Theoretical Study on Anisotropic Magnetoresistance Effects of I//[100], I//[110], and I//[001] for Ferromagnets with A Crystal Field of Tetragonal Symmetry

Using the electron scattering theory, we obtain analytic expressions for anisotropic magnetoresistance (AMR) ratios for ferromagnets with a crystal field of tetragonal symmetry. Here, a tetragonal distortion exists in the [001] direction, the magnetization ${\mbox{\boldmath $M$}}$ lies in the (001) plane, and the current ${\mbox{\boldmath $I$}}$ flows in the [100], [010], or [001] direction. When the ${\mbox{\boldmath $I$}}$ direction is denoted by $i$, we obtain the AMR ratio as ${\rm AMR}^i (\phi_i)= C_0^i + C_2^i \cos 2\phi_i + C_4^i \cos 4 \phi_i \ldots = \sum_{j=0,2,4,\ldots} C_j^i \cos j\phi_i$, with $i=[100]$, $[110]$, and $[001]$, $\phi_{[100]} = \phi_{[001]}=\phi$, and $\phi_{[110]}=\phi'$. The quantity $\phi$ ($\phi'$) is the relative angle between ${\mbox{\boldmath $M$}}$ and the $[100]$ ($[110]$) direction, and $C_j^i$ is a coefficient composed of a spin--orbit coupling constant, an exchange field, the crystal field, and resistivities. We elucidate the origin of $C_j^i \cos j\phi_i$ and the features of $C_j^i$. In addition, we obtain the relation $C_4^{[100]} = -C_4^{[110]}$, which was experimentally observed for Ni, under a certain condition. We also qualitatively explain the experimental results of $C_2^{[100]}$, $C_4^{[100]}$, $C_2^{[110]}$, and $C_4^{[110]}$ at 293 K for Ni.

Using the electron scattering theory, we obtain analytic expressions for anisotropic magnetoresistance (AMR) ratios for ferromagnets with a crystal field of tetragonal symmetry. Here, a tetragonal distortion exists in the [001] direction, the magnetization  ([110]) direction, and C i j is a coefficient composed of a spin-orbit coupling constant, an exchange field, the crystal field, and resistivities. We elucidate the origin of C i j cos jφ i and the features of C i j . In addition, we obtain the relation C at 293 K for Ni.

Introduction
The anisotropic magnetoresistance (AMR) effect for ferromagnets,  in which the electrical resistivity depends on the direction of magnetization M , has been studied extensively both experimentally and theoretically. The efficiency of the effect "AMR * E-mail address: kokado.satoshi@shizuoka.ac.jp ratio" is defined by with ρ i ⊥ = ρ i (π/2). Here, ρ i (φ i ) is the resistivity at φ i in the current I direction, i, where φ i is the relative angle between the thermal average of the spin S (∝−M ) and a specific direction for the case of i.
The AMR ratio AMR i (0) has often been investigated for many magnetic materials.
In particular, the experimental results of AMR i (0) for Ni-based alloys have been analyzed by using the electron scattering theory with no crystal field, i.e., the Campbell-Fert-Jaoul (CFJ) model. 3) We have recently extended this CFJ model to a general model that can qualitatively explain AMR i (0) for various ferromagnets. 25,26) On the other hand, when S lies in the (001) plane and I flows in the i direction, with i = [100] and [110], AMR i (φ i ) has been experimentally observed to be [7][8][9][10][11][12][13][14] AMR i (φ i ) = C i 0 + C i 2 cos 2φ i + C i 4 cos 4φ i + . . .
with φ [100] = φ and φ [110] = φ ′ , where φ is the relative angle between the S direction and the [100] direction (see Fig. 1) and φ ′ is the relative angle between the S direction and the [110] direction (see Fig. 1). In addition, C i 0 is the constant term in the case of i, and C i j is the coefficient of the cos jφ i term in the case of i. The case of Eq. (2) with C i 2 = 0 and C i j = 0 (j ≥ 4) is called the twofold symmetric AMR effect, while the case of Eq. (2) with C i 2 = 0 and C i j = 0 (j ≥ 4) is the higher-order fold symmetric AMR effect. The twofold symmetric AMR effect has often been observed for various ferromagnets and analyzed on the basis of our previous model. 25,26) The higher-order fold symmetric AMR effect of I//[100] and I// [110] has been observed for typical ferromagnets Ni,30,31) Fe 4 N, 7) and Ni x Fe 4−x N (x = 1 and 3). 12) In particular, the relation  (4) has been found in the temperature dependence of the AMR ratio. 7,12,30,31) The AMR ratio of Eq. (2) has sometimes been fitted by using an expression by Döring. This expression consists of an expression for the resistivity, which is based on the symmetry of a crystal (see Appendix A). 14, 32, 33) Döring's expression can be easily applied to the cases of the arbitrary directions of I and M . The expression, however, has been considered unsuitable for physical consideration because it was not based on the electron scattering theory.
To improve this situation, we have recently developed a theory of the twofold and fourfold symmetric AMR effect using the electron scattering theory. Here, we derived an expression for AMR [100] (φ) of Eq. (2) for ferromagnets with a crystal field. As a result, we found that C [100] 4 appears under a crystal field of tetragonal symmetry, whereas it takes a value of almost 0 under a crystal field of cubic symmetry. 27) The expression for AMR [110] (φ ′ ), however, has scarcely been derived.
In the future, not only the expression for AMR [100] (φ) but also expressions for AMR [110] (φ ′ ) and so on will play an important role in theoretical analyses and physical considerations of experimental results. In addition, Eq. (4) should be confirmed by using the electron scattering theory.
In this paper, using the electron scattering theory, we first obtained analytic ex-  1). Second, we elucidated the origin of C i j cos jφ i and the features of C i j . In addition, we obtained the relation C of Eq. (4) under a certain condition. Third, we qualitatively explained the experimental result of C i j at 293 K for Ni using the expression for C i j . The AMR ratios AMR [100] (0) and AMR [110] (0) also corresponded to that of the CFJ model 3) under the condition of the CFJ model.
The present paper is organized as follows. In Sect. 2, we present the electron scattering theory, which takes into account the localized d states with a crystal field of tetragonal symmetry. We first obtain wave functions of the d states using the first-and second-order perturbation theory. Second, we show the expression for the resistivity, which is composed of the wave functions of the d states. In Sect. 3, we describe the expressions for AMR i (φ i ) for ferromagnets including half-metallic ferromagnets. In Sect. 4, we elucidate the origin of C i j cos jφ i and the features of C i j . In Sect. 5, the relation C is obtained under a certain condition. In Sect. 6, we qualitatively explain the experimental result of C i j at 293 K for Ni. The conclusion is presented in Sect. 7. In Appendix A, we report the expression for the AMR ratio by Döring. In Appendix B, we give an expression for a wave function obtained by applying the perturbation theory to a model with degenerate unperturbed systems. In Appendix C, we describe the expressions for resistivities for the present model. In Appendix D, C i j is expressed as a function of the resistivities. In Appendix E, we give the expression for C i j . In Appendix F, we explain the origin of C i j cos jφ i . In Appendix G, we show that the present model corresponds to the CFJ model 3) under the condition of the CFJ model.

Theory
In this section, we describe the electron scattering theory to obtain ρ i (φ i ) and [110], and [001] for the ferromagnets.

Model
For this system, we use the two-current model with the s-s and s-d scatterings. [25][26][27][28] The

Hamiltonian
Following our previous study, 27,28) we consider H as the Hamiltonian of the localized d states of a single atom 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 a distortion in the [001] direction is added to a crystal field of cubic symmetry.
The reason for choosing this crystal field is that C  The relation between φ and φ ′ is given by Eq. (5). Furthermore, the x-, y-, and z-axes are specified to describe the Hamiltonian of Eq. (6). where and The above terms are explained as follows. The term H cubic represents the crystal field of cubic symmetry. The term −S · H is the Zeeman interaction between the spin angular momentum S and the exchange field of the ferromagnet H, where H ∝ −M , H ∝ S , and H > 0. The term V so is the spin-orbit interaction, where λ is the spin-orbit coupling constant 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. 25) The term V tetra is an additional term to reproduce the crystal field of tetragonal symmetry. The state |m, χ σ (φ) is expressed by |m, χ σ (φ) = |m |χ σ (φ) . The state |m is the orbital state, where f (r) is the radial part of the 3d orbital, and Γ and ζ are constants. The states |xy , |yz , and |xz are called dε orbitals and |x 2 − y 2 and |3z 2 − r 2 are dγ orbitals.
On the basis of the relation of the parameters, we consider H 0 of Eq. (7) and V of Eq.

Resistivity
Using Eq. (26), we can obtain an expression for ρ i (φ). The resistivity ρ i (φ) is described by the two-current model, 3) i.e., The quantity ρ i σ (φ) is the resistivity of the σ spin at φ in the case of i, where σ = + (−) denotes the up spin (down spin) for the case in which the quantization axis is chosen along the direction of S [see Eqs. (15) and (16)]. The resistivity ρ i 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. 38,39) The conduction band consists of the s, p, and conductive d states. 25) In addition, 1/τ i σ (φ) is the scattering rate of the conduction electron of the σ spin in the case of i, expressed as Here, 1/τ s,σ is the s-s scattering rate, which is considered to be independent of i. The The conduction state of the σ spin |e ik i σ ·r , χ σ (φ) is represented by the plane wave, i.e., where k i σ [=(k i x,σ , k i y,σ , k i z,σ )] is the Fermi wave vector of the σ spin in the i direction, r is the position of the conduction electron, and Ω is the volume of the system. The quantity 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. 25) The quantity N n is the number of nearest-neighbor host atoms around a single impurity, 25) n imp is the number density of impurities, and is the Planck constant h divided by 2π.
(50) can be calculated by using Eq. (C·1) in Ref. 27. Here, Eq. (50) has been introduced to investigate the relation between the present result and the previous ones 3, 25,26) (also see Appendix G).
We also note that, as found from Eq. (50), ρ s,σ→m,ς of Eq .(49) satisfies This relation is useful to give a physical explanation for C i j .

Application
We apply the theory of Sect. 2 to ferromagnets with D in accordance with our previous study. 26) In addition, we set for simplicity.

Simplified system
On the basis of the above-mentioned C i j , we obtain a simple expression for C i j for the simplified system. In this system, we assume (52)]. This assumption may be valid for the system of D (d) For this system, we consider three types: In Tables I, II, and III, we show C i j for types A, B, and C, respectively. The coefficient C i j for type A is derived by imposing Eq. (60) on Eqs. (E·1)−(E·3), (E·4)−(E·8), (E·9), and (E·10). The coefficient C i j for type B is obtained by imposing r ≪ 1, r s,σ→ε1,− ≪ 1, r s,σ→ε2,− ≪ 1, and r s,σ→γ,− ≪ 1 on C i j for type A in Table I. The coefficient C i j for type C is obtained by imposing r ≫ r s,σ→ε1,− , r ≫ r s,σ→ε2,− , r ≫ r s,σ→γ,− , and r ≫ 1 on C i j for type A in Table I in Table I include r only in the respective denominators and then they become smaller than the other C i j . We also mention that C for type A in Table I are, respectively, the coefficients in our previous study, i.e., Eqs. (61) and (62) in Table I and AMR [110] (0) of Eq. (58) with C in Table I correspond to the CFJ model 3) under the condition of the CFJ model (see Appendix G). 40)

Consideration
We consider the origin of C i j cos jφ i for type A and features of C i j for types A, B, and C.   47). We find that C i 2 cos 2φ i and C i 6 cos 6φ i are related to the probability amplitudes of the slightly hybridized states and C i 4 cos 4φ i is related to the probability of the slightly hybridized state (i.e., |3z 2 − Table I. The coefficient C i j for type A, i.e., the generalized strong ferromagnet with the s-d scattering "s,    Table I. We first find that C i j consists of the terms with (λ/∆) 2 , (λ/H) 2 , and λ 2 /(H∆). Their terms are related to the changes in the d states due to V so , as noted in Sect. 4.1. On the basis of , we show that such terms arise from the following two origins. One is the square of the first-order perturbation terms in the d states such Table III. The coefficient C i j for type C, i.e., the specified strong ferromagnet with the dominant s-d scattering "s, + → d, −". This C i j is obtained by imposing r ≫ r s,σ→ε1,− , r ≫ r s,σ→ε2,− , r ≫ r s,σ→γ,− , and r ≫ 1 on C i j for type A in Table I, where r is set to be large enough for the term including r in the numerator to become dominant in each C i j in spite of ∆/H ≪ 1.    cos 8φ [110] , this |xy, χ − is the slightly reduced state, which is included in |m, χ ς (φ) in Eq. (26). In contrast, the other states in this table are the slightly hybridized states, which are included in |n, χ σ (φ) in Eq. (26).

Coefficients for Ni
Using C i j for type A, we qualitatively explain the experimental results of C at 293 K for Ni (see Table V). In particular, we focus on their signs. The details are described below.
We first note that the experimental values in Table V indicate the estimated values of C i j in the expression for the AMR ratio by Döring in Appendix A. These values are estimated by applying Döring's expression to the experimentally observed AMR ratio. 30) Here, Döring's expression consists of C i j cos jφ i with j =0, 2, and 4; that is, this expression does not take into account higher-order terms of C i j cos jφ i with j ≥ 6. In contrast, our theory produces higher-order terms of C Next, as a model to investigate the experimental result, we choose type A in Table   I. The procedure for choosing type A is as follows: (i) The relatively large value of C ) in Table V  (iii) We choose a type suitable for explaining the experimental result from types A, B, and C. Since the dominant s-d scattering for Ni is considered to be s+ → d−, 26) type C is a prime candidate, while type B is not a candidate. In type C in Table   III, however, we cannot explain the experimental results in For C i j of type A, we roughly determine the parameters. From the previously evaluated ρ s→d↓ /ρ s↑ (∼ 2.5), 25) we first set r s,±→ε1,− = 2.50 and r s,±→ε2,− = r s,±→γ,− = 2.00, where the relation of r s,±→ε1,− > r s,±→ε2,− (= r s,±→γ,− ) results in C  | is also obtained in our theory. Such a difference between the experimental and theoretical results may be a future subject of research.
In particular, C We discuss the dominant s-d scatterings observed in C  Table I  is the term with (λ/∆) 2 , which is positive. In addition, the is the term with (λ/∆) 2 , which is negative. These terms arise from the s-d scattering "s, − → d, −". The dominant s-d scattering observed in C i 4 is thus different from that observed in C i 2 . We also comment on the dominant terms in C where Eqs. (54) and (52)   at 293 K for Ni 30,31) and the theoretical values calculated from C i j for type A in Table I. In this calculation, we use the parameter sets of r s,±→ε1,− = 2.50, 25)

Conclusion
We theoretically studied AMR m,− = 0. The coefficient C i j is composed of λ, H, ∆, δ ε , δ γ , and s-s and s-d resistivities. From such C i j , we obtained a simple expression for C i j for the simplified system with r s,−→3z 2 −r 2 ,− = r s,−→x 2 −y 2 ,− . This system was divided into types A, B, and C. Type A is the generalized strong ferromagnet with the s-d scattering "s, + → d, − and s, − → d, −", type B is the half-metallic ferromagnet with the dominant s-d scattering "s, − → d, −", and type C is the specified strong ferromagnet with the s-d scattering "s, + → d, −". The coefficient C i j for type A includes C i j for types B and C. The AMR ratios AMR [100] (0) and AMR [110] (0) for type A also corresponded to that of the CFJ model 3) under the condition of the CFJ model.
(ii) We found that C i j cos jφ i for type A originates from the changes in the d states due to V so . Concretely, C i j cos jφ i is related to the probability amplitudes and probabilities of the slightly hybridized states or the probability amplitudes of the slightly reduced state in the dominant states.
(iii) For type A, C i j has terms with (λ/∆) 2 , (λ/H) 2 , and λ 2 /(H∆) 2 . In addition, C (v) Using the expressions for C i j for type A, we qualitatively explained the experimental results of C < 0, we also predicted the relation of D

Appendix A: Expression for AMR Ratio by Döring
We report the expression for the AMR ratio by Döring, which consists of the expression for the resistivity based on the symmetry of a crystal. 14, 32, 33) Here, we note that this expression is the same form as an expression for a spontaneous magnetostriction, which minimizes the total energy consisting of the magnetoelastic energy for a spin pair model and the elastic energy for a cubic system. 50) The AMR ratio ∆ρ/ρ is expressed as where ρ is the resistivity for certain directions of I and M ; ρ 0 is the average resistivity for the demagnetized state; α 1 , α 2 , and α 3 indicate the direction cosines of the M direction; β 1 , β 2 , and β 3 denote the direction cosines of the I direction; and k 1 , k 2 , k 3 , k 4 , and k 5 are the coefficients. 32, 33) In this study, M lies in the (001) plane (see Fig. 1); that is, α 3 is set to be 0.
On the basis of Eq. (B·1), we next derive the expression for |m). When a specific state in |m i is written as |m , |m) becomes where, for example, V m,n is given by m|V |n .

Appendix D: Coefficient Expressed by Using Resistivities
We express C i j as a function of ρ i,(v) j,σ . Here, C i j is expressed up to the second order of λ/H, λ/∆, λ/(H ± ∆), δ t /H, δ t /∆, and δ t /(H ± ∆), with t = ε or γ.

(1) I//[100]
Under the condition of the CFJ model, C [100] 2 in Table I  in Table I  , C appear under the crystal field of tetragonal symmetry, whereas they take values of almost 0 under the crystal field of cubic symmetry.