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Magnetic response of zigzag nanoribbons under electric fields
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 216002 (http://iopscience.iop.org/0953-8984/26/21/216002) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.33.16.124 This content was downloaded on 19/06/2014 at 10:40
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 216002 (6pp)
doi:10.1088/0953-8984/26/21/216002
Magnetic response of zigzag nanoribbons under electric fields F J Culchac1, Rodrigo B Capaz1, A T Costa2 and A Latgé2 1
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil 2 Instituto de Física, Universidade Federal Fluminense, Niterói-RJ 24210-340, Brazil E-mail: [email protected] Received 23 November 2013, revised 5 February 2014 Accepted for publication 14 February 2014 Published 8 May 2014 Abstract
Spin excitations in zigzag graphene nanoribbons are studied when the system is subjected to an electric field in the transversal direction. The magnetic properties and the lifetime of the spin excitations are systematically investigated and compared using a tight-binding electron–electron model treated by a mean-field Hubbard model. The effects of electron–hole asymmetry introduced by next-nearest neighbor hopping are also investigated. We show that by increasing the electric field, the antiferromagnetic correlations between the edges of the nanoribbons are decreased due to a reduction of the magnetic moments. The results show that the spin wave lifetime may be controlled by the intensity of the transversal electric field, indicating that zigzag nanoribbons may be considered great candidates for future spintronic applications. Keywords: spin waves, graphene nanoribbons, spintronics (Some figures may appear in colour only in the online journal)
1. Introduction
both cases, a strong damping of the spin waves, thus decreasing their relaxation time considerably. Moreover, as the doping levels increase, the ferromagnetic alignment along the border becomes unstable. The possibility of controlling the spin relaxation times in such systems by doping can be conveniently used in designing new spintronic devices based on nanoribbons. Experimental manipulation of magnetization by applying electric fields on ferromagnetic semiconductor systems [7, 8] and thin-film ferromagnets (FePt and FePd thin layers) [9] has been largely reported in the literature. The search for magnetic systems that are easy to manipulate is the key parameter for spintronic data storage and processing. Similar processes may be designed for graphene systems in an attempt to modify the magnetic responses in a controlled way. Therefore, one additional possibility of manipulating the spin excitation lifetimes is the application of a transverse electric field. Calculations based on density-functional theory (DFT) [14–17] and tight-binding approximation [18] show that ZGNRs present half-metallicity properties, which considerably increases the possibility of using the ZGNRs for spintronic purposes.
Spintronics in graphene-based materials has emerged as an interesting theme in nanoscience mainly due to the variety of potential applications [1]. The presence of edges in graphene significantly alters the low energy spectrum of the πelectrons. In particular, graphene nanoribbons with zigzag edges (ZGNRs) present localized edge states at the Fermi energy [2] that favor the formation of magnetic moments. Experimental evidence of finite magnetization along ZGNR edges has been reported experimentally [3, 4] and has important implications for spin-dependent transport properties in these systems. We have shown in previous works [5, 6] that spin excitations in ZGNRs exhibit energy dispersion relations predominantly linear for large wavelengths, due to antiferromagnetic coupling between the magnetizations on the opposite edges. For infinite ZGNRs, the spin waves have an essentially infinite lifetime due to the presence of an energy gap in neutral nanoribbons [5]. For finite nanoribbons the spin waves are damped at all energies [6]. It was also shown that electron or hole doping caused by electrostatic gating may induce, in 0953-8984/14/216002+6$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 1. Schematic view of zigzag graphene nanoribbon geometry with an electric field applied across the ribbon in the transverse direction (y-direction). The dotted lines encircle two arbitrary unit cells, labeled m and m′. The indices l and l′ refer to atoms inside each unit cell, and ±V is the electric voltage defined by the field intensity.
Figure 2. Magnetic moment at the edges (sites 1 and 16) of an 8-ZGNR as a function of the electric field potential.
i with spin σ, and Vi is the electric potential energy associated with the transverse electric field. The magnetization of the ZGNRs is calculated self-consistently starting from initial values of the occupation numbers (〈niσ〉) that are chosen conveniently. The 〈niσ〉 are found by an interactive procedure until convergence, under the condition of global charge neutrality. For the nearest neighbor hopping case, where electron– hole symmetry is maintained, 〈niσ〉 can be evaluated keeping the Fermi level (EF = 0) fixed. When next-nearest neighbors are included, the electron-hole symmetry is broken and extra care must be taken to keep the Fermi level at a position where global charge neutrality is attained. The spin wave excitations are extracted from the transverse dynamical susceptibility,
In this work we study spin excitations in ZGNRs when a transverse electric field is applied. The electric field lifts the spin degeneracy of the nanostructure and promotes the formation of a half-metallic state beyond a critical field intensity that depends on the width of the ribbon. We are interested in investigating the changes induced by the electric field on the spin dynamic properties of ZGNRs, such as lifetimes and the stability of ferromagnetic order. We describe the electronic structure of the system using a tight-binding model plus a mean-field Hubbard term to take into account the effects of a screened Coulomb repulsion. 2. Theory
χij+− ( t ) = − iΘ ( t ) [ Si+ ( t ), S −j (0)] ,
The π-electrons in graphene are well described by the following Hamiltonian H = −t 0
∑ ∑ ci†σ cjσ − t ′ ∑ ∑ ci†σ cjσ + ∑ Ui(ni ↑ ij
1st
σ
ij
2nd
+ n i, ↑ n i ↓ − n i, ↑ n i, ↓
σ
where S + = c↑†c↓ and S− = (S+)† are spin-raising and lowering operators. For graphene nanoribbon geometry it is convenient to define a mixed Bloch-Wannier basis to describe the electronic states
n i, ↓
i
) + ∑ Vni iσ , iσ
(2)
(1)
cl ( q ) =
where t0 and t′ are the hopping integrals for the nearest and second-nearest neighbors, respectively, and we use t0 = 2.7 eV and t′/t0 = 0.1 following reference [18]. We assume the value of the on-site effective Coulomb interaction as U/t0 = 1, which reproduces the electronic structure of a ZGNR, as provided by first-principle calculations [4] and also correctly describes the magnetic effects of the nanoribbon, derived from the screened Coulomb interaction. The electron–electron interactions have been treated within a mean-field approximation in the description of the ground state. This level of approximation is enough to demonstrate the instability of the spin-unpolarized ground state, thus indicating that the true ground state should be spinpolarized. This conclusion is robust with respect to higherlevel treatments of electron–electron correlation [10–12]. The operator ci†σ creates an electron at the atomic state in the site
1 N
∑ eimkac ( xm , yl ), m
where c(xm, yl) is the annihilation operator for a Wannier state at a site l in unit cell m, a = 3 a 0 is the distance between unit cells and a 0 ≈ 1.42Å is the carbon-carbon distance. The unit cell of the zigzag ribbon is depicted in figure 1. Within the random-phase approximation it is possible to relate the interacting susceptibility to the mean-field susceptibility χ0 [22] 0 +− 0 χll+− ′ ( q , Ω ) = χ ′ ( q , Ω ) + ∑ χln ( q , Ω ) Unχ ′ ( q , Ω ). ll nl n
The spin excitations above the spin-polarized ground state have been treated within the random phase approximation, and may be considered as a kind of dynamical mean-field approximation. Within this level of approximation, the electron and hole of the spin excitation are correlated, forming a bound 2
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 3. Band structure for an 8-ZGNR with only nearest neighbor hoppings (a)–(c) and up to second-nearest neighbors (d)–(f), for three values of electric field. The majority and minority spin sub-bands are shown by the blue and red line, respectively.
the ribbon width is increased [5]. In figure 2 we show the local magnetic moments at the opposite edges of an 8-ZGNR (16 atoms) as a function of the electric field magnitude. As the electric field increases, the magnetic moments at the edges are slightly reduced for increasing field intensities up to a critical intensity, at which a sudden drop occurs and the magnetic moments vanish. The magnetic moment values are almost independent of the presence of second neighbor hopping in the model Hamiltonian. For both descriptions (t′ = 0 and t′ = 0.1t0) the magnetization goes to zero for V > 0.23t0. However, important differences are found in the band structure and in the electronic density of states for the two cases, as depicted in figures 3(a)–(c) and 3(d)–(f). In particular, when V = 0 the majority (up) and minority (down) spin sub-bands are degenerate in both cases, and both display an energy gap. However, the inclusion of next neighbor hoppings breaks the electron– hole symmetry of the original model, as can be verified in the density of states. For increasing electric field values the majority–minority degeneracy is lifted; the gap in the majority spin band begins to close whereas the width of the minority band gap increases. For V = 0.23t0 the gap closes, as a signature of the half-metallic state. For larger fields, the gap eventually reopens and
state. This bound electron–hole pair (of opposite spins) is the magnon of itinerant ferromagnets. The transverse dynamical susceptibility χll'+− (q, Ω ) is a matrix, where l and l′ are sites within a unit cell and q is a wave vector parallel to the length of the ribbon. The spectral density of the spin waves is given by Al = −ℑχll+− ( q, Ω ), with l denoting the imaginary part. It may be interpreted as the local density of states of magnons with wave vector q on site l of the unit cell. The dynamic susceptibility matrix also describes the response of the system to an external field of frequency Ω, applied transversely to the ground state magnetization direction. The spin excitation frequencies are identified by the positions of the peaks of the spectral density. 3. Results First, we analyze the magnetic properties of ZGNRs as an electric field is applied in the transversal direction. When V = 0 (zero electric field), the magnetic moments are localized at the nanoribbon edges and decay rapidly towards the center of the ribbon [14, 19, 20]. There is a ferromagnetic exchange coupling along the edge and an antiferromagnetic exchange coupling between the edges in this ground state. The antiferromagnetic coupling is RKKY-like and tends to decrease as 3
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
The spin wave dispersion relations for the nearest neighbor model (second neighbor results are very similar) are shown in figure 5(b), considering different electric field magnitudes. For V = 0, the nanoribbon presents a magnetic moment of 0.26μB at the edge atoms and the dispersion relation has a quasi-linear behavior. As the electric field increases, the antiferromagnetic coupling begins to decrease because the magnetic moments at the edge sites are reduced, and the quadratic contribution of the ferromagnetic coupling along the edge is more noticeable. The dispersion relation for V = 0.23t0 (curve with down triangle symbols) clearly shows such behavior, being much less linear than the V = 0 case. For this particular potential energy, the edge magnetic moment is equal to 0.2μB. One interesting feature of this graphene system is the possibility of tuning the lifetime of the spin wave through the electric field. It is well known that the main contribution to the damping of spin waves in itinerant magnets comes from the decay in Stoner excitations [21–25]. As infinite ZGNRs are insulating, only spin waves with energies equal to or larger than the half gap energy are damped. In figure 6, we can observe the linewidth as a function of the wave vector for (a) t′ = 0 and (b) t′ = 0.1t0. When V = 0 for the two cases, the linewidths are approximately zero, indicating a very large spin wave lifetime, until they reach the value of half gap energy. As the electric field is increased, the spin wave linewidth with lower energies starts to increase. This is due to the fact that the energy gap is reduced with the electric field (figure 4), giving rise to a significant enhancement in the density of Stoner excitations. The behavior of the density of Stoner modes A0 as a function of the wave vector for t′ = 0 and t′ = 0.1t0 is also shown in figures 6(c) and (d), respectively. In contrast to the case without an electric field, the linewidth closely follows the behavior of A0 up to a certain wave vector, but saturates above this value, while the density of Stoner modes continues to increase. The behavior is qualitatively the same for both first- and second-nearest neighbor Hamiltonians, and it may indicate that, although there are Stoner modes in the same energy range as spin waves, the decay is not allowed. This hypothesis has to be tested by further investigations. Manipulation of the lifetime of the spin wave is an important issue in the development of spintronic devices. We have previously shown [5, 6] that one can manipulate the lifetime of the spin wave through doping, but in that case an instability of the magnetic order in the edge is generated, making it more difficult to use this mechanism in technological applications. For this reason it is important to verify that the application of an electric field does not generate a similar instability. To verify that, we analyze the behavior of the mean-field transverse susceptibility at zero frequency, as a function of the wave vector, A00 ( q ) = χ 0 ( q, Ω = 0). In a stable ferromagnetic system, A00 ( q ) has a maximum at q = 0. In figure 7, we can see that increasing the electric field A00 ( q ) to t′ = 0 and t′ = 0.1t0 always has a maximum at q = 0. The presence of a maximum at any finite q would indicate an instability, as in the case of doping [5].
Figure 4. Energy gap as a function of the electric potential for nearest and second-nearest neighbors. The majority and minority spin energy gap are shown by the blue and red lines, respectively.
magnetic ordering disappears. The gap reopening for larger fields can be understood by recalling results from calculations where electron–electron interactions are not considered. As shown in previous works [13], the electronic structure of ZGNRs is very sensitive to the values of the applied electric field. Differently from the degenerate flat bands exhibited at the Fermi energy for zero field, for finite fields the band structure is modified, exhibiting a degeneracy breaking of the previously flat bands, and the opening of a band gap. The gap increases with the field up to a critical value at which it goes down and closes again. Contrary to the interactive case, this complex evolution of the band gap is not related to magnetic ordering but rather to charge transfer effects and the broken symmetry promoted by the external different electric field. Taking into account second-nearest neighbor hoppings, the half-metallicity is more robust, occurring for a larger range of electric field values, as shown in figure 4. As expected, considering up to second neighbor hopping in the model calculation, a better similarity with the band structure calculated by DFT [14] is achieved. As we shall see, the small differences found in the gap sizes obtained by both model results (t′ = 0 and t′ = 0.1t0) shown in figure 4 will result in small changes in the spin wave lifetime. The main contribution for the spin excitation comes from the edges, where the magnetic moments are much larger than at the interior carbon sites. Thus, we restrict our analysis to the spectral density projected at one of the edges, which is labeled as l = 1 (the behavior of the opposite edge being symmetric). In figure 5(a), the spectral density A1(Ω) of an 8-ZGNR is shown as a function of Ω for three values of the wave vector q. The energy of the spin wave increases roughly linearly with the wave vector. As previously shown [5, 6], spin waves in infinite and finite ZGNRs present a linear energy dispersion relation at small q due to the antiferromagnetic coupling between the edges. This coupling can be reduced, however, when the nanoribbons are subjected to the effects of a transversal electric field. 4
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 5. (a) Spectral densities associated with spin waves projected at the upper edge for an 8-ZGNR system. (b) The spin wave dispersion relations obtained from the peaks of the spectral density for different values of electric fields.
Figure 6. Linewidth (a) and (b), and density of Stoner modes (c) and (d), as a function of the wave vector for different electric field values, considering t′ = 0 [(a), (c)] and t′ = 0.1t0 [(b), (d)] models.
4. Concluding remarks
interactions described by a mean-field Hubbard model. We highlight two important results of our calculations: first, the reduction in the antiferromagnetic coupling and a related decrease in the magnetic moments at the edges. This reduction is manifested in the behavior of the spin wave dispersion
We have investigated electric field effects on the spin excitations of ZGNRs using a tight-binding model with first- and second-nearest neighbor hoppings and electron–electron 5
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
[3] Kobayashi Y, Fukui K, Enoki T and Kusakabe K 2006 Phys. Rev. B 73 125415 [4] Tao C et al 2011 Nat. Phys. 7 616 [5] Culchac F J, Latgé A and Costa A T 2011 New J. Phys. 13 033028 [6] Culchac F J, Latgé A and Costa A T 2012 Phys. Rev. B 86 115407 [7] Ohno H, Chiba D, Matsukura F, Omiya E, Abe E, Dietl T, Ohno Y, Ohtani K 2000 Nature 408 944 Chiba D, Yamanouchi F, Matsukura F and Ohno H 2003 Science 301 943 [8] Chiba D, Sawicki M, Nishitani Y, Nakatani Y, Matsukura F and Ohno H 2008 Nature 455 515 [9] Weisheit M, Fähler S, Souche Y, Poinsignon C and Givord D 2007 Science 315 349 [10] Feldner H, Meng Z Y, Honecker A, Cabra D, Wessel S and Assaad F F 2010 Phys. Rev. B 83 115416 [11] Feldner H, Meng Z Y, Lang T C, Assaad F F, Wessel S and Honecker A 2011 Phys. Rev. Lett. 106 226401 [12] Golor M, Lang T C and Wessel S 2013 Phys. Rev. B 87 155441 [13] Chang C P, Huang Y C, Lu C L, Ho J H, Li T S and Lin M F 2006 Carbon 44 508 [14] Son Y W, Cohen M L and Louie S G 2006 Nature 444 347 [15] Son Y W, Cohen M L and Louie S G 2006 Phys. Rev. Lett. 97 216803 [16] Rudberg E, Salek P and Luo Y 2007 Nano Lett. 7 2211 [17] Kan E, Li Z, Yang J and Hou J G 2007 Appl. Phys. Lett. 91 243116 [18] Wakabayashi K and Dutta S 2012 Solid State Commun. 152 1420 [19] Pisani L, Chan J A, Montanari B and Harrison N M 2007 Phys. Rev. B 75 064418 [20] Jung J and MacDonald A H 2009 Phys. Rev. B 79 235433 [21] Khajetoorians A A, Lounis S, Chilian B, Costa A T, Zhou L, Mills D L, Wiebe J and Wiesendanger R 2011 Appl. Phys. Lett. 106 037205 [22] Barbosa L H, Muniz R B, Costa A T and Mathon J 2001 Phys. Rev. B 63 174401 [23] Muniz R B and Mills D L 2002 Phys. Rev. B 66 174417 [24] Tang H, Plihal M and Mills D L 1998 J. Magn. Magn. Mater. 187 23 [25] Costa A T, Muniz R B and Mills D L 2003 Phys. Rev. B 68 224435
Figure 7. The zero-frequency mean-field spectral density A00 ( q )
for different electric potentials for (a) t′ = 0 (solid line) and (b) t′ = 0.1t0 (dashed line).
relation. Second, the manipulation of the spin wave lifetime via an electric field, without producing magnetic instability, makes these systems extremely promising in the future development of spintronic devices. Acknowledgments The authors acknowledge financial support from CAPES, CNPq, INCT de Nanomateriais de Carbono. AL and RBC would like to thank FAPERJ. References [1] Pesin D and MacDonald A H 2012 Nat. Mater. 11 409 [2] Wakabayashi K, Fujita M, Ajiki H, Sigrist M 1999 Phys. Rev. B 59 8271 Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Japan 65 1920
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Magnetic response of zigzag nanoribbons under electric fields
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 216002 (http://iopscience.iop.org/0953-8984/26/21/216002) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.33.16.124 This content was downloaded on 19/06/2014 at 10:40
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 216002 (6pp)
doi:10.1088/0953-8984/26/21/216002
Magnetic response of zigzag nanoribbons under electric fields F J Culchac1, Rodrigo B Capaz1, A T Costa2 and A Latgé2 1
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil 2 Instituto de Física, Universidade Federal Fluminense, Niterói-RJ 24210-340, Brazil E-mail: [email protected] Received 23 November 2013, revised 5 February 2014 Accepted for publication 14 February 2014 Published 8 May 2014 Abstract
Spin excitations in zigzag graphene nanoribbons are studied when the system is subjected to an electric field in the transversal direction. The magnetic properties and the lifetime of the spin excitations are systematically investigated and compared using a tight-binding electron–electron model treated by a mean-field Hubbard model. The effects of electron–hole asymmetry introduced by next-nearest neighbor hopping are also investigated. We show that by increasing the electric field, the antiferromagnetic correlations between the edges of the nanoribbons are decreased due to a reduction of the magnetic moments. The results show that the spin wave lifetime may be controlled by the intensity of the transversal electric field, indicating that zigzag nanoribbons may be considered great candidates for future spintronic applications. Keywords: spin waves, graphene nanoribbons, spintronics (Some figures may appear in colour only in the online journal)
1. Introduction
both cases, a strong damping of the spin waves, thus decreasing their relaxation time considerably. Moreover, as the doping levels increase, the ferromagnetic alignment along the border becomes unstable. The possibility of controlling the spin relaxation times in such systems by doping can be conveniently used in designing new spintronic devices based on nanoribbons. Experimental manipulation of magnetization by applying electric fields on ferromagnetic semiconductor systems [7, 8] and thin-film ferromagnets (FePt and FePd thin layers) [9] has been largely reported in the literature. The search for magnetic systems that are easy to manipulate is the key parameter for spintronic data storage and processing. Similar processes may be designed for graphene systems in an attempt to modify the magnetic responses in a controlled way. Therefore, one additional possibility of manipulating the spin excitation lifetimes is the application of a transverse electric field. Calculations based on density-functional theory (DFT) [14–17] and tight-binding approximation [18] show that ZGNRs present half-metallicity properties, which considerably increases the possibility of using the ZGNRs for spintronic purposes.
Spintronics in graphene-based materials has emerged as an interesting theme in nanoscience mainly due to the variety of potential applications [1]. The presence of edges in graphene significantly alters the low energy spectrum of the πelectrons. In particular, graphene nanoribbons with zigzag edges (ZGNRs) present localized edge states at the Fermi energy [2] that favor the formation of magnetic moments. Experimental evidence of finite magnetization along ZGNR edges has been reported experimentally [3, 4] and has important implications for spin-dependent transport properties in these systems. We have shown in previous works [5, 6] that spin excitations in ZGNRs exhibit energy dispersion relations predominantly linear for large wavelengths, due to antiferromagnetic coupling between the magnetizations on the opposite edges. For infinite ZGNRs, the spin waves have an essentially infinite lifetime due to the presence of an energy gap in neutral nanoribbons [5]. For finite nanoribbons the spin waves are damped at all energies [6]. It was also shown that electron or hole doping caused by electrostatic gating may induce, in 0953-8984/14/216002+6$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 1. Schematic view of zigzag graphene nanoribbon geometry with an electric field applied across the ribbon in the transverse direction (y-direction). The dotted lines encircle two arbitrary unit cells, labeled m and m′. The indices l and l′ refer to atoms inside each unit cell, and ±V is the electric voltage defined by the field intensity.
Figure 2. Magnetic moment at the edges (sites 1 and 16) of an 8-ZGNR as a function of the electric field potential.
i with spin σ, and Vi is the electric potential energy associated with the transverse electric field. The magnetization of the ZGNRs is calculated self-consistently starting from initial values of the occupation numbers (〈niσ〉) that are chosen conveniently. The 〈niσ〉 are found by an interactive procedure until convergence, under the condition of global charge neutrality. For the nearest neighbor hopping case, where electron– hole symmetry is maintained, 〈niσ〉 can be evaluated keeping the Fermi level (EF = 0) fixed. When next-nearest neighbors are included, the electron-hole symmetry is broken and extra care must be taken to keep the Fermi level at a position where global charge neutrality is attained. The spin wave excitations are extracted from the transverse dynamical susceptibility,
In this work we study spin excitations in ZGNRs when a transverse electric field is applied. The electric field lifts the spin degeneracy of the nanostructure and promotes the formation of a half-metallic state beyond a critical field intensity that depends on the width of the ribbon. We are interested in investigating the changes induced by the electric field on the spin dynamic properties of ZGNRs, such as lifetimes and the stability of ferromagnetic order. We describe the electronic structure of the system using a tight-binding model plus a mean-field Hubbard term to take into account the effects of a screened Coulomb repulsion. 2. Theory
χij+− ( t ) = − iΘ ( t ) [ Si+ ( t ), S −j (0)] ,
The π-electrons in graphene are well described by the following Hamiltonian H = −t 0
∑ ∑ ci†σ cjσ − t ′ ∑ ∑ ci†σ cjσ + ∑ Ui(ni ↑ ij
1st
σ
ij
2nd
+ n i, ↑ n i ↓ − n i, ↑ n i, ↓
σ
where S + = c↑†c↓ and S− = (S+)† are spin-raising and lowering operators. For graphene nanoribbon geometry it is convenient to define a mixed Bloch-Wannier basis to describe the electronic states
n i, ↓
i
) + ∑ Vni iσ , iσ
(2)
(1)
cl ( q ) =
where t0 and t′ are the hopping integrals for the nearest and second-nearest neighbors, respectively, and we use t0 = 2.7 eV and t′/t0 = 0.1 following reference [18]. We assume the value of the on-site effective Coulomb interaction as U/t0 = 1, which reproduces the electronic structure of a ZGNR, as provided by first-principle calculations [4] and also correctly describes the magnetic effects of the nanoribbon, derived from the screened Coulomb interaction. The electron–electron interactions have been treated within a mean-field approximation in the description of the ground state. This level of approximation is enough to demonstrate the instability of the spin-unpolarized ground state, thus indicating that the true ground state should be spinpolarized. This conclusion is robust with respect to higherlevel treatments of electron–electron correlation [10–12]. The operator ci†σ creates an electron at the atomic state in the site
1 N
∑ eimkac ( xm , yl ), m
where c(xm, yl) is the annihilation operator for a Wannier state at a site l in unit cell m, a = 3 a 0 is the distance between unit cells and a 0 ≈ 1.42Å is the carbon-carbon distance. The unit cell of the zigzag ribbon is depicted in figure 1. Within the random-phase approximation it is possible to relate the interacting susceptibility to the mean-field susceptibility χ0 [22] 0 +− 0 χll+− ′ ( q , Ω ) = χ ′ ( q , Ω ) + ∑ χln ( q , Ω ) Unχ ′ ( q , Ω ). ll nl n
The spin excitations above the spin-polarized ground state have been treated within the random phase approximation, and may be considered as a kind of dynamical mean-field approximation. Within this level of approximation, the electron and hole of the spin excitation are correlated, forming a bound 2
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 3. Band structure for an 8-ZGNR with only nearest neighbor hoppings (a)–(c) and up to second-nearest neighbors (d)–(f), for three values of electric field. The majority and minority spin sub-bands are shown by the blue and red line, respectively.
the ribbon width is increased [5]. In figure 2 we show the local magnetic moments at the opposite edges of an 8-ZGNR (16 atoms) as a function of the electric field magnitude. As the electric field increases, the magnetic moments at the edges are slightly reduced for increasing field intensities up to a critical intensity, at which a sudden drop occurs and the magnetic moments vanish. The magnetic moment values are almost independent of the presence of second neighbor hopping in the model Hamiltonian. For both descriptions (t′ = 0 and t′ = 0.1t0) the magnetization goes to zero for V > 0.23t0. However, important differences are found in the band structure and in the electronic density of states for the two cases, as depicted in figures 3(a)–(c) and 3(d)–(f). In particular, when V = 0 the majority (up) and minority (down) spin sub-bands are degenerate in both cases, and both display an energy gap. However, the inclusion of next neighbor hoppings breaks the electron– hole symmetry of the original model, as can be verified in the density of states. For increasing electric field values the majority–minority degeneracy is lifted; the gap in the majority spin band begins to close whereas the width of the minority band gap increases. For V = 0.23t0 the gap closes, as a signature of the half-metallic state. For larger fields, the gap eventually reopens and
state. This bound electron–hole pair (of opposite spins) is the magnon of itinerant ferromagnets. The transverse dynamical susceptibility χll'+− (q, Ω ) is a matrix, where l and l′ are sites within a unit cell and q is a wave vector parallel to the length of the ribbon. The spectral density of the spin waves is given by Al = −ℑχll+− ( q, Ω ), with l denoting the imaginary part. It may be interpreted as the local density of states of magnons with wave vector q on site l of the unit cell. The dynamic susceptibility matrix also describes the response of the system to an external field of frequency Ω, applied transversely to the ground state magnetization direction. The spin excitation frequencies are identified by the positions of the peaks of the spectral density. 3. Results First, we analyze the magnetic properties of ZGNRs as an electric field is applied in the transversal direction. When V = 0 (zero electric field), the magnetic moments are localized at the nanoribbon edges and decay rapidly towards the center of the ribbon [14, 19, 20]. There is a ferromagnetic exchange coupling along the edge and an antiferromagnetic exchange coupling between the edges in this ground state. The antiferromagnetic coupling is RKKY-like and tends to decrease as 3
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
The spin wave dispersion relations for the nearest neighbor model (second neighbor results are very similar) are shown in figure 5(b), considering different electric field magnitudes. For V = 0, the nanoribbon presents a magnetic moment of 0.26μB at the edge atoms and the dispersion relation has a quasi-linear behavior. As the electric field increases, the antiferromagnetic coupling begins to decrease because the magnetic moments at the edge sites are reduced, and the quadratic contribution of the ferromagnetic coupling along the edge is more noticeable. The dispersion relation for V = 0.23t0 (curve with down triangle symbols) clearly shows such behavior, being much less linear than the V = 0 case. For this particular potential energy, the edge magnetic moment is equal to 0.2μB. One interesting feature of this graphene system is the possibility of tuning the lifetime of the spin wave through the electric field. It is well known that the main contribution to the damping of spin waves in itinerant magnets comes from the decay in Stoner excitations [21–25]. As infinite ZGNRs are insulating, only spin waves with energies equal to or larger than the half gap energy are damped. In figure 6, we can observe the linewidth as a function of the wave vector for (a) t′ = 0 and (b) t′ = 0.1t0. When V = 0 for the two cases, the linewidths are approximately zero, indicating a very large spin wave lifetime, until they reach the value of half gap energy. As the electric field is increased, the spin wave linewidth with lower energies starts to increase. This is due to the fact that the energy gap is reduced with the electric field (figure 4), giving rise to a significant enhancement in the density of Stoner excitations. The behavior of the density of Stoner modes A0 as a function of the wave vector for t′ = 0 and t′ = 0.1t0 is also shown in figures 6(c) and (d), respectively. In contrast to the case without an electric field, the linewidth closely follows the behavior of A0 up to a certain wave vector, but saturates above this value, while the density of Stoner modes continues to increase. The behavior is qualitatively the same for both first- and second-nearest neighbor Hamiltonians, and it may indicate that, although there are Stoner modes in the same energy range as spin waves, the decay is not allowed. This hypothesis has to be tested by further investigations. Manipulation of the lifetime of the spin wave is an important issue in the development of spintronic devices. We have previously shown [5, 6] that one can manipulate the lifetime of the spin wave through doping, but in that case an instability of the magnetic order in the edge is generated, making it more difficult to use this mechanism in technological applications. For this reason it is important to verify that the application of an electric field does not generate a similar instability. To verify that, we analyze the behavior of the mean-field transverse susceptibility at zero frequency, as a function of the wave vector, A00 ( q ) = χ 0 ( q, Ω = 0). In a stable ferromagnetic system, A00 ( q ) has a maximum at q = 0. In figure 7, we can see that increasing the electric field A00 ( q ) to t′ = 0 and t′ = 0.1t0 always has a maximum at q = 0. The presence of a maximum at any finite q would indicate an instability, as in the case of doping [5].
Figure 4. Energy gap as a function of the electric potential for nearest and second-nearest neighbors. The majority and minority spin energy gap are shown by the blue and red lines, respectively.
magnetic ordering disappears. The gap reopening for larger fields can be understood by recalling results from calculations where electron–electron interactions are not considered. As shown in previous works [13], the electronic structure of ZGNRs is very sensitive to the values of the applied electric field. Differently from the degenerate flat bands exhibited at the Fermi energy for zero field, for finite fields the band structure is modified, exhibiting a degeneracy breaking of the previously flat bands, and the opening of a band gap. The gap increases with the field up to a critical value at which it goes down and closes again. Contrary to the interactive case, this complex evolution of the band gap is not related to magnetic ordering but rather to charge transfer effects and the broken symmetry promoted by the external different electric field. Taking into account second-nearest neighbor hoppings, the half-metallicity is more robust, occurring for a larger range of electric field values, as shown in figure 4. As expected, considering up to second neighbor hopping in the model calculation, a better similarity with the band structure calculated by DFT [14] is achieved. As we shall see, the small differences found in the gap sizes obtained by both model results (t′ = 0 and t′ = 0.1t0) shown in figure 4 will result in small changes in the spin wave lifetime. The main contribution for the spin excitation comes from the edges, where the magnetic moments are much larger than at the interior carbon sites. Thus, we restrict our analysis to the spectral density projected at one of the edges, which is labeled as l = 1 (the behavior of the opposite edge being symmetric). In figure 5(a), the spectral density A1(Ω) of an 8-ZGNR is shown as a function of Ω for three values of the wave vector q. The energy of the spin wave increases roughly linearly with the wave vector. As previously shown [5, 6], spin waves in infinite and finite ZGNRs present a linear energy dispersion relation at small q due to the antiferromagnetic coupling between the edges. This coupling can be reduced, however, when the nanoribbons are subjected to the effects of a transversal electric field. 4
F J Culchac
J. Phys.: Condens. Matter 26 (2014) 216002
Figure 5. (a) Spectral densities associated with spin waves projected at the upper edge for an 8-ZGNR system. (b) The spin wave dispersion relations obtained from the peaks of the spectral density for different values of electric fields.
Figure 6. Linewidth (a) and (b), and density of Stoner modes (c) and (d), as a function of the wave vector for different electric field values, considering t′ = 0 [(a), (c)] and t′ = 0.1t0 [(b), (d)] models.
4. Concluding remarks
interactions described by a mean-field Hubbard model. We highlight two important results of our calculations: first, the reduction in the antiferromagnetic coupling and a related decrease in the magnetic moments at the edges. This reduction is manifested in the behavior of the spin wave dispersion
We have investigated electric field effects on the spin excitations of ZGNRs using a tight-binding model with first- and second-nearest neighbor hoppings and electron–electron 5
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[3] Kobayashi Y, Fukui K, Enoki T and Kusakabe K 2006 Phys. Rev. B 73 125415 [4] Tao C et al 2011 Nat. Phys. 7 616 [5] Culchac F J, Latgé A and Costa A T 2011 New J. Phys. 13 033028 [6] Culchac F J, Latgé A and Costa A T 2012 Phys. Rev. B 86 115407 [7] Ohno H, Chiba D, Matsukura F, Omiya E, Abe E, Dietl T, Ohno Y, Ohtani K 2000 Nature 408 944 Chiba D, Yamanouchi F, Matsukura F and Ohno H 2003 Science 301 943 [8] Chiba D, Sawicki M, Nishitani Y, Nakatani Y, Matsukura F and Ohno H 2008 Nature 455 515 [9] Weisheit M, Fähler S, Souche Y, Poinsignon C and Givord D 2007 Science 315 349 [10] Feldner H, Meng Z Y, Honecker A, Cabra D, Wessel S and Assaad F F 2010 Phys. Rev. B 83 115416 [11] Feldner H, Meng Z Y, Lang T C, Assaad F F, Wessel S and Honecker A 2011 Phys. Rev. Lett. 106 226401 [12] Golor M, Lang T C and Wessel S 2013 Phys. Rev. B 87 155441 [13] Chang C P, Huang Y C, Lu C L, Ho J H, Li T S and Lin M F 2006 Carbon 44 508 [14] Son Y W, Cohen M L and Louie S G 2006 Nature 444 347 [15] Son Y W, Cohen M L and Louie S G 2006 Phys. Rev. Lett. 97 216803 [16] Rudberg E, Salek P and Luo Y 2007 Nano Lett. 7 2211 [17] Kan E, Li Z, Yang J and Hou J G 2007 Appl. Phys. Lett. 91 243116 [18] Wakabayashi K and Dutta S 2012 Solid State Commun. 152 1420 [19] Pisani L, Chan J A, Montanari B and Harrison N M 2007 Phys. Rev. B 75 064418 [20] Jung J and MacDonald A H 2009 Phys. Rev. B 79 235433 [21] Khajetoorians A A, Lounis S, Chilian B, Costa A T, Zhou L, Mills D L, Wiebe J and Wiesendanger R 2011 Appl. Phys. Lett. 106 037205 [22] Barbosa L H, Muniz R B, Costa A T and Mathon J 2001 Phys. Rev. B 63 174401 [23] Muniz R B and Mills D L 2002 Phys. Rev. B 66 174417 [24] Tang H, Plihal M and Mills D L 1998 J. Magn. Magn. Mater. 187 23 [25] Costa A T, Muniz R B and Mills D L 2003 Phys. Rev. B 68 224435
Figure 7. The zero-frequency mean-field spectral density A00 ( q )
for different electric potentials for (a) t′ = 0 (solid line) and (b) t′ = 0.1t0 (dashed line).
relation. Second, the manipulation of the spin wave lifetime via an electric field, without producing magnetic instability, makes these systems extremely promising in the future development of spintronic devices. Acknowledgments The authors acknowledge financial support from CAPES, CNPq, INCT de Nanomateriais de Carbono. AL and RBC would like to thank FAPERJ. References [1] Pesin D and MacDonald A H 2012 Nat. Mater. 11 409 [2] Wakabayashi K, Fujita M, Ajiki H, Sigrist M 1999 Phys. Rev. B 59 8271 Fujita M, Wakabayashi K, Nakada K, Kusakabe K 1996 J. Phys. Soc. Japan 65 1920
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