⇦Chinese Physics Letters, 2019, Vol. 36, No. 1, Article code 017301 Spin Caloritronic Transport of Tree-Saw Graphene Nanoribbons * Yu-Zhuo LV (吕钰卓), Peng ZHAO (赵朋)** Affiliations School of Physics and Technology, University of Jinan, Jinan 250022 Received 13 August 2018, online 25 December 2018 *Supported by the Natural Science Foundation of Shandong Province under Grant No ZR2016AM11.
**Corresponding author. Email: ss_zhaop@ujn.edu.cn Citation Text: Lv Y Z and Zhao P 2019 Chin. Phys. Lett. 36 017301    Abstract Using density functional theory combined with non-equilibrium Green's function method, we investigate the spin caloritronic transport properties of tree-saw graphene nanoribbons. These systems have stable ferromagnetic ground states with a high Curie temperature that is far above room temperature and exhibit obvious spin-Seebeck effect. Moreover, thermal colossal magnetoresistance up to 10$^{20}$% can be achieved by the external magnetic field modulation. The underlying mechanism is analyzed by spin-resolved transmission spectra, current spectra and band structures. DOI:10.1088/0256-307X/36/1/017301 PACS:73.23.-b, 85.65.+h, 71.15.Mb © 2019 Chinese Physics Society Article Text Spin caloritronics, which focuses on the interplay of spin and heat transport in materials, has attracted great attention because it offers potential applications in thermoelectric waste heat recovery and information-processing technologies.[1-4] The concept of spin caloritronics can be traced back to the late-1980s with Johnson and Silsbee's investigations of non-equilibrium thermodynamics of spin, charge and heat in metallic heterostructures.[5,6] However, this field only grew after the experimental observation of spin-Seebeck effect (SSE) in 2008 by Uchida et al. in ferromagnetic metal NiFe alloy.[7] In SSE, the spin-up and spin-down currents flowing in opposite directions, as well as the associated spin voltage are induced by a temperature gradient, which lays the foundation for applying spin caloritronics in the energy-saving devices. Recent studies have shown that graphene nanoribbons (GNRs) have strong potential for spin caloritronics. For example, Zeng et al. explored the thermally induced spin transport properties of zigzag GNRs (ZGNRs), and observed SSE, thermal spin-filtering and thermal magnetoresistance effects.[8] Ni et al. found the SSE and thermal colossal magnetoresistance effects in a ZGNR heterojunction with a single-hydrogen-terminated ZGNR as the left electrode and a double-hydrogen-terminated ZGNR as the right electrode.[9] Huang et al. reported ZGNR co-doped with nitrogen and boron atoms exhibits excellent thermal spin transport properties.[10] Tang et al. designed a kind of armchair GNRs (AGNR) with special edge hydrogenation, and found SSE and spin-Seebeck diode effects.[11] Recently, Yu et al. proposed two kinds of novel GNRs with sawtooth edges: tree-saw GNRs (TGNRs) and Christmas-tree GNRs (CGNRs).[12] They found that both TGNRs and CGNRs exhibit ferromagnetic (FM) ground states. Especially, TGNRs have more stable FM ground states, which portends much higher Curie temperature and potential applications in the field of spin caloritronics. In the present work we use the density functional theory (DFT) in conjunction with non-equilibrium Green's function (NEGF) technique and explore the spin caloritronic transport properties of TGNRs. Our results show that obvious SSE can be generated in these systems. Moreover, with the help of external magnetic field modulation, a thermal colossal magnetoresistance of up to 10$^{20}$% can be achieved.
cpl-36-1-017301-fig1.png
Fig. 1. (a) Schematic of the TGNR-based spin caloritronic device, which is divided into three parts: left electrode (LE), central scattering region (CSR) and right electrode (RE). Two integers ($n_{1}$, $n_{2}$) indicate the size of TGNR primitive cell, $T_{\rm L}$ and $T_{\rm R}$ are the electron temperature of LE and RE, and $\Delta T=T_{\rm L}-T_{\rm R}$ is the temperature difference. (b) Spin density distribution across the CSR for the P and AP configurations, where the pink and cyan colors denote the spin-up and spin-down components, respectively.
We adopt two integers ($n_{1}$, $n_{2}$) to characterize the size of TGNR primitive cell, where $n_{1}$ is the number of continuous zigzag carbon chains, and $n_{2}$ is the number of hexagonal rings along the $n_{2}$ direction (see Fig. 1(a)), respectively. Figure 1(a) plots the TGNR-based spin caloritronic device with $n_{1}=n_{2}=3$, which is labeled as T(3,3) for simplicity. The integer $n_{2}=3$ for TGNRs corresponds to the lowest numbers meeting the requirement of the sawtooth edges.[12] Our test shows that the TGNRs with larger $n_{1}$ or $n_{2}$ present similar results, thus we focus on T(3,3). As shown in Fig. 1(a), the entire device consists of left electrode (LE), central scattering region (CSR) and right electrode (RE). The electrodes and CSR are represented by one and four T(3,3) primitive cells along the transport ($z$) direction, respectively. Our test shows that the longer CSR gives similar results. Large enough vacuum layers in the $x$ (25 Å) and $y$ (20 Å) directions are added to eliminate the interaction between adjacent ribbons. Our calculations are all performed by the Atomistix Toolkit (ATK) package,[13-16] which is based on the DFT+NEGF method. Structural relaxation is carried out for the TGNR primitive cell until the residual force on each atom is less than 0.01 eV/Å and the total energy is converged to 10$^{-5}$ eV. The spin generalized gradient approximation with Perdew–Burke–Ernzerhof form of functions (SGGA.PBE) is employed for the exchange-correlation potential.[17] The core electrons are described by Troullier–Martins norm-conserving pseudopotentials,[18] while the valence electron wave functions are expanded by double-zeta plus polarization (DZP) basis set. The real space grid technique is used with the energy cutoff of 200 Ry in numerical integration. The $k$-point grids 1, 1 and 100 are adopted in the $x$, $y$ and $z$ directions, respectively. The thermal spin currents flow through the device are calculated by the Landauer–Büttiker formula[19] $I_{\sigma} =(e/h)\int_{-\infty }^\infty {T_{\sigma } (E)[f_{\rm L} (E,T_{\rm L})-f_{\rm R} (E,T_{\rm R})]{\rm d}E}$, where $e$ is the electron charge, $h$ is Planck's constant, $T_{\sigma }(E)$ is the transmission function of carriers with energy $E$ and spin index $\sigma$ ($\sigma$=up and dn for spin-up and spin-down), and $T_{\rm L/R}$ and $f_{\rm L/R}$ are the temperature and Fermi-Dirac distribution function of LE/RE, respectively. To study the magnetic ground state, three different magnetic states, FM state, antiferromagnetic (AFM) state, and non-magnetic (NM) state are considered. We use the labels $E_{\rm FM}$, $E_{\rm AFM}$ and $E_{NM}$ to denote the total energies per primitive cell for the FM, AFM and NM states of T(3,3), respectively. The calculated energy differences $E_{\rm AFM}-E_{\rm FM}=130.74$ meV and $E_{NM}-E_{\rm FM}=262.13$ meV, indicating that the FM state is the ground state. Using the mean field theory, the Curie temperature ($T_{\rm C}$) is estimated,[20] namely, $E_{\rm AFM}-E_{\rm FM}=\gamma k_{\rm B}T_{\rm C}$/2, where $\gamma$ and $k_{\rm B}$ represent the dimension of the system ($\gamma=1$ for one-dimensional TGNR system) and the Boltzmann constant, respectively. We then obtain a $T_{\rm C}$ up to 3029 K, which is far above room temperature and indicates that TGNRs are potential materials for spin caloritronic devices. Because the magnetization of electrodes can be controlled by applying external magnetic fields, two distinct magnetic configurations are considered: parallel (P) and anti-parallel (AP) configurations.[21-23] To be specific, for the P configuration, no external magnetic field is needed due to the high $T_{\rm C}$, while for the AP configuration, external magnetic fields at two electrodes pointing in opposite ($+y$, $-y$) directions are needed. Figure 1(b) plots the spin density ($\Delta \rho =\rho_{\rm up}-\rho_{\rm dn}$) distribution across the CSR for the P and AP configurations, where the pink and cyan colors denote the spin-up and spin-down components, respectively. Evidently, the spin density distribution verifies two magnetic configurations setups: the whole CSR are spin-up polarized under the P configuration, while the left and right regions of CSR are spin-up and spin-down polarized under the AP configuration, respectively.
cpl-36-1-017301-fig2.png
Fig. 2. Calculated thermal spin currents versus $\Delta T$ with $T_{\rm L}=300$, 400 and 500 K.
We first consider the P configuration. Figure 2 plots the calculated thermal spin currents versus temperature difference ($\Delta T$) between two electrodes, i.e., $\Delta T=T_{\rm L}-T_{\rm R}$, with $T_{\rm L}$ varying from 300 to 500 K. As one can see, the spin-up ($I_{\rm up}$) and spin-down ($I_{\rm dn}$) currents are positive and negative, respectively, indicating that they flow in opposite directions. This is an obvious SSE because both $I_{\rm up}$ and $I_{\rm dn}$ are induced by temperature gradient rather than by bias gradient. Moreover, both $I_{\rm up}$ and $I_{\rm dn}$ are very weak when $T_{\rm L}=300$ K, and both of them increase with $T_{\rm L}$ or $\Delta T$.
cpl-36-1-017301-fig3.png
Fig. 3. (a) Difference in Fermi–Dirac distribution function of two electrodes, ($f_{\rm L}-f_{\rm R}$), with $T_{\rm L}=400$ K and $T_{\rm R}=340$ K. (b) Spin-resolved transmission spectra. (c) Spin-resolved current spectra for fixed $\Delta T=60$ K with $T_{\rm L}=300$, 400 and 500 K. (d) Spin-resolved current spectra for fixed $T_{\rm L}=400$ K with $\Delta T_{\rm L}=20$, 40 and 60 K.
According to the Landauer–Büttiker formula, the thermal spin currents are the joint results of $T_{\sigma}$ and $f_{\rm L}-f_{\rm R}$. To clarify the underlying mechanism of the observed SSE, we first consider $f_{\rm L}-f_{\rm R}$, which is dependent on $T_{\rm L}$ and $T_{\rm R}$ since two electrodes are composed of the same material. Figure 3(a) plots $f_{\rm L}-f_{\rm R}$ with $T_{\rm L}=400$ K and $T_{\rm R}=340$ K as a function of $E-E_{\rm F}$, where $E_{\rm F}$ is the Fermi level. As can be seen, $f_{\rm L}-f_{\rm R} < 0$ or $>0$ when $E < E_{\rm F}$ or $E>E_{\rm F}$, indicating that both the hole and electron carrier concentrations in LE are higher than those in RE. Consequently, both hole and electron carriers diffuse from the hot LE to the cold RE, leading to a positive hole current $I_{\rm h}$ and a negative electron current $I_{\rm e}$ in the opposite direction. Moreover, as shown in Fig. 3(a), $f_{\rm L}-f_{\rm R}$ is perfect symmetric with respect to $E_{\rm F}$, and it also exhibits a typical exponential decaying nature characteristics. Thus, only the transmissions close to $E_{\rm F}$ have contribution to the thermal current. However, $I_{\rm h}$ and $I_{\rm e}$ will cancel out each other if the transmissions around $E_{\rm F}$ are also symmetry. Figure 3(b) plots the spin-resolved transmission spectra. Clearly, both $T_{\rm up}$ and $T_{\rm dn}$ are asymmetric about $E_{\rm F}$. Specifically, a broad spin-up transmission band lies in the energy region $-$0.68 eV$\, < E-E_{\rm F} < -0.2$ eV, while two narrow spin-down transmission bands lie in the energy regions of 0.22 eV$\, < E-E_{\rm F} < 0.3$ eV and 0.36 eV$\, < E-E_{\rm F} < 0.48$ eV, respectively. Therefore, nonzero net thermal current can be obtained due to the break of electron-hole symmetry. To further understand the combined effects of $T_{\sigma}$ and $f_{\rm L}-f_{\rm R}$ on the thermal spin currents, as shown in Figs. 3(c) and 3(d), we plot the spin-resolved current spectra $J_{\sigma }=T_{\sigma}(f_{\rm L}-f_{\rm R}$) as a function of $E-E_{\rm F}$. Figure 3(c) corresponds to $J_{\sigma}$ for fixed $\Delta T=60$ K with $T_{\rm L}=300$, 400 and 500 K. Figure 3(d) corresponds to $J_{\sigma}$ for fixed $T_{\rm L}=400$ K with $\Delta T=20$, 40 and 60 K. Evidently, as shown in Figs. 3(c) and 3(d), the peaks of $J_{\rm up}$ and $J_{\rm dn}$ occur below and above $E_{\rm F}$, giving rise to positive $I_{\rm up}$ and negative $I_{\rm dn}$. Moreover, the magnitude of thermal spin currents is determined by the cover area of $J_{\sigma}$ according to the Landauer–Büttiker formula. As shown in Fig. 3(c), both the cover areas of $J_{\rm up}$ and $J_{\rm dn}$ are very small when $T_{\rm L}=300$ K, leading to the very weak $I_{\rm up}$ and $I_{\rm dn}$. As $T_{\rm L}$ goes up, both the cover areas of $J_{\rm up}$ and $J_{\rm dn}$ increase obviously, resulting in the significant increases of $I_{\rm up}$ and $I_{\rm dn}$. Similarly, as shown in Fig. 3(d), both the cover areas of $J_{\rm up}$ and $J_{\rm dn}$ increase with $\Delta T$, resulting in the gradual increases of $I_{\rm up}$ and $I_{\rm dn}$. The origin of transmission bands around $E_{\rm F}$ can be elucidated by the spin-resolved band structures of two electrodes. Figure 4(a) plots the spin-resolved band structures of LE (left panel) and RE (right panel), as well as the spin-resolved transmission spectra (middle panel). Clearly, the band structures of LE are the same as those of RE under the P configuration since two electrodes are composed of the same material. Around $E_{\rm F}$, two spin-up bands $V_{1}$ and $V_{2}$ appear in energy region $-$0.68 eV$\, < E-E_{\rm F} < -0.36$ eV and $-$0.46 eV$\, < E-E_{\rm F} < -0.2$ eV, while two spin-down bands $C_{1}$ and $C_{2}$ appear in the energy regions 0.22 eV$\, < E-E_{\rm F} < 0.3$ eV and 0.36 eV$\, < E-E_{\rm F} < 0.48$ eV, respectively. The perfect overlapping of $V_{1}$, $V_{2}$, $C_{1}$ and $C_{2}$ between LE and RE causes the spin-resolved transmission bands around $E_{\rm F}$.
cpl-36-1-017301-fig4.png
Fig. 4. Spin-resolved band structures of LE (left panel) and RE (right panel), as well as the spin-resolved transmission spectra (middle panel) under (a) P and (b) AP configurations.
cpl-36-1-017301-fig5.png
Fig. 5. Total thermal current under the P and AP configurations for fixed $T_{\rm L}=400$ K with different $\Delta T$. The inset shows the magnetoresistance ratio (MR) as a function of $\Delta T$ with fixed $T_{\rm L}=400$ K.
Next, we consider the AP configuration. Figure 4(b) plots the spin-resolved band structures of LE (left panel) and RE (right panel), as well as the spin-resolved transmission spectra (middle panel) under the AP configuration. As we can see, $C_{1}$, $C_{2}$ and $V_{1}$, $V_{2}$ in RE are now spin-up and spin-down bands, which are exactly interchanged compared with those in LE. Thus, a large transmission gap around $E_{\rm F}$ occurs due to the lack of overlapping between bands with the same spin index. As a result, $I_{\rm up}$ and $I_{\rm dn}$ can hardly pass through the device under the AP configuration. Figure 5 plots the total thermal current ($I_{\rm up}+I_{\rm dn}$) under the P ($I_{\rm P}$) and AP ($I_{\rm AP}$) configurations for fixed $T_{\rm L}=400$ K with different $\Delta T$. Clearly, obvious $I_{\rm P}$ can flow through the device while $I_{\rm AP}$ is strongly blocked, resulting in a thermal colossal magnetoresistance effect when the magnetic configuration goes from P to AP. To quantify the predicted thermal colossal magnetoresistance effect, we define the magnetoresistance ratio (MR) as MR = $[(|I_{\rm P}|-|I_{\rm AP}|)/|I_{\rm AP}|]\times100{\%}$. As shown in the inset of Fig. 5, MR is as large as 10$^{20}$% and increases gradually with $\Delta T$. This large MR is much greater than those reported previously.[8,9] In conclusion, we have investigated the spin caloritronic transport properties of TGNRs using the DFT+NEGF method. Our results show that TGNRs have stable FM ground states with high $T_{\rm C}$ far above room temperature. Under the P configuration, these systems exhibit obvious SSE due to the break of electron-hole symmetry originated from the asymmetry of spin-resolved transmission spectra around $E_{\rm F}$. Moreover, large thermal colossal magnetoresistance of up to 10$^{20}$% can be realized when the magnetic configuration switches between P and AP. These results demonstrate that TGNRs hold great potential for spin caloritronics. References Electron spins blow hot and coldSpin caloritronicsVoltage tuning of thermal spin current in ferromagnetic tunnel contacts to semiconductorsGiant Thermoelectric Effect in Graphene-Based Topological Insulators with Heavy Adatoms and NanoporesThermodynamic analysis of interfacial transport and of the thermomagnetoelectric systemSpin-injection experimentObservation of the spin Seebeck effectGraphene-based Spin CaloritronicsSpin Seebeck Effect and Thermal Colossal Magnetoresistance in Graphene Nanoribbon HeterojunctionThermal spin filtering effect and giant magnetoresistance of half-metallic graphene nanoribbon co-doped with non-metallic Nitrogen and BoronMetal-free magnetism, spin-dependent Seebeck effect, and spin-Seebeck diode effect in armchair graphene nanoribbonsA unified geometric rule for designing nanomagnetism in grapheneAb initio modeling of open systems: Charge transfer, electron conduction, and molecular switching of a C 60 deviceAb initio modeling of quantum transport properties of molecular electronic devicesDensity-functional method for nonequilibrium electron transportThe SIESTA method for ab initio order- N materials simulationGeneralized Gradient Approximation Made SimpleEfficient pseudopotentials for plane-wave calculationsGeneralized many-channel conductance formula with application to small ringsSiN-SiC nanofilm: A nano-functional ceramic with bipolar magnetic semiconducting characterPrediction of very large values of magnetoresistance in a graphene nanoribbon deviceStructure and binding of alkanethiolates on gold and silver surfaces: implications for self-assembled monolayersSpin polarization and giant magnetoresistance effect induced by magnetization in zigzag graphene nanoribbons
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