Impossibility of Increasing N\acute{\textrm{e}}el Temperature in Zigzag Graphene Nanoribbon by Electric Field and Carrier Doping
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Impossibility of Increasing N´eel Temperature in ZigzagGraphene Nanoribbon by Electric Field and Carrier Doping
Teguh Budi Prayitno a, ∗ a Physics Department, Faculty of Mathematics and Natural Science, Universitas Negeri Jakarta, Kampus AJl. Rawamangun Muka, Jakarta Timur 13220, Indonesia
Abstract
We investigated the dependence of N´eel temperature as a critical temperature onthe electric field and hole-electron doping in the antiferromagnetically ordered zigzaggraphene nanoribbon. The temperature was calculated by averaging the magnon energyin the Brillouin zone within the mean-field approximation. We employed the general-ized Bloch theorem instead of the supercell approach to reduce the computational costsignificantly to obtain the magnon spectrum. We showed that the N´eel temperaturereduces when increasing both the electric field and the hole-electron doping, thus thesetreatments will never enhance the N´eel temperature.
Keywords:
Graphene nanoribbon, Critical temperature, Spin sti ff ness
1. Introduction
Recently, exploring the electric and magnetic properties in the low-dimensionalmaterials gives significant impacts in the condensed matter subject. It was started fromthe discovery of graphene as a two-dimensional material composed of carbon atoms byNovoselov et al . [1, 2, 3]. Besides the abundance of carbon in nature, it was verified thatgraphene has high electrical and thermal conductivities, exhibiting interesting physicalproperties. Previous works reported that graphene can be applied well such as for theoptoelectronics [4, 5] or transistor [6, 7]. Next, experimental results and theoreticalstudies of replicas of graphene such as germanene and silicene also show promisingmaterials for the future nanoelectronic devices. Compared to the graphene, germanene ∗ Corresponding author
Preprint submitted to Journal of L A TEX Templates February 15, 2021 nd silicine have intrinsic gap [8, 9] which can be controlled by strain [10] or elec-tric field [11]. From this benefit, germanene and silicine can be more applicable forthe logic-based devices such as transistor. Then, performing the experiments or den-sity functional theory (DFT), the future applicable devices based on low-dimensionalmaterials are explored such as for semiconductors [12, 13] or thermoelectric materials[14, 15].The main question regarding the low-dimensional materials is related to the criticaltemperature (Curie or N´eel temperature), at which the magnetism in any materials islost. In the bulk materials, such as 3 d ferromagnetic metals [16, 17], Heusler alloys[18, 19], or 3 d transition metal oxides [20], the critical temperatures are always higherthan the room temperature, thus any practical devices based on these materials willoperate properly. On the contrary, within the DFT, the critical temperature in the low-dimensional systems, such as 1-T transition metal dihalides monolayer [21, 22] andmost of transition metal dichalcogenides monolayer [23], are predicted to be lowerthan the room temperature. So, the magnetism for these low-dimensional materialsshould be lost at room temperature.Regarding the low-dimensional materials, the critical temperature in the zigzaggraphene nanoribbons (ZGNR), a one-dimensional structure of graphene, is not thor-oughly elucidated. In the previous DFT calculations, Yazyev and Katsnelson [24] withthe supercell approach stated that the critical temperature in the ZGNR only reachesthe room temperature if the order of spin correlation length is only a few nanometers,a subtle feature that is very di ffi cult to realize now. At the same time, Kunstmann etal . [25] also claimed that the magnetism in the ZGNR only preserves at a very lowtemperature. Based on their reports, the magnetism in the ZGNR is only stable belowroom temperature. As a consequence, any practical devices based on ZGNR will neverfunction well.The purpose of this paper is to investigate the influence of the N´eel temperature as acritical temperature of ZGNR with respect to the electric field and hole-electron dopingbased on the spin-waves excitations within frozen magnon method. The calculation ofthe N´eel temperature in the antiferromagnetic edge state ZGNR will be performed bythe mean-field approximation (MFA) within the generalized Bloch theorem (GBT).2he benefit of using the GBT rather than the supercell approach is the e ffi ciency toobtain not only the N´eel temperature but also the spin sti ff ness through the primitivecell. As reported in the previous paper [26], in an sp -electron system as in the ZGNR,the Stoner excitations may not be neglected. If so, the calculation of spin sti ff ness willonly possible in the low magnon energy close to Γ point, thus it is very di ffi cult torealize through the supercell approach.To obtain the N´eel temperature, we average the magnon energies for a set of spiralvectors in the Brillouin zone. Here, we exploit the conical spiral instead of the flatspiral to obtain constant magnetic moments during the self-consistent calculation. Thisapproach was successfully employed to estimate critical temperatures in some mate-rials [27, 28, 29, 30]. We prove that both the electric field and hole-electron dopingreduce the N´eel temperature as well as the spin sti ff ness for all ribbon widths, makingthe impossibility to reach the room temperature in the ZGNR with these treatments.These findings are caused by the small magnetic moments of magnetic carbonatoms at the edges by applying both the electric field and the hole-electron doping.By using the Hubbard approach, Kunstmann et al . [25] showed that at the such con-dition the magnetism in the ZGNR becomes unstable, thus disappearing the magneticproperties at room temperature. So, our results are in good agreement with the formerprediction. This means that even though the electric field and hole-electron doping cangenerate some magnetic properties for spintronic applications, it will not operate wellat room temperature.
2. Structure Model and Calculation Method
We applied the GBT within the first-principles calculation as implemented in theOpenMX code [31], a DFT package exploiting the localized basis function [32] andnorm-conserving pseudopotentials [33], with a 150 Ryd cuto ff energy and employedthe generalized gradient approximation (GGA) [34] as the exchange-correlation func-tional for the electron-electron interaction. The implementation of GBT is to expressthe non-collinear wavefunction as the linear combination of pseudo-atomic orbitals3LCPAOs) by including the spiral wavevector q ψ ν k ( r ) = √ N N X n e i ( k − q ) · R n X i α C ↑ ν k , i α φ i α ( r − τ i − R n ) + N X n e i ( k + q ) · R n X i α C ↓ ν k , i α φ i α ( r − τ i − R n ) . (1)Here, the localized orbital function φ i α can be generated as many as possible by meansof the confinement technique [35].To evaluate the N´eel temperature and the spin sti ff ness, we employed the frozenmagnon method and mapped the total energy di ff erence in the self-consistent calcula-tion of the spiral magnetic configurations [36] M i ( r + R i ) = M i ( r ) cos ( ϕ + q · R i ) sin θ i sin ( ϕ + q · R i ) sin θ i cos θ i (2)onto the Heisenberg Hamiltonian model. Within the frozen magnon method, the magnonenergy of the ZGNR can be formulated as [37] ~ ω q = µ B M E ( q , θ ) − E ( , θ )sin θ . (3)Meanwhile, the N´eel temperature within the MFA can be estimated by [38] k B T MFA C = M µ B N X q ~ ω q , (4)with N denotes the number of q .We applied the conical spiral ( θ = ◦ ) to fix the antiferromagnetically orderedmagnetic moments of carbon atoms at the edges, as shown in Fig. 1. This can berealized by introducing the penalty functional if the magnetic moments start to deviate[39]. For the atomic structure, an experimental lattice of graphite of 2.46 Å as a unitcell in the x -axis was applied while the vacuums in the non-periodic cells in the otheraxes were set to 50 Å. For the basis sets, we used two s - and two p - orbitals for thecarbon atoms, and two s - and a p - orbitals for the hydrogen atoms. At the same time,the boundary cuto ff radii were assigned to 4.0 a.u. and 6.0 a.u. for the carbon andhydrogen atoms, respectively. 4 igure 1: (Color online) Top view (a) and side view (b) of N -ZGNR antiferromagnetic edge states withthe ribbon width N . The large and small filled spheres denote carbon and hydrogen atoms, respectively.Meanwhile, the primitive cell is pointed out by a dashed line.
3. Results and Discussions
We divide this section into two subsections exploring the influence of electric fieldand hole-electron doping on the ZGNR. Here, we provide the magnon spectra for N -ZGNR ( N = , ,
12) in the Brillouin zone, where N is the ribbon width. First ofall, for the non-electric-field and non-doped cases, the magnon energy increases as N increases, thus increasing the N´eel temperature and the spin sti ff ness. However, sincethere are flat high energy dispersions around 300 meV, the N´eel temperatures will notreach the room temperature. The application of electric field in the ZGNR is very important to study the mag-netic features. When the transverse electric field is applied along the ribbon width, thehalf-metallic feature is induced [40, 41, 42]. Even though this feature is very usefulfor developing spintronic devices, however, we prove that the ZGNR-based applicable5evices cannot operate well at room temperature since the N´eel temperature reducesdue to transverse electric field. G X (a) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (b) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm 0 50 100 150 200 250 300 G X (c) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (d) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm 0 50 100 150 200 250 300 G X (e) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (f) E ne r g y ( m e V ) q (2 p /a) E = 0 V/nmE = 1 V/nm
Figure 2: (Color online) Magnon dispersions of ZGNR in the Brillouin zone (a, c, e) and low magnonenergies close to Γ point (b, d, f) under electric field E . The solid lines in (b, d, f) represent the fittingfunction ~ ω q = Dq (1 − β q ). Here, 6-ZGNR, 10-ZGNR, and 12-ZGNR are depicted by figures (a, b), (c,d), and (e, f), respectively. Here, we apply the transverse electric field E along N in the y -axis and plot themagnon spectra for E = / nm and E = / nm, as shown in Fig. 2. As immediatelyobserved, the applied E reduces all the magnon spectra for each N . We also see that6here are still flat dispersions for all N in the one-third of the Brillouin zone at the highenergies near X point as E increases. Meanwhile, the low energies near Γ point alsoreduce for all N as E increases, reducing the spin sti ff ness. T C ( K ) D ( m e VÅ ) E (V/nm)
Figure 3: (Color online) Electric field E dependence of N´eel temperature T C and spin sti ff ness D . Based on Figs. 2(a), 2(c), and 2(e), we calculate the N´eel temperature T C by averag-ing all the magnon energies in the Brillouin zone by means of the MFA approach in Eq.(4). Meantime, the spin sti ff ness D is evaluated in the low magnon energies near Γ pointas shown in Figs. 2(b), 2(d), and 2(f) through the least-square fit ~ ω q = Dq (1 − β q ).To view the reduction more clearly, we provide Fig. 3 to show the reductions of T C and D as E increases. Thus, the applied E cannot enhances the T C up to the roomtemperature. 7 .2750.280.2850.29 0 0.2 0.4 0.6 0.8 1 M ( m B ) E (V/nm)6−ZGNR10−ZGNR12−ZGNR
Figure 4: (Color online) Magnetic moment M as a function of electric field E . As shown in Fig. 3, both the T C and D reduce as N increases when E increases.We notice that the reductions of T C as well D are more rapid for the large N that thosefor the small N as E increases. If we consider that the exchange interaction J i j dependsonly on the distance between two edge carbon atoms, 6-ZGNR should have the largest J i j . In this case, the electron from one edge carbon atom hops more easily to the otheredge carbon atom in the small N than that in the large N . As a consequence, the large J i j may prohibit the rapid reduction for the T C and D which is caused by E .The origin of impossibility of increasing the T C is caused by the small magneticmoments of the edge carbon atoms. Our calculation finds the magnetic moment ofeach edge carbon atom is about 0.3 µ B . When E is applied, the magnetic momentgenerally decreases for all N as shown in Fig. 4, in good agreement with Ref. [43].In addition, we also see that the reduction of magnetic moment under E for the large N is more rapid than that for the small N , the same tendency as in the reduction of T C and D . According to Kunstmann et al . [25], this small magnetic moment in the ZGNRyields magnetic instability, namely, the magnetism in the ZGNR cannot hold at roomtemperature. It is also supported by the previous authors who reported the reduction of D under E [44, 45, 46]. It has been reported that the hole-electron doping can induce the magnetic phasetransition from ferromagnetic-canted-antiferromagnetic states [47]. So, the implemen-8ation will be important to control the magnetic state for the applicable devices. Ap-plying the hole-electron doping can be realized by employing the chemical doping orfield e ff ect transistor (FET) doping. In this calculation, we exploit the Fermi level shift(FLS) approach where the system is neutralized by inserting the uniform backgroundcharge. Here, we also show that the hole-electron doping also cannot increase the T C . G X (a) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (b) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) 0 50 100 150 200 250 300 G X (c) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (d) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) 0 50 100 150 200 250 300 G X (e) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) 0 5 10 15 20 0 0.005 0.01 0.015 0.02 0.025 0.03 (f) E ne r g y ( m e V ) q (2 p /a) x = 0 ( e /cell) x = 0.0098 ( e /cell) x = −0.0098 ( e /cell) Figure 5: (Color online) Magnon dispersions of ZGNR in the Brillouin zone (a, c, e) and low magnonenergies close to Γ point (b, d, f) under doping x . The solid lines in (b, d, f) represent the fitting function ~ ω q = Dq (1 − β q ). Here, 6-ZGNR, 10-ZGNR, and 12-ZGNR are depicted by figures (a, b), (c, d), and (e,f), respectively.
9e plot the magnon spectra via self-consistent calculation as performed in the E case. As shown in Fig. 5, we also observe the reduction of magnon spectra for each N as the doping x increases. In addition, the flat dispersions are still observed for all N inthe one-third of the Brillouin zone X point as the doping increases. This indicates thatthe flat dispersions may be the natural feature of magnon dispersion in the ZGNR. Wealso notice that increasing x will also reduce the T C and D .By applying the same way to calculate the T C and D as in the E case, both the T C and D incline to reduce as x increases. Figure 6 shows the reduction of T C and D whenincreasing x . When we consider the trends of magnetic moments of edge carbon atoms,we also see the asymmetric reduction of magnetic moment as shown in Fig. 7, similarto Ref. [37]. This means that taking the doping into account also yields the magneticinstability as in the E case. Unlike the E -field case, no rapid reduction of the T C and D as x increases, indicating that x does not influence the rapid reduction of the T C and D for the large N .The linear dependence of magnetic moment M on x can be explained as follows.The calculations of magnon energy apply the Heisenberg model in which the totalenergy di ff erence ∆ E = E ( q ) − E ( q =
0) in the self-consistent calculation is mappedonto the Heisenberg Hamiltonian, as stated in Eq. 3. In this case, ∆ E is proportional to M . We then find that ∆ E decreases linearly as the doping increases, the same tendencywith Ref. [47]. The reduction of ∆ E leads to a loss of magnetism gradually in ZGNRdue to low concentration of doping. So, the reduction should be linear. At the sametime, due to proportionality between ∆ E and M , M should also reduce linearly as thedoping increases. When the doping is su ffi ciently high, the ZGNR should becomenon-magnetic. Note that introducing hydrogen passivation at the edge will removethe dangling bond state, thus reducing the magnetic moment of edge carbon atoms, aspointed out by Song et al . [48]. This means that introducing the doping or electric fieldreduces the magnetism in the ZGNR.Regarding the doping case, we give some comments on the possibility to increasethe T C . Since the main problem of small magnon energy is the small magnetic momentof each edge carbon atom, the most possible way is to introduce the metal atom espe-cially with the large magnetic moment. As reported in Ref. [25], the small magnetic10 T C ( K ) D ( m e VÅ ) x ( e /cell) Figure 6: (Color online) Doping x dependence of N´eel temperature T C and spin sti ff ness D . moment leads to unstable magnetism that vanishes the magnetism at room temperature.When the metal atom is introduced, it forms a strong bonding between the edge carbonatom and metal atom which transfer the charge from the metal atom to the edge atom,thus increasing the magnetic moment of edge carbon atom.For the related experiment, Magda et al . [49] grew the ZGNR onto Au(111) sub-strate by chemical deposition. They justified that this material will be stable at roomtemperature. In the computational framework, previous authors also showed that therobust magnetism in the ZGNR can be achieved by introducing the metal atoms whenconsidering the spiral density waves [50, 51, 52]. They found the large scale energyof spiral states that can be observed at room temperature. Since the spiral spin densitywaves are manifestation of spin-wave excitations / magnon, introducing the metal atoms11an enhance the T C up to the room temperature. M ( m B ) x ( e /cell)6−ZGNR10−ZGNR12−ZGNR Figure 7: (Color online) Magnetic moment M as a function of doping x .
4. Conclusions
We have performed the self-consistent non-collinear spiral calculations to inves-tigate the e ff ect of N´eel temperature T C in ZGNR under the electric field E and thehole-electron doping x . We show that the T C cannot be increased by introducing E and x . In addition, the reductions of T C are also followed by the reductions of spin sti ff ness D , thus there is a close relationship between the T C and D . These features are causedby the small magnetic moment of magnetic edge carbon atoms, making the magneticinstability.We also show that the T C and D reduce more rapidly in the large ribbon width N than those in the small N under E but not under x . These features are caused by theexchange interaction J i j between two edge carbon atoms, i.e., the small ribbon widthgets the largest J i j . So, the large J i j can compensate the rapid reductions of the T C and D at the large E . On the contrary, no rapid reductions of T C and D are observed as x increases. Based on the results, introducing E and x never gives the T C close to theroom temperature. 12 cknowledgments A personal high computer has been used to performed the computations. We herebystate that this is an independent research without any fundings.
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