χ σ (φ) and |x 2 − y 2 , χ σ (φ) as noted in (ii) of Sec. 2.5. Also, c ′ 4 of Eq. (63) arises from high-order processes of dγ − dε − dγ ′ , in which the dγ states are hybridized to the dγ ′ states via the dε states. Such processes reflect the fact that there are no off-diagonal matrix elements in the subspace of the dγ states (see Table I).

Various features of C 2 and C 4
We investigate various features of C 2 and C 4 for a strong ferromagnet with H=1 eV and λ=−0.01 eV. We here use C 2 of Eq. (61) and C 4 of Eq. (71) for PT and C 2 of Eq.
(58) and C 4 of Eq. (59) for the EDM, where |λ|/δ ε =|λ|/δ γ =1/2 is set for C 2 and C 4 for the EDM. We also utilize Eqs. (69) and (70). As a particularly important result, we find that C 4 appears under the crystal field of tetragonal symmetry, whereas it vanishes under the crystal field of cubic symmetry. 44) Using the EDM, we obtain the r ε /r γ dependences of C 2 and C 4 for a system with the crystal field of cubic symmetry, where ∆=0.1 eV, δ ε =δ γ =0, r=0, r γ =0.01, and η=0  (see Fig. 7). We find that C 2 can be expressed as a linear function of r ε /r γ . The sign of C 2 changes in the vicinity of r ε /r γ ∼1. Furthermore, C 4 takes a value of almost 0. Figure 8 shows the r ε /r γ or η dependences of C 2 and C 4 for a system with the crystal field of tetragonal symmetry, where ∆=0.1 eV, r=0, and η=0 and 1. From the results of PT, we find C 2 ∼0 for the system with r ε1 /r γ =r ε (1 + η)/r γ =1 and C 2 =0 for that with r ε1 /r γ =r ε (1 + η)/r γ =1. This feature mainly reflects Eq. (72). We also obtain C 4 =0 for the system with η=0 and C 4 =0 for that with η =0 because of C 4 ∝ r ε1 − r ε2 (= r ε η). The coefficients C 2 and C 4 obtained by PT qualitatively agree well with those obtained by the EDM.
In Fig. 9, we show C 2 for systems with the crystal field of tetragonal symmetry, where r ε /r γ =1, 1.2, and 2 and η=0. Here, C 4 for PT takes a value of 0 because of η=0, and |C 4 | for the EDM is much smaller than |C 2 |. The upper panel shows the r γ dependence of C 2 for systems with ∆=0.1 eV and r=0. The middle panel shows the r dependence of C 2 for systems with ∆=0.1 eV and r γ =0.01. The lower panel shows the ∆ dependence of C 2 for systems with r=0 and r γ =0.01. In the upper panel, when r ε /r γ =1, C 2 for PT is close to that for the CFJ model, i.e., with α=r ε =r γ (see Appendix E.2). In the middle and lower panels, C 2 for PT takes a value of almost 0 in the case of r ε /r γ =1. The sign of C 2 for PT is negative in the case of r ε /r γ =1.2 or 2. In addition, |C 2 | for PT increases with decreasing r or ∆ and with increasing r ε /r γ . These features mainly reflect Eq. (72). In all panels, C 2 for PT qualitatively agrees well with that for the EDM. C 2 for PT is negative, while that of C 4 for PT is positive. In addition, |C 2 | and C 4 for PT increase with decreasing r γ , r, or ∆ and with increasing η. Such features are mainly due to Eqs. (72) and (73). The coefficients C 2 and C 4 for PT qualitatively agree well with those for the EDM.  The quantity r γ dependences of C 2 and C 4 for the systems with ∆=0.1 eV and r=0. Middle panel: The quantity r dependences of C 2 and C 4 for the systems with ∆=0.1 eV and r γ =0.01. Lower panel: The energy ∆ dependences of C 2 and C 4 for the systems with r=0 and r γ =0.01. The solid curves show i,+ ∼0. 19) In addition, we mainly focus on the effect of the PDOSs of the dε states on C 4 . Note that we do not take into account the realistic crystal structure of Fe 4 N (i.e., a perovskite-type structure 45) ) for simplicity. 48) From Eqs. (72) and (73), we first obtain simple expressions for C 2 and C 4 for Fe 4 N.
By taking into account the relation for Fe 4 N, i.e., r ≪ 1 and r γ ≪ 1, 19) C 2 and C 4 are given by with ρ s→dγ ,− =ρ s→d x 2 −y 2 ,− =ρ s→d 3z 2 −r 2 ,− . Here, ∆R ε is proportional to D ε2,− , which is the difference in the PDOSs at E F among the dε states.
We next determine parameter sets for λ, ∆, R, R ε1 , R ε2 , and ∆R ε that can reproduce the experimental result for the T dependences of C 2 and C 4 . The quantity λ is set to λ=−0.013 eV for Fe. 36) The quantity ∆ is assumed to be ∆=0.1 eV. 37) Here, the T dependence of ∆ is considered to be negligibly small, because the decrease in the lattice constant due to a decrease in T is less than 0.5%, 22) where ∆ of 0.1 eV is due to the Coulomb interaction between a magnetic ion and the surrounding ions. We accordingly adopt the T dependences of R, R ε1 , R ε2 , and ∆R ε . The T dependence of R (R ε1 , R ε2 , and ∆R ε ) is shown in the middle (lower) panel of Fig. 11. Details of the parameter sets are given below.
Here, ρ imp s,− /ρ imp s→dγ,− is set so that ρ imp The parameter sets in the high-and low-temperature ranges are noted below.
(i) In the high-temperature range of T > 35 K, we have with Θ=0.0270, T * =35 K, R * =ΘT * , and C is the experimental value of C 2 at T =T * . 22) The procedure for determining this parameter set is as follows: First, C 4 is experimentally observed to be almost 0. Since C 4 ∝ ∆R ε [see Eq. (75)], we assume ∆R ε =0 (or R ε1 =R ε2 ); that is, the PDOSs of the dε states at E F take the same value.
From the viewpoint of the crystal structure of Fe 4 N, this assumption may imply that the crystal exhibits cubic symmetry. 48) Next, since |C 2 | gradually decreases with increasing T , we straightforwardly take into consideration the T dependence of R of Eq. (77), where R is included in the denominator of C 2 of Eq. (74). The denominator 1 + R is expressed as by using Eqs. (77) and (78). Here, ρ ph s,− is assumed to be proportional to T on the basis of the experimental result for the T dependence of the total resistivity. 50) Thereby, R (≈ ρ ph s,− /ρ imp s→dγ,− ) is given by where θ is a constant number. On the other hand, R ε2 is determined so that Eq.
(ii) In the low-temperature range of T ≤ 35 K, we have The procedure for determining this parameter set is as follows: We first adopt R=ΘT , which is the same as Eq. (79) in the high-temperature range, on the basis of the expeimental result of the T dependence of the total resistivity. 50) Second, since C 4 was experimentally observed to be a linear function of T , we assume C 4 to be C 4 =pT + q, where p and q are constants. The constants p and q are determined so that Eq. (75) satisfies the condition (T, C 4 )=(T l , C (T l ) 4 ), (T * , 0). As a result, C 4 is expressed as . From this C 4 and Eq. (75), we obtain ∆R ǫ of Eq. (86). The obtained ∆R ǫ may indicate the following two properties: One is that the crystal has tetragonal symmetry, which generates ∆R ε =0 due to the difference of the PDOS at E F among the dε states. The other is that the tetragonal distortion increases with decreasing T . Third, R ε2 is assumed to be R ε2 =p ′ T + q ′ as a simple form, where p ′ and q ′ are constants. The constants p ′ and q ′ are determined so that Eq. (74) satisfies the condition (T, C 2 )=(T l , C  (75), we obtain C 2 and C 4 for Fe 4 N, where C 4 in the low-temperature range was described above. In the upper panel of Fig. 11, we show the T dependences of C 2 and C 4 . We find that C 2 and C 4 for PT successfully reproduce the experimental results. In particular, the experimental results in the range 4 K ≤ T ≤ 35 K, in which the change of |C 2 | is about four times as large as that of |C 4 |, can be explained by the ratio of the coefficients of ∆R ε between Eqs. (74) and (75).
Finally, we comment on the above-mentioned T dependence of ∆R ε , i.e., the difference in the PDOSs at E F among the dε states. The T dependence of ∆R ε has been assumed to arise from the increase of the tetragonal distortion due to a decrease in T . The tetragonal distortion may originate from the anisotropic thermal compression of the lattice. This compression is considered to be due to the adhesion between the Fe 4 N film and the MgO substrate. We expect that such an assumption will be verified experimentally in the future.

Conclusions
We theoretically studied the twofold and fourfold symmetric AMR effects of ferromagnets. In particular, we obtained the coefficients of the twofold symmetric term (cos 2φ term) and the fourfold symmetric term (cos 4φ term) in the AMR ratio, denoted as C 2 and C 4 , respectively. We used the two-current model for the system consisting of the conduction state and localized d states. The localized d states were obtained from the Hamiltonian with the spin-orbit interaction, the exchange field, and the crystal field. Details are given as follows: (i) We performed the numerical calculation of C 2 and C 4 for a strong ferromagnet using d states, which were obtained by applying the EDM to the Hamiltonian. The result revealed that C 4 appears under the crystal field of tetragonal symmetry, whereas it vanishes under the crystal field of cubic symmetry.
(ii) We derived general expressions for the resistivity, C 2 , and C 4 for ferromagnets with the tetragonal field using the d states, which were obtained by applying first-and second-order PT to the Hamiltonian. From the expressions, we obtained expressions for C 2 and C 4 for the strong ferromagnet with the tetragonal field. The result showed that C 2 cos 2φ is related to the real part of the probability amplitudes of the specific hybridized states |3z 2 −r 2 , χ σ (φ) and |x 2 −y 2 , χ σ (φ) and C 4 cos 4φ is related to the probabilities of |3z 2 − r 2 , χ σ (φ) and |x 2 − y 2 , χ σ (φ) . In addition, we investigated various features of C 2 and C 4 obtained by PT and found that they qualitatively agreed well with those obtained by the EDM.
(iii) We analyzed the experimental results of the T dependences of C 2 and C 4 for an Fe 4 N film on a MgO substrate using the dominant terms in C 2 and C 4 obtained by PT. The dominant term in C 2 was proportional to the difference in the PDOSs at E F between the dε and dγ states, and that in C 4 was proportional to the difference in the PDOSs at E F among the dε states. The experimental results in the hightemperature range (35 K < T ≤ 300 K) were well reproduced by taking into account the T dependence of the s-s resistivity and by assuming that the PDOSs of the dε states at E F took the same value. This assumption might imply that the crystal structure of Fe 4 N exhibits cubic symmetry. Also, the experimental results in the lowtemperature range (4 K ≤ T ≤ 35 K) were successfully reproduced by assuming that the difference in the PDOSs at E F among the dε states increased with decreasing T .
Equations (A·1) and (A·2) play an important role in C 2 and C 4 , as described when we discuss the φ dependence of the wave functions (see Sec. 2.2).
Here, the dε states for each spin at the corner site are considered to be degenerate.
The three face-center sites are in the xy, yz, and xz planes and are denoted F xy , F yz , and F xz , respectively. We can then specify the ground and excited states of the dε orbitals at F xy , F yz , and F xz taking into account the effect of N at the body center site. At F xy , the ground state is the xy orbital, while the excited states are the yz and xz orbitals. At F yz , the ground state is the yz orbital, while the excited states are the xy and xz orbitals. At F xz , the ground state is the xz orbital, while the excited states are the xy and yz orbitals. In this system, the PDOSs of the xy, yz, and xz states for each spin at E F take the same value. In contrast, when the system has tetragonal distortion in the z direction, the PDOS of the xy state at E F is different from that of the yz or xz state at E F . 49) We obtain ρ imp s,− ∝ D (s) − from Eqs. (17) and (19) in Ref. 19. 50) From Fig. 2 in Ref. 22, we roughly evaluate the total resistivity ρ total to be ρ total ≃T /3 + 14 µΩ · cm. This ρ total is expressed as ρ total = ρ + ρ − /(ρ + + ρ − ) ≃ ρ − owing to the relation for Fe 4 N, i.e., ρ − /ρ + ≪ 1. 19) Here, ρ − is simply given by