Spin polarization ratios of resistivity and density of states estimated from anisotropic magnetoresistance ratio for nearly half-metallic ferromagnets
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Japanese Journal of Applied Physics
BRIEF NOTE
Spin polarization ratios of resistivity and density of states estimatedfrom anisotropic magnetoresistance ratio for nearly half-metallicferromagnets
Satoshi Kokado ∗ , Yuya Sakuraba , and Masakiyo Tsunoda Graduate SchoolofIntegrated Scienceand Technology,ShizuokaUniversity, Hamamatsu432-8561,Japan NationalInstitutefor MaterialsScience(NIMS), Tsukuba 305-0047,Japan Departmentof ElectronicEngineering,Graduate SchoolofEngineering,Tohoku University, Sendai980-8579,JapanWe derive a simple relational expression between the spin polarization ratio of resistivity, P ρ , and theanisotropic magnetoresistance ratio ∆ ρ/ρ , and that between the spin polarization ratio of the density ofstates at the Fermi energy, P DOS , and ∆ ρ/ρ for nearly half-metallic ferromagnets. We find that P ρ and P DOS increase with increasing | ∆ ρ/ρ | from 0 to a maximum value. In addition, we roughly estimate P ρ and P DOS for a Co FeGa . Ge . Heusler alloy by substituting its experimentally observed ∆ ρ/ρ into the respectiveexpressions. The anisotropic magnetoresistance (AMR) e ff ect, in which the electrical resistivitydepends on the magnetization direction, has been investigated using relatively easy experi-mental techniques for the last 160 years. The e ffi ciency of the e ff ect “AMR ratio” is generallydefined by ∆ ρ/ρ = ( ρ k − ρ ⊥ ) /ρ ⊥ , (1)where ρ k ( ρ ⊥ ) is the resistivity in the case of the electrical current parallel (perpendicular)to the magnetization. We recently derived the general expression of ∆ ρ/ρ and found that ∆ ρ/ρ<
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The HMF isdefined as having a finite density of states (DOS) at the Fermi energy E F in one spin channeland zero DOS at E F in the other spin channel [see Fig. 1(c)]. Namely, the magnitude of thespin polarization ratio of the DOS at E F , | P DOS | , is 1, where P DOS is P DOS = ( D ↑ − D ↓ ) / ( D ↑ + D ↓ ) , (2)with D ↑ ( D ↓ ) being the DOS of the up spin (down spin) at E F . The above condition has been ∗ E-mail: [email protected]
BRIEF NOTE experimentally verified for Heusler alloys.
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On the other hand, in recent years, a current-perpendicular-to-plane giant magnetoresis-tance (CPP-GMR) e ff ect for ferromagnet / nonmagnetic-metal / ferromagnet pseudo spin valveshas been actively studied for application to read sensors of future ultrahigh-density magneticrecording. In particular, studies to enhance the magnitude of the GMR e ff ect are being carriedout intensively. Here, the magnitude of this e ff ect is represented by the resistance change areaproduct ∆ RA , with ∆ R = R P − R AP , where R P ( R AP ) is the resistance of the parallel (antiparallel)magnetization and A is the area of the sample. According to the CPP-GMR theory by Valetand Fert, ∆ RA is expressed by the spin polarization ratio of the resistivity of ferromagnets(the so-called bulk spin asymmetry coe ffi cient), P ρ , and so on. Here, P ρ is defined as P ρ = ( ρ ↓ − ρ ↑ ) / ( ρ ↑ + ρ ↓ ) , (3)where ρ ↑ ( ρ ↓ ) is the resistivity of the up spin (down spin) of ferromagnets. The increase in P ρ tends to increase ∆ RA .
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For example, when ferromagnets are Heusler alloys, ∆ RA becomes relatively large. Recently, Sakuraba et al . have experimentally observed the positive correlation between | ∆ ρ/ρ | of the Co FeGa . Ge . (CFGG) Heusler alloy and ∆ RA of CFGG / Ag / CFGG pseudospin valves. Here, this CFGG was regarded as a nearly HMF, in which there is a low DOS ofthe down spin at E F [see Fig. 1(c)]. The correlation was considered on the basis of the relationbetween ∆ ρ/ρ and P ρ mediated by D ↓ / D ↑ . A relational expression between ∆ ρ/ρ and P ρ and that between ∆ ρ/ρ and P DOS , however, have scarcely been derived. Such expressionsmay make it possible to estimate P ρ and P DOS from the relatively easy AMR measurements.In this paper, we derived a simple relational expression between P ρ and ∆ ρ/ρ and thatbetween P DOS and ∆ ρ/ρ for nearly HMFs using the two-current model. We found that P ρ and P DOS increased with increasing | ∆ ρ/ρ | . We also estimated P ρ and P DOS for CFGG bysubstituting its experimentally observed ∆ ρ/ρ into the respective expressions.We first report the general expression of ∆ ρ/ρ , which was previously derived by using thetwo-current model with the s - s and s - d scatterings.
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Here, s denotes the conduction stateof s, p, and conductive d states, and d represents localized d states.
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The localized d stateswere obtained from a Hamiltonian with a spin–orbit interaction and an exchange field H ex .The AMR ratio ∆ ρ/ρ was finally expressed as ∆ ρ/ρ = − c (1 − x d ) h − Z β ↓ x s / ( β ↑ r m ) i . (1 + Z x s / r m ) , (4)where c = γ/ [( β ↑ y ↑ ) − +
1] ( > Z = (1 + β ↑ y ↑ ) / [1 + ( x d / x s ) β ↓ y ↑ ], γ = (3 / λ/ H ex ) , r m = m ∗↓ / m ∗↑ , BRIEF NOTE x s = D s ↓ / D s ↑ , x d = D d ↓ / D d ↑ , y ↑ = D d ↑ / D s ↑ , β σ = n imp N n | V s σ → d σ | / ( n imp | V imp s | + | V ph s | ), and σ = ↑ or ↓ . Here, λ is the spin–orbit coupling constant, n imp is the impurity density, N n is the numberof nearest-neighbor host atoms around the impurity, V s σ → d σ is the matrix element for the s – d scattering due to nonmagnetic impurities, V imp s is that for the s – s scattering due to theimpurities, and V ph s is that for the s – s scattering due to phonons. The quantity D s σ is thepartial DOS of the conduction state of the σ spin at E F and D d ς ( ς = ↑ or ↓ ) is the partial DOSof the localized d state of the magnetic quantum number M and the ς spin at E F [see Fig.1(a)]. In addition, m ∗ σ is an e ff ective mass of electrons in the conduction band of the σ spin,which is expressed as ~ ( d E σ / dk σ ) − , where E σ is the energy of the conduction state of the σ spin, k σ is the wave vector of the σ spin in the current direction [see Fig. 1(b)], and ~ is thePlanck constant h divided by 2 π . Note that Eq. (4) was derived under the assumption thatthe s – s scattering rate is proportional to D s σ (i.e., the magnitude of the Fermi wave vector ofthe σ spin). In the metallic case of Fig. 1(a), therefore, Eq. (4) is e ff ective at 0 K and in thetemperature T range of the T -linear resistivity including 300 K. On the other hand, in theHMF case of Fig. 1(c), Eq. (4) is e ff ective at 0 K and for k B T ≪ E c − E F and k B T ≪ E F − E v ,where E c ( E v ) is the energy at the bottom of the conduction band (at the top of the valenceband) of the down spin and k B is the Boltzmann constant. This restriction reflects that Eq. (4)does not take into account the thermal excitation of carriers.From Eq. (4), we next obtain a simple expression of ∆ ρ/ρ with x s = x d ≡ x to clearly showthe e ff ect of the DOS on ∆ ρ/ρ . Here, x is assumed to be 0 ≤ x <
1, where x = x ,
0) correspondsto the HMF (non-HMF) [see Fig. 1(c)]. In addition, we set β ↑ = β ↓ ≡ β for simplicity. Suchsimplifications permit only a rough estimation of ∆ ρ/ρ . Equation (4) then becomes ∆ ρ/ρ = − c (1 − x )( r m − x ) / ( r m + x ) . (5)Figure 1(d) shows the x dependence of ∆ ρ/ρ of Eq. (5), with r m = ∆ ρ/ρ takes − c at x = x = r m ( r m ,
1) and becomes closer to 0as x approaches 1. This behavior indicates that ∆ ρ/ρ< ∆ ρ/ρ< that is, ∆ ρ/ρ is set to be −| ∆ ρ/ρ | . Utilizing Eq. (5) with ∆ ρ/ρ = −| ∆ ρ/ρ | , we derive the relational expression between P DOS and ∆ ρ/ρ , and that between P ρ and ∆ ρ/ρ . The details are written as (i)–(iii):(i) The quantity x is obtained as solutions of Eq. (5), i.e., x = , for | ∆ ρ/ρ | / c = , (6) x = a − b ( , , for 0 < | ∆ ρ/ρ | / c < , (7) BRIEF NOTE where a = (1 / r m + / (1 − | ∆ ρ/ρ | / c ), b = (1 / √ d , and d = [( r m + / (1 − | ∆ ρ/ρ | / c )] − r m .Equations (6) and (7) correspond to the HMF and nearly HMF cases, respectively. As to Eq.(7), we originally obtain x ± = a ± b , where 0 < x − < x + >
1. From the assumption of 0 ≤ x < x − , i.e., Eq. (7). The range 0 < | ∆ ρ/ρ | / c < d ≥ ≤| ∆ ρ/ρ | / c ≤ + ( r m + / (2 r m ) and x > < | ∆ ρ/ρ | / c <
1, where | a | > b .(ii) The spin polarization ratio P DOS of Eq. (2) is written as P DOS = (1 − x ) / (1 + x ) , (8)with D ↑ ( ↓ ) = D s ↑ ( ↓ ) + P M = − D d ↑ ( ↓ ) = D s ↑ ( ↓ ) + D d ↑ ( ↓ ) and x s = x d ≡ x . In the HFM case of Eq. (6), P DOS becomes 1. In the nearly HMF case of Eq. (7), P DOS is obtained by substituting x of Eq.(7) into Eq. (8): P DOS = h ( r m +
1) (2 − | ∆ ρ/ρ | / c ) i − " (1 − | ∆ ρ/ρ | / c ) (1 − r m ) + q ( r m + − r m (1 − | ∆ ρ/ρ | / c ) . (9)(iii) The spin polarization ratio P ρ of Eq. (3) is obtained by using ρ ↑ = ρ s ↑ + ρ s ↑→ d ↑ and ρ ↓ = ρ s ↓ + ρ s ↓→ d ↓ in the two-current model, where ρ s σ ( ρ s σ → d ς ) is the resistivity due to the s – s scattering ( s – d scattering). In ρ ↑ and ρ ↓ , terms with γ are ignored because the e ff ect of γ on P ρ is negligibly small. As a result, P ρ is written as P ρ = r m (1 + β ↓ y ↓ ) − x s (1 + β ↑ y ↑ ) r m (1 + β ↓ y ↓ ) + x s (1 + β ↑ y ↑ ) , (10)where y ↓ = D d ↓ / D s ↓ , ρ s ↓ /ρ s ↑ = r m / x s , ρ s σ → d ς /ρ s σ = β σ ( D d ς / D s σ ), and ρ s σ ′ → d ς /ρ s σ = ( ρ s σ ′ /ρ s σ ) β σ ′ ( D d ς / D s σ ′ ) in Ref. 8, with σ , σ ′ , and ς = ↑ or ↓ . When x s = x d ≡ x (i.e., y ↑ = y ↓ ) and β ↑ = β ↓ ≡ β , Eq. (10) is rewritten as P ρ = ( r m − x ) / ( r m + x ) . (11)In the HMF case of Eq. (6), P ρ becomes 1. In the nearly HMF case of Eq. (7) (i.e., metalliccase), P ρ is obtained by substituting x of Eq. (7) into Eq. (11): P ρ = ( r m + − q ( r m + − r m (1 − | ∆ ρ/ρ | / c ) . (12)In Figs. 2(a) and 2(b), we show the | ∆ ρ/ρ | / c dependences of P ρ of Eq. (12) and P DOS ofEq. (9), respectively, where r m = P ρ and | ∆ ρ/ρ | / c , and that between P DOS and | ∆ ρ/ρ | / c . Namely, P ρ and P DOS increase to 1 withincreasing | ∆ ρ/ρ | / c from 0 to 1 (maximum value). The reason for this is that the increase in | ∆ ρ/ρ | / c decreases x [see Fig. 2(c)] and then the decrease in x increases P ρ and P DOS [seeFig. 2(d)]. Furthermore, P ρ and P DOS increase with decreasing r m . The reason for this is that BRIEF NOTE the decrease in r m reduces the maximum value of x [see Fig. 2(c)] and narrows the range of x , and then that feature of x increases P ρ and P DOS [see Fig. 2(d)].As an application, we investigate P ρ and P DOS for CFGG. Regarding parameters, we firstset γ = The quantity y ↑ is roughly estimated to be 10 from the par-tial DOSs of similar Heusler alloys. Next, we consider the uncertain parameter β , whichincludes information on impurities and phonons. Although β actually depends on materials,we determine β from the β dependence of ∆ ρ/ρ for Fe, Co, Ni, and Fe N in Fig. 1(e), wherethe respective parameters are noted in Table I. By comparing the calculation results of Eq.(4) with the experimental results of ∆ ρ/ρ at 300 K in Table I, β is roughly evaluated to be0.1 [see Fig. 1(e)]. This β = c is thus deter-mined to be 0.005; that is, | ∆ ρ/ρ | can take c = x = This c increases with decreasing T due to the decrease in | V ph s | . Judging from the experimentalresult of ∆ ρ/ρ ∼− | ∆ ρ/ρ | < T ann dependences of P ρ and P DOS for CFGG by substituting its experimental result of ∆ ρ/ρ at 300 K [see triangles in Fig.3(a)] into Eqs. (12) and (9), respectively. The white circles in Figs. 3(a) and 3(b) indicate the T ann dependences of P ρ and P DOS , respectively, where r m = We findthat P ρ and P DOS increase with increasing | ∆ ρ/ρ | and decreasing r m in the same trend as theresults in Fig. 2. Such P ρ is compared with the previous values at T ann =
500 and 600 ◦ C in Ta-ble II [see black dots in Fig. 3(a)], which were evaluated by fitting Valet–Fert’s expression to the experimental results of the CFGG thickness dependence of ∆ RA at 300 K.
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Since P ρ at r m = r m =
25, 26)
Table II also shows P DOS ( ,
1) at r m = P DOS , the decrease in | H ex | , and so on. The originof the present P DOS ,
1, however, has not yet been identified.In summary, we derived the simple relational expression between P ρ and ∆ ρ/ρ , and thatbetween P DOS and ∆ ρ/ρ for nearly HMFs. In these expressions, P ρ and P DOS increased to 1with increasing | ∆ ρ/ρ | / c from 0 to 1 (maximum value). In addition, we roughly estimated P ρ and P DOS for CFGG using the respective expressions.
BRIEF NOTE
References
1) W. Thomson, Proc. R. Soc. London , 546 (1856-1857).2) T. R. McGuire, J. A. Aboaf, and E. Klokholm, IEEE Trans. Magn. , 972 (1984).3) T. Miyazaki and H. Jin, The Physics of Ferromagnetism (Springer, New York, 2012)Sect. 11.4.4) M. Tsunoda, H. Takahashi, S. Kokado, Y. Komasaki, A. Sakuma, and M. Takahashi,Appl. Phys. Express , 113003 (2010).5) F. Yang, Y. Sakuraba, S. Kokado, Y. Kota, A. Sakuma, and K. Takanashi, Phys. Rev. B , 020409 (2012).6) Y. Sakuraba, S. Kokado, Y. Hirayama, T. Furubayashi, H. Sukegawa, S. Li, Y. K.Takahashi, and K. Hono, Appl. Phys. Lett. , 172407 (2014).7) S. Li, Y. K. Takahashi, Y. Sakuraba, N. Tsuji, H. Tajiri, Y. Miura, J. Chen, T.Furubayashi, and K. Hono, Appl. Phys. Lett. , 122404 (2016).8) S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J. Phys. Soc. Jpn. , 024705(2012).9) S. Kokado and M. Tsunoda, Adv. Mater. Res. , 978 (2013).10) S. Kokado and M. Tsunoda, J. Phys. Soc. Jpn. , 094710 (2015).11) T. Valet and A. Fert, Phys. Rev. B , 7099 (1993).12) H. S. Goripati, T. Furubayashi, Y. K. Takahashi, and K. Hono, J. Appl. Phys. ,043901 (2013).13) S. Li, Y. K. Takahashi, T. Furubayashi, and K. Hono, Appl. Phys. Lett. , 042405(2013).14) B. S. D. C. S. Varaprasad, A. Srinivasan, Y. K. Takahashi, M. Hayashi, A. Rajanikanth,and K. Hono, Acta Mater. ,6257 (2012).15) The s – s scattering rate due to phonons is proportional to D s σ T in the high temperaturerange including 300 K, with T being the temperature. We here have | V ph s | ∝ T . As to thescattering rate due to phonons, see H. Nishimura, KisoKotaiDenshiRon (Basic ElectronTheory of Solids) (Gihodo Shuppan, Tokyo, 2003) p. 137 [in Japanese].16) C. Kittel,
Introduction to Solid State Physics (Wiley, New York, 1986) 6th ed., p. 193.17) As seen from Fig. 1(d), systems with r m < ∆ ρ/ρ< x ≪
1, whichcorresponds to nearly HMFs. In contrast, systems with r m ∼ ∆ ρ/ρ ∼ x . Only these systems should actually be called non-HMFs.18) Note that x s in Eq. (10) [i.e., x of Eq. (7)] contains not γ but | ∆ ρ/ρ | / c with BRIEF NOTE < | ∆ ρ/ρ | / c < P ρ . T because the down-spin electrons arethermally excited from the valence band to the conduction band. Therefore, in the HMFcase of x =
0, Eq. (11) is valid at 0 K and appropriate for k B T ≪ E c − E F and k B T ≪ E F − E v .20) S. Sharma and S. K. Pandey, J. Phys.: Condens. Matter , 215501 (2014).21) D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum, New York, 1986) p. 95 and 111.22) We utilize the same y ↑ as that of the fcc Ni because of the same crystal structure and thesmall di ff erence in the number of electrons.23) A. Sakuma, J. Phys. Soc. Jpn. , 2007 (1991).24) The CFGG with the L structure has E c − E F ∼ E F − E v ∼ which aremuch larger than k B T = T =
300 K.25) In this study, we consider the case of r m ≤ P DOS with r m > P ρ with r m = T ann = ◦ C, P DOS with r m = | P DOS | ( ∼ P DOS ( = which is regarded as the nearlyHMF.26) As a di ff erent method from the present one, we note that r m may be roughly evaluatedfrom the above-mentioned equation, m σ = ~ ( d E σ / dk σ ) − , where E σ is obtained by, forexample, first-principles calculation. BRIEF NOTE E or E E › fl (a) (b) (c) EE F ds, p or k k › fl E › E fl E F E F d s, p x –1012345 Dr / r / c r m =1 r m =0.7 r m =0.5 r m =0.3 (d) b –0.0100.010.020.030.040.050.06 Dr / r FeCo NiFe N (e)
Fig. 1. (Color online) (a) Partial DOSs of the s, p, and d states for the usual ferromagnets. (b) E σ - k σ curveof the s and p states in (a). (c) Partial DOSs of the s, p, and d states for half-metallic Heusler alloys. In the caseof the nearly HMF, there is a low DOS of the down spin at E F . (d) x dependence of ∆ ρ/ρ/ c of Eq. (5) with x s = x d ≡ x , β ↑ = β ↓ ≡ β , and r m = β dependence of ∆ ρ/ρ of Eq. (4) for Fe, Co, Ni, and Fe Nis shown by solid curves. Here, we set r m = ∆ ρ/ρ at 300 K for Ni, Co, Fe, and Fe N, respectively (see Table I).
BRIEF NOTE | Dr / r |/ c P r r m =10.70.50.3 (a) | Dr / r |/ c P DO S r m =10.70.50.3 (b) | Dr / r |/ c –5 –4 –3 –2 –1 x r m =10.70.50.3 (c) x P r o r P DO S r m =10.70.50.3 Solid curves: P r Dashed curve: P DOS (d)
Fig. 2. (Color online) (a) | ∆ ρ/ρ | / c dependence of P ρ of Eq. (12). (b) | ∆ ρ/ρ | / c dependence of P DOS of Eq.(9). (c) | ∆ ρ/ρ | / c dependence of x of Eq. (7). (d) x dependences of P ρ of Eq. (11) and P DOS of Eq. (8). Here, weset r m = x s = x d ≡ x . T ann ( o C) P r -5-4-3-2-10 ( Dr / r ) / . r m =0.87 (a)0.50.70.3 T ann ( o C) P DO S r m =0.87 (b)0.50.30.7 Fig. 3. (Color online) (a) The white circles show the T ann dependence of P ρ of Eq. (12) for CFGG, where r m = P ρ at T ann = and 600 ◦ C in Table II. The triangles show the experimental result of the T ann dependence of ∆ ρ/ρ at 300 K forCFGG. (b) The T ann dependence of P DOS of Eq. (9) for CFGG with r m = BRIEF NOTETable I.
Parameters x s , x d , and y ↑ , and experimental values of ∆ ρ/ρ at 300 K for bcc Fe, fcc Co, fcc Ni, andFe N. Each x s is evaluated from the values of ρ s ↓ /ρ s ↑ and ρ s ↓ /ρ s ↑ = r m ( D s ↑ / D s ↓ ) (Ref. 8) with r m = x s x d y ↑ ∆ ρ/ρ (experiment)bcc Fe 1.6 0.50 25 (Ref. 21) 0.0030 (Ref. 2)fcc Co 0.37 10 3.5 (Ref. 22) 0.020 (Ref. 2)fcc Ni 0.32 10 3.5 (Ref. 21) 0.022 (Ref. 2)Fe N 25 5.0 20 (Ref. 23) − Table II.
Spin polarization ratios P DOS of Eq. (9) and P ρ of Eq. (12) at T ann =
500 and 600 ◦ C for CFGG.They are the respective values at r m = P ρ , which wereevaluated on the basis of ∆ RA at 300 K,
12, 13) are also noted. T ann ( ◦ C) P DOS of Eq. (9) P ρ of Eq. (12) P ρ (previous values)500 0.54 0.73 0.73 ± ±0.02 (Ref. 13)