The electrical resistance of spatially varied magnetic interface. The role of normal scattering

We investigate the diffusive electron transport in conductors with spatially inhomogeneous magnetic properties taking into account both impurity and normal scattering. It is found that the additional interface resistance that arises due to the magnetic inhomogeneity depends essentially on their spat...

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Datum:	2011
Hauptverfasser:	Gurzhi, R.N., Kalinenko, A.N., Kopeliovich, A.I., Pyshkin, P.V., Yanovsky, A.V.
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Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2011
Schriftenreihe:	Физика низких температур
Schlagworte:	Электронные свойства проводящих систем
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spelling	irk-123456789-1184932017-05-31T03:08:56Z The electrical resistance of spatially varied magnetic interface. The role of normal scattering Gurzhi, R.N. Kalinenko, A.N. Kopeliovich, A.I. Pyshkin, P.V. Yanovsky, A.V. Электронные свойства проводящих систем We investigate the diffusive electron transport in conductors with spatially inhomogeneous magnetic properties taking into account both impurity and normal scattering. It is found that the additional interface resistance that arises due to the magnetic inhomogeneity depends essentially on their spatial characteristics. The resistance is proportional to the spin flip time in the case when the magnetic properties of the conducting system vary smoothly enough along the sample. It can be used to direct experimental investigation of spin flip processes. In the opposite case, when magnetic characteristics are varied sharply, the additional resistance depends essentially on the difference of magnetic properties of the sides far from the interface region. The resistance increases as the frequency of the electron-electron scattering increases. We consider also two types of smooth interfaces: (i) between fully spin-polarized magnetics and usual magnetic (or non-magnetic) conductors, and (ii) between two fully oppositely polarized magnetic conductors. It is shown that the interface resistance is very sensitive to appearing of the fully spin-polarized state under the applied external field. 2011 Article The electrical resistance of spatially varied magnetic interface. The role of normal scattering / R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, A.V. Yanovsky // Физика низких температур. — 2011. — Т. 37, № 2. — С. 186–194. — Бібліогр.: 13 назв. — англ. 0132-6414 PACS: 72.25.Mk, 73.40.Cg http://dspace.nbuv.gov.ua/handle/123456789/118493 ru Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	Russian
topic	Электронные свойства проводящих систем Электронные свойства проводящих систем
spellingShingle	Электронные свойства проводящих систем Электронные свойства проводящих систем Gurzhi, R.N. Kalinenko, A.N. Kopeliovich, A.I. Pyshkin, P.V. Yanovsky, A.V. The electrical resistance of spatially varied magnetic interface. The role of normal scattering Физика низких температур
description	We investigate the diffusive electron transport in conductors with spatially inhomogeneous magnetic properties taking into account both impurity and normal scattering. It is found that the additional interface resistance that arises due to the magnetic inhomogeneity depends essentially on their spatial characteristics. The resistance is proportional to the spin flip time in the case when the magnetic properties of the conducting system vary smoothly enough along the sample. It can be used to direct experimental investigation of spin flip processes. In the opposite case, when magnetic characteristics are varied sharply, the additional resistance depends essentially on the difference of magnetic properties of the sides far from the interface region. The resistance increases as the frequency of the electron-electron scattering increases. We consider also two types of smooth interfaces: (i) between fully spin-polarized magnetics and usual magnetic (or non-magnetic) conductors, and (ii) between two fully oppositely polarized magnetic conductors. It is shown that the interface resistance is very sensitive to appearing of the fully spin-polarized state under the applied external field.
format	Article
author	Gurzhi, R.N. Kalinenko, A.N. Kopeliovich, A.I. Pyshkin, P.V. Yanovsky, A.V.
author_facet	Gurzhi, R.N. Kalinenko, A.N. Kopeliovich, A.I. Pyshkin, P.V. Yanovsky, A.V.
author_sort	Gurzhi, R.N.
title	The electrical resistance of spatially varied magnetic interface. The role of normal scattering
title_short	The electrical resistance of spatially varied magnetic interface. The role of normal scattering
title_full	The electrical resistance of spatially varied magnetic interface. The role of normal scattering
title_fullStr	The electrical resistance of spatially varied magnetic interface. The role of normal scattering
title_full_unstemmed	The electrical resistance of spatially varied magnetic interface. The role of normal scattering
title_sort	electrical resistance of spatially varied magnetic interface. the role of normal scattering
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2011
topic_facet	Электронные свойства проводящих систем
url	http://dspace.nbuv.gov.ua/handle/123456789/118493
citation_txt	The electrical resistance of spatially varied magnetic interface. The role of normal scattering / R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, A.V. Yanovsky // Физика низких температур. — 2011. — Т. 37, № 2. — С. 186–194. — Бібліогр.: 13 назв. — англ.
series	Физика низких температур
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fulltext	© R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky, 2011 0Fizika Nizkikh Temperatur, 2011, v. 37, No. 2, p. 186–194 The electrical resistance of spatially varied magnetic interface. The role of normal scattering R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: kopeliovich@mail.ru Received May 26, 2010 We investigate the diffusive electron transport in conductors with spatially inhomogeneous magnetic proper- ties taking into account both impurity and normal scattering. It is found that the additional interface resistance that arises due to the magnetic inhomogeneity depends essentially on their spatial characteristics. The resistance is proportional to the spin flip time in the case when the magnetic properties of the conducting system vary smoothly enough along the sample. It can be used to direct experimental investigation of spin flip processes. In the opposite case, when magnetic characteristics are varied sharply, the additional resistance depends essentially on the difference of magnetic properties of the sides far from the interface region. The resistance increases as the frequency of the electron-electron scattering increases. We consider also two types of smooth interfaces: (i) be- tween fully spin-polarized magnetics and usual magnetic (or non-magnetic) conductors, and (ii) between two ful- ly oppositely polarized magnetic conductors. It is shown that the interface resistance is very sensitive to appear- ing of the fully spin-polarized state under the applied external field. PACS: 72.25.Mk Spin transport through interfaces; 73.40.Cg Contact resistance, contact potential. Keywords: diffusive electron transport, spin flip processes, magnetic characteristics. 1. Introduction The well-known and highly-applied giant magnetoresis- tance effect [1] is one of the effects which arise at contact between conductors with different magnetic properties. Really, the interface between two fully opposite polarized ferromagnetic conductors is an opaque obstacle for carriers as their spin polarizations are specified rigidly by the mag- netization of the corresponding regions. The lesser effect arises at contact of a fully polarized magnetic conductor with a non-magnetic conductor as it was discussed in Ref. 2. Inserting a non-magnetic conductor between fully polarized magnetic sides (see Fig. 1) double its resistance (when the length of the non-magnetic part is less than the spin flip relaxation length λ ). The reason is that one of the spin channels is cut off due to full polarization of the mag- netic sides. Note, one may detect a non-equilibrium spin- polarization that exists in the magnetically inhomogeneous circuit [2], measuring its resistance. Really, spin polariza- tion of a non-magnetic section disappears under demagne- tization of the magnetic, and the interface resistance disap- pears too. Naturally, a contact between different magnetics is a source of the additional resistance. Spin-accumulation ef- fects were investigated in the presence of the spin- dependent scattering early [3–6] in magnetic layered struc- tures. These effects cause the interfacial resistance but electron-electron scattering, which conserves the total momentum of the system of interacting particles, is not to be used. As it was firstly demonstrated in Ref. 7, the electron- electron scattering increases the interfacial resistance es- sentially. The physical reason is the mutual friction be- tween “spin-up” and “spin-down” electrons. Let's assume that electric current flows from a non-magnetic conductor into a fully polarized magnetic region where all electrons Fig. 1. M–N–M contact. M — fully polarized magnetic conduc- tor, N — non-magnetic conductor. M N M The electrical resistance of spatially varied magnetic interface. The role of normal scattering Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 187 are fully spin-polarized (“spin-up”, for example, see Fig. 2). Then, “spin-down” electrons have no possibility to pass into the magnetic conductor and they are accumulated near the interface. In other words, a non-equilibrium additional spin density ↓δρ arises in the non-magnetic conductor at the length-scale of the order of λ from the interface. The value of ↓δρ is determined by the following condition. The sum of the ohmic “spin-down” current and the diffu- sion “spin-down” current should be zero at the interface, i.e. the “drift” velocity of the “spin-down” component is zero. Thus, the crowd of “spin-down” unmovable elec- trons, which are accumulated near the interface, “slows down” the flow of “spin-up” electrons. In other words, the crowd of “spin-down” electrons plays the role of “effec- tive” scattering impurities. As a result, the interfacial resistance increases indefi- nitely (and not just a twice) with increasing of the frequen- cy of electron-electron collisions. Here we should note that electron-electron collisions are the “normal” collisions that conserve the total momentum of the system of interacting particles. So, they do not provide the resistance of the ho- mogeneous parts of the electrical circuit but they give the essential contribution into the interface resistance. The contribution is proportional to the frequency of electron- electron collisions. Thus, the relative contribution of the interface resistance into the total circuit resistance increas- es with the electron-electron frequency increase. Clearly, any normal collisions will play the same role (e.g., elec- tron-phonon collisions, when phonons are tightly coupled to the electron system). Note, that the role of electron- electron collisions was discussed in [7] for a particular case when uniform magnetic conductors contact with non- magnetic conductors. Below we analyze the problem for the general case of non-homogeneous conductors. In this paper we propose some new possibilities for in- vestigation of the role of a non-equilibrium spin density for electron transport properties. As we demonstrate below, a number of effects arises when electron spectrum is varied smoothly in space. Thus, a such type conductors are quite perspective for direct experimental research of non- equilibrium spin density effects. Firstly, one can exclude the influence of electrical contact barriers. Secondly, varia- tions of magnetic properties can be easy induced by apply- ing spatially inhomogeneous gate voltage in the case of a two-dimensional electrons in heterostructures (see Ref. 8), by variation of an external magnetic field or by space- dependent distributions of doping impurities. Note, we suppose a collinear magnetization of an inhomogeneous magnetic system. The additional source of spin mixing due to non-collinear spins has not being included into our con- sideration. The resistance of a smooth interface, that is a magnetic layer varied in spatial like a domain wall beha- vior, was calculated in [9]. The paper is organized as follows. The general equa- tions for the resistance of a magnetically inhomogeneous conductor are derived in Sec. 2. The resistance is given in terms of the entropy production at electron diffusion and spin-flip processes. The case of relatively frequent spin- flip scattering is given in Sec. 3. Here we discussed the role of electron-electron collisions and their interplay with spin-flip scattering. The case of relatively rare spin-flip scattering can be found in Sec. 4. Electron transport through smooth almost fully spin-polarized interfaces is discussed in Sec. 5. A detailed derivation of the expres- sions of Sec. 4 is given in the Appendix. In this paper we consider the diffusion transport regime and apply the modified set of equations were derived by Flensberg (see Ref. 10). We do not consider the influence of current flow on the electron spectrum, i.e. we do not suppose spin-torque effects [11] assuming that electric current is weak enough. 2. The electrical resistance of a smooth magnetic interface. The general approach The role of electron-electron collisions for the interface resistance was demonstrated in Ref. 7 for the case of the zero-length interface between a non-magnetic conductor and a fully polarized magnetic material. Below we derive some general formulas for the electrical resistance of a conductor with an arbitrary spatial variation of the electron spectrum. Let us rewrite the set of equations (1a), (1b) given in Ref. 10 in the following “vector” form for our one- dimensional task ˆ' = ,−βjμ (1) ' = ( ) ,f− ⋅j a aμ (2) 2 2 1 0 1 1 1 0 = ( , ), = ( , ), = (1, 1), = ( ), = ( ), = = ( ) , = , = . i i sf j j e An e An e eA n n f ↑ ↓ ↑ ↓ − − ↑↑ ↑ ↓↓ ↓↑ ↓ − ↑↓ ↓↑ ↑ ↓ − − − ↑ ↓ μ μ − β ρ + β ρ + Π β β − τ Π Π +Π j aμ (3) Here ↑μ , ↓μ , j↑ , j↓ are the spin components of the elec- trochemical potential and the current density, correspon- dingly. A prime denotes differentiation with respect to the coordinate along the conductor, x; n↑ , n↓ are densities of “spin-up” and “spin-down” components, correspondingly; i↑ρ , i↓ρ are the corresponding resistivities due to the Fig. 2. The “crowd” effect: “spin-down” electrons, which are accumulated near the interface, “slow down” the flow of “spin- up” electrons. I N M R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky 188 Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 processes with momentum loss, e is the electron charge. ,↑ ↓Π are the spin-dependent densities of states on the Fermi surface, sfτ is the spin-flip characteristic time. The coefficient A is proportional to the frequency of normal collisions Nν . In the case of the electron-electron scatter- ing we have (see Ref. 7) 2 1 1 1, = ,ee m mA e m n n n n− − − − ↑ ↓ ≈ ν + (4) where m is a current carriers mass, eeν is the frequency of electron-electron collisions. Let us specify the problem of the resistance of the in- homogeneous conductor in the following way. Let L is the characteristic scale of the inhomogeneous part of the conductor which is homogeneous at \| \|>x L . The “resistiv- ities” of its sides, ˆ lβ at < / 2x L and ˆ rβ at > / 2x L , may be different from one another. The resistance of the con- ductor section between points =x M , =x M− , >>M L , obviously, is given by = [ ( ) ( )] /R M M ejsμ − −μ . Here, s is the area of the conductor cross-section(consider it to be constant), μ is the electro-chemical potential which is the same for the both spin components far from the inhomo- geneity region; js is the total current across the conductor cross-section. Evidently, by reason of electrical charge conservation js does not depend on the x-coordinate. It is convenient to write that current in the following form: =j ⋅j n , where = (1,1)n and conservation of j is seen in equation (2) scalary multiplied by n . Let us demonstrate that there is a relation between the electrical resistance of a magnetically homogeneous con- ductor and the entropy production rate which is similar to that in the case of a homogeneous conductor [12]. After integration by parts β̂j j and taking into account Eqs. (1) and (2), we obtain the following 2 2 1 ˆ= ( ) . M M R f dx ej s − ⎡ ⎤β + ⋅⎣ ⎦∫ j j aμ (5) As it is easy to check, the quadratic form ˆ / eβj j is an essentially positive. Consequently, R is positive too. The first term in subintegral expression of Eq. (5) corresponds to the entropy production at Joule heating. The second one corresponds to the entropy production due to the spin-flip scattering. Let magnetic characteristics are varied rather smooth with x . So, the local equilibrium between spin-up and spin down-components had time to be established, i.e. = =e eμ nμ μ , where ( )e xμ is an equilibrium electro- chemical potential. In this case, the “equilibrium” current, as it follows from Eq. (1), is given by 1ˆ=e e −′−μ βj n . As we consider the total current to be fixed, =e j⋅j n . So, we may express the derivative e′μ through the total current density directly 1 1 1 ˆ= , = .e nn nn j − − − ′μ − β β β n n (6) Thus, we the “equilibrium” resistance is given by 1 1= . M e nnM dxR es − − β∫ (7) On the other hand, integrating the quadratic form ˆ e eβj j in parts and taking into account Eq. (1) and = ( ) = 0e e e e′ ′⋅ μ ⋅j n jμ yields 2 1 ˆ= . M e e e M R dx ej s − β∫ j j (8) Equations (7) and (8) are equivalent. It is easy to see that the mentioned equilibrium state could be established in the case when the diffusion spin-flip length λ is much less than the characteristic length-scale of the inhomoge- neous interface region, i.e. << Lλ . Here sfvλ ≈ ττ , v is the carriers velocity, τ is the relaxation time which cor- responds to scattering processes that change the momen- tum of an electron essentially (either with respect to the normal collisions or to the collisions that do not conserve the quasi-momentum of the electron system). Let us define the addition to the total resistance, = eR R RΔ − , which arises due to the non-equilibrium spin density. The addition is a direct analog of the interface resistance between two homogeneous magnetic conduc- tors. From Eqs. (5) and (8) we get 2 2 1 ˆ= ( ) , = .e R f dx ej s ∞ −∞ ⎡ ⎤Δ Δ βΔ + ⋅⎣ ⎦ Δ − ∫ j j a j j j μ (9) Here we take into account that ˆ ˆ= =e e e′βΔ Δ β −Δ ⋅ μ =j j j j j n = 0. The first equality is due to the fact that operator β̂ is a self-adjoint operator. The second one is valid because Δj is an antisymmetric vector as to respect of the spin compo- nents: = jΔ Δj a , = 0Δ ⋅j n when the total current density j is fixed. We extend integration in Eq. (9) until infinite limits because Δj and ⋅aμ are vanishing in the homoge- neous sides. Note, that RΔ is positive as it follows from Eq. (9). In other words, the total resistance increases be- cause a non-equilibrium spin density arises due to the magnetic inhomogeneity. Here we should note that diffusion current jΔ arose due to the non-equilibrium spin density. Let us define the diffusion coefficient for the non-equilibrium spin density. From Eq. (1) we get that = aa j′ ⋅ −β Δaμ . Taking into ac- count the well-known relations between the electro- chemical potentials and electron densities, , , ,=n↑ ↓ ↑ ↓ ↑ ↓δ Π μ , and the condition of the electric neutrality, =n n↓ ↑δ −δ , we obtain the diffusion coefficient ( )21 1 0 1= , = .aa i i aa D e A n n e − − ↑ ↓ ↑ ↓ ⎡ ⎤β ρ +ρ + +⎢ ⎥Π β ⎣ ⎦ The electrical resistance of spatially varied magnetic interface. The role of normal scattering Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 189 Diffusive spin relaxation length is the following 1= = .sf aa D f λ τ β (10) Equations (7) and (9) turn out to be more convenient for calculations of the resistance as to compare with the direct calculations of the electrochemical potential with the needed accuracy. Below we investigate theoretically the “non- equilibrium” addition to the resistance RΔ mainly. The reason is that one can separate the contributions of eR and RΔ into the total resistance, R , as they have different tem- perature and interface length-scale dependences. Moreover, in certain cases RΔ gives the main contribution. 3. A case of strong spin-flip scattering At / << 1Lλ , the spin equilibrium is established during the time when an electron passes diffusively the inhomo- geneous interface region. Solving Eqs. (7) and (3) yields the “equilibrium” resistance 2 2 1 1 2 ( )1= . ( ) M i i i i e i iM A n n R dx se A n n − − ↑ ↓ ↑ ↓↓ ↑ − − ↑ ↓− ↑ ↓ ρ ρ + ρ +ρ ρ +ρ + +∫ (11) This result demonstrates the following well-known fact [13]. The electron-electron scattering increases with tem- perature increasing and resistivity increases too. However, this transition does not change the order of magnitude of the resistivity while its value increases. Physically, the electron distribution transforms from the impurity formed one to the drift distribution which is typical for the strong electron-electron scattering. In the case, which is described by Flensberg's approximation [10], the impurity scattering forms a non-drift distribution, i.e. spin com- ponents have different drift velocities if i in n↑ ↑ ↓ ↓ρ ≠ ρ . Thus, in two opposite limiting cases the resistivity is given by different formulas in subintegral function of Eq. (11): ( ) / ( )i i i i↑ ↓ ↑ ↓ρ ρ ρ +ρ and 2 2 1 1 2( ) / ( )i in n n n− − − − ↑ ↓↓ ↑ ↑ ↓ ρ +ρ + , correspondingly. Therefore, one can observe this transition experimentally when the temperature increases. However, there is no transition if one of the spin component is fully depleted (e.g., 0n↓ → ) by applying of an external mag- netic field or by electrical gating. The last statement is va- lid within the Flensberg's model which assumes that: (I) the electron spectrum is isotropic and (II) there are no groups of current carriers with different characteristics but with the same spin polarization. There is no temperature dependence of the resistance of a non-magnetic conductor within this model too. This is the consequence of spin de- generacy ( =i in n↑ ↑ ↓ ↓ρ ρ ). Now, let us calculate the contribution of the non- equilibrium spin density into the resistance, i.e. RΔ . From Eq. (2) we get the antisymmetrical part of the electro- chemical potential, ⋅aμ , which is due the non-equilibrium spin density. To a first approximation, we put the “equili- brium” current in the right-hand-side of Eq. (2) 1= ( ) . 2 ef ′⋅ − ⋅a j aμ (12) Here we take into account that 2 = 2a . As vector Δj , is antisymmetrical, Eq. (1) yields ( )= . aa j ′⋅ Δ − β aμ (13) Here and below matrix elements of operators in the basis of a and n are defined in the same way as in Eq. (6). It follows from Eqs. (12) and (13) that <<j jΔ at <<sfD Lτ . Thus, the approximation method used above for solving Eqs. (1) and (2) is verified when << Lλ . As it follows from Eq. (13), within the same accuracy, the first term in the quadratic brackets in Eq. (9) is vanishing as to compared with the second one 2 2( )ˆ = << ( ) . aa f ′⋅ Δ βΔ ⋅ β aj j aμ μ Therefore, from Eq. (9) we have 2 2 ( )1= , 4 eR dx fej s ∞ −∞ ′⋅ Δ ∫ j a where the equilibrium current ej can be found from Eqs. (1) and (6) 1 1 ˆ = .e nn j − − β β nj (14) Finally, taking into account 1 1/ = /an nn an aa − −β β −β β we get the non-equilibrium spin density contribution to the resis- tance 2 1 1= 4 an aa R dx es f ∞ −∞ ′⎛ ⎞β Δ =⎜ ⎟β⎝ ⎠ ∫ 22 2 2 1 1 2 0 ( )1= . 4 ( ) sf i i i i A n n dx e s A n n − −∞ ↑ ↓ ↑ ↓ − − ↑ ↓−∞ ↑ ↓ ′⎛ ⎞ρ −ρ + −τ ⎜ ⎟ ⎜ ⎟Π ρ + ρ + +⎝ ⎠ ∫ (15) Note, unlike the “equilibrium” resistance given by Eq. (11), the non-equilibrium spin density contribution to the resis- tance, RΔ , does not tend to zero at , 0i↑ ↓ρ → (if ).n n↑ ↓≠ The physical reason is the same that was described in Ref. 7: while normal collisions give no contribution into the resistance of homogeneous conductors, they cause the “crowd effect” in inhomogeneous system and, thus, in- crease the resistance. (Here we should note, that Eq. (11) corresponds to the approximation when the conductor is locally homogeneous). R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky 190 Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 Thus, Eq. (15) gives the main contribution to the resis- tance of the sample for the case when normal collisions dominate over other scattering processes: 2<<i An−ρ . (Here we neglected by the “side resistance” assuming that the length of the sample be of the order of the length-scale of the inhomogeneous interface region: M L≈ ). Then we rewrite Eq. (15) in the following simple form 2 2 0 1= . 4 sf n n R dx n ne s ∞ ↑ ↓ −∞ ↑ ↓ ′τ ⎛ ⎞− ⎜ ⎟⎜ ⎟Π +⎝ ⎠ ∫ (16) Summarize, Eq. (15) and, especially, Eq. (16) open a way for investigation of spin-flip processes by measuring the resistance of a conductor which magnetic properties are varied smoothly enough. 4. A case of weak spin flip scattering In the case <<L λ , we may neglect by the spin flip scattering inside the interface region within the main ap- proximation. Then, the interface plays the role of a spin non-equilibrium density generator while spin relaxation occurs in the sides, out of interface. In other words, we may suppose the current density in the interface region, cj , is constant. Correspondingly, the electro-chemical poten- tial is given by 0 ˆˆ ˆ= ( ) , ( ) = ( ) , x c cB x B x x dx′ ′− β∫jμ μ (17) where cμ is a constant vector. Solutions of Eqs. (1) and (2) in homogeneous sides around the interface are the fol- lowing / /ˆ= e , = ( ) e ,x xc bx c± λ ± λΔ μ − λ βj a n amμ , 1 , = .l r nnl r jb −β (18) Here index “l” and sign “+” corresponds to the left side, index “r” and sign “–” to the right side, respectively; ,l rc , ,l rμ are arbitrary constants that are defined below, the diffusive spin relaxation length ,l rλ is given by Eq. (10). One may match functions μ in the interface region with its side asymptotes and find out the resistance. This seemingly simple task turns out to be relatively difficult. It is much easier to keep an accuracy of approximation ap- plying Eq. (9). In the main approximation on the small parameter /L λ we have to match currents = = .c l el r erc c+ +j a j a j (19) Thus, we obtain from Eq. (9) 2 2 2= .l l aal r r aarRej s c c PΔ λ β + λ β + (20) The first and second terms in Eq. (20) are the contributions of the boundary regions which are out of the interface. The length-scales of these regions are of the order of λ . To calculate these terms we used 2 1 2( ) = ( ')f f −⋅a jμ (as it follows from Eq. (2)) and Eq. (10). The third term in Eq. (20) is the contribution of the interface region into the integral (9) ( )2 2= .aa c e aasd sdaP c dx ∞ −∞ ⎡ ⎤β − −β⎢ ⎥⎣ ⎦∫ j j (21) Here subscripts “a” at vectors denote their components along the ort a: = 2 ac⋅a c ; index “sd” marks the value of the given function in the corresponding boundary = at > 0, = at < 0.aasd aar aasd aalx xβ β β β (22) We have subtracted 2 aasd sdcβ from the integrand in Eq. (21) to keep integral convergence. In this way we sepa- rate the intrinsic and extrinsic (as to the interface region) contributions in Eq. (20). In Appendix we give equations for ,l rc , caj , see Eqs. (A.2), (A.3). Thus, to obtain the non-equilibrium ad- dition to the resistance in the case of the “short” interface ( >> )Lλ it is enough to put Eq. (A.2) and (A.3) into the Eqs. (20) and (21). Let us suppose that << /aa aasd Lβ β λ , i.e. there is no leap of the electrical resistance in the interface region (for the both spin components). Then, from Eqs. (20), (21) we obtain (see Appendix) = ,j tR R RΔ Δ + Δ (23) [ ]2( / ) ( / ) = , 4 ( ) l r aal aar an aa r an aa l j l aal r aar R es λ λ β β β β − β β Δ λ β + λ β (24) 2 1= . 4 ansd an t aa aasd aa R dx es ⎡ ⎤β β Δ β −⎢ ⎥ β β⎢ ⎥⎣ ⎦ ∫ (25) Here tRΔ is the direct contribution of the interface region. Generally speaking, sdβ depends on the origin of the coordinate. But it is a technique only that allows us to keep the convergence of the integral in Eq. (25). Thus, the choice of the origin does not affect the result in the main approximation on the small parameter /L λ . The reason is that tRΔ is not small as to compare with jRΔ when ˆ rβ and ˆ lβ are close to each other. Resistance jRΔ (see Eq. (23)) arises due to the differ- ence in magnetic properties between interface sides. This contribution was calculated [3,4] for the case when elec- tron-electron scattering is neglecting. Note, Eq. (24) is va- lid even if >L λ in the case, when the interface region has a complex structure and includes a number of narrow sub- interfaces which length trL is much less than the diffusion spin flip length λ : <<trL λ . It is the case, in Eqs. (15) and (16) one should exclude from integration that regions where derivative with respect to x-coordinate is diverged. Then, like to the general case of short interfaces, the con- The electrical resistance of spatially varied magnetic interface. The role of normal scattering Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 191 tribution of an subinterface is given by Eq. (24) where in- dexes “l” and “r” correspond to the sub-interface bounda- ries. (Note, the given approach is valid when distances between interfaces exceed 2λ , see [5].) As it follows from Eqs. (23)–(25), the electron-electron scattering gives contribution into the interface resistance ( ,aa an i eeβ β ∝ ν + ν , see Eq. (3), where iν and eeν are the electron-impurity and the electron-electron frequencies of scattering). Moreover, the relative contribution of the spin non-equilibrium density, / eR RΔ , rises with the elec- tron-electron scattering increasing as it gives no contribu- tion into the “equilibrium” resistance eR . Thus, the “crowd effect” [7] exists in the case of a smooth interface at an arbitrary relation between L and λ (the case when >>L λ was discussed in Sec. 3). In the case of different magnetic sides, a difference between drift velocities of the spin components disappears at the length-scale of the order of λ deep into the side. There is a strong mutual friction between “spin-up” and “spin-down” components that leads to appearing of the resistance. In the case of the same magnetic sides, the electron-electron scattering equilibrates the difference in drift velocities mainly over a length of the interface. Let us analyze the temperature dependence of the resis- tance. Taking into account that the diffusion length de- creases at temperature increase (see Eq. (10)), we get eeRΔ ∝ ν , when the electron–electron scattering domi- nates over the electron-impurity scattering and sides have different magnetic characteristics. On the other hand, eeRΔ ∝ ν when magnetic characteristics of the sides are identical to each other. RΔ increases with eeν until <<L λ . In the opposite limit case, Eqs. (15) and (16) are valid. Note, the experimental results like that were obtained in Ref. 2 (fully-polarized magnetic sides which are separated by a non-magnetic insertion) can be described by equation (25) for the case of continuous transition along coordinate between magnetic materials and the non-magnetic inser- tion ( / = 1ansd aasdβ β ). However, when that contact is “sharp”, Eq. (25) gives the same result as Ref. 7. Here we have to stress on the difference between this result and results given below in Sec. 5. In contrast to the results of Sec. 5, the result discussed above is not so sensitive to ap- pearing of the fully spin-polarized state. In conclusion of this Section let us demonstrate the de- pendence of the additional resistance on the length of the interface, L , for the case when magnetic characteristics are varied weakly. Let “electroconductivity tensor” be written as 1 0 ˆˆ ˆ ˆ=−α ≡ β α + δα , where 0α̂ does not depend on the coordinate x and 0ˆ ˆ<<δα α . It is easy to get solu- tions of Eqs. (1), (2) which are approximately valid for any relation between L and λ . Let, our conductor is a non- magnetic in the main approximation, i.e. 0 0= = 0an naα α . Then we obtain the first order correction to the aμ on the small parameter ˆδα \| \|/ 0 0 ( ) = e ( ) . 2 x x a an nn aa jx x dx′− − λλ ′ ′ ′δμ − δα α α ∫ (26) In order to find nδμ we have to put zero the symmetrical correction to the total current with the accuracy to the second order terms: 0 0 = 0nn n an a nn n′ ′ ′α δμ + δα δμ + δα μ , where 0 0= /n nnj′μ − α . As a result, the resistance = eR R Rδ δ + Δ is given by 2 0 1= ( ) ,e nn nn R x dxδ − δα α ∫ \| \|/ 2 0 0 = e ( ) ( ) . 2 x x an an nn aa R x x dxdx′− − λλ ′ ′ ′ ′Δ δα δα α α ∫ (27) In Fig. 3 we plotted the additional resistance RΔ as a function of the length L for the cases of identical (a) and different (b) sides of the interface. Note, 1R L−Δ ∝ at >>L λ in both cases. Magnetic properties of the interface were modelled by the following way: a) 2 2/ 0, = (1 0.5 e )x Ln n − ↑ ↓ m , b) 2 21/2 / 0, = (1 0.5 e ) . x y Ln n dy− − ↑ ↓ −∞ π ∫m Solid curves were calculated from the solutions given by Eq. (27). The dashed line (Fig. 3,a) represents the depen- Fig. 3. The relative interfacial resistance / eR RΔ as a function of the normalized length of the interface /L λ for the cases of iden- tical (a) and different (b) sides of the interface. The length of the sample is assumed as 10λ and = 5ee iν ν a b L/� L/� � R R/ e � R R/ e 0.10 0.08 0.06 0.04 0.02 2 4 6 8 100 2 4 6 8 100 0.10 0.08 0.06 0.04 0.02 R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky 192 Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 dence ( )R LΔ in the limit <<L λ . It was calculated from the solution given by Eq. (25). Dotted lines (Fig. 3,a and Fig. 3,b) correspond to the limit >>L λ . These curves were calculated for the given model from Eq. (15). The result for the “zero-length” interface (shown by the triangle in Fig. 3,b) was calculated from Eq. (24). The additional equilibrium resistance eRδ is, obviously, proportional to the L for the case of identical sides and it does not depend on the electron-electron scattering. Thus, it is almost tem- perature independent and one could eliminate this contri- bution by the special choice of ˆδα : it should be “anti- symmetric” either as to the coordinate or as to the spin projection (at = , = 0nn↓ ↑δα −δα δα ). 5. “Spin-stop” interfaces In this section we discuss the case when the interface forms an essential barrier for electron transport (at least for one of the spin components). In the case when aaLβ (which is proportional to the sum of interface resistivities for both spin channels) becomes of the order of the “boun- dary resistances” out of the interface, aasdλβ (which are due to the spin non-equilibrium densities), we can't neglect by the terms of the order of BΔ in Eqs. (A.2), (A.3) in spite of / << 1L λ . In the case when sides around interface are identical to each other and >>aa aasdLβ λβ , we obtain from Eqs. (20), (21) and Eqs. (A.2), (A.3) 2 = 2 s aasd ansd an aasd aa B R es B ⎡ ⎤λ β β Δ Δ − +⎢ ⎥ β Δ⎢ ⎥⎣ ⎦ 2 1 ( ) , 4 an an aa aasd aa aa B dx es B ∞ −∞ ⎡ ⎤β Δ + β −β −⎢ ⎥β Δ⎣ ⎦ ∫ ˆ ˆ ˆ= ( ) ( ).B B BΔ Δ ∞ − Δ −∞ (28) Here, the second term does not increase infinitely with aaβ and anβ increase. The reason is that the equality / = /an aa an aaB Bβ β Δ Δ is valid with the good accuracy in that region where aaβ and anβ are large enough. That is why the second term in Eq. (28) is an essential only in that case, when the first term is vanishing (i.e. when the differ- ence in quadratic brackets is neglecting). Thus, as follows from (A.1) and (28), the “leap” of resistivity causes the appearance of non-small spin non-equilibrium density out of the interface region at the length-scale of the order of λ around it. It is the effect that gives the main contribution into RΔ . Equation (28) gives the same result as correspon- ding equations in Ref. 7 for the case when fully polarized magnetic region ( 0, / / 1an aa an aan B B↑ → β β →Δ Δ → ) is sided by non-magnetic conductors ( = 0ansdβ ). Let us discuss a gated magnetic interface with con- trolled density of the spin components. Here one may achieve a fully polarized magnetic state by applying an external field to the gate thus depleting one of the spin components, 0n↑ → (we call it as a “spin-stop” inter- face). As it follows from Eqs. (3), (4), the corresponding resistivity increases 1n−↑↑ ↑ β ∝ (in the two-dimensional case both impurity assisted resistance, /i n↑ ↑ ↑β ∝Π , and electron-electron contribution are proportional to 1n− ↑ ). Consequently, 1,aa an n− ↑ β β ∝ . Thus, within the validity of Eq. (25) the contribution of the each point of the interface region into the RΔ rises as 1n− ↑ . The increase is limited by the value which is given by Eq. (28) when aaLβ is of the order of aasdλβ . The reason is that “spin-up” current can't enter into the fully “spin- down” polarized region when >>aa aasdLβ λβ , so it have to be converted into the “spin-down” component out that region. Meanwhile, at <<aa aasdLβ λβ the region does not limit “spin-up” current flow. Figure 4 demonstrates the increase of the interfacial resistance at depleting of the “spin-up” component due to the electrical gating in a two-dimensional magnetic con- ducting heterostructure ( 0n↑ → at = 0x ). Here we as- sume that temperature is zero and use the following model for the electrical potential: 2 2 0/ = exp ( / )Fe U x L↑ϕ ε − . Here F↑ε is the Fermi energy for “spin-up” electrons and full depleting is achieved at = 0x . For the sake of speci- ficity, let us assume that equilibrium spin densities for Fig. 4. The relative interfacial resistance as a function of the rela- tive gate voltage 0 max= / FU e ↑ϕ ε (a). / eR RΔ plotted as a function of 01 1 U− − to reveal the characteristic features at depleting one of the spin components. The length of the sample is 2λ and assumed =ee iν ν (b). 0.04 0.03 0.02 0.01 0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0 0.8 0.9 1.0 L = 0.1� L = 0.03� L = 0.1� L = 0.03� L n= 0.1 ( = 0)� ↑ L n= 0.03 ( = 0)� ↑ U0 1 – √1 – U0 � R R/ e � R R/ e a b The electrical resistance of spatially varied magnetic interface. The role of normal scattering Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 193 “spin-up” and “spin-down” electrons are related to each other as / =1/ 2n n↑ ↓ far from the gate ( \| \|>>x λ ). Then we may write the x-dependencies of the spin densities in the following way: 2 2/ 0 0= (1 e )Lxn n U↑ −− , 2 2/ 0 0= (2 e )Lxn n U↓ −− . Both solid and dotted curves were calculated from the so- lution given by Eq. (20). Note, at 0 1U → , these curves tends to the values (that marked by the triangle and square, correspondingly) which were calculated from Eq. (28) for the limit case of the zero-spin density for one of the spin components. It is easy to see, the resistance is very sensi- tive to formation of the fully spin-polarized state. Thus, it can serve as a marker of this state appearing. In the opposite case, when the spatial transition to the fully spin-polarized magnetic is smooth enough ( <<L λ ) the spin-flip scattering converts current with one spin pola- rization to a current of the opposite polarization with pene- tration current deep into the fully spin-polarized magnetic region. Thus, there is no large additional resistance of the interface. There are no divergences in Eqs. (11), (15) and (16) at 0n↑ → . Within the frames of our approach we may calculate al- so the resistance of the smooth interface between two op- posite fully spin-polarized magnetic sides (GMR contact). Let the left side is “spin-up” polarized ( = 0n↓ ) and the right one is “spin-down” polarized ( = 0n↑ ). Let us sup- pose also that there is an interface region between spin- polarized sides which length is tr <<L λ where “spin-up” and “spin-down” densities are not zero. Within the main approximation on the small parameter tr /L λ the electro- chemical potential aμ does not depend on the coordinate x as it has enough time to be balanced inside the interface region while spin-flip process occurs. Thus, within this approximation we get the following solution of Eq. (2) 1= ( ) , = [ ( ) ] , x r a a a l l j f x dx j j f x dx −′ ′ ′ ′−μ + μ∫ ∫ tr ˆ= ( ) << / .n n a L′μ − β μj (29) Here, we take into account the following boundary condi- tions: = 0j↓ at = lx x , and = 0j↑ at = rx x . As it fol- lows from Eq. (1), x-dependent parts of the electro- chemical potentials ,↑ ↓μ , are of the order of 2 tr( / )a Lμ λ . Really, while ↓↓β increase rapidly near the left boundary, the product j↓↓ ↓β remains bounded above as j↓ tends to zero. Thus, our assumption on the x -independence of the aμ is valid. Within the accuracy of the model we have take into account the second term only in the subintegral function in Eq. (9). Thus we get for the resistance of the interface 1 0 inter = . r sfl R es dx − ⎡ ⎤Π⎢ ⎥ τ⎢ ⎥⎣ ⎦ ∫ (30) 6. Summary In summary, we have investigated the diffusive electron transport in conductors with spatially inhomogeneous magnetic properties taking into account both impurity and normal scattering. We have obtained the general equations for the electrical resistance of spatially inhomogeneous magnetic interfaces with collinear magnetization(see Sec. 2). The equations open an effective way to calculate the interfacial resistance. We found that spatial magnetic inhomogeneity causes the additional interfacial resistance which depends essentially on the spatial characteristics. If interfacial inhomogeneity is smooth enough, the spin non- equilibrium density causes the additional resistance the value of which is determined by spin-flip processes. It can be used for direct experimental investigation of spin flip processes (see Sec. 3). The simplest relation between spin flip processes and the additional interfacial resistance arises when electron- electron scattering dominates over other scattering processes. In the case when the length-scale of an inhomo- geneous interface region is short enough, we found sepa- rately both the contribution of the interface region into the resistance and the contribution of homogeneous sides (see Section 0.4). The interfacial resistance increases with the increase of the electron-electron scattering frequency. The reason is the mutual friction between “spin-up” and “spin- down” electron subsystems and the “crowd” effect. We have found also the resistance of smooth “spin- stop” interface that means an essential barrier for electron transport of one of the spin components. We have demon- strated the sensitivity of the interfacial resistance to forma- tion of a fully spin-polarized magnetic under the influence of applied external fields. We have demonstrated also that resistance measurements provide direct information on the frequency of spin-flip processes when both sides have an- tiparallel spin orientation. It's shown also in Sec. 3 that the formation of a fully spin-polarized magnetic shows itself in the temperature dependence of the “spin-equilibrium resis- tance". The work was supported in part by NanoProgram of the NAS of Ukraine and NASU Grant F26-2. Appendix To find constants ,l rc in Eqs. (20), (21) we have to write the antisymmetrical part of the difference of the matching equations for the electro chemical potentials in the sides of the interface region ˆ ˆ( ) = 0 , r r aar l l aal aar r aal l r er l el c c B c B c B B λ β + λ β + Δ + Δ + + Δ + Δa j j (A.1) where 0 0 ˆ ˆ ˆ ˆˆ ˆ= ( ) , = ( ) .l l r rB x dx B x dx ∞ −∞ ⎡ ⎤ ⎡ ⎤Δ β −β Δ β −β⎣ ⎦ ⎣ ⎦∫ ∫ R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, P.V. Pyshkin, and A.V. Yanovsky 194 Fizika Nizkikh Temperatur, 2011, v. 37, No. 2 Equation (10) is written in the main approximation on the small parameter /L λ . Here we have taken into account , , , ˆ = = 0r l er l er lβ − ⋅a j a μ . Note that the form of equation does not depend on the choice of the x-coordinate of mat- ching points out of the interface region. Really, expanding the exponent in Eq. (18) into the series near the interface region we obtain /= x aa aa aac e c xc± λ⋅ λ β ≈ λ β − βa m mμ . At matching the second term will be cancelled by the x-depen- dent part of μ from Eq. (17). For example, at matching in the right-hand side we get ˆ ( ) = aar r r aarB x c B c xc− −Δ − βa a . There is no need to keep next order terms on the parameter /L λ . Solving (A.1) together with (19) yields , , , , ˆ( ) = ,l r aal r er el a er l r l r aar l aal aa B c B ±λ β − + Δ − λ β + λ β + Δ j j a j (A.2) ˆ ˆ ˆ= .r lB B BΔ Δ + Δ Thus, we get for caj from Eq. (19) the following 2 2 = , 2( ) r aar ear l aal eal an ca r aar l aal aa j j B j j B λ β + λ β − Δ λ β + λ β + Δ (A.3) = .na ea aa j j β − β Here we take into account = = / 2enr enlj j j . Substituting Eqs. (A.2), (A.3) into Eqs. (20) and (21) give us a cumbersome equation for RΔ . However, it can be simplified essentially when there is no “leap” of the resistance in the interface region, i.e. one may neglect by the BΔ in Eqs. (A.2), (A.3)) as to compare with aaλβ . Then, the contribution of the interface region, P , is not vanishing as to compare with contribution of the regions of the diffusion spin non-equilibrium density, 2 s aasd jλ β , in that case only when side conductivities of are close to one another: 2(\| \| / ) /ear ealj j j L− λ . Then, one can neglect both by the difference between earj and ealj and by the value \| \| /s ear ealc j j j−∼ . As a result, after substituting Eqs. (A.2), (A.3) in Eqs. (20), (21) we get the formulas for the addition to the resistance, i.e. Eqs. (23)–(25). 1. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chaze- las, Phys. Rev. Lett. 61, 2472 (1988). 2. M.J.M. de Jong and L.W. Molenkamp, Phys. Rev. B49, 5038 (1994); Phys. Rev. B51, 13389 (1995). 3. M. Jonson and R.H. Silsbee, Phys. Rev. B35, 4959 (1987); Phys. Rev. Lett. 58, 2271 (1987). 4. P.C. van Son, H. van Kempen, and P. Wyder, Phys. Rev. Lett. 58, 2271 (1987). 5. T. Valet and A. Fert, Phys. Rev. B48, 7099 (1993). 6. A. Shpiro and P. Levy, Phys. Rev. B63, 014419 (2000). 7. R.N. Gurzhi, A.N. Kalinenko, A.I. Kopeliovich, A.V. Yanov- sky, E.N. Bogachek, and Uzi Landman, J. Supercond.: Inc. Nov. Magn. 16, 201 (2003). 8. T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). 9. P. Levy and S. Zhang, Phys. Rev. Lett. 79, 5110 (1997). 10. K. Flensberg, T.S. Jensen, and N.A. Mortensen, Phys. Rev. B64, 245308 (2001). 11. Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, and B.I. Halpe- rin, Rev. Mod. Phys. 77, 1375 (2005). 12. J.M. Ziman, Electrons and Phonons: The Theory of Trans- port Phenomena in Solids, Clarendon Press, Oxford, UK (1960). 13. R.N. Gurzhi, Sov. Phys. Usp. 11, 255 (1968).

The electrical resistance of spatially varied magnetic interface. The role of normal scattering

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