Ballistic and diffuse transport through a ferromagnetic domain wall

Ballistic and diffuse transport through a ferromagnetic domain wall

Arne Brataas*

Department of Applied Physics and Delft Institute of Microelectronics and Submicrontechnology, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

Gen Tatara

Graduate School of Sciences, Osaka University, Toyonaka, Osaka 560, Japan Gerrit E. W. Bauer

Department of Applied Physics and Delft Institute of Microelectronics and Submicrontechnology, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 16 February 1999兲

We study transport through ballistic and diffuse ferromagnetic domain walls in a two-band Stoner model with a rotating magnetization direction. For a ballistic domain wall, the change in the conductance due to the domain wall scattering is obtained from an adiabatic approximation valid when the length of the domain wall is much longer than the Fermi wavelength. In diffuse systems, the change in the resistivity is calculated using a diagrammatic technique to the lowest order in the domain-wall scattering and taking into account spin dependent scattering lifetimes and screening of the domain-wall potential.关S0163-1829共99兲09829-X兴

I. INTRODUCTION

In a ferromagnet, domains with different directions of the magnetization are favored by the long-range magnetic dipo-lar interaction. The boundary between the domains, the do-main walls 共DW’s兲, are a source of magnetoresistance that recently has attracted experimental1–5 and theoretical interest.6–8

For ballistic systems, where the electron mean free path is longer than the system size, first-principles band-structure calculations have shown that the DW resistance is enhanced due to the nearly degenerate bands at the Fermi energy.8The rotating magnetization direction causes an effective potential barrier for the electrons which increase the resistance. Re-cently large DW resistance in ballistic Ni nanocontacts has been measured.9 The appearance of large DW resistance in small contacts is in agreement with the results of Ref. 8.

In the diffuse transport regime Cabrera and Falicov10 in-terpreted transport through a single DW as a tunneling pro-cess and the corresponding MR was found to be exponen-tially small. Berger11modeled the domain-wall scattering as a force on the magnetic moment of the conduction electrons. Tatara and Fukuyama6 calculated the DW conductivity for spin-independent scattering lifetimes. Levy and Zhang7 pointed out that spin dependent impurity scattering can strongly enhance the DW resistivity.

Beyond the semiclassical transport theories, Tatara and Fukuyama6 predict a negative DW resistivity as a result of the reduced weak-localization correction due to the decoher-ence of the electrons by the scattering off the domain wall. However, quantum interference effects do not explain the experiments in Refs. 3 and 4, where the negative DW resis-tance persists up to high temperature where the inelastic-scattering length is shorter than the mean free path. It has been suggested by Ruediger et al.12 that the experimentally observed negative DW resistance is an extrinsic effect

caused by the interplay between orbital effects due to the internal magnetic fields and surface scattering. Recently, it was also demonstrated that the large negative domain-wall resistance of Co films1is due to MR resistivity anisotropy.13 It is the purpose of the present paper to give a detailed account of the transport through a domain wall both in the ballistic and the diffuse regime in a two-band Stoner model. In the ballistic transport regime, the transport through the magnetic domain wall can be treated by an adiabatic ap-proximation similar to the one used for transport through a quantum point contact. In the diffuse regime we will use the diagrammatic technique introduced in Ref. 6 and generalize it to the case of asymmetric impurity scattering lifetimes共but without localization effects6兲 and screening of the domain-wall potential. Our results, although more general, reduce in the case of strong spin splitting to results that are very simi-lar to those obtained in Ref. 7 using a Boltzmann equation. We explain why the results of the two methods differ. Some results have been published already in a brief report and a conference proceedings.8,14Here we give an in-depth discus-sion of the results including the technical details of the deri-vations.

The paper is organized in the following way. The two-band Stoner model for the ferromagnet with a rotating mag-netic field and how it can be reduced to a more tractable form by a local gauge transformation is discussed in Sec. II. The ballistic transport regime is discussed in Sec. III and the diffuse transport regime in Sec. IV. We give our conclusions in Sec. V. The appendixes include the adiabatic approxima-tion which can be used in the ballistic situaapproxima-tion, recipes for the calculation of the frequency summation of the Feynman diagrams, and the spin-spiral case which can be exactly di-agonalized.

II. MODEL

Throughout, we will use an effective two-band model to describe the ferromagnet with the Hamiltonian

PRB 60

H⫽

冕

dr⌿₀†共r兲

冋

⫺ ប

2

2mⵜ

2_⫹␮

BH共r兲•␴

册

⌿0共r兲, 共1兲

where␮Bis the Bohr magneton,␴x,␴y, and␴zare the Pauli

spin matrices, H(r) is the effective magnetic field arising from the electronic exchange interaction, the magnetic dipole interaction, and the external magnetic field, and⌿0(r) is the

two-component spinor wave function. The direction of the effective magnetic field is represented by a rotation angle

␪(z) which varies along the z direction. The spin-orbit inter-action and the Lorentz force due to the internal magnetiza-tion are disregarded, since experimentally the DW magne-toresistance can be separated from the anomalous magnetoresistance共AMR兲 and the orbital magnetoresistance 共OMR兲.4 _{We use a local gauge transformation}6 _⌿

0(r)

⫽U(r)⌿(r), where

U共r兲⫽cos共␪/2兲␴z⫹sin共␪/2兲␴x 共2兲

and introduce the Fourier transform of the change of the direction of the magnetic field a(z)⬅d␪(z)/dz ⫽兺qexp(iqz)aq. After the gauge transformation 共2兲 the

Hamiltonian 共1兲 becomes H˜⫽U†HU⫽H0⫹V. The

unper-turbed term is H0⫽

兺

ks 共⑀k s_⫺␮兲c ks † cks, 共3兲

where s⫽⫹ (s⫽⫺) denotes spin-up 共spin-down兲 states,

⑀k

s_⫽ប2_k2_{/2m⫺s⌬, and the spin splitting is ⌬⫽␮}

B兩H兩. The

interaction with the DW is6

V⫽ ប 2 2m 1 4 _kqq

兺

_⬘saq⫺q⬘aq⬘ck⫹qzs † cks ⫹ប 2 2m _kqss

兺

_⬘

冉

kz⫹ q 2

冊

aqck⫹qzˆs † 共␴y兲s,s⬘cks⬘, 共4兲

where zˆ is a unit vector in the z direction and kz⫽kzˆ.

III. BALLISTIC TRANSPORT

Transport through the domain wall is ballistic when the system size is smaller than the mean free path. In this re-gime, the transport properties can be described by the Land-auer conductance, G⫽e 2 h _k

兺

储ss⬘ T_k 储 ss⬘_{, 共5兲} where T_k 储 ss⬘

is the transmission probability for an electron in the transverse mode k_储 to pass the domain wall from spin state s

⬘

to spin state s. Domain walls in transition metals are much thicker than the Fermi wavelength 共the length of the domain wall is ␭w⫽40 nm 共Fe兲, ␭w⫽100 nm 共Ni兲, ␭w⫽15

nm共Co兲, and the Fermi wavelength is roughly ␭F⬃0.2 nm兲.

The transmission probability can therefore be calculated with the aid of an adiabatic approximation on the eigenstates of the Hamiltonian after the gauge transformation 共2兲 共see Ap-pendix A兲. The domain wall is then equivalent to an effective potential barrier for the electrons. The conductance is deter-mined by the minimum number of propagating modes, which

is where the gradient of the rotating magnetic field has its maximum value, a(z)⬅d␪(z)/dz˜amax 共for details, see

Appendix A兲. The conductance can then be found from the conductance of a spin spiral with gradient amax, which has

the dispersion E_max⫾ ⫽ប2/(2m)关k2⫹a_max2 /4⫾(k_z2a_max2 ⫹p4)1/2兴, and the conductance is

G⫽e

2

2

兺

ks 兩vs兩␦共Emax

s _⫺E

F兲, 共6兲

where vs⫽⳵Es/(ប⳵kz) is the group velocity. Carrying out

the integration, we find the domain-wall resistance Rw/R0

⫽R⫺R0 关1/R⫽G, 1/R0⫽G0⫽(e2Ak¯F 2_)/(2␲h)]: Rw R0 ⫽

再

amax 2 /共4k¯F 2_{兲 a} max 2 _⭐2p2 p2/k¯F 2_⫺p4_/共a max 2 k ¯ F 2_{兲 a} max 2 _⬎2p2_, 共7兲

where ប2¯k_F2/(2m)⬅EF and ⌬⬅ប2p2/(2m). The screening

of the domain-wall potential discussed below is not impor-tant for the calculation of the conductance, since by a calcu-lation following the lines in Sec. IV, we find that the shift in the chemical potential due to the rotating magnetization is

␦␮⬇⫺(1/48)Ew(⌬/EF)2⫹O(⌬/EF)3. The conductance is

G⬃k_F2⬃␮, and therefore screening gives a vanishing small contribution to the change in the resistance when the splitting is sufficiently small, ␦R_␮/R₀⬇(1/48)(E_w/E_F)(⌬/E_F)2.

Using parameters for Fe, Ni, and Co (␭w⫽40 nm, ␭w ⫽100 nm, ␭w⫽15 nm, respectively兲, we find Rw/R ⫽0.0008%, Rw/R0⫽0.0001%, Rw/R0⫽0.008%,

respec-tively. Within the two-band model, the ballistic domain-wall scattering is thus very weak. In first-principles band-structure calculations these small numbers are enhanced by orders of magnitude due to the共near兲 degeneracy of the energy bands at the Fermi level.8

IV. DIFFUSE TRANSPORT

When the system size is much larger than the mean free path, the transport is in the diffuse regime. We assume that the electrons are subject to spin-dependent scattering, which is modeled by short-range scatters giving rise to spin-dependent lifetimes ␶_⫹ and ␶_⫺ for spin-up and spin-down states, respectively, which will be treated as adjustable pa-rameters.

We study the current in the z direction. The current op-erator transformed by the local spin rotation U(r) in Eq.共2兲 is J˜⫽U†JU⫽J₀⫹J_g. The unperturbed current operator is

J0⫽ eប m

兺

ks kzcks † cks 共8兲

and due to the local gauge transformation6

Jg⫽

eប

2m _kqss

兺

_⬘aqck⫹qzs共␴y兲s,s⬘cks⬘. 共9兲 The conductivity is calculated from the Kubo formula:

␴共␻兲⫽_␻i

冋

⌸共␻兲⫹n0e 2

where n0⫽N/V is the electron density, N is the number of

electrons, and V is the volume of the system. The current-current correlation function⌸(␻) is obtained by an analyti-cal continuation (i␻l˜␻⫹i␦, ␦˜0⫹) from the Matsubara

correlation function ⌸l_⫽⫺1 V 1 ប

冕

0 ␤ប ei␻_lt

_具

_兩T ␶˜J共␶兲J˜共0兲兩

典

, 共11兲

where 1/␤⫽k_BT, k_B is the Boltzmann constant, T is the temperature, and T_␶is the␶-ordering operator. We will only study the dc conductivity at low temperatures by letting ␻ ˜0 and T˜0. The relevant Feynman diagrams to the low-est order in the scattering by the domain wall were identified in Ref. 6 and are shown in Fig. 1. Diagram 0 represents the zeroth-order Drude contribution, diagram 1 is due to the cor-relation of the correction to the conductivity operator 共9兲, diagrams 2 and 4 are self-energy corrections from the inter-action Hamiltonian共4兲 to the electron Green’s function, dia-gram 5 is a vertex correction, and diadia-gram 3 is the correla-tion of the change in the current operator 共9兲 and the interaction Hamiltonian 共4兲. The electron Green’s function appearing in the Feynman diagrams in Fig. 1 is a configura-tional average over impurity positions, e.g., the retarded Green’s function is G_ksR共␻兲⫽ 1 ប␻⫺⑀k s_⫹iប/2␶ s . 共12兲

The scattering lifetimes of the states at the Fermi energy due to the impurity scattering␶s are assumed to be isotropic but

may be spin dependent.15

The dc conductivity of a single domain ferromagnet is

␴0⫽e 2 V

兺

ks

冉

⳵⑀ks ⳵kz

冊

2 ␦共⑀ks⫺␮0兲␶s 共13兲 ⫽e 2 m共n⫹␶⫹⫹n⫺␶⫺兲, 共14兲 where ␮0 is the bulk chemical potential, n_⫹ (n_⫺) is the electron density of spin-up共spin-down兲 states, and␶_⫹ (␶_⫺)

is the scattering lifetime of spin-up 共spin-down兲 states. There are two contributions to the conductivity which to the lowest order in the domain-wall scattering are additive. First, screening shifts the chemical potential and induces a DW conductivity in Eq. 共13兲. Second, the electrons are di-rectly scattered by the domain wall by the interaction term 共4兲 and the gauge transformed current operator 共9兲.

Since the width of domain walls in transition metals is much larger than the screening length, electroneutrality dic-tates that the electron density to a good approximation is the same in the presence or absence of the DW, but the chemical potential differs. This is in contrast to the treatment in Refs. 6 and 7, where the chemical potential is assumed to be the same in the presence or absence of the DW. The change in the conductivity due to the chemical potential shift can be found from Eq. 共13兲 setting␮0˜␮0⫹␦␮:

␦␴0⫽␦␮e 2

m

兺

s

Ns␶s, 共15兲

which to lowest order in the domain-wall scattering may be added to the DW conductivity. Here N_s⫽mk_Fs/(2␲2ប2) is the electron density of states at the Fermi energy, k_Fs is the spin dependent Fermi wave vector related to the spin depen-dent electron density by ns⫽(kF

s

)3/(6␲2), and we also in-troduce⑀_Fs⬅ប2(k_Fs)2/(2m).

We proceed by calculating the chemical potential shift due to the rotating magnetization. The zeroth-order contribu-tion to the electron density is n0⫽兺ks␪(␮⫺⑀k

s

)/V, where

␪(x) is the Heaviside step function. The Feynman diagrams of the contributions to the electron density in the lowest-order interaction with the domain wall are shown in Fig. 2. Diagram A is due to the first term in Eq.共4兲 and diagram B is due to the second-order contribution of the second term in Eq. 共4兲. Combining the two terms, the second-order contri-bution to the electron density is

n2⫽ ប2 2mV

兺

kqs 兩aq兩 2

冉

ប 2_k z 2 /2m 2⌬共kq兲 ⫺ 1 4

冊

␦共⑀k_⫺ s _{⫺␮兲, 共16兲} where 2⌬(kq)⫽⑀_k ⫹ ⫺s_⫺⑀ k_⫺ s _{and k} ⫾⫽k⫾(q/2)zˆ. Since the

DW is much thicker than the Fermi wavelength, we disre-gard the wave-vector dependence共q兲 on the electron states at the Fermi level and introduce the energy parameter for the domain wall, Ew⫽兺qប2兩aq兩2/(2m), e.g., with cos␪ ⫽tanh(z/␭W), Ew⫽␲ប2/(L␭Wm), where nW⫽1/L is the

FIG. 1. Feynman diagrams of the contributions to the conduc-tivity to the lowest order in the domain-wall scattering. Solid lines indicate the electron Green’s function and the dashed lines the in-teraction with the domain wall. The vertex ⫻ arises from the un-perturbed current operator 共8兲, the vertex 䊊 is due to the gauge transformation on the current operator共9兲, the vertex 〫 is due to the first term in the interaction Hamiltonian共4兲, and the vertex 䊐 is due to the second term in the interaction Hamiltonian共4兲.

FIG. 2. Feynman diagrams of the contributions to the electron density to the lowest order in the domain-wall scattering. Solid lines indicate the electron Green’s function and the dashed lines the in-teraction with the domain wall. The vertex 〫 is due to the first term in the interaction Hamiltonian共4兲 and the vertex 䊐 is due to the second term in the interaction Hamiltonian共4兲.

‘‘density’’ of the domain wall. The chemical potential shift follows from n0⫹n2⫽n, where n is the electron density,

␦␮⫽Ew

冋

1 4⫺

兺

s sN_s⑀_Fs 6⌬

兺

s Ns

册

. 共17兲

The correction in the conductivity 共15兲 due to the chemical potential shift becomes

␦␴0⫽e 2_E w m

兺

s Ns␶s

冋

1 4⫺

兺

s sNs⑀F s 6⌬

兺

s Ns

册

. 共18兲

The corrections to the current-current correlation function from diagrams 1–5 are

⌸1 l_⫽e 2_ប2 4m2V

兺

kqs 兩aq兩 2_␲1l_{共kqs兲, 共19a兲} ⌸2 l_⫽e 2_ប4 8m3V

兺

kqs 兩aq兩 2_k z 2_␲2l_{共kqs兲, 共19b兲} ⌸3 l_⫽e 2_ប4 2m3_V

兺

_kqs 兩aq兩 2

冉

_k z⫺ q 2

冊

kz␲3 l_{共kqs兲, 共19c兲} ⌸4 l_⫽e 2_ប6 4m4V

兺

kqs 兩aq兩 2

冉

_k z⫺ q 2

冊

2 k_z2␲4l共kqs兲, 共19d兲 ⌸5 l_⫽e 2_ប6 4m4V

兺

kqs 兩aq兩 2_k z 2

冉

k_z2⫺q 2 4

冊

␲5 l 共kqs兲, 共19e兲 where the frequency summations are defined by

␲1l_⫽1 ␤

兺

n G_k ⫺s n⫹l G_k ⫹⫺s n , 共20a兲 ␲2l_⫽1 ␤

兺

n 关Gks n⫹l G_ksn⫹lG_ksn ⫹共l˜⫺l兲兴, 共20b兲 ␲3l_⫽1 ␤

兺

n 关Gk⫺s n⫹l_G k_⫹⫺s n⫹l _G k_⫺s n _{⫹共l˜⫺l兲兴,} 共20c兲 ␲4l_⫽1 ␤

兺

n 关Gk⫺s n⫹l_G k_⫹⫺s n⫹l _G k_⫺s n⫹l_G k_⫺s n _{⫹共l˜⫺l兲兴,} 共20d兲 ␲5l_⫽1 ␤

兺

n Gk_⫺s n⫹l Gk_⫹⫺s n⫹l Gk_⫺s n Gk_⫹⫺s n . 共20e兲

In the low impurity density limit the energy splitting be-tween the bands is larger than the broadening of the bands

due to the impurity scattering, ⌬␶/បⰇ1. The frequency sums 共20兲 are evaluated in Appendix B, where the weak wave-vector dependence on the electron states at the Fermi level is disregarded consistent with the treatment of the chemical-potential shift above. Carrying out the Matsubara frequency sums over the internal energies at low tempera-ture, we find that the correction to the dc conductivity due to the DW is ␴1⫽0, 共21a兲 ␴2⫽⫺e 2_E w m

兺

s Ns␶s 1 4, 共21b兲 ␴3⫽⫺e 2_E w m

兺

s Ns␶s 2 3s ⑀F s ⌬ , 共21c兲 ␴4⫽e 2_E w m

兺

s Ns␶s

冋

s⑀_Fs 2⌬ ⫺

冉

⑀F s ⌬

冊

2 ␶⫹⫹␶⫺ 10␶_⫺s

册

, 共21d兲 ␴5⫽e 2_E w m

兺

s Ns␶s 1 5

冉

⑀F s ⌬

冊

2 . 共21e兲

The contribution to the conductivity from the diagrams 1–5 in Fig. 1 is

兺

i⫽1 5 ␴i⫽⫺ e2_E w m

兺

s Ns␶s ⫻

冋

1 4⫹ 1 6s ⑀F s ⌬ ⫺ 1 10

冉

⑀F s ⌬

冊

2

冉

1⫺ ␶s ␶⫺s

冊

册

. 共22兲

The DW resistivity can be found from ␳_w⫽⫺␴_w␳₀2, where the DW conductivity change due to the rotating magnetic field is␴_w⫽␦␴0⫹兺_i5_⫽1␴_i: ␳w⫽ e2␳02Ew 6m

兺

s N_s␶_s

冋

␦⑀F ⌬ ⫹ s⑀_Fs ⌬ ⫺ 3 5

冉

⑀F s ⌬

冊

2

冉

1⫺ ␶s ␶⫺s

冊

册

, 共23兲 where ␦⑀F⫽兺ssNs⑀F s_/兺

sNs. The first term in Eq. 共23兲 is

always positive, but the second and the third terms can be negative when the relaxation time of the minority-spin elec-trons is longer than the relaxation time for the majority-spin electrons. However, as will be demonstrated below, the domain-wall resistivity given by Eq.共23兲 is always positive. Our speculation about the possibility of a negative domain-wall resistance in Ref. 14 is thus not justified from Eq. 共23兲 only. The result 共23兲 differs from that obtained in Ref. 6 for spin independent relaxation times ␶s⫽␶, where screening

was not taken into account, i.e., a constant chemical potential and not a constant electron density was assumed. The result also differs from the calculation in Ref. 7 based on the Bolt-zmann equation. We believe that this latter discrepancy is because in Ref. 7 screening as well as the effect of the gauge transformation on the current operator共9兲 are neglected. The latter corresponds to the neglect of the diagrams 1 and 3 in Fig. 1. For sufficiently weak impurity scattering and/or a large spin splitting (⌬␶/បⰇ1) the contribution from

dia-gram 1, Eq.共21a兲, vanishes, so only in this limit the omission of this diagram is justified. Furthermore, the difference in the Fermi wave vectors for the spin-up and spin-down electrons was disregarded in parts of the calculations in Ref. 7 and an approximation was introduced in order to solve the integral equation for the Boltzmann equation. Indeed, assuming small spin splittings (⌬/⑀FⰆ1, but ⌬␶/បⰇ1) in Eq. 共23兲, we

ob-tain ␳w ␳0 ⬇3EwEF 20⌬2 共␶⫹⫺␶⫺兲2 ␶⫹␶⫺ 共24兲

which is very similar, but not identical to the result in Ref. 7. In this limit, the domain-wall resistivity increases quadrati-cally with the asymmetry in the spin-up and spin-down scat-tering lifetimes as pointed out in Ref. 7. For larger spin split-ting, Eq. 共23兲 should be used.

It is also interesting to study the domain-wall resistivity when the spin splitting is large, 2⌬⬎␮, which is the case for a half metallic ferromagnet in which the minority-spin den-sity of states vanishes, N_⫺⫽0. In this regime we find

␳w ␳0 ⫽Ew ␮

冋

␮ 2⌬ ⫺ 3 5

冉

␮ 2⌬

冊

2

冉

1⫺␶⫹ ␶⫺

冊册

. 共25兲

The first term in Eq. 共25兲 can be interpreted as additional intraband scattering in the majority-spin channel due to the rotating magnetization and the second term as virtual trans-port in the minority-spin channel which has a negative con-tribution when ␶_⫺⬎␶_⫹. The domain-wall resistivity 共25兲 is always positive. In the limit of large spin splittings⌬Ⰷ␮the domain-wall resistivity vanishes, since the coupling between the bands becomes vanishingly small. Note that the present formalism is valid only for wide walls, since the domain-wall scattering is treated as a perturbation.

Our perturbative result 共23兲 can be checked against an exact calculation for a spin spiral, d␪(z)/dz⫽a0, aq ⫽a0␦q,0with spin independent lifetimes␶s˜␶. The detailed

calculation is shown in Appendix C. The Hamiltonian is di-agonalized in spin space by u_⫾⫽N_⫾关1,i(1⫿

冑

1⫹␣2)/␣兴T, where N_⫾ is a normalization constant,␣⫽kza0/ p2, and⌬

⫽ប2_p2_{/2m. The corresponding eigenvalues are E}

k

⫾

⫽(ប2_/2m)(k2_⫹a 0 2_⫿

冑

k_z2a₀2⫹p4). The 共Drude兲 resistivity can be calculated from the Kubo formula,

␳w⫽

e2␳₀2Ew

2m⌬ ␶共n⫹⫺n⫺兲. 共26兲 This is in exact agreement with Eq.共23兲 for aq⫽a0␦q,0when

␶⫹⫽␶⫺; a good indication of the correctness of our

pertur-bation approach. The calculations in Refs. 6 and 7 disagree with the exact result共26兲 presumably due to the reasons out-lined above.

The result for the domain-wall resistivity共23兲 can be ana-lyzed by introducing k_F⫹⫽

冑

␥kF, kF⫺⫽kF/

冑

␥, ␶⫹⫽

冑

␩␶,

␶⫺⫽␶/

冑

␩, where␥⫽kF⫹/kF⫺is a measure of the polarization

of the ferromagnet,␩⫽␶_⫹/␶_⫺is a measure of the asymme-try of the scattering lifetimes, kF is the average Fermi wave

vector, and␶ is the average scattering lifetime. The domain-wall resistivity is proportional to ␬⫽EW/4EF. Typically in

Fe k_F⬃1.7 Å , ␭_W⬃300 Å , and n_w⬃2.5 ␮m⫺1 giving␬

⬃10⫺6_{, which means that the domain-wall scattering for}

symmetric scattering lifetimes is very weak. However, as pointed out in Ref. 7, the domain-wall resistivity can become appreciably larger when taking into account the lifetime asymmetry of the carriers. We show in Fig. 3 the scaled domain-wall resistivity␳w/(␬␳0) as a function of the

asym-metric scattering lifetimes␶_⫹/␶_⫺in the case of a small spin polarization ␥⫽1.01 共solid line兲, intermediate spin polariza-tion ␥⫽1.20 共dashed line兲, and large spin polarization ␥ ⫽10.0 共dotted line兲. For a larger spin polarization, the domain-wall resistivity naturally becomes asymmetric in the relative difference in the scattering lifetimes␶_⫹and␶_⫺. The domain-wall resistivity becomes noticeable for asymmetric lifetimes and can become of the order␳w/␳0⬃1%.

V. CONCLUSIONS

We studied the contribution of domain-wall scattering on the transport properties of a ferromagnet using an effective two-band model. In a diffuse ferromagnet, the domain-wall resistivity is calculated from the Kubo formula. The domain-wall resistivity is found to be strongly enhanced when the scattering lifetimes of the majority spins and minority spins are different, in agreement with the results in Ref. 7. In the ballistic regime, we have demonstrated how the domain-wall scattering creates an effective barrier that the electrons must pass. The results from the two-band model give only very small corrections to the resistance of the system.

First-principle band-structure calculations have shown that the domain-wall resistance can be increased by orders of FIG. 3. The relative change in the resistivity␳w/(␬␳0) due to

the scattering by the domain wall as a function of the asymmetry in the scattering lifetimes ␶_⫹/␶_⫺. The solid line is for ␥⫽kF⫹/kF⫺

⫽1.01, the dashed line is for␥⫽kF⫹/kF⫺⫽1.20, and the dotted line

magnitude in the ballistic regime.8It would be interesting to perform a realistic band-structure calculation also for diffuse systems. However, the generalization of our two-band results turns out to be cumbersome.14

ACKNOWLEDGMENTS

This work is part of the research program for the ‘‘Stich-ting voor Fundamenteel Onderzoek der Materie’’ 共FOM兲, which is financially supported by the ‘‘Nederlandse Organi-satie voor Wetenschappelijk Onderzoek’’ 共NWO兲. We ac-knowledge benefits from the TMR Research Network on ‘‘Interface Magnetism’’ under Contract No. FMRX-CT96-0089 共DG12-MIHT兲 and support from the NEDO joint re-search program 共NTDP-98兲. We acknowledge stimulating discussion with Jaap Caro, Ramon P. van Gorkom, Junichiro Inoue, Paul J. Kelly, and Andrew D. Kent.

APPENDIX A: ADIABATIC APPROXIMATION

In the ballistic regime, it is most convenient to start with the Hamiltonian in its first quantized form, which after the gauge transformation共2兲 reads H˜⫽H0⫹V:

7 H0⫽⫺ ប2 2mⵜ 2_⫹⌬␴ z, 共A1兲 V⫽ ប 2m␴ya共z兲pz⫺ iប2 2m␴ya

⬘

共z兲⫹ ប2 8ma 2_{共z兲, 共A2兲}

where a(z)⫽d␪(z)/dz is the gradient of the rotating angle of the magnetization and a

⬘

(z)⫽d2␪(z)/dz2. The wave function can be written as

⌿共r兲⫽␾共␳兲

冋

Ak共z兲

冉

1

0

冊

⫹Bk共z兲

冉

0

1

冊册

, 共A3兲 where ␾(␳)⫽A⫺1/2exp(ik_储␳) is the transverse part of the wave function关k⫽(k_储,kz)兴, Ak(z) is the spin-up-like

longi-tudinal amplitude and Bk(z) is the spin-down-like

longitudi-nal amplitude. The Schro¨dinger equation then becomes

冉

⫺ d 2 dz2⫺␻⫹ 0 _⫺a d dz⫺ a

⬘

2 a⫹a

⬘

2 ⫺ d2 dz2⫺␻⫺ 0

冊

•

冉

A_Bk k

冊

⫽

冉

0 0

冊

, 共A4兲 where ␻⫾0⫽k⬜2⫺a共z兲2/4⫿p2, 共A5兲 k_⬜2⫽2mE_F/ប2_⫺k 储 2_{, and E}

F is the Fermi energy. The

off-diagonal terms in Eq.共A4兲 describe the coupling between the spin-up- and spin-down-like states. In the case of a spin spi-ral, a

⬘

(z)⫽0, the eigenstates can be found to be Ak

⫽Ak

0

exp(ikzz), Bk⫽Bk

0

exp(ikzz), where the dispersion of the

modes kz is determined by

k_⬜2⫽k_z2⫹a2/4⫾

冑

p4⫹q2k_z2. 共A6兲 The coupling is weak when the gradient of the spin rotation gradient is slow compared to the Fermi wavelength. This

permits us to make use of a multiple scale analysis共or adia-batic approximation兲.16This analysis is done by introducing the small parameter ⑀, so that ␻(z)˜␻(⑀z), a(z)˜a(⑀z), and da/dz˜⑀da/dz and we introduce the new variable Z ⫽g(⑀z)/⑀, where g(⑀z) is a scaling function.16We expand the longitudinal function A_k(z) and B_k(z) to the lowest order in the small parameter ⑀, Ak(z,Z)⫽ak

0

(Z,z)⫹O(⑀). To the lowest order in the small parameter ⑀, the equation to solve is thus

冉

d2 dZ2⫹ ␻_⫹0 共g

⬘

兲2 a g

⬘

⫺ a g

⬘

d2 dZ2⫹ ␻_⫺0 共g

⬘

兲2

冊

•

冉

ak 0 b_k0

冊

⫽

冉

0 0

冊

. 共A7兲 We now make the ansatz a_k0(z,Z)⫽a_k0,0(z)exp(iZ) and find that the scaling function g is chosen such that

关共g

⬘

兲2_⫺␻

⫺兴关共g

⬘

兲2⫺␻⫹兴⫽p4⫹共ag

⬘

兲2 共A8兲

and Z⫽(1/⑀)兰zdx g

⬘

(x), so that the adiabatic solution is Ak共z兲⬃a0共z兲exp

冉

冕

z

dx g

⬘

共x兲

冊

. 共A9兲 Similarly we can find a solution for Bk(z). Disregarding

tun-neling states which only give an exponentially small contri-bution to the conductance, the number of propagating modes is determined by the condition Im关g

⬘

(x)兴⫽0. From Eq. 共A8兲, we see that the number of propagating modes is deter-mined by the position where a(z) attains its maximum, i.e., the conductance can be calculated as for a spin spiral with a(z)˜a_max.

APPENDIX B: FREQUENCY SUMMATIONS

The typical frequency sum to be performed is

␲l_⫽1

␤

兺

n

Xn⫹lYn⫹共l˜⫺l兲, 共B1兲 where Xnand Ynare Matsubara Green’s functions. They can be written in the spectral representation

Xn⫽

冕

⫺⬁ ⬁ d⑀ 2␲ SX共⑀兲 i␻_n⫺⑀, 共B2兲

where the spectral function is determined by the retarded and the advanced function

SX共⑀兲⫽i关XR共⑀兲⫺XA共⑀兲兴. 共B3兲

Performing the frequency summation in Eq.共B1兲, we get

␲l_⫽

冕

⫺⬁ ⬁ d⑀1 2␲

冕

_⫺⬁ ⬁ d⑀2 2␲SX共⑀1兲SY共⑀2兲 ⫻

冋

f共⑀2兲⫺ f 共⑀1兲 i␻l⫺共⑀1⫺⑀2兲 ⫺ f共⑀1兲⫺ f 共⑀2兲 i␻l⫺共⑀2⫺⑀1兲

册

. 共B4兲 The dc conductivity is obtained by an analytical continuation

I⬅⫺ lim

␻˜0 ␲l_共i␻

l˜␻⫹i␦兲

␻ 共B5兲

and we consider the limit of zero temperature (T˜0): I⫽ ប

2␲SX共0兲SY共0兲 共B6兲

⫽ប

␲Re关XR共0兲YA共0兲⫺XR共0兲YR共0兲兴. 共B7兲

The product of the two retarded 共advanced兲 Green’s func-tions vanishes when integrating over the energy since the poles are on the same side of the imaginary plane. The sum can then be simplified to

I⫽ប

␲Re关XR共0兲YA共0兲兴. 共B8兲

This relation will be used in the following in order to calcu-late the contributions from the diagrams 1–5 共Fig. 1兲.

We use G_s ⬘ R 共0兲Gs A_共0兲⫽ Gs⬘ R 共0兲⫺Gs A_共0兲 ⫺i共␦s⫹␦s⬘兲⫺共⑀s⫺⑀s⬘兲 , 共B9兲 where␦s⫽ប/(2␶s) and obtain in the limit␦sⰆ␮ the

contri-bution from ␲1 to the conductivity I₁⬇

冋

1⫹

冉

⑀ s_⫺⑀s⬘ ␦s⫹␦s⬘

冊

2

册

⫺1 _␶ s␶s⬘ ␶s⫹␶s⬘ 关␦共␰s⬘_{兲⫹␦共␰}s_兲兴, 共B10兲 where ␰s⫽⑀s⫺␮ is the quasiparticle energy relative to the Fermi level. In the case of no spin splitting (␰s˜␰) and (␦s˜␦), the result is I1⬇␶␦(␰). In the limit of strong spin

splitting, the result is vanishingly small 共of order ប/⌬␶ small兲,

I1⬇0. 共B11兲

The sum␲2 gives a contribution I2⫽ ប 2␲ ⳵ ⳵␰s 1 共␰s_兲2_⫹␦ s 2⬇␶s ⳵ ⳵␰s␦共␰ s_{兲. 共B12兲}

The sum␲3 gives a contribution I₃⫽⫺ប ␲ 1 共␰s_兲2_⫹␦ s 2 ␰⫺s 共␰⫺s_兲2_⫹␦ ⫺s 2 . 共B13兲

In the limit of vanishing spin splitting and equal lifetimes, the contribution is I3⬇␶(⳵/⳵␰)␦(␰), which agrees with the

result of␲2 as it should. When⌬␶s/បⰇ1 共large spin

split-ting兲 it is

I3⬇⫺

2␶_s

⑀⫺s_⫺⑀s␦共␰

s_{兲. 共B14兲}

In the case of no spin splitting, the sum ␲4 gives a con-tribution I4⬇ប

冋

⫺ 1 4␦共␰兲␦ ⫺3_⫹1 8␦

⬙

共␰兲␦ ⫺1

册

_{. 共B15兲}

In the general case, we use

G_sR共0兲G_⫺sR 共0兲⫽ Gs R_共0兲⫺G ⫺s R _共0兲 i共␦_⫺s⫺␦s兲⫺共⑀⫺s⫺⑀s兲 . 共B16兲 For large splitting the result is then

I4⬇ ⫺␶s ⑀⫺s_⫺⑀s␦

⬘

共␰ s_兲⫺ 2␶s 共⑀⫺s_⫺⑀s_兲2␦共␰ s_兲 ⫹ ␶s

冉

1⫺ ␶s ␶⫺s

冊

共⑀⫺s_⫺⑀s_兲2 ␦共␰ s_{兲. 共B17兲}

Finally, the sum ␲5 gives a contribution I5⫽ ប 2␲ 1 共␰s_兲2_⫹␦ s 2 1 共␰⫺s_兲2_⫹␦ ⫺s 2 . 共B18兲

In the case of no spin splitting the sum is

I5⬇ប

冋

1 4␦共␰兲␦ ⫺3_⫹1 8␦

⬙

共␰兲␦ ⫺1

册

_{. 共B19兲}

For large spin splitting, we have

I5⬇

1

共⑀s_{⫺⑀⫺s兲}2关␶s␦共␰

s_兲⫹␶

⫺s␦共␰⫺s兲兴. 共B20兲 APPENDIX C: SPIN SPIRAL

The spin-spiral system has a constant gradient of the ro-tating magnetization direction (aq⫽a0␦q,0). We perform the

local gauge transformation共2兲. The transformed Hamiltonian is H˜⫽ ប 2 2m

兺

k c_k†

冉

k2⫺␴zp2⫹a0kz␴y⫹ 1 4a0 2

冊

ck, 共C1兲

where ckis an annihilation operator in the spinor spin space

and the exchange splitting ⌬⬅ប2p2/(2m) has been intro-duced. The transformed current operator is

J⫽eប m

兺

k ck †

冉

kz⫹ a0 2 ␴y

冊

ck. 共C2兲 The Hamiltonian 共C1兲 can be exactly diagonalized, and the eigenvalues are E_k⫾⫽ប 2 2m

冉

k 2_⫹1 4a0 2_⫿

冑

_k z 2_a 0 2_⫹p4

冊

_共C3兲

with the corresponding eigenvectors

u_⫾⫽N_⫾

冉

1

i共1⫿

冑

1⫹␣2兲/␣

冊

, 共C4兲 where the parameter␣⬅kza0/ p2 is introduced and the

N_⫾2_⫽ ␣ 2

2

冑

1⫹␣2共⫿1⫹

冑

1⫹␣2兲. 共C5兲 The annihilation operators are transformed as c⫽(u_⫹,u_⫺)a. In the new basis, the current operator is

J ˜_⫽eប m

兺

k a_k†

冉

kz⫺ a0␣ 2

冑

1⫹␣2 a0 2

冑

1⫹␣2 a0 2

冑

1⫹␣2 kz⫹ a0␣ 2

冑

1⫹␣2

冊

ak. 共C6兲 The dc conductivity is ␴⫽ 1 4␲ ប V _kss

兺

_⬘兩Jss⬘兩 2_A k s_A k s⬘_{, 共C7兲}

where the electron spectral function (A⫽⫺2 Im GR) at the Fermi level is

A_ks⫽ ប/␶ 共Ek

s_⫺␮兲2_{⫹共ប/␶兲}2. 共C8兲

Here we have inserted a phenomenological scattering life-time, which is identical for the two eigenstates. Note that we cannot treat different scattering lifetimes for the minority and majority states in the bulk ferromagnet with the method out-lined in this appendix, since the lifetimes appearing in Eq. 共C7兲 are the lifetimes for the exact eigenstates in the spin spiral. In order to determine the relation between the differ-ent lifetimes, the general method described above in our

pa-per should be used. From Eqs.共C7兲 and 共C8兲 it can be seen that the off-diagonal terms in the conductivity A_⫹A_⫺ are in the order 1/(␶⌬/ប)2 smaller than the diagonal terms. We further assume that the scattering by the domain wall is weak, i.e., a₀2Ⰶp2 and a0kF

s_Ⰶp2 _{and expand the result for}

the conductivity to the second order in a0. The conductivity

becomes ␴⫽ e 2_␶ 6␲2m

兺

s

冉

k_␮2⫺a0 2 4 ⫹sp 2

冊

3/2

冉

1⫺sEw 4⌬

冊

, 共C9兲 where ប2k_␮2/(2m)⫽␮ and Ew⫽ប2a0 2

/(2m) are used. The conductivity should be related to the electron density by eliminating any reference to the chemical potential which may change in the presence of the domain wall,

ns⫽ 1 6␲2

冉

k␮ 2_⫺a0 2 4 ⫹sp 2

冊

3/2

冉

1⫹sEw 4⌬

冊

. 共C10兲 Inserting Eq.共C10兲 into Eq. 共C9兲, the conductivity can there-fore be written as

␴⫽␴0

冉

1⫺n⫹⫺n⫺ n_⫹⫹n_⫺

E_w

2⌬

冊

, 共C11兲 where␴0⫽e2(n_⫹⫹n_⫺)␶/m is the Drude conductivity. The

domain-wall resistivity, ␳_w⫽⫺␦␴_w/␴₀2, is thus

␳w⫽

e2␳₀2E_w

2m⌬ 共n⫹⫺n⫺兲, 共C12兲 where␳0⫽1/␴0.

*Also at Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands.

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