OFFPRINT Landau-Zener and Rabi oscillations in the -dependent conductance

L. J. Fernandez-Alc´ azar´ and H. M. Pastawski EPL, 105 (2014) 17005

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Landau-Zener and Rabi oscillations in the spin-dependent conductance

L. J. Fernandez-Alc´ azar´ and H. M. Pastawski Instituto de F´ısica Enrique Gaviola, CONICET-UNC - C´ordoba, Argentina and Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba Ciudad Universitaria, 5000 C´ordoba, Argentina

received 18 November 2013; accepted in final form 3 January 2014 published online 3 February 2014

PACS 72.25.-b – Spin polarized transport PACS 72.25.Ba – Spin polarized transport in metals PACS 85.75.-d – Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields

Abstract – We describe the spin-dependent quantum conductance in a wire where a magnetic field is spatially modulated. The change in direction and intensity of the magnetic field acts as a perturbation that mixes spin projections. This is exemplified by a ferromagnetic nanowire. There the local field varies smoothly its direction generating a domain wall (DW) as described by the well-known Cabrera-Falicov model. Here, we generalize this model to include also a strength mod- ulation. We identify two striking diabatic regimes that appear when such magnetic inhogeneity occurs. 1) If the field strength at the DW is weak enough, the local Zeeman energies result in an avoided crossing. Thus, the spin-flip probability follows the Landau-Zener formula. 2) For strong fields, the spin-dependent conductance shows oscillations as a function of the DW width. We interpret them in terms of Rabi oscillations. Time and length scales obtained from this simplified view show an excellent agreement with the exact dynamical solution of the spin-dependent trans- port. These results remain valid for other situations involving modulated magnetic structures and thus they open new prospects for the use of quantum interferences in spin-based devices.

This paper is dedicated to the memory of the lifelong collaborator Patricia Rebeca Levstein.

Copyright c EPLA, 2014

Introduction. – The control and design of spin- semiconducting spin valves [8] showed magnetic com- dependent electronic transport in magnetically modulated mensurabilities as well as regimes compatible with a devices represents a promising technological challenge [1]. Landau-Zener problem. Thus, quantum transport through Spintronic devices switch the spin state or filter electrons magnetic inhomogeneities becomes a promising tool in by spin. The most direct way to tune the transport, apart spintronics, where the different characteristic times and from spin-orbit effects [2,3], is to use designed magnetic lengths should be identified. These scales should be com- inhomogeneities which couple directly to the spin. The pared with the electron’s Fermi wavelength. The tunnel- most prominent devices are those based on the giant mag- ing adiabaticity is given by the electron’s speed. In this netoresistance [4]. Other recent developments involve spin context, a dynamical description of the transport process valves based on organic molecules [5,6], and quasi–one- would improve the comprehension of these phenomena. dimensional spin transistors [7,8]. Since quantum effects In this letter we consider a variant of the Cabrera and become relevant, transport is based on Landauer’s motto: Falicov [12] model for spin-dependent electronic transport “conductance is transmittance” [9]. through a soft magnetic DW. It is representative of a Previous studies of spin-dependent quantum transport wide class of magnetic inhomogeneities. In the original suggest the presence of interesting physical phenom- model the field just rotates along the DW. We extend it ena. For example, conductance through ferromagnetic allowing a modulation in the field strength. This simple nanowires with a domain wall (DW) shows some Fabry- variation will have nontrivial consequences on transport. P´erot like interferences, which were not fully under- We show that in a weak perturbation regime, the spin- stood [10,11]. Also, transport on magnetically modulated dependent conductance through the DW can be described

17005-p1 L. J. Fern´andez-Alc´azar and H. M. Pastawski by the Landau-Zener (LZ) formula. In contrast, for a where ε is the energy associated with the dynamics along strong perturbation regime, we find well-defined interfer- x, identifying, ences as a function of the DW width, which are interpreted x as Rabi oscillations. This interpretation is confirmed by μBBz(x)=E↓(x)=Δ0 tanh , (5) an analysis of the wave packet dynamics. The physics and W   computational strategies described here could help in the and design of better spintronic devices. x μ B (x)=V (x)=αΔ0/ cosh , (6) Hamiltonian of the conduction electrons. – We B x ↑↓ W consider a single-spin channel [10]:   where E↓(x)=−E↑(x). The local states become mixed 2 d2 Hˆ = − − μ · B (x). (1) by V↑↓(x) while the electron moves through the DW. 2m dx2 Evaluation of the conductance. – To evaluate the The first term is the kinetic energy along x of electrons quantum conductance we will use the Landauer-B¨uttiker with effective mass m. The second term is the Zeeman in- equation [14]. There, different conductances are given by teraction between the spin magnetic moment μ and B (x), the transmittances between states of definite momentum the effective magnetic field at x. Here, μ = −μ σ ,where B and spin projection at the contacts [9]. μ is the Bohr magneton and σ =(σ ,σ ,σ ) is the vec- B x y z In a tight-binding representation [15], the spatial coordi- tor of the . In particular, the dependence on nate takes discrete values x in a grid of unit a, x → na. x of the magnitude and direction of B (x) may cause the n n Every site in the grid has an associated normalized local spin-dependent scattering. These inhomogeneities may be wave function |n , which will be called n-th orbital as in natural, as in ferromagnetic DWs [13], or artificially gen- a LCAO scheme. Each orbital has an energy given by the erated, as in magnetic semiconducting waveguides [7]. We local potential E (x )=E ,wheres is either ↑ or ↓,and will express our results in the concrete language of ferro- s n n,s the transverse field yields V (x )=V . Any electronic magnetic nanowires. ↑↓ n n,↑↓ wave function with well-defined spin s is now written in Electronic structure in modulated magnetic terms of a discrete sum: fields. – The Cabrera and Falicov soft-DW model [12] considers a quantum spin channel laid along x,anda |ϕs→ un,s |n, s , (7) n magnetic field whose orientation rotates as it progress  along the DW. We generalize this description by in- cluding a modulation in the field intensity. The vector where, according to eq. (4), the spin-orbital amplitudes un,s must satisfy f(x)=(fx(x),fy(x),fz(x)) = −B (x)/B∞, describes the DW shape, and satisfies |f(x)|→1asx −→ ± ∞ .The −V [u +1 − 2u + u 1 ]+E u asymmetry in the modulation strength is described by the n ,↑ n,↑ n− ,↑ n,↑ n,↑ + Vn,↑↓un+1,↓ = εun,↑, parameter α = B(0)/B∞.Usingtheaxisz as quantiza- ⎧ ⎪ (8) tion direction for the spin, ⎪ ⎨⎪ −V [un+1,↓ − 2un,↓ + un−1,↓]+En,↓un,↓  x x + Vn,↑↓un+1,↑ = εun,↓. f(x)= α/ cosh , 0, tanh . (2) ⎪ W W ⎪      The⎩⎪ unit of energy is given by the hopping strength Here, W is the half-width of the DW. Obviously, α =1 V = 2/(2ma2). We consider energies at the band cen- corresponds to a field of constant strength that rotates ter (ε ≃ 2V ). Each spin orientation is represented by across the DW. The Zeeman term is μ·B (x)=Δ0σ ·f(x). achainwithN ≫ 1 orbitals that comprises the whole Here, Δ0 = μ B . The wave function for a conduction B ∞ DW. L and R are sites indices symmetrically arranged electron has components along both spin directions, |↑ at the left and right sides of the DW and satisfying and |↓, referred to the quantization axis parallel to the (R − L)a = Na ≫ W .ThenE ≡ E = ±Δ0 for field at the left domain (laboratory frame), as n,s L,s n ≤ L and En,s ≡ ER,s = ∓Δ0 for n ≥ R. Since far

|ψ(x) = ϕ↑(x) |x, ↑ + ϕ↓(x) |x, ↓ . (3) away from the DW, V↑↓ ≡ 0, the asymptotic eigenvalues, with wave vector k, are εk,s = ±Δ0 +2V − 2V cos (ka). Therefore, the equations governing the electron tunneling In the region of the DW both spin orientations become and the spin flip are coupled by the perpendicular component of the magnetic field, represented by the hopping element V . 2 d2 n,↑↓ − ϕ (x)+E (x)ϕ (x)+V (x)ϕ (x)=εϕ (x), The magnetic domains, which play the role of contacts, 2m dx2 ↑ ↑ ↑ ↑↓ ↓ ↑ ⎧ are described through a renormalization procedure [16]. ⎪ 2 2 In an open system it leads to a non-Hermitian effective ⎨⎪ d − ϕ↓(x) − E↓(x)ϕ↓(x)+V↑↓(x)ϕ↑(x)=εϕ↓(x), Hamiltonian [17]: 2m dx2 ˆ ˆ ˆ ⎪ (4) HT = H + Σ, (9) ⎩⎪

17005-p2 Landau-Zener and Rabi oscillations in the spin-dependent conductance

a) b)

α<1 α>1 Zeeman Energy Zeeman Energy x x

Fig. 1: (Color online) Solid black lines are the local eigenenergies for electrons with ↑ and ↓ spin in the presence of a locally rotated field which is schematized on top. (a) Weak (α<1) or (b) strong (α>1) field strength at the DW center. Dashed lines are the unperturbed Zeeman energies. where

Σ(ˆ ε)=ΣL↑(ε) |L ↑ L ↑| +ΣL↓(ε) |L ↓ L ↓|

+ΣR↑(ε) |R ↑ R ↑| +ΣR↓(ε) |R ↓ R ↓| . (10)

Σj are the self-energies that satisfy the Dyson equation in the magnetic domains:

V 2 Σj (ε)= =ReΣj (ε) − iΓj (ε) (11) ε − Ej − Σj (ε)

≃−iΓj, at the band center. (12)

The double spin-orbital subscript j = L ↑,L↓,R↑ or R ↓ indicates the left (L) or right (R) channels inside the mag- netic domains with the corresponding spin orientation. 2aΓj/ is the group velocity at the spin channels j con- Fig. 2: (Color online) Scheme of the eigenenergies of a two-level nected to each spin orbital at the sides, i.e. L ↑,L↓,R ↑ system, |↑ and |↓, as functions of the dynamical parameter, and R ↓ in a four-terminal circuit. x = vt, that controls the Landau-Zener transition. The en- We obtain the retarded and advanced Green functions ergy levels show an avoided crossing due to the presence of a R −1 A perturbation that couples the states. The unperturbed ener- from HˆT as Gˆ (ε) = limη→0+ (ε +iη − HˆT ) and Gˆ = gies are shown with dashed lines. Energy and coordinate have GˆR†. The transmittance is [15] arbitrary dimensions. R 2 Tij (ε)=2Γi(ε) Gij (ε) 2Γj(ε). (13) time x(t)≡x ≃ vF t. The spin-dependent mean poten-

Here, Γj =Im(Σj )andi, j = L ↑,L ↓,R ↑,R ↓,be- tial energy will change as the wave packet moves through ing j and i the electronic input and output spin-orbital the DW and starts to be mixed with that of the oppo- channels, respectively. When the evaluated channels cor- site spin by the V↑↓ term. These potential energies are respond to opposite spin projections in opposite sides of assimilable to the time-dependent local energies of a two the DW we call them spin-flip transmittances, e.g., T↓↑. level system in a LZ problem [18]. This last describes the transition between two states when their unperturbed Transport in the regime of α<1 as a Landau- energies are swept across a degeneracy point while a con- Zener problem. – In eq. (4) V is responsible for the ↑↓ stant perturbation produces an avoided crossing. In fig. 1 mixture of the spin orientations. While V (x) vanishes ↑↓ we show the local Zeeman levels for the DW region. The within the domains, it is roughly constant at the DW cen- comparison between the levels of fig. 1(a) and the energy ter. Besides, E and E account for the Zeeman energy in ↑ ↓ diagram of the two-level system, shown in fig. 2, evidences the laboratory frame and they have the meaning of effec- the analogy between the α<1 regime and the LZ prob- tive potentials for those electrons oriented along the field. lem. Thus, the LZ equation predicts that the probability While, inside the domains, E − E =2Δ0 quantifies the ↑ ↓ to exit in the state |↓ to the right, provided that it entered Zeeman splitting, at the DW center both energies intersect from the left in the state |↑,is E↑ = E↓ =0.

Consider an electron wave packet with a given spin pro- 2 jection that moves with definite momentum towards the 2π |V↑↓| P↓↑ =1− exp −  . (14) DW. Its mean position results proportional to the elapsed (dE(0)/dt)

17005-p3 L. J. Fern´andez-Alc´azar and H. M. Pastawski

the giant magnetoresistance arises from the scattering at the DW [4]. On the other hand, if the DW width is broad enough, the electrons tend to change their spin orientation T↓↑ ≃ 1, preventing the magnetoresistance. Considering the overall dependence on W for α ≤ 1, we appreciate that there is a fair correspondence between the transmit- tance and the LZ probability that improves as α becomes smaller. Even when there are small discrepancies, these can be attributed to the complexity inherent in our prob- lem, where the “perturbation” V↑↓ gradually turns on while the levels become degenerate at the DW center. This exceeds the simplicity of the LZ model. The two lower panels of fig. 3 correspond to a α>1 max max regime, with V↑↓ =3Δ0 and V↑↓ =5Δ0.Inboth cases the transmittances oscillate as a function of W .This might suggest an analogy to the Fabry-P´erot interferences in tunneling problems. In the case of ref. [10] this phe- nomenon is justified because their potentials have slopes with discontinuities at the DW. However, here we con- sider a high-energy problem with smooth potential barri- Fig. 3: (Color online) The transmittance T↓↑ vs. the DW half- ers. Therefore, it always results that T↓↑ +T↑↑  1andthe width W for different α. The energy is measured in units of reflectances are nearly zero. Hence, the Fabry-P´erot inter- V . The Zeeman splitting is 2Δ0 =0.2V and V↑↓ = αΔ0.The ferences are discarded as possible origin of the observed transition probabilities predicted by the Landau-Zener formula oscillations. Instead, while the electron moves across the PLZ are shown with dashed lines. DW its spin oscillates between the states |↑ and |↓ driven

Here, we can use E(x)=E↑(x) − E↓(x)andx ≃ vF t by the “perturbation” V↑↓. Thisisconsistentwiththefact to evaluate the derivative. The adiabaticity parameter that the spin precesses around the local field. This is called 2  for a semiclassical spin and Rabi oscilla- 2π |V↑↓| /( dE/dt) describes a fully adiabatic transition 2 0  if it is much greater than 1 resulting in P↓↑  1, while the tion for a spin-(1/2) [19]. This π (a/W ) |V↑↓| / (Δ V ) 1 opposite limit is a diabatic process where P↓↑  0. regime, contrasts with the adiabatic transition where the We will consider a wave packet with εF ≫ Δ0 and electron’s spin simply remain aligned with the local mag- vF (x) ≃ 2aV/. Hence, the time of transit through the netic field while it crosses the DW. However, since both DW is τW =2W/vF . We can relate the DW crossing with DWs are smooth, the oscillation frequency varies contin- the LZ problem identifying uously and thus it is not obvious that well-defined Rabi oscillations would show up. dE dE dx = Transport in the regime of : interferences dt dx dt α>1 as Rabi oscillations. – In the DW, the Zeeman energies 2Δ0 ≃ vF , (15) in the laboratory frame are degenerate while the coupling W V↑↓(x)ismaximumV↑↓ = V↑↓(x)|x=0 = αΔ0. Locally, where E(x) is calculated as in the LZ formula, being this can be seen as a two-level system undergoing a Rabi E↑ and E↓ obtained from eq. (5). Thus, the adiabatic- oscillation with period 2 ity parameter results as π (a/W ) |V | / (Δ0V ), where ↑↓ π V↑↓ = αΔ0. We will choose to control the adiabaticity τR = . (16) of the crossing by changing W . V↑↓ Numericalresults.–We will use eq. (13) to evaluate Therefore, the length traveled by the electron during that the spin-dependent transmittance, T↓↑, that describes the Rabi cycle is πaV spin-flip process. In fig. 3 we compare T↓↑ with the spin- LR = τRvF =2 , (17) flip probability of the LZ problem. Both are shown as αΔ0 a function of the parameter W . We show four different where vF ≃ 2aV/. We adopt the term “Rabi oscilla- DW characterized by α =0.5, 1, 3 and 5. In all cases we tion” to emphasize that the spin-(1/2) is in an oscillating consider Δ0 =0.1andV =1. superposition of its two possible projections. The two upper panels of fig. 3 may be associated to the We analyze the spin-flip transmittances in terms of the

α ≤ 1 regime. At the center of the DW, V↑↓(x)|x=0 = length scales estimated above. In the case of the two up- max max V↑↓ =0.5Δ0 and V↑↓ =1Δ0. If the DW is abrupt, per panels of fig. 3, α =0.5andα =1, the Rabi os- i.e. W ≃ 0, the electrons keep their initial spin orienta- cillation might have, according to eq. (17), characteristic tion, thus T↓↑ ≃ 0. This coincides with the regime where lengths LR ≃ 125.6a and LR ≃ 62.8a, respectively. These

17005-p4 Landau-Zener and Rabi oscillations in the spin-dependent conductance

Fig. 4: (Color online) Probability (color scale) for a ↓ spin Fig. 5: (Color online) Probability (color scale) for a ↓ spin orientation (upper panel) or ↑ spin orientation (lower panel) as orientation (upper panel) and ↑ spin orientation (lower panel) a function of time and position, given that the initial electron as a function of time and position, for Δ0 =0.1V , W =8a and state on the left has ↑ spin and moves with the Fermi velocity α =0.5. The dashed lines indicate the effective width of the  vF =2aV/ . Here, Δ0 =0.1V , W =8a and α =5.The DW, 2W . The Rabi oscillations are not developed. midpoint of the DW is located at x =0a. The probability oscillations confirm that Rabi oscillations are present. The vertical and horizontal short-dashed lines come into contact at the domains is 2Δ0,withΔ0 =0.1V . The DW center is the maxima and minima of the oscillations. From these, the placed at x =0a and its width is 2W =16a, which implies magnitudes of the period and oscillation characteristic length can be inferred. These coincide with those predicted by the an adiabaticity parameter of about 1. According to fig. 3, eqs. (16) and (17), τR =6.3/V and LR =12.6a as shown by this W ensures some oscillations in the transmittance. The the white arrows. intensity plot is consistent with the fact that as the elec- tron moves through the DW, the probability of finding the electron with the spin-up projection decreases while the complementary spin-down density increases. Subse- are much longer than the DW width needed for an adi- quently, an increase in the spin-up probability is produced abatic spin-flip. This explains the absence of oscillations while a decrease occurs for the opposite spin projection. and the applicability of the LZ formula. In contrast, the This cycle is repeated until the electron reaches the end of α =3andα = 5 cases, shown in the two lower panels, the DW. This oscillation between the two spin projections the spin-flip transmittances present oscillations with char- is identified with a Rabi oscillation. The observed period acteristic lengths L ≃ 8a and L ≃ 4a, respectively. N N and characteristic length are in full agreement with those According to our hypothesis of transmittances modulated given by eqs. (16) and (17), τ =6.3/V and L =12.6a, by Rabi oscillations, the spacing between two consecutive R R respectively. These magnitudes are drawn in fig. 4. local minima must be L ≃ 10.5a and L ≃ 6.3a, respec- tively. The discrepancy between our na¨ıve prediction and The same analysis is performed for a α =0.5DWand the numerical results is justified by the fact that the Rabi shown in fig. 5. This α, together with 2W =16a, implies a length is not a perfectly defined magnitude in our smooth low adiabaticity parameter of about 0.01. Again, the final DW model. This is because V↑↓, and hence the involved electronic state |ϕR in the right domain, after traversing periods, change as the electron moves through the DW. the DW, is a superposition of the two spin projections. In order to confirm the Rabi oscillation hypothesis, we The probability of finding the state |↓, is consistent with the transmittance shown in fig. 3 and the LZ prediction: analyze the dynamical evolution of the electron’s spin im- 2 2 plementing a unitary algorithm based on the Trotter ap- T↓↑ ≡| ↓|ϕR| = P↓↑ and T↑↑ = | ↑ |φR| = P↑↑ ≃ proximation [20]. We consider an initial Gaussian wave 1 − T↓↑. As consequence of the DW smoothness there packet with spin |↑ and a well-defined momentum at the are no significative reflections. In contrast to the previous band center. This last condition avoids undesired effects case, we see that the time oscillations are not developed. of dispersion. During the electron’s transit through the While in the present work we just analyzed a single DW, the spin projection tries to follow the magnetic field incoming wave vector, considering a metallic wire would and thus the final spin projection depends on W . involve integrating, up to the Fermi energy, over transver- In fig. 4 we show the probability densities P↓↑(x, t)and sal channels equivalent to those as described here. This P↑↑(x, t)forα = 5. Here, the second subscript is the initial could smear out the Rabi oscillations reported here and spin, while the first one indicates the corresponding spin one would need a proper design to overcome this diffi- projection at time t. The upper panel shows the proba- culty. However, in magnetic semiconducting waveguides, bility for the |↑ spin projection, while the lower panels the relevant role of lateral quantization leaves the consid- show the |↓ spin projection. The Zeeman splitting inside ered model as a realistic description [8].

17005-p5 L. J. Fern´andez-Alc´azar and H. M. Pastawski

Conclusion. – In this letter we explored the quan- ∗∗∗ tum phenomena associated to spin-dependent transport in the presence of a smooth magnetic inhomogeneity, much This work was performed with the financial support as a DW in a magnetic nanowire. For this purpose, we from CONICET, ANPCyT, SeCyT-UNC and MinCyT- extended the Cabrera-Falicov model to account for mod- Cor. ulations on the magnetic-field intensity at the DW. The physics we described is not restricted to this case. In- deed, our results and strategies remain valid for other sit- REFERENCES uations, such as magnetically modulated semiconducting [1] Zutiˇ cI.,FabianJ.´ and Das Sarma S., Rev. Mod. Phys., structures [7]. There, spectral modulations are described 76 (2004) 323. by variants of eq. (4). In ref. [8] a situation assimilable [2] Schliemann J., Egues J. C. and Loss D., Phys. Rev. to our α<1 is presented for a spin transistor based on Lett., 90 (2003) 146801. helical magnetic fields. [3] Reynoso A. A., Usaj G., Balseiro C. A., Fein- We showed that, for α ≤ 1, the spin-dependent trans- berg D. and Avignon M., Phys. Rev. Lett., 101 (2008) port across the magnetically modulated region is fairly 107001. 80 described by the LZ formula. LZ applies to the whole dy- [4] Fert A., Rev. Mod. Phys., (2008) 1517. [5] Steil S., Großmann N., Laux M., Ruffing A., namical range, from diabatic to fully adiabatic crossing. Steil D., Wiesenmayer M., Mathias S., Monti O., We showed that, by performing the appropriate mapping Cinchetti M. and Aeschlimann M., Nat. Phys., 9 of the relevant variables, LZ yields a quite fair descrip- (2013) 242. tion under well-defined conditions for the perturbation. [6] Dediu V. A., Nat. Phys., 9 (2013) 210. For α>1, we found that conductance has quantum in- [7] Betthausen C., Dollinger T., Saarikoski H., terferences which manifest as oscillations as a function of Kolkovsky V., Karczewski G., Wojtowicz T., the DW width. These cannot be assigned to Fabry-P´erot Richter K. and Weiss D., Science, 337 (2012) 324. interferences. By performing a dynamical study of the [8] Saarikoski H., Dollinger T. and Richter K., Phys. tunneling process, we showed that a spin-polarized wave Rev. B, 86 (2012) 165407. 71 packet propagating across the DW, can be seen as a two- [9] Imry Y. and Landauer R., Rev. Mod. Phys., (1999) level system undergoing Rabi oscillations. S306. [10] Gopar V. A., Weinmann D., Jalabert R. A. and A possible experimental set up to test the discussed Stamps R. L., Phys. Rev. B, 69 (2004) 014426. effects in all regimes could be a linear semiconducting [11] Falloon P. E., Jalabert R. A., Weinmann D. and waveguide in the presence of a locally modulated field. Stamps R. L., Phys. Rev. B, 70 (2004) 174424. In such a case, few conducting channels are enabled by [12] Cabrera G. G. and Falicov L. M., Phys. Status Solidi a gate voltage that also controls the carrier’s wavelength. B, 61 (1974) 539. Finally, the dynamical description of the transport prob- [13] Czerner M., Yavorsky B. Yu. and Mertig I., Phys. 77 lem as presented here, may prove useful for the converse Rev. B, (2008) 104411; Burton J. D., Sabirianov R.F.,JaswalS.S. Tsymbal E. Y., Mryasov O. problem: i.e. evaluating the dynamics of a DW under and N., Phys. Rev. Lett., 97 (2006) 077204. pulsed electrical currents. This may extend the interest [14] Buttiker¨ M., Phys. Rev. Lett., 57 (1986) 1761. of our strategy to study a problem of growing interest: [15] Pastawski H. M. and Medina E., Rev. Mex. Fis., 47 electrically driven domain-wall–based memories in quasi– (2001) 1. one-dimensional (1D) magnetic wires [21]. [16] Levstein P. R., Pastawski H. M. and D’Amato J. L., In summary, for the perturbative regime (α<1), we J. Phys.: Condens. Matter, 2 (1990) 1781. 398 probed a definite connection between steady-state spin- [17] Pastawski H. M., Physica B, (2007) 278. 137 dependent transport across a magnetically modulated re- [18] Zener C., Proc. R. Soc. Lond. A, (1932) 696. [19] Cohen-Tannoudji C., Grynberg G. and Dupont-Roc gion and the time-dependent Landau-Zener problem. In J., - Interactions: Basic Processes and Appli- the strong perturbation regime (α>1), we showed that cations (Wiley, New York) 1992, p. 10. the steady-state conductance presents interferences. We [20] De Raedt H., Annu.Rev.Comput.Phys., 4 (1996) 107. probed that they arise from Rabi oscillations, by perform- [21] Thomas L., Hayashi M., Jiang X., Moriya R., Ret- ing a time-dependent calculation. tner C. and Parkin S., Science, 315 (2007) 1553.

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