Role of electron correlations in transport through domain walls in magnetic nanowires M. A. N. Ara´joa,b∗ , V. K. Dugaevb,c,d, V. R. Vieirab , J. Berakdard, and J. Barna´e,f u s a arXiv:cond-mat/0602399v1 [cond-mat.str-el] 16 Feb 2006 b Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139, U.S.A. CFIF and Departamento de F´ isica, Instituto Superior T´cnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal e c Frantsevich Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, Vilde 5, 58001 Chernovtsy, Ukraine d Max-Planck Institut f¨r Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany u e Department of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´, Poland and n f Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna´, Poland n (Dated: April 28, 2017) The transmission of correlated electrons through a domain wall in ferromagnetic quasi-onedimensional systems is studied theoretically in the case when the domain wall width is comparable with the Fermi wavelength of the charge carriers. The wall gives rise to both potential and spin dependent scattering. Using a poor man’s renormalization group approach, we obtain scaling equations for the scattering amplitudes. For repulsive interactions, the wall is shown to reflect all incident electrons at the zero temperature fixed points. In one of the fixed points the wall additionally flips the spin of all incident electrons, generating a finite spin current without associated charge current. PACS numbers: 73.63.Nm, 71.10.Pm, 75.70.Cn, 75.75.+a Introduction Electronic properties of magnetic wires with domain walls (DWs) attract much interest because of possible applications of the associated magnetoresistance effect [1]. So far, the puzzlingly huge magnetoresistance up to thousands of percents in Ni wires and structured magnetic semiconductors is not well understood [2, 3, 4]. Most of the existing theories of the DW resistance do not take into account the electron correlations [5, 6, 7, 8, 9], except for the case of a wide DW (adiabatic regime) [10]. However, in effectively one-dimensional (1D) structures one should take into account the tendency to the non-Fermi-liquid behavior [11], possible enhancement of the electron-impurity interaction [12] and of the localization effects [13]. Moreover, non-adiabatic DWs (smaller or comparable to the Fermi wavelength) can be achieved in ferromagnetic semiconductor wires with constrictions. In this Letter we study the effect of electron correlations on the resistance of a ferromagnetic wire with a non-adiabatic DW. Since the DW in a 1D wire acts as a localized spin-dependent scattering center, a strong influence of electron correlations is expected [12]. We apply the renormalization method [14, 15] for the logarithmically diverging terms in the perturbation theory of the electron interactions. We find the zero-temperature fixed points for repulsive interactions, in which the wall reflects all incident electrons. This might explain the giant magnetoresistences observed in magnetic nanowires. Moreover, we show that for one fixed point the electron spin is reversed at the reflection. This leads to a nonzero spin current with no charge current, which is of high in- ´ leave from Departamento de F´ ısica, Universidade de Evora, ´ P-7000-671, Evora, Portugal ∗ On terest for applications in spintronics devices. Model and method We consider a magnetic 1D system with a local exchange coupling between conduction electrons and a spatially varying magnetization M(r). The wire itself defines the easy axis (z-axis), and a DW centered at z = 0 separates two regions with opposite magnetizations: Mz (z → ±∞) = ±M0 . Assuming M(r) to lie in the xz plane and the DW width to be smaller than the Fermi wavelength, one can write the single-particle Hamiltonian as (J > 0): h ¯ 2 d2 ˆ H0 = − + ¯ V δ(z) + JMz (z)ˆz + ¯ λδ(z)ˆx , (1) h σ h σ 2m dz 2 where the term hλδ(z)ˆx describes spin dependent ¯ σ scattering due to the Mx (z) component [9], ¯ λ = h ∞ J −∞ Mx (z)dz, and V is a potential (spin independent) scattering term. We assume that spin-↑ electrons are spin majority ones for z < 0 and spin-minority for z > 0. An incident electron from the left with momentum k and spin ↑ (or ↓) can be transmitted to the z > 0 region preserving its spin, but changing momentum from k to k − (or k + ), given by k ± = k 2 ± 4JM0 m/¯ 2 . If the h transmission occurs with spin reversal, the momentum k remains unchanged. The reflection amplitudes for spin-σ electrons with or without spin reversal are denoted by ′ rσ and rσ , respectively. The same convention applies to the transmission amplitudes t′ and tσ . The scattering σ amplitudes are given by: 2 (v + v ∓ + 2iV ) v = r↑(↓) (k) + 1 , (2) (v + v ∓ + 2iV )2 + 4λ2 4iλv ′ (3) = r↑(↓) (k) , t′ (k) = ↑(↓) ∓ + 2iV )2 + 4λ2 (v + v t↑(↓) (k) = with v = ¯ k/m, v ± = ¯ k ± /m, where the upper (lower) h h sign refers to ↑ (↓). We shall henceforth denote by 2 ǫ(±p, σ) the energy of a scattering state with momentum +p (or −p) and spin σ, incident from the left (or right). The scattering amplitudes satisfy general relations that can be found from a generalization of the Wronskian theorem [16, 17] to spinor wave-functions: the Wronskian of any two states having the same energy is a constant, t dψ1 dψ2 − ψ2 (z) = const, (4) dz dz where ψ t denotes the transpose of the spinor ψ. In order to deal with the electron interactions, it is convenient to rewrite the scattering states in second quantization form, making use of right (ˆqσ ) and left (ˆqσ ) a b moving plane wave states. We introduce operators ck,σ ˆ ˆ and dk,σ for the eigenstates corresponding to electrons incident from the left and right, respectively. The plane wave operators are linear combinations of the operators of scattering states. Electron interactions are then introduced through the Hamiltonian dk1 dq † ˆ† ˆ Hint = g1,α,β b a ˆ b ak +q,β ˆk1 −q,α ˆ (2π)2 k1 ,α k2 ,β 2 dk1 dq † ˆ† ˆ a ˆ b bk +q,β ak1 −q,α , (5) ˆ + g2,α,β (2π)2 k1 ,α k2 ,β 2 t W (ψ1 , ψ2 ) ≡ ψ1 (z) where the Greek letters denote spin indices, and the summation convention over repeated indices is used. The coupling constants g1 and g2 describe back and forward scattering processes between electrons moving in opposite directions, respectively. Because the Fermi momentum is spin dependent, we distinguish between g1(2)↑ , which describe interaction between spin-majority particles (that is spin-↑ on the left and spin-↓ on the right of the barrier) and g1(2)↓ , which describe interaction between spin-minority particles. We use g1(2)⊥ to denote interaction between particles with opposite spin. The forward scattering process between particles which move in the same direction will not affect the transmission amplitudes, although it will renormalize the Fermi velocity [11, 14]. This effect is equivalent to an effective mass renormalization and the electrons with different spin orientations may turn out to have different effective masses, in which case our calculations remain valid [17]. Following Ref. [14], the corrections to the transmission amplitudes are calculated to first order in the perturbaˆ tion Hint . Such corrections diverge logarithmically near the Fermi level and will be dealt with in a poor man’s renormalization approach. The perturbative correction to tσ (p′ ) can be obtained from the perturbation expansion for the Matsubara propagator Gσ (τ ) = − Tτ e− ˆ Hint (τ ′ )dτ ′ ap,σ (τ )ˆ† ′ ,σ ˆ cp 0 , (6) where ... 0 denotes the average in the Fermi sea of scattering states. The zero-order propagator for σ =↑ is: (0) G↑ (iω) i it↑ (p′ ) 1 , − = ′ , ↑) p − p′ + i0 iω − ǫ(p p − p′− − i0 (7) where 0 denotes a positive infinitesimal. The poles in the denominators identify the semi-axis on which the electron behaves as a right moving plane wave. The transmission amplitude appears associated with the denominator p − p′− − i0 which, for the variable p, gives a pole in the upper half plane. The meaning of this pole is that the transmitted particle is right-moving in the z > 0 half-axis. Our strategy is to calculate the first order correction term to G, in which a pole in p − p′− − i0 will appear with the residue −iδt↑ (p′ ), which is the transmission amplitude correction. (1) The diagrammatic representation of G↑ is shown in Fig. 1. The horizontal lines represent the electron scattered by the Hartree-Fock potential of the Fermi sea. Only processes where the electron is back-scattered by the Fermi sea are logarithmically large [14]. Consider, for instance, the upper left diagram – an electron, initially in state cp′ ,↑ close to the Fermi level passes through the DW as a right-moving (ˆ) particle. Then, it is reflected (from a ˆ particle) while exchanging momentum q with the a to b ˆ Fermi sea on the z > 0 semi-axis. Finally, it is reflected by the DW again, becoming a spin-up right moving particle of momentum p. A logarithmic divergence occurs if the polarized Fermi sea can provide exactly the momentum that is required to keep the electron always near the Fermi level in the intermediate virtual steps. According to the physical interpretation of the diagrams, we always know on which side of the DW the interaction with the Fermi sea (closed loop in the diagram) is taking place. FIG. 1: Feynman diagrams for the first order contribution G (1) to the propagator (6). The scattering state is represented by a double line, the a (ˆ particle is represented by a conˆ b) tinuous (dashed) line. The loop represents the Hartree-Fock potential of the Fermi sea. The scattered electron exchanges momentum q with the Fermi sea. We use Fermi level velocities v± and wavevectors kF ± for majority or minority spin particles, henceforth. It can be seen that g1⊥ terms are proportional to log |kF + − kF − |, so they do not diverge. Logarithmic divergence 3 would be restored in a spin degenerate system (kF + = kF − ). This can be understood from the diagrams in Fig. 1 as follows: the electron with spin α is reflected by a polarized Fermi sea with spin −α. The momentum provided by the Fermi sea is 2kF −α , while the momentum required by the electron is 2kF α . The g2⊥ terms produce logarithmic divergences that would not exist in the absence of spin-flip scattering (t′ = r′ = 0). For the calculation of δt′ (p′ ) the propagators we need to consider σ are − Tτ ap,−σ (τ )ˆ† ′ ,σ . ˆ cp The expression for δt↑ (p′ ) is directly proportional to log(|ǫ′ |/D), where ǫ′ denotes the energy of the scattered electron and D is an energy scale near the Fermi level where the electron dispersion can be linearized [δt↑ (p′ )/ log(|ǫ′ |/D) is given by the right-hand side of equation (8) below]. The logarithmically divergent perturbation can be dealt with using a renormalization procedure [14]; reducing step by step the bandwidth D and removing states near the band edge is compensated by renormalization of t↑ . Applying this procedure and noting that t↑ + δt↑ remains invariant as D is reduced, one finds the following differential equation: dt↑ + ∂ δt↑ dD = 0 . ∂D Now, we introduce the variable ξ = log(D/D0 ), which is integrated from 0 to log(|ǫ′ |/D0 ), corresponding to the fact that the bandwidth is progressively reduced from D0 to |ǫ′ | (which will be taken as temperature: |ǫ′ | = T ). It is convenient to rewrite the interaction parameters as g↑(↓) = (g2↑(↓) − g1↑(↓) )/4hv+(−) , g⊥ = g2⊥ /2h(v+ + v− ). The scaling differential equations for the transmission amplitudes are The scaling equations for spin-↓ amplitudes follow from the above by simply inverting the spin and velocity indices. Equations (8)-(11) are the central result of this ′ ′ paper. Theorem (4) gives v− /v+ = t↓ /t↑ = r↓ /r↑ . A standard second-order renormalization group treatment shows that in a 1D magnetized system the coupling constants in equation (5) are not renormalized because the logarithmically divergent contributions cancel each other. Fixed points We have performed a numerical analysis of the scaling equations using the DW model (1), with V = 0, for the initial parameters. We now analyze the nature of the fixed points predicted by the scaling equations. The parameters of the model which enter the scaling equations are g↑ , g↓ , g⊥ , and the ratio v− /v+ . For repulsive interactions (g↑ , g↓ , g⊥ > 0) the system flows to insulator fixed points. For λ/v+ larger than about 0.1, all transmission amplitudes vanish faster than any reflection amplitude as T → 0. In this limit we may rewrite ′ the scaling equations for r↑ , r↑ and the theorem (4) neglecting the small transmission amplitudes. Theorem (4) ∗ ∗ ′ ′∗ for W (ψk,↑ , ψk− ,↓ ) tells us that r↓ r↓ + r↑ r↓ = 0. The charge conservation condition is satisfied solely by the reflections, v− ′ 2 v− ′ 2 |r | = |r↓ |2 + |r | , (12) 1 = |r↑ |2 + v+ ↑ v+ ↑ from which we conclude that |r↑ | = |r↓ | at the fixed point. Equations (10) and (11) now take the form dr↑ v− ( 2g⊥ − g↑ − g↓ ) 1 − |r↑ |2 r↑ , = dξ v+ ′ dr↑ v− ′ 2 ′ |r | r↑ . = ( g↑ + g↓ − 2g⊥ ) 1 − dξ v+ ↑ (13) (14) In the derivation of (13) and (14) only the smallness of the transmissions amplitudes was assumed. The phases dt↑ ∗ ∗ ′ ∗ ∗ ′ ′ = g↓ r↓ r↓ t↑ + r↓ r↑ t′ + g↑ r↑ r↑ t↑ + r↑ r↑ t′ of the complex numbers r↑ , r↑ are unchanged during scal↑ ↓ dξ ing. The two fixed points we may consider correspond to ′∗ ′ ′∗ ′∗ ′∗ ′ + g⊥ r↓ r↑ t↑ + r↑ r↓ t′ + r↑ r↑ t′ + r↓ r↑ t↑ , (8) ↑ ↓ r↑ approaching 0, or |r↑ | approaching 1. The situation |r↑ | → 0 requires 2g⊥ − g↑ − g↓ > 0 and, ′ by charge conservation we have |r↑ | → v+ /v− . Upon ′ dt↑ ′∗ ′∗ ′ ′ ∗ ′ ∗ ′ = 2g↓ r↓ r↓ t↑ + 2g↑ r↑ r↑ t↑ + 2g⊥ r↓ r↑ t↑ + r↑ r↓ t↑ .(9) integrating (14) with ξ going from 0 to log(T /D0 ), the dξ ′ ′ ′ amplitude r↑ will vary from r↑,0 to r↑ (T ). Introducing ′ ′ the reflection coefficient R↑ = (v− /v+ )|r↑ |2 , we obtain Equations for the reflection amplitudes rσ (p′ ) ′ and rσ (p′ ) can be obtained from the propagators 2(g↑ +g↓ −2g⊥ ) R′ ↑,0 T b cp − Tτ ˆp,±σ (τ )ˆ† ′ ,σ . The equation for r↑ (p′ ) is 1−R′ D0 ↑,0 R′ (T ) = . (15) ↑ 2(g↑ +g↓ −2g⊥ ) R′ ↑,0 T 1 + 1−R′ dr↑ D0 ∗ ∗ ′ ′ ∗ ∗ ↑,0 = g↑ r↑ r↑ r↑ + r↑ t′ t′ + g↓ r↓ t↑ t↓ + r↓ r↓ r↑ ↑ ↑ dξ If 2g⊥ − g↑ − g↓ > 0 then R′ (T ) → 1 as T → 0. The ↑ ′∗ ′ ′∗ ′ ′∗ ′∗ +g⊥ r↑ r↓ r↑ + r↓ r↑ r↑ + r↑ t↓ t′ + r↓ t↑ t′ − g↑ r↑ (10) ↑ ↑ DW reflects all incident electrons while additionally reversing their spin – it is a 100% “spin-flip reflector” at ′ and the equation for r↑ (p′ ) is zero temperature, generating a finite net spin current but no charge current. In order to find the low T behavior of ′ dr↑ ′ ∗ ′ ∗ ′ ∗ ′ ∗ ′ transmissions we put rσ = 0, |r↑ | = v+ /v− in the scal= g↑ r↑ r↑ r↑ + r↑ t↑ t↑ + g↓ r↓ t↑ t↓ + r↓ r↓ r↑ dξ ing equations for the transmission amplitudes and obtain ′ ′∗ ′∗ ′ ′ ′∗ ′∗ |t↑ | ∼ |t′ | ∼ |t′ | ∼ T 2g⊥ . +g⊥ r↑ r↓ r↑ + r↓ r↑ r↑ + r↑ t′ t′ + r↓ t↑ t↑ − g⊥ r↑ . (11) ↓ ↑ ↑ ↓ 4 In the regime where g↑ +g↓ −2g⊥ > 0 we have R′ (T ) → ↑ 0, R↑ (T ) → 1. The DW reflects then all incident electrons while preserving their spin. The scaling equations for the transmission amplitudes yield |t↑ | ∼ T g↑ +g↓ , |t′ | ∼ T 2gσ . σ ′ If g↑ + g↓ − 2g⊥ = 0 then both rσ (T ) and rσ (T ) tend to finite values. The scaling equations for t↑ , t′ , with σ constant reflection amplitudes, become a linear algebraic 3 by 3 system. The eigenvalues of the matrix give the temperature exponents and each transmission amplitude will be a linear combination of the three powers of T . For decreasing temperature there may be crossovers from one exponent to the other and the lowest one dominates as T → 0. For smaller values of λ/v+ (i.e., smaller than about 0.1) in the Hamiltonian (1), the system flows to a fixed ′ point where r↑ vanishes about as fast as the transmissions and |rσ | → 1. The transmission amplitudes still scale to zero as |t↑ | ∼ T g↑ +g↓ , |t′ | ∼ T 2gσ . The scaling equation σ ′ ′ for r↑ can be linearized in r↑ . Neglecting the second ′ order terms in t, t , and considering that |rσ | → 1, we obtain |R′ | ∼ T 2(g↑ +g↓ ) . In the scaling equation (10) for ↑ r↑ we cannot neglect the terms containing transmission amplitudes on the right hand side. The behavior of rσ as T → 0 can be found from the charge conservation condition: 1 − |rσ |2 ∼ T min{2(g↑ +g↓ ),4gσ } . We may estimate the λ parameter in Eq.(1) by asˆ ˆ suming that M (z) = M0 cos θ(z)z + M0 sin θ(z)x with cos θ(z) = tanh(z/L), where L is the length of the DW [10]. We then have λ = πM0 L/¯ , implying that h JM0 λ = πm 2 2 (LkF + ). v+ h ¯ kF + (16) The condition for the DW to be smaller than the Fermi wavelength is LkF + < 2π. The smaller Fermi wavelength is that of spin-majority electrons, 2π/kF + . The ratio v− /v+ depends on the degree of polarization of the electron system. In a 1D nonmagnetic system there is a single Fermi momentum, kF , for up and down electrons and a Fermi energy EF = ¯ 2 kF /(2m). Once the system beh 2 comes magnetized, there is a Zeeman energy shift of the bands, ∆E/2 = JM0 , and the two new Fermi momenta, kF ± , satisfy ∆E kF ± =1± . kF 4EF (17) Inserting this result in Eq.(16) above, we obtain (∆E/4EF ) λ =π 2 (Lk+ ) . v+ [1 + (∆E/4EF )] (18) In the full polarization limit kF − = 0, kF + = 2kF , and Eq. (18) gives λ/v+ ≈ 0.79 LkF + . Typical values for a non-fully polarized system are EF = 90 meV and ∆E = 30 meV [4]. In this case we have v− /v+ = 0.84 and Eq. (18) gives λ/v+ ≈ 0.22LkF +. Therefore, if LkF + is smaller than about 2π, the system can flow to any of the fixed points described above. Lateral quantization may produce several channels. The possibility of inter-channel scattering then arises due to two causes: (i) electron interactions (which would require a modification of our theory to allow for interchannel scattering); (ii) impurity scattering. For the latter to be negligible the electron mean free path must be larger than the size of the constriction pinning the DW. The above spin-flip reflector DW was not found in Ref [10]. This is because the adiabatic DW considered in Ref [10] is a poor spin-flip reflector at the noninteracting level already – as in the regime of small λ/v+ above. Discussions with P. A. Lee, A. H. Castro Neto and P. Sacramento are gratefully acknowledged. M.A.N.A. is grateful to Funda¸ao para a Ciˆncia e Tecnoloc˜ e gia for a sabbatical grant. This research was supported by Portuguese program POCI under Grant No. POCI/FIS/58746/2004, EU through RTN Spintronics (contract HPRN-CT-2000-000302), Polish State Committee for Scientific Research under Grant No. 2 P03B 053 25, and by STCU Grant No. 3098 in Ukraine. ˇ c [1] S. A. Wolf, et al. Science 294, 1488 (2001); I. Zuti´, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] N. Garc´ et al, Phys. Rev. Lett. 82, 2923 (1999); G. ia Tatara, et al, Phys. Rev. Lett. 83, 2030 (1999). [3] H. D. Chopra et al, Phys. Rev. B 66, 020403(R)(2002); H. D. Hua et al, Phys. Rev. B 67, 060401(R) (2003). [4] C. R¨ster, et al, Phys. Rev. Lett. 91, 216602 (2003). u [5] G. Tatara, et al, Phys. Rev. Lett. 78, 3773 (1997). [6] R. P. van Gorkom et al., 83, 4401 (1999). [7] L. R. Tagirov et al, J. Magn. Magn. Mater. 258-259, 61 (2003). [8] M. Ye. Zhuravlev, et al, Appl. Phys. Lett. 83, 3534 (2003). [9] V. K. Dugaev, et al, Phys. Rev. B 68, 104434 (2003); V. K. Dugaev, et al, Phys. Rev. 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