We study the effect of magnetic anisotropy in a single electron transistor
with ferromagnetic electrodes and a non-magnetic island. We identify the
variation $\delta \mu$ of the chemical potential of the electrodes as a
function of the magnetization orientation as a key quantity that permits to
tune the electrical properties of the device. Different effects occur depending
on the relative size of $\delta \mu$ and the charging energy. We provide
preliminary quantitative estimates of $\delta \mu$ using a very simple toy
model for the electrodes.Comment: Conference article presented at PASPS IV, Sendai, August 200

Abolfath

Baibich

Bolotin

Brey

Deshmukh

Dietl

Fernández-Rossier

Garcia

Giddings

Gould

Jacob

Julliere

Maekawa

Pasupathy

Ruster

Saito

Strandberg

van Houten

Velev

Viret

Wunderlich

English

arXiv

We study the effect of magnetic anisotropy in a single electron transistor with ferromagnetic electrodes and a non-magnetic island. We identify the variation δμ of the chemical potential of the electrodes as a function of the magnetization orientation as a key quantity that permits to tune the electrical properties of the device. Different effects occur depending on the relative size of δμ and the charging energy. We provide preliminary quantitative estimates of δμ using a very simple toy model for the electrodes.JFR acknowledges funding from Generalitat Valenciana (GV05-152), Spanish MEC (Grants FIS200402356) and Programa Ramón y Cajal. RA and LB acknowledge funding from the Spanish MEC (grant MAT2005-07369-C03-03)

Fernández-Rossier, Joaquín

Aguado Sola, Ramón

Brey Abalo, Luis

RUa Reposity University of Alicante

pss header will be provided by the publisherAnisotropic magnetoresistance in single electron transportJ. Ferna´ndez-Rossier ∗1, R. Aguado 2, and L. Brey21 Department of Applied Physics, University of Alicante, 03690 Spain2 Instituto de Ciencia de Materiales, CSIC, Cantoblanco, Madrid 28049, SpainReceived 15 July 2006Key words Coulomb Blockade, Anisotropic Magnetoresistance, Single electron transportPACS 04A25We study the effect of magnetic anisotropy in a single electron transistor with ferromagnetic electrodes anda non-magnetic island. We identify the variation δµ of the chemical potential of the electrodes as a functionof the magnetization orientation as a key quantity that permits to tune the electrical properties of the device.Different effects occur depending on the relative size of δµ and the charging energy. We provide preliminaryquantitative estimates of δµ using a very simple toy model for the electrodes.Copyright line will be provided by the publisherIntroduction Spin polarized transport through carefully designed ferromagnet-non magnetic -ferromagnet(FM1-NM-FM2) nanostructures can be very sensitive to the relative orientation of their magnetic moments[1],~Ω1 and ~Ω2 . The latter are controlled by external magnetic fields that result in the so called tunnelmagnetoresistance[2] (TMR), Giant magnetoresistance[3] (GMR) and Ballistic magnetoresistance[4] (BMR)when the NM layer is a tunnel barrier, a metal and a geometrical nanoconstriction respectively.In bulk ferromagnets, the dependence of resistance on the angle between the magnetization ~Ω and thecurrent ~j gives rise to the so called anisotropic magneto-resistance (AMR). The microscopic origin of thisphenomenon is the spin orbit interaction, which also accounts for the stability of the magnetization orien-tation (magnetic anisotropy). Recently, the concept of Tunneling Anisotropic Magnetoresistance (TAMR)has been proposed theoretically [5] and independently verified in experiments [6, 7, 8] in tunnel junctionswith GaAsMn electrodes. The related concept of Ballistic Anisotropic Magneto Resistance (BAMR) hasbeen proposed theoretically [9] and observed in atomic sized Nickel [10] and Iron nanocontacts [11]. Asopposed to TMR and BMR, where the high and low resistance states are related to variations in ~Ω1 · ~Ω2,TAMR and BAMR effects occur for ~Ω1 = ~Ω2 ≡ ~Ω and depend on the angle between the transport directionand ~Ω, which is controlled by an external field.The microscopic origin of BAMR and TAMR can be traced back to the dependence of the electronicstructure on the angle between ~Ω and the crystal lattice, originated by the spin orbit (SO) coupling. Inideal 1-dimensional chains, BAMR occurs if the number of bands at the Fermi energy is different for themagnetization parallel and perpendicular to current flow [9]. In real metallic nanocontacts, the transmissionof the different channels is not perfect and an ab-initio approach [12] extended to include SO interactionwould be necessary to account for the experimental reports. In the case of TAMR the relevant quantity is thetransmission [5], which is related to the density of states at the Fermi energy. The size of the AMR effectsdepends on the relative ratio of the spin orbit interaction ∆SO , the Fermi energy ǫF and the exchangesplitting J . Not surprisingly, TAMR has been reported first in III-V ferromagnetic semiconductors, whereSO coupling is the largest energy scale in the system.Motivated by recent experimental results [13], we consider a different kind of AMR effect, which takesplace in a single electron transistor (SET) with ferromagnetic electrodes. Although in the experiments[13]both the electrodes and the dot are made of ferromagnetic GaMnAs, here we consider a simpler SET with∗ Corresponding author: e-mail: jfrossier@ua.es, Phone: +34 96 590 3400-2019,pss data will be provided by the publisher2 J. Ferna´ndez-Rossier: TAMR in Coulomb Blockadea non-magnetic island (NMI). To the best of our knowledge AMR effect in this kind of SET have not beenexplored so far [14, 15]. The NMI is influenced by the magnetization orientation of the electrodes, ~Ω, bothbecause the tunneling rates Γ and the electrode chemical potential µ depend on ~Ω. Whereas the changein Γ is related to TAMR, the effect described here is related also to the change in chemical potential, andtherefore closer to the so called magneto Coulomb Blockade effect [16].In the rest of this paper we give a preliminary account of how the magnetization orientation of ferromag-netic electrodes affects the properties of a SET. The latter are described with a very simple toy model withRashba spin orbit and Stoner magnetism. We discuss how, depending on the ratio between the differentenergy scales of the device, different magneto-transport effects are possible.Theory. We consider single electron tranport through a NMI in the Coulomb Blockade (CB) regime,weakly coupled to two identical ferromagnetic electrodes, denoted by left (L) and right (R). The reversesituation, a magnetic dot with strong anisotropy coupled to non-magnetic electrodes, has been consideredelsewhere [17]. The island is capacitively coupled to a gate electrode, which is able to change electrostaticpotential, and thereby the charge, inside the NMI. The Hamiltonian of the system reads:H =∑k,ν,η=L,REηkνη†k,νηk,ν+∑n(ǫn + eφext) d†nσdnσ+Qˆ22C+∑σ,kν,η〈kνη|V |σn〉η†kνdnσ+h.c. (1)The first term describe the ferromagnetic electrodes in a mean field approximation that yields single particlestates labeled with |kν〉, where ν is a band index that includes spin. Because of spin orbit interaction, thespin σ of the electrons in the electrodes is not conserved. The next 2 terms describe the single particlepart, including the coupling to the gate potential φext, and the electrostatic Coulomb repulsion of theNMI Hamiltonian in the orthodox model [18]. Since we neglect spin-orbit interactions in the NMI, thesingle particle levels n are also eigenstates of the spin operator, which we quantize parallel to the electrodemagnetic moment orientation, ~Ω. The last term describes the tunneling of electrons between the electrodesand the NMI. This tunneling occurs via a spin conserving operator, V . The single particle energies of theNMI are denoted by ǫn, and Qˆ = e∑n d†ndn is NMI charge operator. C is the capacitance of the island.The most fundammental result of orthodox CB theory states that the number of electrons in the islandchanges from N to N + 1 when the gate electrode sets into resonance the ground state energies with Nand N + 1 electrons. In the framework of othodox CB theory this occurs when [18]:ǫN+1 + EC(N +12)− eφext = µ(~Ω)(2)where µ is the chemical potential of the left and right electrodes, eφext is the gate potential and EC ≡ e2Cis the SET charging energy. Since the chemical potential of the electrodes depends on the orientation oftheir magnetic moment, ~Ω, it is apparent that the charging curve N(φext, ~Ω) of the SET depends on ~Ω.As the gate is ramped, peaks in the zero bias conductance appear at the degeneracy point betweenground states with N and N + 1 electrons. The width of the peaks is related to the tunneling rates Γηnσ,which also depend on ~Ω. Since Γηnσ can be made arbitrarily small, we can in a first stage ignore itsdependence on ~Ω and discuss the effects related to a change in the chemical potential of the electrodes, as~Ω is rotated with a magnetic field. For the sake of the discussion we assume a uniaxial magnet, so thatthe angle θ characterizes the angle between the magnetization and the axis zˆ: ~Ω = (sin(θ), 0, cos(θ)).We consider two orientations of the magnetization of the electrodes, yielding different chemical potentials,µ1 = µ(θ = 0) and µ2 = µ(θ = π/2). We define δµ ≡ µ1 − µ2. Depending on the relative value ofδµ, Γ and EC , we distinguish 3 different regimes: (i) Standard. This occurs if δµ << Γ. In this case theeffect of the change of chemical potential is negligible in both the charging state and the conductance ofthe device.(ii) Magnetoresistive Γ < δµ << EC . In this case the change of the chemical potential is big enoughas to detune the single electron device from resonance. If the gate potential is tuned to set the device into ac© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimpss header will be provided by the publisher 3peak of conductance at a given ~Ω1, a large change in the conductance will take place when the orientationof the electrode magnetization is changed to ~Ω2 so that δµ > Γ. The change in resistance can be positive ornegative depending on whether the single electron transistor gets closer or further away from the CB peak.In this regime the charge state of the dot is not affected, or at least weakly affected but the dependence ofΓ on ~Ω will play an important role, as it happens in the case of TAMR.(iii) MagnetocapacitiveΓ << EC < δµ In this case the change of the chemical potential is big enoughas to change the charge state of the dot. Neglecting the single-particle spacing, the charging energy forδN extra electrons is δNEC . Therefore, the number of carriers in the NMI can change by as manyas δN = δµEC. This anisotropic magneto-capacitive effect is different from TAMR and specific of SETwith ferromagnetic electrodes. The number of electrons δN injected or extracted from the NMI could bemonitored by the number of peaks in the zero bias conductance as an external magnetic field rotates ~Ω sothat µ crosses the SET charging boundaries defined by (2).Whereas there is plenty of experimental information about typical charging energies of SET, whichrange from µeV to meV, we are not aware of reports of the dependence of chemical potential on mag-netization orientation. Calculations of this quantity using realistic models or ab-initio are necessary.Here we use a very simple model to explore this problem. We describe the electrodes as a one dimen-sional electron gas with parabolic dispersion and spin splitting J modified by a Rashba spin orbit term:Hel =p22mδσ,σ′ −J2~σ · ~Ω+λpσz . Here m is the effective mass (in units of the free electron mass) and λ isthe strength of the spin-orbit interaction. We take h¯ = 1. and λ has dimensions of velocity. A dimension-less ratio of Stoner and spin orbit interactions is given by (λh¯kf )/J , where kf is the Fermi wave-vector.In magnetic semiconductors like GaMnAs the exchange splitting is comparable or even smaller than theSO coupling. In metals like Nickel the exchange splitting is more than 10 times larger than the SO orbitinteraction.The eigenstates |kν〉 and eigenvalues Ekν of Hel can be obtained analytically. In figure 1 we plot Ekνfor J = 1, m = λ = 0.5. For this rather large value of the spin orbit interaction, the bands changesignificantly as the ~Ω rotates. In this sense the situation is similar to the case of GaMnAs [19] Upon simplequadratures we can obtain µ(~Ω) for fixed values of J ,m and λ as a function of the electron density. Infig. 1d we see that the µ(n) curve is different for different values of θ. The jump in µ(n) is related tothe occupation of the upper band, which occurs at a value of the density that depends on ~Ω. Assuminga constant electronic density we can invert the curve µ(n) and obtain δµ (fig. 1e). We see that δµ canbe either positive or negative, depending on n, and its absolute value ranges 0 and 0.15 J , which wouldlargerly exceed the change in ν when the magnetic field is applied parallel to ~Ω [16]. The change in sizeand magnitude of δµ as a function of n are related to the change of the ratios between exchange energy,band energy and spin orbit coupling. In fig. 1f and 1g we show µ(θ) for n = 0.5 and n = 0.3 respectively.The horizontal lines in 1g stand for values of µ at which the number of electrons in the NMI changes,assuming EC = 0.05J , we see how the dot would gain up to 2 extra electrons as the θ is varied fromπ/2 (the easy axis in this case) to 0 or π. The vertical lines in 1g stand for the values of θ at which aconductance peak would be observed.We now discuss briefly how the tunneling rates are affected by the rotation of the electrode magnetiza-tion. The scattering rate for the n level of the QD, with spin σ along the ~Ω axis reads:Γηnσ(~Ω) =1h¯∑k,ν=±|〈kη|V |n〉|2|Ck,ν,σ(~Ω)|2δ(ǫn − Eην,k(~Ω))(3)where |kν〉 =∑σ Ck,ν,σ(~Ω)|σ〉|k〉. It is apparent that Γnσ depends on ~Ω both through the density of statesand throug the mixing coefficients Ck,ν,σ . The tunability of Γ can bring and additional knob to study theSET in the Kondo regime [15].In summary, we have provided a simple conceptual framework to understand the different effectsthat occur in a non-magnetic single electron island coupled to ferromagnetic electrodes with magneticc© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim4 J. Ferna´ndez-Rossier: TAMR in Coulomb Blockade-2 -1 0 1 201234E(k)0 0.4 0.8 µ0.20.40.60.8nθ =0θ=45θ =90-2 -1 0 1 201234E(k)-2 -1 0 1 201234E(k)0 0.5 1n-0.0500.050.10.15δµ (a)(b)(c)(d)(e)0 1 2 3 θ0.6550.660.665 µ(θ)0 1 2 3 θ0.20.250.3 µ(θ) N+1N(f)(g)N+2Fig. 1 . 1a-1c: Bands for three different magnetization orientations, θ = 0 (a), θ = pi4(b) and θ = pi2(c). 1d: densityvs chemical potential for the 3 same values of θ. 1e: difference in chemical potential for µ(0) − µ(pi/2).1f,1g µ(θ)for n = 0.5 (f) and n = 0.3 (g). In the latter we also show the charge in the dot for different values of µ.anisotropy. The sensitivity of a single-electron transistor device to the chemical potential of the electrodesresults in new physical effects when these are ferromagnetic. Both the single particle lifetimes Γ and thecharge vs gate curve depend on the orientation of the magnetic moment ~Ω with respect to the easy axis.This effect can be used to probe the chemical potential of ferromagnetic electrodes and could have practicalapplications. Further theoretical work is necessary to provide realistic description of the electrodes as wellas to consider the case of ferromagnetic island, either metallic [20] or semiconducting [21].Acknowledgements We acknowledge fruitful discussions with J. J. Palacios, D. Jacob and C. Tejedor. JFR acknowl-edges funding from Generalitat Valenciana (GV05-152), Spanish MEC (Grants FIS200402356) and Programa Ramo´ny Cajal. RA and LB acknowledge funding from the Spanish MEC (grant MAT2005-07369-C03-03).References[1] S. Maekawa and T. Shinjo (Eds.), Spin Dependent Transport in Magnetic Nanostructures (Taylor and Francis,New York, 2002)[2] M. Julliere Physics Letters A 54,225 (1975)[3] M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988)[4] N. Garcia, M. Munoz, and Y. W. Zhao, Phys. Rev. Lett. 82, 2923 (1999)[5] L. Brey, C. Tejedor, J. Ferna´ndez-Rossier, Appl. Phys. Lett. 1996 (2004)[6] C. Gould et al., Phys. Rev. Lett. 93 117203 (2004). C. Ruster et al., Phys. Rev. Lett. 94 027203 (2005)[7] A. D. Giddings, et al., Phys. Rev. Lett. 94 127202 (2005).[8] H. Saito, S. Yuasa and K. Ando , Phys. Rev. Lett. 95 086604 (2005).[9] J. Velev et al., Phys. Rev. Lett. 94, 127203 (2005)[10] K. Bolotin, , F. Kuemmeth and D. C. Ralph, Phys. Rev. Lett. 97, 127202 (2006)[11] M. Viret et al., Eur. Phys. J. B 51, 1 (2006)[12] D. Jacob, J. Ferna´ndez-Rossier, J. J. Palacios, Phys. Rev. B.71 220403(2005)[13] J. Wunderlich et al., Phys. Rev. Lett. 97, 077201 (2006)[14] M. Deshmukh and D. Ralph, Phys.Rev. Lett. 89, 266803 (2002)[15] A. N. Pasupathy et al., Science 306, 86 (2004)[16] K. Ono, H. Shimada, and Y. Ootuka, J. Phys. Soc. Jpn.66, 1261 (1997).[17] J. Ferna´ndez-Rossier and R. Aguado, cond-mat/0604437[18] H. van Houten, C. Beenakker, A. A. M. Staring, Single Charge Tunneling, edited by H. Grabert and M. H.Devoret, NATO ASI Series B294, Plenum, New York, 1992.[19] M. Abolfath et al., Phys. Rev. B63, 054418 (2001). T. Dietl, et al., Phys. Rev. B 63, 195205 (2001)[20] T. O. Strandberg, C. M. Canali and A. H. MacDonald, Phys. Rev. B73, 144415 (2006).[21] J. Ferna´ndez-Rossier and L. Brey, Phys. Rev. Lett. 93 117201 (2004)c© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Anisotropic magnetoresistance in single electron transport

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Anisotropic magnetoresistance in single electron transport

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