Abstract
We demonstrate that the giant magnetoresistance can be switched off (on) in even (odd) width zigzag graphenelike nanoribbons by an atomistic gate potential or edge disorder inside the domain wall in the antiparallel (ap) magnetic configuration. A strong magnetothermopower effect is also predicted that the spin thermopower can be greatly enhanced in the ap configuration while the charge thermopower remains low. The results extracted from the tightbinding model agree well with those obtained by firstprinciples simulations for edge doped graphene nanoribbons. Analytical expressions in the simplest case are obtained to facilitate qualitative analyses in general contexts.
Introduction
The giant magnetoresistance (GMR) effect, which is discovered in sandwiched structures of magnetic and nonmagnetic materials in 1988^{1,2}, means that a large conductance difference turns up when the relative spin orientation in adjacent magnetic materials change from parallel (p) to antiparallel (ap). GMR is essential in some spintronic devices that manipulate electron spin rather than charge^{3} and has led to the explosive enlargement of storage capability on hard disk in the last decades, reflecting its important value in commercial applications.
Charge and spin thermopower of a material describes the ability to produce charge and spin current, respectively, from a temperature gradient rather than a voltage one. Charge thermopower devices such as thermocouple have been widely used in our daily life. Spin thermopower has been observed in magnetic^{4} and nonmagnetic^{5} materials and should have application potential in spintronics. Similar to the GMR effect, the thermopower in magnetic junctions can change when the magnetic configuration changes from p to ap and lead to the magnetothermopower phenomena^{6}. The GMR and thermopower effects are both intrinsically relevant to the transmission spectrum in tunneling junctions. Recently, magnetoresistance^{7,8,9,10,11,12,13} and spin thermopower^{13,14,15,16} in graphenelike nanoribbons have become one of the focuses.
The discovery of graphene and its many outstanding properties^{17} has inspired enormous attentionon on graphenelike twodimensional (2D) materials. The long spin relaxation time and length in graphene indicates it a prospective material for spintronics^{18}. Specially, experimental observations have confirmed the existence of magnetism on zigzag edges of graphene^{19,20}. Both theoretical models^{20,21,22} and computational simulations based on ab initio density functional theory^{23,24} have shown that the magnetism origins from the spin polarized edge atoms. Zigzag graphene nanoribbons (ZGNRs) have two zigzag edges and can be in ferromagnetic (FM) or antiferromagnetic (AFM) state classified by the relative spin orientations on the two edges^{20,24}. External magnetic field can convert a ZGNR from its ground insulator AFM state to its metallic FM state. The consequent large colossal magnetoresistance was observed persistent up to room temperature^{7}. In addition, GMR in FM ZGNRs of even width has been predicted very high^{10} as well as the colossal magnetoresistance^{8}. ZGNR based GMR devices are believed very useful based on the fact that ZGNRs can be synthesized and relevant devices fabricated in atomic precision with the stateoftheart technology^{19,20}. Traditional chemical and physical edge functionalizations in atomistic scale such as atom adsorption, doping, vacancy and local lattice distortion have been proposed to manipulate further the properties and design diversified devices^{25,26}. Specifically, the edge disorders or defects can break geometry symmetry of ZGNRs and enhance the spin thermopower effect. In addition, scanning tunneling microscopy (STM) tip and atomic force microscopic techniques can be used as atomistic gate to apply local electrostatic potential for graphene transistors and manually introduce and control the disorders^{27,28}. In this work, based on the tightbinding model and the firstprinciples simulations, we propose a mechanism to switch on and off GMR and spin thermopower by applying an atomistic extrinsic potential on edge of FM zigzag nanoribbons (ZNRs) of twodimensional (2D) graphenelike honeycomb lattice. Our prediction may be confirmed via an experimental setup schemed in Fig. 1 by combining the techniques for measuring the currentvoltage curve of narrow nanoribbons^{7,29,30} with the STM techniques for controlling the potential in atomistic scale^{27,28}. In addition, the effects discussed in the following for single local potential or impurity might be enhanced in disordered systems^{31}.
Models and Methods
As schemed in Fig. 1(a), we partition a FM ZNR into twoprobe devices with left (L) and right (R) electrodes and a central (C) device region (domain wall in the ap configuration). A simplified tightbinding Hamiltonian can be used to describe the system^{9}
here (c_{i,σ}) is the creation (annihilation) operator of spin σ (↑ or ↓) on site i. t is the nearestneighbor hopping integral and is chosen as the unit of energy throughout the paper. The uniform onsite energy of the corresponding pristine 2D materials gives the Dirac point energy and is set to zero. U_{i} and λ_{σ}M_{i} (λ_{↑} = −1 and λ_{↓} = +1) are the onsite extrinsic potential energy from gate or disorder and the local magnetization, respectively. The magnetizations on the two edges are in parallel and linearly decay to zero from the edges to the center along the y direction. Along the x direction, λ_{σ}M_{i} is constant λ_{σ}M (full magnetization) in the electrodes and varies linearly in region C as depicted in Fig. 1(b,c) for the ap and p magnetic configurations of electrodes, respectively.
In the LandauerButtiker formalism with noncoherent effects neglected, the current of spin σ read where is the FermiDirac distribution function in electrode L (R). T and μ_{σ} are the electron temperature and the Fermi energy. τ_{σ}(ε) = Tr[Γ_{L}(ε)G^{r}(ε)Γ_{R}(ε)G^{a}(ε)]_{σ} is the electron transmission calculated by nonequilibrium Green’s function (NEGF) method^{32}. Here G^{r}(ε) = [G^{a}(ε)]^{+} = [ε − h_{C} − ∑_{L} − ∑_{R}]^{−1} is the retarded Green’s function corresponding to the Hamiltonian h_{C} in region C and is the broadening function. The selfenergy function ∑_{L(R)} due to the coupling between the device and electrode L (R) is obtained via the iterative procedure.
In the linear response regime of small voltage bias ΔV_{σ} and temperature difference ΔT between the electrodes, we express in Taylor expansion^{33} and have I_{σ} = G_{0}K_{0σ}(μ_{σ}, T)ΔV_{σ} + G_{0}K_{1σ}(μ_{σ}, T)ΔT/(eT) with K_{νσ} = ∫dε(ε − μ_{σ})^{ν}τ_{σ}(ε)(∂f_{σ}/∂ε) for ν = 0, 1 and the conductance quantum G_{0} = e^{2}/h. The tunneling magnetoresistance of the device is then evaluated^{34} from the total linear conductance in the p (ap) configuration. The charge and spin Seebeck coefficients are defined by S_{C(S)} = (S_{↑} ± S_{↓})/2 with when I_{σ} = 0^{35,36,37,38}. At low temperature the Mott formula^{35,36} applies and analytical results can be obtained in simple cases.
More realistic Hamiltonians for ZNRs of specific materials can be obtained from the density functional theory (DFT) based on firstprinciples. As an example, we have carried out simulations on magnetoresistance and thermopower for doped nZGNRs with width n employing the Atomistic Toolkits (ATK) package^{39,40}. The doubleζ plus polarization (DZP) basis set and an energy cutoff of 150 Ry within the local spin density approximation (LSDA) are used in the DFT simulation and a force tolerance of 0.03 eV/Å is set on each atom during the geometry relaxation. We find good agreement between results from the tightbinding model and from the firstprinciples simulations.
Results and Discussion
We consider ZNR systems of m = 5 in the linear response regime with the Fermi energy at the neutral point, i.e. μ_{σ} = 0 throughout the manuscript. The effect of one single impurity on the Fermi energy is assumed negligible. For systems with Fermi energy away from the neutral point, the conductance of the system can be estimated from the energy dependence of transmission. In the p configuration, perfect ZNRs are uniformly magnetized and translationally symmetric along the x direction. The extended edge states have flat bands at the full magnetization energy λ_{σ}M near the boundary of the first Brillouin zone. There is one transport channel for each spin σ at ε = 0^{9} so for both even and odd n at zero temperature. In the presence of a local extrinsic potential U_{i} at any edge site i, a bound state of energy appears around the site with . The bound state can slightly reduce if is close to zero^{41} as illustrated by the transmission spectra in the left insets of Fig. 1(f,g) where U_{i} > 0. The charge and spin thermopower are both very limited according to the Mott formula.
In the ap configuration the conductance becomes much more sensitive to U_{i} and n. We define the site i = 0 as the edge site where the magnetization reverses direction in the domain wall. For a geometrically leftright symmetric junction as schemed in Fig. 1(a), the residual magnetization vanishes at site 0, i.e. M_{0} = 0. In the absence of extrinsic potential U_{i}, we have for even n and for odd n at zero temperature due to geometry symmetry as indicated in the right insets of Fig. 1(f,g). The corresponding magnetoresistance R_{M} is about 100% for even n and very small for odd n^{10,11,12,13,17}. In the presence of U_{i}, we will present the results for U_{i} > 0 in two typical cases, i = 0 and i > 0 in the following since those for U_{i} < 0 or i < 0 can be then deduced based on the symmetry of the systems.
At first we apply a local potential U_{0} only at edge site 0 of an nZNR. For even n, as shown in the right inset of Fig. 1(f), a transmission peak emerges at ε = 0 with approaching to unit as U_{0} increases. The corresponding R_{M}, as plotted in Fig. 1(f), then decreases from 100% to near zero and can become negative if is reduced to by the bound state. In contrast, for odd n as illustrated in Fig. 1(g), decreases inversely with U_{0} from unit and approaches to zero, which leads to a jump of R_{M} from zero to near 100%. When the dip bottom of due to the bound state passes through the Fermi energy near U_{0} = 0.5 t, a R_{M} minimum appears for both even and odd n.
The conductance and magnetoresistance are also relevant to the full magnetization M which can vary for different materials and/or substrates. In Fig. 2(a,b), we plot versus M under U_{0} = t for various even and odd n, respectively. In nonmagnetic ZNRs (M = 0), or R_{M} = 0. As M increases, decreases gradually from 2G_{0} but remains high for even n. In contrast, for odd n, is very sensitive to M and jumps down to zero once M becomes finite, similar to the behavior of perfect evenwidth ZNRs^{10} as shown in the inset of Fig. 2(a). It then increases monotonically with M but remains in low values. In the range of M ∈ [0, 0.2]t, in 2ZNRs while in 3ZNRs and the difference between of even and oddwidth ZNRs enlarges as the width increases. In the insets of Fig. 2(a,b), we plot also the variation of versus M under U_{0} of different strengths to show its dependence. At any finite M, increases (decreases) from zero (2G_{0}) and saturates to 2G_{0} (zero) for even (odd) n as U_{0} increases (see also the insets of Fig. 1(f,g) and Figure S1).
The corresponding magnetoresistances R_{M} versus M are shown in Fig. 2(c,d) for even and odd n, respectively. Under U_{0} = t, R_{M} is well below 10% for all even n and above 60% for all odd n in the range M ∈ [0, 0.2]t. When the energy of the bound state confined by U_{0} in the p configuration is close to the Fermi energy, R_{M} can be negative for even n and shows an extra dip for odd n.
Using the density functional theory combined with the nonequilibrium Green’s function implemented in the ATK package^{39,40}, we have simulated the transmission spectrum of hydrogenpassivated 4ZGNRs and 5ZGNRs substitutionally doped by a boron atom at edge site 0. The result agrees well with that from the tightbinding model of parameters M = 0.07 t, U_{0} = 1.15 t for 4ZGNR and M = 0.08 t, U_{0} = 1.20 t for 5ZGNR with t = 2.7 eV as shown in Figure S2. The magnetoresistance switch effect can be quite robust in real materials like ZGNRs. In the insets of Fig. 2(c,d), we plot R_{M} versus the temperature T for 4ZNRs and 5ZNRs with parameters close to those of the above boron doped ZGNRs, i.e. U_{0} = t and M = 0.1 t. The behaviors of perfect/doped and even/odd ZNRs remains well distinguished from each other up to the room temperature T = 0.01 t/k_{B}.
In the simplest case of 2ZNR structure with m = 0, we have the analytical transmission expressions and with and . They may be helpful for qualitative analysis in more general contexts. For M = 0.1 t, the zero temperature magnetoresistance decreases from R_{M} = 100% at U_{i} = 0 to R_{M} = −22.1% at U_{0} = t similar to the result presented in Fig. 2(c) with m = 5 (see Figure S3 for the details).
If the potential shifts from site 0 to a site i > 0 where a downward residual magnetization M_{i} exists, the twofold rotation symmetry of the system is broken and the transmission becomes spin dependent. The spin thermopower in the ap configuration can be greatly enhanced and shows strong evenodd effects on the nanoribbon width.
In evenwidth nanoribbons the spinup (down) transmission peak shifts accordingly from to as shown in Fig. 3(a) for U_{2} = t, n = 4, i = 2, and M_{2} = iM/(m + 1) = 0.033 t with M = 0.1 t and m = 5. The transmission spectrum then has a positive (negative ) which results in a negative S_{↑} (positive S_{↓}) at low temperature according to the Mott formula. Interestingly, the spin Seebeck coefficient appears usually much greater than the charge one, i.e. S_{S} ≫ S_{C}, because the transmission peaks of up and down spins are located almost symmetrically beside the Fermi energy, i.e. and τ_{↑}(0) ≈ τ_{↓}(0). S_{S} can be further enhanced in wider nanoribbons where the transmission peaks become sharper with bigger τ′(0) and smaller τ(0).
The temperature dependent S_{S} and S_{C} plotted in Fig. 3(b,c), respectively, indicate that the Mott formula is valid below the critical temperature T_{c} = τ(1−τ )/τ′k_{B}_{ε=0} which is determined by the width Δ_{σ} and the position of the transmission peak. At high temperature the Seebeck coefficients decay gradually with the temperature due to the nonlinearity of the transmission spectrum.
The Seebeck coefficients can also be manipulated by the full magnetization M since both and Δ_{σ} increase with M. As illustrated in Fig. 3(d,e) for i = 2, the variation of τ(0) is limited due to the opposite effects from increase of and Δ_{σ} but τ′(0) decreases quickly and results in the decrease of S_{σ}. On the other hand, if the potential shifts from site 2 to site 3 while M remains fixed, increases since M_{3} > M_{2}, S_{S} then increases with the decrease of τ(0).
In oddwidth nanoribbons, the spinup (down) transmission dip shifts from to when the potential moves from site 0 to a site i > 0 as shown in Fig. 3(f) for n = 5. The dip width Δ_{σ} increases with but decreases with n. We have and then a positive S_{S} rather than the negative one in evenwidth nanoribbons. S_{S} decreases with i since the corresponding increase of enhances τ(0). S_{S} decreases also with M because the increase of both Δ_{σ} and reduces τ′(0) as shown in Fig. 3(j). If n increases, S_{S} becomes weaker further as illustrated in Fig. 3(g) when the decrease of Δ_{σ} leads to competing larger τ′(0) and larger τ(0). Different from the cases in evenwidth ZNRs, as shown in Fig. 3(i,j), and are more sensitive to n in oddwidth ZNRs.
Summary
Zigzag nanoribbons of graphenelike materials are expected very useful for spintronics due to their edge spin polarization or magnetism. Giant magnetoresistance exists in pristine evenwidth ferromagnetic nanoribbons. With the help of an atomistic gate potential or edge disorder, the giant magnetoresistance can be switched off (on) in even (odd) width nanoribbons. This originates from the jump of electronic transmission at the Fermi energy from zero to near 100% or verse vise in the antiparallel magnetic configuration if the potential is located at the transition interface of magnetization. If the potential shifts from the interface, the transmission peaks or dips of opposite spins split symmetrically beside the Fermi energy. The spin thermopower then becomes very large according to the Mott formula showing strong magnetothermopower effect. This suggests that spin current can be produced from temperature gradient in the material.
Additional Information
How to cite this article: Zhai, M.X. and Wang, X.F. Atomistic switch of giant magnetoresistance and spin thermopower in graphenelike nanoribbons. Sci. Rep. 6, 36762; doi: 10.1038/srep36762 (2016).
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Acknowledgements
We appreciate Hong Guo and YuShen Liu for helpful discussion. This work was supported by National Natural Science Foundation of China (Grant Nos 61674110 and 91121021).
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Affiliations
Jiangsu Key Laboratory of Thin Films, College of Physics, Optoelectronics and Energy, Soochow University, 1 Shizi Street, Suzhou 215006, China
 MingXing Zhai
 & XueFeng Wang
Hongzhiwei Technology Co. Ltd., 1888 Xinjinqiao Road, Pudong, Shanghai 201206, China
 MingXing Zhai
Key Laboratory of Terahertz SolidState Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, China
 XueFeng Wang
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M.X.Z. and X.F.W. designed research, performed research, analyzed data, and wrote the paper.
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The authors declare no competing financial interests.
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Correspondence to XueFeng Wang.
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