Electric-field-induced spin spiral state in bilayer zigzag graphene nanoribbons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Electric-field-induced spin spiral state in bilayerzigzag graphene nanoribbons
Teguh Budi Prayitno
Department of Physics, Faculty of Mathematics and Natural Science, UniversitasNegeri Jakarta, Kampus A Jl. Rawamangun Muka, Jakarta Timur 13220, IndonesiaE-mail: [email protected]
Abstract.
We investigated the emergence of spin spiral ground state induced bythe electric field in the bilayer zigzag graphene nanoribbons for the ferromagneticedge states. To do that, we employed the generalized Bloch theorem to createflat spiral alignments for all the magnetic moments of carbon atoms at the edgeswithin a constraint scheme approach. While the small ribbon width can preserve theferromagnetic ground state, the large one shows the spiral ground state starting froma certain value of the electric field. We also pointed out that the spiral ground state iscaused by the reduction of spin stiffness. In this case, the energy scale exhibits a subtlenature that can only be considered at the low temperature. For the last discussion, wealso revealed that the spin spiral ground state appears more rapidly when the thicknessincreases. Therefore, we justify that the large ribbon width and large thickness cangenerate many spiral states induced by the electric field.
Keywords : graphene nanoribbons, spin spiral, spin waves, spin stiffness
1. Introduction
The investigation on spin spiral ground state in the materials may be triggered by theexperimental series by Tsunoda and Tsunoda et al. [1, 2] when they observed a spiralground state on γ − Fe, an fcc phase of iron. In the experiments, they stabilized theprecipitates of γ − Fe at the low temperature in a fcc Cu matrix. So, the spiral groundstate in the γ − Fe is a consequence of stabilizing the γ − Fe at the low temperature. Afterthat, the theoretical discussions on this subject become popular and are still interesting.All the related authors found that the ground state of γ − Fe is very sensitive to the latticeparameter [3, 4, 5, 6, 7].The spiral (SP) state is a special case of helimagnetic state with a fixed cone angle[8]. Some papers reported that the magnetic domain wall can be portrayed by theSP formation to exploit physical features such as ferroelectricity or magnetoresistancefor spintronic applications [9, 10, 11, 12, 13]. Beside the bulk materials, furtherinvestigations also show that the SP state may emerge in the lower-dimensionalmaterials, such as two-dimensional metal dihalides [14, 15] and one-dimensionalmonoatomic chains [16, 17]. So, we expect to find interesting physical properties inducedby the SP formation in the low-dimensional materials.Intensive studies of exploiting the electronic and magnetic properties in the low-dimensional materials are initially driven by the discovery of graphene [18, 19, 20], atwo-dimensional sheet of hexagonal carbon lattice. Previous studies show that graphenecan be utilized for future devices, ranging from the electronic devices [21, 22], opticaldevices [23, 24], to photonic devices [25, 26]. Nevertheless, the magnetism in graphenecan only be considered if its dimensionality is reduced into one-dimensional structurebased on the theoretical study proposed by Fujita et al. [27] to become either the zigzaggraphene nanoribbon (ZGNR) or the armchair graphene nanoribbon (AGNR). Here, weonly consider the magnetic properties in the ZGNR.It has been reported that the simplest way to achieve a metallic or an insulatingproperty in the ZGNR is to arrange the magnetic moments of carbon atoms at theedges [28]. Interestingly, this way can be used to create spin-wave excitations where themagnetic moments of carbon atoms along the edge are continuously rotated with thefixed cone angle. Previous studies exploit the spin-wave excitations in the monolayerZGNR by using the supercell within the Hubbard model to investigate the criticaltemperature [29, 30], the lifetime of spin excitation [31, 32], and the SP state [33, 34]. Forthe latter, the other authors also observed the SP state in the monolayer ZGNR withinthe supercell approach by exploiting the non-equilibrium Green’s function method[35, 36] or by inserting the transition metal atoms [37]. However, the wavevector, atwhich the SP state becomes a ground state, was not obviously elucidated.The above previous methods to find the SP state is not absolutely simple becausethe origin structure of ZGNR should be modified. Here, we present the simplest waybut powerful to find the SP state in the ZGNR. Based on our previous results in themonolayer case [38], we continue to investigate the SP state in the bilayer ZGNR forthe ferromagnetic edge states under the transverse electric field by using the generalizedBloch theorem (GBT). Due to the crystal structure, the explorations on the magneticproperties in the bilayer ZGNR are much more than those in the monolayer case[39, 40, 41, 42].There are two main reasons why the GBT is more powerful than the otherapproaches for investigating the SP state. The main advantage of using the GBT insteadof the supercell is the lowest computational cost because it only requires the primitiveunit cell. Even we do not need the additional atom to generate the SP state as inRefs. [36, 37]. Beside the efficiency, the GBT can also give the SP ground state moreaccurately than the supercell when the SP state appears in the very small wavevector(long period) near Γ point. Even when the supercell combined with the non-equilibriumGreen’s function method or the Hubbard approach, it is still very difficult to determinethe wavevector which gives the SP ground state.We found that the SP state appears due to the reduction of spin stiffness startingfrom a certain value of electric field, similar to the monolayer case [38]. Here, the spinstiffness was calculated by using a least-square fit from the self-consistent total energydifference for a set of wavevectors. So, when the spin stiffness could be still calculatedby this fit, the ground state is a ferromagnetic (FM) state. This means that there wouldbe a phase transition from the FM state to the SP state as the spin stiffness can nolonger be calculated by the fit. We also showed that this trend only occurs for the largeribbon width while the small one tends to preserve the FM state. Note that the SP statecan only be induced if the initial state is the FM state whereas the antiferromagnetic(AFM) state is the most stable state in the multilayer ZGNR [41]. This means that theSP state happens due to instability of the FM state under the electric field.For the last session, we also discussed the influence of thickness on the SP state.When the thickness was taken into account, we observed a shift of electric field at whichthe SP state emerges for the first time. As the thickness increases, the electric fielddecreases, thus accelerating the emergence of SP state in terms of the electric field. Forthe larger ribbon width, the wavevector, at which the SP state emerges at a certainelectric field, was larger than that for the smaller one for all the thicknesses. We alsofound that the SP state only emerges for the large ribbon width for all the thicknesses.This implies that not only the ribbon width but also the thickness can control the phasetransition from the FM state to the SP state. Based on the results, we claim that thesimilar phase transition under electric field should also occur in the multilayer ZGNR.
2. Computational Method
We used the OpenMX code [43] to perform the first-principles non-collinear calculationsby implementing the GBT. Here, the wavefunction was expanded by the numericallinear combination of pseudo-atomic orbitals (LCPAO) as basis functions, which areproduced within a confinement method [44, 45]. To do the efficient calculation, thenorm-conserving pseudopotential [46] was used to represent the core Coulomb potential.For employing the GBT, the spiral wavevector q was inserted in the phase term ofLCPAO written as [47] ψ ν k ( r ) = 1 √ N " N X n e i ( k − q ) · R n X iα C ↑ ν k ,iα φ iα ( r − τ i − R n ) ! + N X n e i ( k + q ) · R n X iα C ↓ ν k ,iα φ iα ( r − τ i − R n ) ! , (1)where the localized function φ iα is well defined within a cutoff radius as a boundary inthe real space. Meanwhile, the flat spiral configuration was governed by the rotation ofthe magnetic moment M with a fixed cone angle θ M i ( t ) = M i cos ( ϕ i + q · R i + ω q t ) sin θ i sin ( ϕ i + q · R i + ω q t ) sin θ i cos θ i . (2)In this paper, we selected an AB-stacking primitive bilayer ZGNR due to itsstability, as shown in Fig. 1. We set the experimental lattice parameter of 2.46 ˚Ain the x- axis as a periodic lattice and thickness of 3.35 ˚A from graphite in the z- axis asa non-periodic direction. Then, we initially set an FM alignment of magnetic momentsof carbon atoms at the four edges with θ = π/ E was appliedalong y- axis, parallel to the ribbon width N . Note that the applied E can be used toinduce the half-metallic property within B3LYP exchange-correlation functional [48, 49].For the detailed computation, two valence s -orbitals and two valence p -orbitals werespecified for the carbon atoms while two valence s -orbitals plus one valence polarization p -orbital were set for the hydrogen atoms. At the same time, the cutoff radii forthe carbon and hydrogen atoms are 4.0 Bohr and 6.0 Bohr, respectively. The non-collinear self-consistent calculation was then performed by using the Perdew, Burke,and Ernzerhof exchange-correlation functional [50] within 65 × × k -point samplingand cutoff energy of 150 Ryd. Figure 1. (Color online) Crystal structure of AB-stacking bilayer ZGNR from topview (a) and side view (b). The initial FM state in (b) is set to create the flat spiralduring the calculation. Here, the carbon and hydrogen atoms are depicted by the largeand small spheres, respectively. Meanwhile, d denotes the thickness, N represents theribbon width, and the dashed rectangle means the primitive cell.
3. Results and Discussions
First, we consider the experimental thickness d =3.35 ˚A. Figure 2 shows the FM groundstate ( q = 0) for the non-electric-field ( E = 0 V/nm) for all N while the appearancesof SP ground state occur at E = 0 . E = 1 . E . Note that E = 0 . E = 1 . E . E ne r g y ( m e V ) q (Å −1 )6−ZGNR10−ZGNR12−ZGNR−0.5 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08(b) E ne r g y ( m e V ) q (Å −1 )6−ZGNR10−ZGNR12−ZGNR−0.5 0 0.5 1 1.5 2 0 0.02 0.04 0.06 0.08(c) E ne r g y ( m e V ) q (Å −1 )6−ZGNR10−ZGNR12−ZGNR Figure 2. (Color online) Total energy difference ∆ E = E ( q ) − E ( q = 0) as a functionof q along x- axis (periodic direction) for d =3.35 ˚A and N = 6 , ,
12 with the appliedelectric field E = 0 V/nm (a), E = 0 . E = 1 . y- axis.Here, the SP ground states appear at q = 0 .
024 for 12-ZGNR in (b), and at q = 0 . q = 0 .
064 for 12-ZGNR in (c).
When we check the spin stiffness D with respect to E , we find that D reduces as E increases, as shown in Fig. 3 for all N . This similar tendency was also reported byRhim and Moon by applying the Hubbard Hamiltonian [51]. Note that D is obtainedby fitting the total energy difference through the equation ∆ E = Dq (1 − βq ) in Figs.3(a-c), where q is defined in units of ˚A − . In this case, D can only be evaluated for theFM state. We also show that for E = 0 V/nm, D increases as N increases, as shown inFig. 3(d). However, at the same time in Fig. 3(d), the reduction of D for the large N is more rapid than that for the small N , similar to the monolayer case [38]. E ne r g y ( m e V ) q (Å −1 )E = 0 V/nmE = 0.3 V/nmE = 0.6 V/nmE = 0.9 V/nm 0 1 2 3 4 5 6 7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(b) E ne r g y ( m e V ) q (Å −1 )E = 0 V/nmE = 0.3 V/nmE = 0.6 V/nmE = 0.9 V/nm 0 1 2 3 4 5 6 7 8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(c) E ne r g y ( m e V ) q (Å −1 )E = 0 V/nmE = 0.3 V/nmE = 0.6 V/nm 300700110015001900 0 0.3 0.6 0.9(d) D ( m e VÅ ) E (V/nm)6−ZGNR10−ZGNR12−ZGNR
Figure 3.
Total energy difference ∆ E = E ( q ) − E ( q = 0) under E for 6-ZGNR (a),10-ZGNR (b), and 12-ZGNR (c) while the dependence of D on E is presented in (d)for d =3.35 ˚A. Here, D in (d) is obtained by fitting ∆ E = Dq (1 − βq ) from (a-c). The above trend can only be explained if the electron-electron interaction J ij amongthe edges only comes from the electron hopping. When N increases, the energy increases,thus D increases. At the same time, the electron hops easily from one edge to the otheredges as N decreases. This means that the large N requires more energy to excite thespin waves than the small N as the electron hops from one edge to the other edges. So,the value of D is caused by the electron hopping, namely, the small N gives the small D . Nevertheless, the smallest N , which has the strongest J ij , tends to preserve the FMstate when E is applied. On the contrary, the large N cannot preserve the FM state ata certain E , thus there is a limit value of E to preserve the FM state. This implies thatthe strongest J ij in 6-ZGNR, even having the smallest D , can overcome the emergenceof SP state due to the reduction of D . Therefore, we justify that the emergence of SPstate is a consequence of reduction of D and instability of FM state under E . D E ( m e V ) q l o w e s t ( Å − ) E (V/nm)
Figure 4.
Dependence of total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) on E (a) and dependence of q on E (b) for d =2.95 ˚A. Here, q lowest means q having thelowest energy as the most stable state. For the last discussion, we show the phase transition from the FM state to the SPstate with respect to E for several thicknesses d , as shown in Figs. 4-8. For each d ,we plot the total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) as well as q lowest with respect to E , where q lowest is addressed to the most stable state. Our findings show D E ( m e V ) q l o w e s t ( Å − ) E (V/nm)
Figure 5.
Dependence of total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) on E (a) and dependence of q on E (b) for d =3.15 ˚A. Here, q lowest means q having thelowest energy as the most stable state. that the 6-ZGNR still preserves the FM state as E is applied. Contrarily, the 10-ZGNRand 12-ZGNR exhibit the SP state starting from a critical E , which also leads to aphase transition from the FM state to the SP state. We also observe that there areenhancements of q lowest and ∆ E as E increases, where the enhancements of the large N are more rapid than those of the small N for each d . However, each d gives the different q lowest as well as the critical E . D E ( m e V ) q l o w e s t ( Å − ) E (V/nm)
Figure 6.
Dependence of total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) on E (a) and dependence of q on E (b) for d =3.35 ˚A. Here, q lowest means q having thelowest energy as the most stable state. For d = 2 .
95 ˚A as shown in Fig. 4, the initial SP state is observed at E = 1 . q lowest = 0 .
016 for 10-ZGNR while the initial SP state in 12-ZGNR occursat E = 0 . q lowest = 0 . d increases up to 3.15 ˚A as shownin Fig. 5, the initial SP state is still observed at E = 1 . q lowest = 0 . E = 0 . q lowest = 0 . d as shown in Fig. 6, E shifts to 1.6V/nm for the initial SP state in 10-ZGNR at the same q lowest = 0 .
024 while the initialSP state in 12-ZGNR still occurs at E = 0 . q lowest = 0 . d increases up to 3.55 ˚A as shown in Fig. 7, there are displacements of E and q lowest for both 10-ZGNR and 12-ZGNR. The initial SP state in 10-ZGNR happens at E = 1 . q lowest = 0 .
008 while the initial SP state in 12-ZGNR occurs at E = 0 .
75 V/nm and at q lowest = 0 . d = 3 .
85 ˚A as shown in Fig.08, E shifts to 1.05 V/nm for the initial SP state in 10-ZGNR at q lowest = 0 .
016 whilethe initial SP state in 12-ZGNR still occurs at E = 0 .
75 V/nm and at the different q lowest = 0 . D E ( m e V ) q l o w e s t ( Å − ) E (V/nm)
Figure 7.
Dependence of total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) on E (a) and dependence of q on E (b) for d =3.55 ˚A. Here, q lowest means q having thelowest energy as the most stable state. As immediately seen in Fig. 9 (a), E inclines to reduce when the SP state initiallyhappens as d increases. This trend is similar to the case of ribbon width N . Thus, E ,which gives the SP state for the first time, only depends on the J ij between the twolayers. By the same analogy in the case of N , the largest d possess the weakest J ij ,thus the SP state occurs at the small E . Notice that this tendency also depends on N ,where the given E to achieve the SP state for the first time occurs earlier for the large N . On the contrary, we observe different trend for the q lowest when the SP state happensfor the first time, as shown in Fig. 9 (b). We find the q lowest inclines to decrease up to1 d = 3 .
55 ˚A and then increases. This means that d = 3 .
55 ˚A is a critical d at which theSP state happens in the long wavelength (small q ). Then, the short wavelength (large q ) for the SP state appears in the large d . So, we justify that the ribbon width N aswell as the thickness d give an important role to control the magnetic properties in thebilayer ZGNR due to the interactions between the magnetic carbon atoms at the edges. D E ( m e V ) q l o w e s t ( Å − ) E (V/nm)
Figure 8.
Dependence of total energy difference ∆ E = E ( q = 0) − E ( q = q lowest ) on E (a) and dependence of q on E (b) for d =3.85 ˚A. Here, q lowest means q having thelowest energy as the most stable state. Now, we would like to give some comments on the obtained scale of energy. Asshown in Figs. 4-8, we can see that all the phase transitions from the FM state to theSP state happen in the order of few of meV. Compared to the thermal energy around26 meV at the room temperature, this scale of energy should be sensitive to the thermalexcitations. This implies that the resulting phase transitions under the electric field aresubtle properties, which can only be investigated at the low temperature. This is dueto a small magnetic moment ( ≈ . µ B ) of each magnetic carbon atom at the edge in2our calculations, which yields magnetic instability at the edge [52]. According to theprevious reports [34, 36, 37], the scale of energy of SP state can be increased up tothe room temperature if the metal atoms are included. The existence of metal atomgenerates the charge transfer from the metal atom to the edge carbon atom, thus creatinga strong bonding. This bonding will induce the robust magnetism, thus enhancing thescale of energy of SP state. E ( V / n m ) q l o w e s t ( Å − ) d (Å) Figure 9.
Dependence of critical E (a) and q lowest (b) on d at which the SP stateinitially emerges. Now, we would like to reveal some possible applications of SP state in the ZGNR.Previous papers reported that an SP configuration can form a domain wall that can beutilized for the spin transport in the spintronic devices by introducing a magnetic field[35] or an atom doping [36]. This spiral domain wall relies on the initial magnetizationdetermined by the cone angle θ . On the other hand, Zhang et al. [53] stated that the SPconfiguration in the ZGNR induced by the Dzyaloshinskii-Moriya interaction may be3utilized for the spin filters in the spintronic devices. This phenomenon can be realizedwhen the ZGNR is grown on the topological insulator substrates. Table 1.
Initial E -induced spin spiral state with respect to d for several N . N d ( ˚A)2.95 3.15 3.35 3.55 3.856 - - - - -8 - - - - 1.6 V/nm10 1.5 V/nm 1.5 V/nm 1.6 V/nm 1.2 V/nm 1.05 V/nm12 0.9 V/nm 0.9 V/nm 0.9 V/nm 0.75 V/nm 0.75 V/nm
4. Conclusions
We prove the existence of the SP state induced by the electric field E in the bilayerZGNR by using the GBT. The observed wavevectors, at which the SP ground stateappears, are absolutely small so that it is very difficult to generate the SP state usingthe supercell. The consequence of the small vector leads to a small energy scale thatis very sensitive to the thermal excitations. In this case, the SP state induced by theelectric field should be a subtle feature that cannot be observed at the room temperature.We also notice that the small ribbon width N in the bilayer ZGNR preserves the FMground state due to the strongest J ij while the large N cannot maintain the FM groundstate so that the ground state changes to the SP state, generating a phase transitionfrom the FM state to the SP state.We also show that not only the ribbon width N but also the thickness d can controlthe SP state as E is applied. Here, we see the dependence of E and q lowest on d withdifferent tendencies, except the small N . As d increases, E inclines to reduce while q lowest tends to decrease until a certain d and start to increase. This condition also generatesa phase transition from the FM state to the SP state. Note that all the energy scalesfor all d are very small, too. Thus, the SP state for all d are also subtle.We also believe that the SP states can also exist in the multilayer ZGNR as E is introduced. They coexist with the FM edge states at the low temperatures. Thisis also due to the small magnetic moment of each edge carbon atom that yields themagnetic instability. In addition, as mentioned previously, since the most stable statein the ZGNR for any layer is always the AFM state, the FM state is unstable under E for the large N and large d . Therefore, the existence of SP state in the ZGNR under E is the consequence of the instability of FM state.4 Acknowledgments
All the detailed calculations were performed by using personal high computer atUniversitas Negeri Jakarta. No funds are available in this research.
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