Spin-Orbit Qubit on a Multiferroic Insulator in a Superconducting Resonator
SSpin-Orbit Qubit on a Multiferroic Insulator in a Superconducting Resonator
P. Zhang, Ze-Liang Xiang, and Franco Nori
1, 2 Center for Emergent Matter Science, RIKEN, Saitama 351-0198, Japan Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: November 14, 2018)We propose a spin-orbit qubit in a nanowire quantum dot on the surface of a multiferroic insula-tor with a cycloidal spiral magnetic order. The spiral exchange field from the multiferroic insulatorcauses an inhomogeneous Zeeman-like interaction on the electron spin in the quantum dot, produc-ing a spin-orbit qubit. The absence of an external magnetic field benefits the integration of suchspin-orbit qubit into high-quality superconducting resonators. By exploiting the Rashba spin-orbitcoupling in the quantum dot via a gate voltage, one can obtain an effective spin-photon couplingwith an efficient on/off switching. This makes the proposed device controllable and promising forhybrid quantum communications.
PACS numbers: 81.07.Ta, 71.70.Ej, 75.85.+t
I. INTRODUCTION
Spin-based qubits, owing to their long coherence timesand individual coherent manipulation, are promisingcandidates for building blocks of quantum informationprocessors.
A conventional spin qubit can be simplyrealized via Zeeman splitting of two Kramers-degeneratestates by a static magnetic field and controlled by anac magnetic field.
However, its application is limiteddue to the difficulty in generating and localizing an acmagnetic field at the nanoscale. Owing to the interplaybetween spin and orbital degrees of freedom, the spin-orbit qubit allows the possibility for manipulating spinsvia an easily-accessible ac electric field, i.e., by means ofthe electric-dipole spin resonance (EDSR).
Intuitively,the interplay between spin and orbit can arise from thespin-orbit coupling (SOC), e.g., the Rashba or Dressel-haus type, which couples the electron spin σ to the mo-mentum p . The SOC-mediated EDSR has been widelystudied in the literature. Instead of invoking SOC,an alternative way to achieve the interplay between spinand orbit is coupling the electron spin σ to the coordinate r . This spin-coordinate coupling can be accomplished by,e.g., an inhomogeneous Zeeman-like interaction ora fluctuating hyperfine interaction. Apart from coherent manipulation, scaling up the spin-orbit-qubit architecture also involves quantum informa-tion storing and transferring. Embedding the spin-orbitqubit into a cavity resonator to achieve spin-photon cou-pling seems particularly attractive, as the mobile pho-tons in the cavity can store and transfer quantum infor-mation with little loss of coherence. Indeed, in viewof their energy scales, the semiconductor-based spin-orbit qubit is compatible with the superconducting mi-crowave resonator. Moreover, integrating the spin-orbitqubit into the superconducting cavity promotes hybridquantum communications, e.g., in combination with su-perconducting qubits or charge qubits.
Several pro-posals for coupling spin-orbit qubits to superconductingcavities have been reported.
However, the spin-orbit qubit invoking SOC requires an external static mag- netic field, which is not naturally compatiblewith superconducting cavities of high quality factors. Therefore, a spin-orbit qubit without an external mag-netic field is preferred for constructing a hybrid system.It has been proposed that, by using an inhomogeneousZeeman-like interaction induced by ferromagnetic con-tacts or micromagnets, one can realize spin-orbitqubits in the absence of a magnetic field and effectivelycouple them to superconducting cavities.
In this work, we propose a spin-orbit qubit mediatedby the spin-coordinate coupling and study its couplingto a superconducting coplanar waveguide resonator. Dif-ferent from previous studies, our proposal relieson the inhomogeneous exchange field arising from themultiferroic insulators with a cycloidal spiral magneticorder.
These multiferroic insulators provide a uniqueopportunity for the design of functional devices owing tothe cycloidal spiral magnetic order as well as the mag-netoelectric coupling.
In our setup for the spin-orbitqubit, as illustrated in Fig. 1(a), a gated nanowire witha quantum dot is placed on top of a multiferroic insu-lator. The spiral exchange field arising from the mag-netic moments in the multiferroic insulator causes an in-homogeneous Zeeman-like interaction on the quantum-dot spin. Therefore, a spin-orbit qubit is produced inthe nanowire quantum dot. The absence of an externalmagnetic field facilitates the integration of the spin-orbitqubit into the superconducting coplanar waveguide, asillustrated in Fig. 1(b). In this hybrid circuit, both thelevel spacing of the spin-orbit qubit and the spin-photoncoupling depend on the ratio between the dot size and thewavelength of the spiral magnetic order in the substrate.When the Rashba SOC is introduced into the nanowire,the level spacing and spin-photon coupling can be ad-justed by tuning the Rashba SOC via a gate voltage onthe nanowire. With the modulation of the Rashba SOC,we can obtain an effective spin-photon coupling with anefficient on/off switching. This is promising for manipu-lating, storing and transferring quantum information inthe data bus provided by the circuit cavity.This paper is organized as follows. First, we establish a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec the spin-orbit qubit on the surface of a multiferroic insu-lator. After that, we integrate the spin-orbit qubit into asuperconducting coplanar waveguide and study its spin-photon coupling. We further study the modulation of theRashba SOC on the hybrid system. At last we discussthe experimental realizability of the proposed device. II. SPIN-ORBIT QUBIT ON A MULTIFERROICINSULATOR
Our study starts from the device schematically shownin Fig. 1(a). In this setup, a nanowire lies on thesurface of a multiferroic insulator, e.g., TbMnO orBiFeO , and is aligned parallel to the propaga-tion direction of the spiral magnetic moments in the sub-strate. The nanowire is gated by two electrodes produc-ing a quantum dot, which is assumed to be subject to a1D parabolic potential.We consider a single electron in the quantum dot. Inthe coordinate system with the x -axis along the nanowireand the z -axis perpendicular to the top surface of themultiferroic insulator, the electron is described by theHamiltonian, H = p m e + 12 m e ω x + J ( x ) · σ . (1)Here m e is the effective electron mass and p = − i (cid:126) ∂ x isthe momentum operator. The second term in the Hamil-tonian is the parabolic potential. The last term depictsthe interaction between the electron spin σ and the ex-change field from the cycloidal spiral magnetic moments, J ( x ) = J (sin( χqx + φ ) , , cos( χqx + φ )) . (2)In writing this term we have assumed the dot size x = (cid:112) (cid:126) / ( m e ω ) to be much larger than the distance( ∼ q = 2 π/λ is the wavevector of the spi-ral order corresponding to a wavelength λ and φ (0 ≤ φ < π ) is the phase of the exchange field at x = 0.The spiral helicity χ (= ±
1) of the magnetic order is re-versable by a gate voltage on the multiferroic insulator[as illustrated by V c in Fig. 1(a)] due to the magnetoelec-tric coupling. The strength of the exchange coupling J (we assume J >
0) between the electron spin and themagnetic moments, depending on their distance and thespecific hosts, is weak and assumed to be of the order of1-10 µ eV. Due to the spiral geometry of the magnetic order,the macroscopic magnetism of the multiferroic insula-tor is zero, while the exchange coupling still breaks thetime-reversal symmetry locally and causes an inhomoge-neous Zeeman-like interaction on the quantum-dot spin.In the presence to this inhomogeneous Zeeman-like in-teraction, a spin-orbit qubit is realizable in the quan-tum dot. One can also understand the availability ofa spin-orbit qubit in our setup in the spiral frame with
FIG. 1: (Color online) (a) Schematic of the proposed spin-orbit qubit: a gated nanowire on the surface of a multiferroicinsulator. The nanowire is aligned along the propagation di-rection of the spiral magnetic order in the multiferroic insula-tor, indicated by the series of rotating arrows. Two gate elec-trodes (indicated by the two dark blue ring-shaped contacts)supply a parabolic confining potential and form a quantumdot in between. The gate electrode on the top of the quan-tum dot with voltage V g controls the Rashba SOC. Two gateelectrodes on the top and bottom of the multiferroic insula-tor supply a voltage V c which controls the spiral helicity ofthe magnetic order. (b) Schematic of the integration of thespin-orbit qubit into a superconducting coplanar waveguideresonator. The nanowire is placed parallel to the electric fieldbetween the center conductor and the ground plane and lo-cated at the maximum of the electric field. Note that in themultiferroic-insulator substrate, only the magnetic momentsnear the nanowire are schematically shown by the rotatingarrows. the spin z -axis along the local magnetic moment. Us-ing a unitary transformation ˜ H = U † ( x ) HU ( x ), where U ( x ) = exp [ − i ( χqx + φ ) σ y / one arrives at˜ H = p m e + 12 m e ω x − α pσ y + Jσ z + (cid:126) q m e , (3)where α = χ (cid:126) q/ (2 m e ). This Hamiltonian evidentlyindicates that in the spiral frame, the exchange fieldsupplies not only the homogeneous Zeeman-like inter-action, Jσ z , but also an effective Rashba-like SOC, − α pσ y . This Hamiltonian is equivalent to the onestudied in Ref. 16, where a spin-orbit qubit was realizedby virtue of an external magnetic field and the genuineRashba/Dresselhaus SOC.Now we demonstrate the realization of a spin-orbitqubit by studying the low-energy bound states in thequantum dot. For the reasonable case with J/ ( (cid:126) ω ) (cid:46) . H = H + H , where H = J ( x ) · σ . The eigenstates of H , describing a har-monic oscillator, can be written as | n ±(cid:105) = | n (cid:105)|±(cid:105) withthe eigenenergies ε n = (cid:0) n + (cid:1) (cid:126) ω ( n = 0 , , ... ). Here | n (cid:105) is the orbital eigenstate of the harmonic oscillator and | + (cid:105) ( |−(cid:105) ) is the spin-up (-down) eigenstate of σ z . Wefocus on the n = 0 Hilbert subspace which is two-folddegenerate. First-order degenerate perturbation theorygives the lowest two bound states of H with energies ε ± = ε ± (cid:126) ∆ /
2, where∆ = ∆ exp ( − η ) , (4)with ∆ = 2 J/ (cid:126) and η = χπx /λ . The correspondingwavefunctions are | (cid:102) ±(cid:105) = e − iφσ y / (cid:110) | ±(cid:105) − Je − η (cid:126) ω + ∞ (cid:88) m =1 (cid:104) ± ( i √ η ) m m (cid:112) (2 m )! | m ±(cid:105)− i ( i √ η ) m − (2 m − (cid:112) (2 m − | m − ∓(cid:105) (cid:105)(cid:111) . (5)The two lowest bound states | (cid:102) ±(cid:105) , spaced by (cid:126) ∆ andabout (cid:126) ω away from the nearest higher-energy state,can be used to encode the spin-orbit qubit. As a re-sult, with the aid of the spiral exchange field supplied bya multiferroic insulator, we realize a spin-orbit qubit inthe absence of an external magnetic field as well as theRashba/Dresselhaus SOC. III. SPIN-PHOTON COUPLING IN ASUPERCONDUCTING CAVITY
The spin-orbit qubit can respond to an ac electric field,via EDSR.
Due to the small level spacing, the spin-orbit qubit is controllable by low-temperature microwavetechnology. This can be accomplished by virtue of a su-perconducting resonator, which works at temperatures ∼ mK with resonance frequencies ∼ GHz. Indeed, inte-grating spin-orbit qubits into superconducting resonatorshas recently attracted much interest, to explorenovel hybrid quantum circuits. Moreover, the spin-orbitqubit proposed here, which is external-magnetic-field-free, is naturally compatible with superconducting res-onators of high quality factors.As schematically shown in Fig. 1(b), we embedthe spin-orbit qubit into a superconducting coplanarwaveguide, with the nanowire aligned parallel tothe electric field between the center conductor and theground plane. The resonant photon energy ( ∼ GHz) istoo low to excite magnons in the multiferroic-insulatorsubstrate, and we assume that the spiral magnetic or-der keeps steady during the operation of the spin-orbitqubit. The spin-orbit qubit, photons, as well as their coupling, can be described by the Hamiltonian, H eff = (cid:126) ∆2 s z + (cid:126) ω r (cid:18) a † a + 12 (cid:19) + (cid:126) g ( a † s − + as + ) . (6)Here a ( a † ) is the annihilation (creation) operator forphotons with frequency ω r in the cavity, and s x,y,z arethe Pauli matrices in the | (cid:102) ±(cid:105) subspace with s ± = ( s x ± is y ) /
2. The spin-photon coupling strength g = (cid:104) (cid:102) | x | (cid:102) −(cid:105) Ee/ (cid:126) , (7)where E is the cavity electric field on the spin-orbit qubit.Up to first order in J/ ( (cid:126) ω ), g = g ( x /λ ) η exp ( − η ) , (8)where g = − eEm e Jλ / (cid:126) .Note that in this device, both the level spacing ∆ andthe spin-photon coupling g are independent of the phase φ and proportional to the exchange coupling strength J .Also, both ∆ and g strongly depend on the ratio betweenthe dot size x and the wavelength λ of the spiral mag-netic order. In Fig. 2, we plot the dependence of ∆ / ∆ and | g/g | on the parameter x /λ . One finds that when x /λ is close to 1, both ∆ and | g | approach zero, hin-dering the device operation. This is because when x /λ is large, the exchange field, oscillating with a high fre-quency in the scale of the dot size, has quite small ma-trix elements between the | n ±(cid:105) and | n (cid:48) ±(cid:105) states. Thisleads to a vanishing Zeeman-like splitting and spin-orbitmixing of the harmonic oscillator states. However, in the x /λ = 0 limit, ∆ reaches its maximum while | g | againapproaches zero. In fact, in this regime, with the approx-imately homogeneous exchange field experienced by thequantum-dot electron, the spin-orbit interplay becomesquite weak and a nearly pure spin qubit with the largestZeeman-like splitting is obtained. ∆ / ∆ | g / g | x / λ FIG. 2: (Color online) Dimensionless level spacing ∆ / ∆ ofthe spin-orbit qubit and dimensionless spin-photon coupling | g/g | versus the dimensionless dot size x /λ . IV. MODULATION BY THE RASHBA SOC
Although the spin-orbit qubit proposed here is avail-able without employing the Rashba/Dresselhaus SOC,in reality the SOC may be present and even important.Nonetheless, the Rashba SOC is controllable, e.g., by agate voltage applied to the nanowire [as illustrated by V g from the gate electrode on top of the nanowire inFig. 1(a)]. Below we introduce the Rashba SOC into thenanowire, supplying an effective channel to modulate thespin-orbit qubit as well as its coupling to photons.With the Rashba SOC included, the Hamiltonian givenby Eq. (1) becomes H α = p m e + 12 m e ω x + αpσ y + J ( x ) · σ , (9)where α is the Rashba SOC strength. We now applythe unitary transformation ˜ H α = U † α ( x ) H α U α ( x ) with U α ( x ) = exp ( − im e αxσ y / (cid:126) ), and obtain ˜ H α = p m e + 12 m e ω x + J α ( x ) · σ + m e α , (10)where J α ( x ) = J (sin( χq α x + φ ) , , cos( χq α x + φ )) with q α = (1 − α/α ) q . The Hamiltonian ˜ H α has exactlythe same form as in Eq. (1). Therefore, one can obtainthe low-energy eigenstates of ˜ H α immediately based onthe results given previously. By noting that the elec-tric dipole moment commutes with the unitary operator U α ( x ), one straightforwardly obtains the level spacing ofthe spin-orbit qubit and the spin-photon coupling in thepresence of the Rashba SOC,∆ α = ∆ exp ( − η α ) , (11) g α = g ( x /λ ) η α exp ( − η α ) , (12)with η α = (1 − α/α ) η .The above results can be understood by consideringthe Rashba SOC to superimpose on the effective Rashba-like SOC from the spiral geometry, or, in other words, toequivalently modulate the wavelength of the spiral mag-netic order. This feature allows to control both the levelspacing of the spin-orbit qubit and the spin-photon cou-pling by adjusting the Rashba SOC via the gate volt-age. To show the modulation of the Rashba SOC onthe hybrid system, in Figs. 3(a, b) we plot the dimen-sionless level spacing, ∆ α / ∆ , and dimensionless spin-photon coupling, | g α /g | , versus the parameters x /λ and α/α . Those calculations indicate that when x ∼ λ and α ∼ (1 ± . α , the spin-orbit qubit can be effectivelycoupled to photons, as indicated by the area near the“on” points in Fig. 3(b). Moreover, by tuning α to α ,the spin-photon coupling is completely switched off dueto the decoupling of the spin to the orbit, as indicatedby the area near the“off” point in Fig. 3(b). During thisswitch process, the level spacing of the spin-orbit qubitchanges by about 30%. These features are promising for (a) α x / λ
0 0.5 1 1.5 2 α / α ∆ α / ∆ (b) α x / λ
0 0.5 1 1.5 α / α -6 -4 -2 |g α /g |
0 0.2 0.4 0.6 0.8 1 on . . off . FIG. 3: (Color online) (a) Dimensionless level spacing ∆ α / ∆ of the spin-orbit qubit and (b) dimensionless spin-photon cou-pling | g α /g | (in log-scale) versus x /λ and α/α . manipulating, storing and transferring information in thehybrid quantum systems. Note that for a particular Rashba SOC, its modulationdepends on the spiral helicity in the substrate, as α de-pends on χ . This feature supplies another control chan-nel of the device via the gate voltage V c on the substrate,and also in turn provides the possibility to determinethe exchange coupling strength J as well as the RashbaSOC strength α . By measuring the level spacings of thespin-orbit qubit corresponding to opposite spiral helici-ties, which satisfy ln[∆ α ( χ = 1) / ∆ α ( χ = − qα/ω ,one can obtain the Rashba SOC strength α with theknowledge of the confining potential of the quantum dot.Here α is assumed to be marginally affected by the rever-sal of the spiral helicity. Further, the exchange couplingstrength J is available based on the known ∆ α and α . V. EXPERIMENTAL REALIZABILITY
Let us now discuss the experimental realizability of theproposed spin-orbit qubit and its coupling to the super-conducting coplanar waveguide. We consider a (cid:104) (cid:105) -oriented Ge nanowire on the surface of the mul-tiferroic insulator BiFeO . In the (cid:104) (cid:105) -orientedGe nanowire, the electron effective mass m e = 0 . m ,where m is the free electron mass. For BiFeO , thewavelength of the spiral magnetic order λ = 62 nm, while the magnon frequency is of the order of 100 GHz. The exchange coupling strength is set as J = 5 µ eV,smaller than the estimated interface exchange coupling(16 µ eV) induced by the ferromagnetic-insulator con-tacts in Ref. 23. The electric field in the supercon-ducting coplanar waveguide has the typical maximalstrength E = 0 . = (2 π )2 . | g | = ( π )0 . | α | =7 . × m/s. Moreover, even when x ∼ λ , the or-bital splitting in the quantum dot is (cid:126) ω ∼ J .In addition to the availability of an effective spin-photoncoupling with an efficient on/off switching, the proposeddevice has another advantage. That is, in isotopically-purified Ge samples, the hyperfine interaction can bemarkedly suppressed and hence the coherence time ofthe spin-orbit qubit in the zero-temperature limit can bequite long. This feature benefits the application of theproposed device.
VI. CONCLUSION
In conclusion, we have proposed a spin-orbit qubitbased on a nanowire quantum dot on the surface of amultiferroic insulator, and designed a hybrid quantumcircuit by integrating this spin-orbit qubit into a super- conducting coplanar waveguide.The spiral exchange field from the magnetic momentsin the multiferroic insulator causes an inhomogeneousZeeman-like interaction on the electron spin in the quan-tum dot. This effect assists the realization of a spin-orbitqubit in the quantum dot. In this approach, no externalmagnetic field is employed, benefitting the on-chip fabri-cation of the spin-orbit qubit in a superconducting copla-nar waveguide. Our study reveals that both the levelspacing of the spin-orbit qubit and the spin-photon cou-pling are proportional to the exchange coupling strengthand depend on the ratio of the dot size to the wave-length of the spiral magnetic order. We further considerthe effect of the Rashba SOC, which is controllable by agate voltage on the nanowire. It is found that by invok-ing the Rashba SOC, one is able to obtain an effectivespin-photon coupling with an efficient on/off switching,making the device promising for applications. The pro-posed spin-orbit qubit may be experimentally realizableby placing a (cid:104) (cid:105) -oriented Ge nanowire on the surfaceof the multiferroic insulator BiFeO . Acknowledgments
The authors gratefully acknowledge E. Ya. Shermanand X. Hu for valuable discussions and comments. F.N.is partially supported by the ARO, RIKEN iTHESProject, MURI Center for Dynamic Magneto-Optics,JSPS-RFBR contract No. 12-02-92100, Grant-in-Aid forScientific Research (S), MEXT Kakenhi on Quantum Cy-bernetics, and the JSPS via its FIRST program. D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998). R. Hanson, J. R. Petta, S. Tarucha, and L. M. K. Vander-sypen, Rev. Mod. Phys. , 1217 (2007). M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. ,61 (2010). I. Buluta, S. Ashhab, and F. Nori, Rep. Prog. Phys. ,104401 (2011). F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Sim-mons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N.Coppersmith and M. A. Eriksson, Rev. Mod. Phys. ,961 (2013). H. A. Engel and D. Loss, Phys. Rev. Lett. , 4648 (2001). F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M.K. Vandersypen, Nature , 766 (2006). D. D. Awschalom, L. C. Bassett, A. S. Dzurak, E. L. Hu,and J. R. Petta, Science , 1174 (2013). E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. , 126405(2003). E. I. Rashba, J. Supercond. , 137 (2005). V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B , 165319 (2006). K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Science , 1430 (2007). S. N. Perge, V. S. Pribiag, J. W. G. van den Berg, K. Zuo,S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L.P. Kouwenhoven, Phys. Rev. Lett. , 166801 (2012). D. V. Khomitsky, L. V. Gulyaev, and E. Ya. Sherman,Phys. Rev. B , 125312 (2012). X. Hu, Y. X. Liu, and F. Nori, Phys. Rev. B , 035314(2012). R. Li, J. Q. You, C. P. Sun, and F. Nori, Phys. Rev. Lett. , 086805 (2013). A. F. Sadreev and E. Ya. Sherman, Phys. Rev. B ,115302 (2013). C. Echeverr´ıa-Arrondo and E. Ya. Sherman, Phys. Rev. B , 155328 (2013). Y. Kato, R. C. Myers, D. C. Driscoll, A. C. Gossard, J.Levy, and D. D. Awschalom, Science , 1201 (2003). Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha,Phys. Rev. Lett. , 047202 (2006). M. Pioro-Ladri`ere, T. Obata, Y. Tokura, Y. S. Shin, T.Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nat.Phys. , 776 (2008). T. Obata, M. Pioro-Ladriere, Y. Tokura, Y. S. Shin, T.Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Phys.Rev. B , 085317 (2010). A. Cottet and T. Kontos, Phys. Rev. Lett. , 160502(2010). E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P.Hanson, and A. C. Gossard, Phys. Rev. Lett. , 246601(2007). M. Shafiei, K. C. Nowack, C. Reichl, W. Wegscheider,and L. M. K. Vandersypen, Phys. Rev. Lett. , 107601(2013). Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod.Phys. , 623 (2013). J. Q. You and F. Nori, Physics Today , 42 (2005); Na-ture , 589 (2011). N. Lambert, C. Flindt, F. Nori, Euro. Phys. Lett. ,17005 (2013). G. Burkard and A. Imamoglu, Phys. Rev. B , 041307(2006). P. Q. Jin, M. Marthaler, A. Shnirman, and G. Sch¨on, Phys.Rev. Lett. , 190506 (2012). D. Lebeugle, D. Colson, A. Forget, M. Viret, P. Bonville, J. F. Marucco, and S. Fusil, Phys. Rev. B , 024116 (2007). T. Kimura, Annu. Rev. Mater. Res. , 387 (2007). Y. Yamasaki, H. Sagayama, T. Goto, M. Matsuura, K.Hirota, T. Arima, and Y. Tokura, Phys. Rev. Lett. ,147204 (2007). D. Khomskii, Physics , 20 (2009). P. Rovillain, R. de Sousa, Y. Gallais, A. Sacuto, M. A.M´easson, D. Colson, A. Forget, M. Bibes, A. Barth´el´emy,and M. Cazayous, Nat. Mater. , 975 (2010). R. Thomas, J. F. Scott, D. N. Bose, and R. S. Katiyar, J.Phys.: Condens. Matter , 423201 (2010). P. Zhang and M. W. Wu, Phys. Rev. B , 014433 (2011). F. Zhai and P. Mu, Appl. Phys. Lett. , 022107 (2011). Y. Wu and P. Yang, J. Am. Chem. Soc. , 3165 (2001). A. B. Greytak, L. J. Lauhon, M. S. Gudiksen, and C. M.Lieber, Appl. Phys. Lett. , 4176 (2004). Y. M. Niquet and C. Delerue, J. Appl. Phys.112