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Ferromagnetic properties in low-doped zigzag graphene nanoribbons
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2016 J. Phys.: Condens. Matter 28 086001 (http://iopscience.iop.org/0953-8984/28/8/086001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.122.230.132 This content was downloaded on 11/02/2016 at 18:23
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 28 (2016) 086001 (5pp)
doi:10.1088/0953-8984/28/8/086001
Ferromagnetic properties in low-doped zigzag graphene nanoribbons Shuaiyu Li1, Lin Tian1, Lingli Shi1, Lin Wen2 and Tianxing Ma1,2 1
Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China Beijing Computational Science Research Center, Beijing 100094, People’s Republic of China
2
E-mail: [email protected] Received 12 September 2015, revised 6 December 2015 Accepted for publication 5 January 2016 Published 29 January 2016 Abstract
The temperature-dependent edge magnetic susceptibility χe and the uniform magnetic susceptibility χ in zigzag graphene nanoribbons is studied within the Hubbard model on a honeycomb lattice. By using the determinant quantum Monte Carlo (DQMC) method, it is found that the ferromagnetic fluctuations at the zigzag edge dominate around half-filling, and that the fluctuations are strengthened markedly by the on-site Coulomb interaction U, which may lead to a possible high-temperature edge ferromagnetic behaviour in low-doped zigzag graphene nanoribbons. Keywords: graphene, magnetism, determinant quantum Monte Carlo method (Some figures may appear in colour only in the online journal)
confirmation, and further studies in this area are promptly needed. In graphene-based materials, it is particularly interesting that their chemical potential can be tuned and that hence it is possible to induce electron or hole doping in such mat erials; this renders graphene susceptible to interaction-driven instabilities, and also opens the door for carbon-based electronics [18]. Regarding graphene nanoribbons, one of their most interesting aspects is that their possible magnetism may be linked with the electron filling and the boundary [21–24]. The temperature-dependent magnetic susceptibility plays a key role in understanding the behaviour of magnetism and is used in this paper as a probe to magnetic correlations. In present paper, we mainly studied the temperature-dependent uniform magnetic susceptibility χu and the edge magnetic susceptibility χe in doped zigzag graphene nanoribbons within the Hubbard model on a honeycomb lattice. By using the determinant quantum Monte Carlo (DQMC) method, our non-perturbative calculations reveal that a ferromagnetic-like behaviour may be induced by a relatively large interaction at the zigzag edge in the graphene nanoribbons. The proposed edge magnetism is important for the realisation of graphenebased nanoelectronics [25, 26]. The sketch for the zigzag graphene nanoribbons has been shown in figure 1, in which the blue and white circles
The graphene-based system is an emerging topic of research which attracts impressive interest in various fields, including physics, chemistry, materials, and engineering [1–8]. Perfect graphene consists of a layer of carbon atoms arranged in a honeycomb lattice, and each unit cell of the hexagonal Bravais lattice contains two carbon atoms, which give rise to two sublattices [1]. Due to this bi-particle nature, graphene nanoribbons can have two different kinds of edge: zigzag or armchair; see figure 1, where a graphene nanoribbon with a zigzag edge is produced if we consider the system is infinite in the x-direction while posessing a finite length along the y-direction. The possible magnetism in graphene nanoribbons may hold the promise of many applications in the design of nanoscale magnetic and spintronics devices [9–11]. It has been shown that the zigzag graphene nanoribbons exhibit a ferromagnetic correlation along the edge at half-filling due to electron–electron correlation [12–15], and a room temperature magnetic order on zigzag edges of narrow graphene nanoribbons is proposed [16]. Armchair graphene nanoribbons, may have ferromagnetic fluctuations in a heavily doped region around the nearly flat band [17]; as well as the perfect graphene [18]. The ferromagneticlike behaviour is also predicted in the graphene quantum dot under strain [19, 20]. The already reported possible magnetism referenced above is waiting for experimental 0953-8984/16/086001+5$33.00
1
© 2016 IOP Publishing Ltd Printed in the UK
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
transverse width and also the edge topology [31–33]. The basic algorithm of the DQMC method is to supply the partition function as a high-dimensional integral over a set of random auxiliary fields, and then the integral is accomplished by Monte Carlo techniques, where 8000 sweeps were used to equilibrate the system, and an additional 30 000 sweeps were then made, each of which generated a measurement. These measurements were separated into ten bins which provide the basis of coarse-grain averages and errors were estimated based on standard deviations from the average. To explore the magnetic properties in zigzag graphene nanoribbons, we calculate the magnetic susceptibility in the z direction at zero frequency,
Figure 1. A piece of a honeycomb lattice displaying zigzag edges with Ly = 4 which defines the width of the ribbon in the transverse direction and Lx = 12 indicating the length in the longitudinal direction. The lattice size N = 2 × 4 × 12 = 96 where the blue and white circles indicate the A and B sublattices respectively, and the sites at the top edge (white circles, Ni = 73, 75, ..., 95) and the bottom edge (blue circles, Ni = 2, 4, ..., 24) are plotted with larger size.
χ=
respectively represent sublattices A and B, and the transverse integer index y = 1, 2, ..., Ly defines the width of the ribbon while x = 1, 2, ..., Lx at the zigzag edge defines the length. We construct the ribbon to be infinite in the x direction and finite in the y direction. Recently, the Peierls–Feynman–Bogoliubov variational principle shows that a generalised Hubbard model with non-local Coulomb interactions is mapped onto an effective Hubbard model with on-site effective interaction U only, which is a good starting point for graphene-based material [27]. Thus the physics of zigzag graphene nanoribbons can be described by the Hubbard model [8, 12, 18, 30, 33], ijσ
+µ ∑(naiσ + nbiσ )
dτ
∑ ∑ eiq⋅ (i − j ′ )〈mi (τ ) ⋅ m j ′ (0)〉 d
d , d ′ = a, b i , j
d
d
d
where mia(τ ) = e Hτ mia(0)e−Hτ with mia = a†i ↑ai ↑ − a†i ↓ai ↓ and mib = b†i ↑bi ↑ − b†i ↓bi ↓. We measure χ in units of 1/ | t|, and in the following we shall study not only the uniform magnetic susceptibility χu, but also the magnetic susceptibility at different edges, especially the edge magnetism at the zigzag edge. The uniform magnetic susceptibility χu refers to the bulk system, and the summation in equation (2) is made over every site in the simulations. For the edge magnetic susceptibility χe at the zigzag edge, the summation is made over the sites along a single zigzag edge firstly, and then an average over the results from both the top edge and the bottom edge is made. The top (white circles, Ni = 73, 75, ..., 95) and the down (blue circles, Ni = 2, 4, ..., 24) zigzag edges, have been marked with a larger size in figure 1. For comparison, the edge magnetism at the armchair edge, χa is simulated as a summation of sites at a single armchair edge; for example, the sites marked by 23, 48, 71, 96 at the right armchair edge. We also define the inter-edge magnetic susceptibility χin which refers to the summation in equation (2), indicating that it is made over every site except the sites at both the top and bottom zigzag edges. Firstly, we present the temperature-dependence of the edge magnetic susceptibility χe in figure 2 around half-filling with U = 3.0| t| for L x = 12, L y = 4. The hole doping level studied here δ = 1.00 − n is no larger than 5 percent which could be realised in experiments either by the electric gate and/or chemical doping methods [1, 2]. It is interesting to see that the χe increases as the temperature decreases, especially in the low temperature region, which shows that the ferromagnetic fluctuations may dominate in the behaviour of the edge magnetic susceptibility. Comparing results with different electron fillings, one may also see that the χe is suppressed as the system is doped away from half-filling. To qualitatively estimate the behaviour of the temperature dependence of the edge magn etic susceptibility, we plot the function y = 1/x based on the Curie–Weiss law χ = C /(T − Tc ) which can describe the magnetic susceptibility χ for a ferromagnetic material in the temperature region above the Curie temperature Tc. It is important to see that the χe at half-filling ( blue line with circles ) is beyond the function y = 1/x, which indicates that the χe tends to be divergent at some low temperatures, and a ferromagnetic behaviour could be expected.
H = −t ∑(a†iσbi + jσ + h.c.) + U ∑(nai ↑nai ↓ + nbi ↑nbi ↓)
∫0
β
i
(1)
iσ
where t is the nearest hopping integrals, μ is the chemical potential and U is the on-site repulsion. Here, aiσ (a†iσ) annihilates (creates) electrons at the site Ri with spin σ (σ =↑, ↓) on sublattice A, as well as biσ (b†iσ) acting on the electrons of sublattice B, naiσ = aiσ a†iσ and nbiσ = biσ b†iσ. In the simulations, we apply periodic boundary conditions at the x direction and open boundary conditions at the zigzag edge, and we assume that the hoping integral is the same for all atoms in our model. This assumption is a reasonable approximation because the providing conditions are at the low-energy region so that the disturbance of the atoms has little influence on other additional atoms [28, 29]. On the honeycomb network for graphene nanoribbons, the nearest-neighbour hopping energy t reported in the literature ranges from 2.5 to 2.8 eV, and the value of the onsite repulsion U can be taken from the estimation from 6.0 to 16.93 eV in polyacetylene [30]. From the Peierls–Feynman– Bogoliubov variational results [27] and following the latter reference we study the model Hamiltonian in the range of U / | t | = 1.0 ∽ 5.0, which puts graphene-based material in the region of a moderate electronic correlation system. Thus, we treat the electronic interactions in such a system by using the DQMC method, which is very suitable for simulating and investigating the nature of magnetic correlation in the presence of moderate Coulomb interactions; especially when treating the band structure changes with respect to the finite 2
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
Figure 3. The temperature-dependent edge magnetic susceptibility χe at n = 1.00 with various on-site repulsion.
Figure 2. The temperature-dependent edge magnetic susceptibility
for different electron fillings at U = 3.0 | t|.
The ferromagnetic-like behaviour in the edge magnetic susceptibility at half-filling is exciting as it is important for realising the graphene-based materials into the application of spintronics [34]. We further our story with different interaction, and the temperature-dependent edge magnetic susceptibility with different U at half-filling have been shown in figure 3. The edge magnetic susceptibility is enhanced as the interaction increases, which leads to a possible divergence of edge magnetic susceptibility at higher temperature. And we fit the DQMC data of U = 3.0| t| with a formula of χe(T ) = a /(T − Tc ) + b (2)
as shown (dashed lines) in figure 3, which allows us to estimate the transition temperature Tc. From equation (2), one may see that at T = Tc, the χe(T ) tends to diverge, and we may estimate the Tc from the fitting data where the χe may diverge at some temperatures. For the data shown in figure 3 for U = 3.0| t| and n = 1.00, Tc is estimated about ∼0.013t (roughly ∼ 325 K) according to our fitting. One can notice significant error bars on the susceptibility, related to the Monte Carlo sampling in lower temperatures. To estimate the error bar of the Tc, we use the standard rule for estimating errors of indirect measurement, and, we use the susceptibility at the lowest temperature Tlowest to estimate δTc = aδχe(Tlowest )/χ2e (Tlowest ) 0.004 | t|, which indicates that the value of Tc should be statistically distinguishable from zero. In graphene-based material, one of its important properties is that its chemical potential can be tuned and hence it is possible to induce electron or hole doping in graphene; this opens the door for carbon-based electronics [18]. From the results shown in figure 2, it is likely that the edge magnetic susceptibility at n = 0.98 has no potential to divergence at U = 3.0| t|. A system with mobile electrons or holes shows that a ferromagnetic behaviour may also be important. We study the edge magnetic susceptibility χe at n = 0.98 with different interactions U in figure 4, in which the χe reveals that it is also possible to show a ferromagnetic behaviour as the interaction is large enough, for example, U ⩾ 4.0| t|.
Figure 4. The temperature-dependent edge magnetic susceptibility χe at n = 0.98 with various on-site repulsion.
The temperature-dependent uniform magnetic susceptibility is shown in figure 5, in which results for n = 0.80 (dark line with circle), n = 0.75 (red line with down triangle) and n = 0.70 (green line with up triangle) are shown. The ferromagnetic fluctuations may dominate in the heavily doped region, where the electron filling is below n = 0.80. This result is similar to that of the armchair graphene nanoribbons [17] or perfect graphene [18], which indicates that uniform magnetism is difficult to realise in graphene-based material as the doping level is far from current experimental ability. Comparing results present in figures 2–5, we argue here that the edge magnetism is more easily achieved experimentally [35, 36]. Finally, we compare the uniform magnetic susceptibility χu, the magnetic susceptibility χe at the zigzag edge, the magnetic susceptibility χa at the armchair edge, and also the inter-edge magnetic susceptibility χin in figure 6 for different fillings and lattice widths. Clearly, the χe (solid lines with circles) at the zigzag edge is much larger than the uniform magnetic susceptibility χu (dashed red lines with square) for the bulk system, 3
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
needed for nonvolatile transistors, many different characteristics are considered including high transition temperature, high carrier mobility, and intrinsic insulation conditions for carrier doping [37]. The proposed possible ferromagnetic-like behaviour here should be important for such kinds of searching, and hence should be useful for future nanodevices. Acknowledgments T Ma thanks Chinese Academy of Engineering Physics (CAEP) for partial financial support. This work is supported by National Natural Science Foundation of China (NSFC)s (Grant. Nos. 11374034 and 11334012), and the Fundamental Research Funds for the Central Universities (No. 2014KJJCB26). This research work is supported by a Tianhe-2JK computing time award at the Beijing Computational Science Research Center (CSRC).
Figure 5. The temperature-dependent uniform magnetic susceptibility χu for several values of n at U = 3.0 | t|.
References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [2] Geim A K and Novoselov K S 2007 Nat. Mater. 6 183 [3] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 [4] Tombros N, Josaz C, Popinciue M, Jonkman H T and van Wees B J 2007 Nature 448 571 [5] Peres N M R 2010 Rev. Mod. Phys. 82 2673 [6] Castro E V, Novoselov K S, Morozov S V, Peres N M R, Lopes dos Santos J M B, Nilsson J, Guinea F, Geim A K and Castro Neto A H 2007 Phys. Rev. Lett. 99 216802 [7] Ma T, Liang C, Wang L-G and Lin H-Q 2012 Appl. Phys. Lett. 100 252402 Li C, Cheng H, Chen R, Ma T, Wang L-G, Song Y and Lin H-Q 2013 Appl. Phys. Lett. 103 172106 Cheng H, Li C, Ma T, Wang L-G, Song Y and Lin H-Q 2014 Appl. Phys. Lett. 105 072103 Fan X, Huang W, Ma T, Wang L-G and Lin H-Q 2015 Europhys. Lett. 112 58003 [8] Lin H, Hikihara T, Jeng H T, Huang B L, Mou C Y and Hu X 2009 Phys. Rev. B 79 035405 [9] Apel W, Pal G and Schweitzer L 2011 Phys. Rev. B 83 125431 [10] Bao W, Zou L-j and Lin H-Q 2015 J. Appl. Phys. 117 17D905 [11] Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Molnar S, Roukes M L, Chtchelkanova A Y and Treger D M 2001 Science 294 1488 [12] Feldner H, Meng Z Y, Lang T C, Assaad F F, Wessel S and Honecker A 2011 Phys. Rev. Lett. 106 226401 [13] Miyamoto Y, Nakada K and Fujita M 1999 Phys. Rev. B 59 9858 [14] Jung J and MacDonald A H 2009 Phys. Rev. B 79 235433 [15] Güçlü A D, Grabowski M and Hawrylak P 2013 Phys. Rev. B 87 035435 [16] Magda G Z, Jin X Z, Hagymási I, Vancsó P, Osváth Z, Nemes-Incze P, Hwang C, Biró L P and Tapasztó L 2014 Nature 514 608 [17] Ma T, Liu S, Gao P, Huang Z B and Lin H Q 2012 J. Appl. Phys. 112 073922 [18] Ma T, Hu F, Huang Z and Lin H Q 2010 Appl. Phys. Lett. 97 112504 [19] Viana-Gomes J, Pereira Vitor M and Peres N M R 2009 Phys. Rev. B 80 245436
Figure 6. The temperature-dependent χu, χe, χin, and χa for several
values of n at U = 3.0 | t| on different lattice widths.
and also larger than the χa (dark lines with solid circle) at the armchair edge. Compared with different lattice width, Ly = 4 (blue lines) and Ly = 6 (pink lines), the χe is slightly enhanced as the width increases. It is, however, difficult to estimate the possible difference between divergence depends on different width, due to the notorious sign problem in DQMC methods, which prevent us from having accurate results in the lower temperature region and with larger system sizes. The interedge magnetic susceptibility χin for n = 0.98 (dotted dark lines) and n = 0.95 (dotted green lines) are shown for a lattice with 2 × 4 × 12 sites. The χin is much lower than χe , and it increases slightly as the electron filling decreases. In summary, we have studied the ferromagnetic properties in doped zigzag graphene nanoribbons by using the DQMC method. Our unbiased numerical results show that a ferromagn etic-like behaviour is induced by relative large interactions at the zigzag edge in the graphene nanoribbons. In nanoelectronics, microelectronic circuits that retain their logic state when the power is off would permit entirely new kinds of computers, and ferromagnetic semiconductors might make this technology possible. When searching for the ferromagnetic semiconductors 4
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
[29] Son Y W, Cohen M L and Louie S G 2006 Phys. Rev. Lett. 97 216803 [30] Neto A H C, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [31] Blankenbecler R, Scalapino D J and Sugar R L 1981 Phys. Rev. D 24 2278 [32] Hirsch J E 1985 Phys. Rev. B 31 4403 [33] Ma T, Huang Z-B, Hu F and Lin H-Q 2011 Phys. Rev. B 84 121410(R) [34] Jiang J, Lu W and Bernholc J 2008 Phys. Rev. Lett. 101 246803 [35] Li G, Luican A, Lopes dos Santos J M B, Castro A H, Reina A, Kong J and Andrei E Y 2010 Nat. Phys. 6 109 [36] Bhowmick S and Shenoy V B 2008 J. Chem. Phys. 128 244717 [37] Ando K 2006 Science 312 1883
[20] Cheng S, Yu J, Ma T and Peres N M R 2015 Phys. Rev. B 91 075410 [21] Fujita M, Wakabayashi K, Nakada K and Kusakabe K 1996 J. Phys. Soc. Japan 65 1920 [22] Nakada K, Fujita M, Dresselhaus G and Dresselhaus M S 1996 Phys. Rev. B 59 17954 [23] Okada S and Oshiyama A 2001 Phys. Rev. Lett. 87 146803 [24] Lee H, Son Y-W, Park N, Han S and Yu J 2005 Phys. Rev. B 72 1744 31 [25] Yan X, Cui X, Li B and Li L-S 2010 Nano Lett. 10 1869 [26] Sepioni M, Nair R R, Rablen S, Narayanan J, Tuna F, Winpenny R, Geim A K and Grigorieva I V 2010 Phys. Rev. Lett. 105 207205 [27] Schüler M, Rösner M, Wehling T O, Lichtenstein A I and Katsnelson M I 2013 Phys. Rev. Lett. 111 036601 [28] Fernéndez-Rossier J 2008 Phys. Rev. B 77 075430
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Ferromagnetic properties in low-doped zigzag graphene nanoribbons
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2016 J. Phys.: Condens. Matter 28 086001 (http://iopscience.iop.org/0953-8984/28/8/086001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.122.230.132 This content was downloaded on 11/02/2016 at 18:23
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 28 (2016) 086001 (5pp)
doi:10.1088/0953-8984/28/8/086001
Ferromagnetic properties in low-doped zigzag graphene nanoribbons Shuaiyu Li1, Lin Tian1, Lingli Shi1, Lin Wen2 and Tianxing Ma1,2 1
Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China Beijing Computational Science Research Center, Beijing 100094, People’s Republic of China
2
E-mail: [email protected] Received 12 September 2015, revised 6 December 2015 Accepted for publication 5 January 2016 Published 29 January 2016 Abstract
The temperature-dependent edge magnetic susceptibility χe and the uniform magnetic susceptibility χ in zigzag graphene nanoribbons is studied within the Hubbard model on a honeycomb lattice. By using the determinant quantum Monte Carlo (DQMC) method, it is found that the ferromagnetic fluctuations at the zigzag edge dominate around half-filling, and that the fluctuations are strengthened markedly by the on-site Coulomb interaction U, which may lead to a possible high-temperature edge ferromagnetic behaviour in low-doped zigzag graphene nanoribbons. Keywords: graphene, magnetism, determinant quantum Monte Carlo method (Some figures may appear in colour only in the online journal)
confirmation, and further studies in this area are promptly needed. In graphene-based materials, it is particularly interesting that their chemical potential can be tuned and that hence it is possible to induce electron or hole doping in such mat erials; this renders graphene susceptible to interaction-driven instabilities, and also opens the door for carbon-based electronics [18]. Regarding graphene nanoribbons, one of their most interesting aspects is that their possible magnetism may be linked with the electron filling and the boundary [21–24]. The temperature-dependent magnetic susceptibility plays a key role in understanding the behaviour of magnetism and is used in this paper as a probe to magnetic correlations. In present paper, we mainly studied the temperature-dependent uniform magnetic susceptibility χu and the edge magnetic susceptibility χe in doped zigzag graphene nanoribbons within the Hubbard model on a honeycomb lattice. By using the determinant quantum Monte Carlo (DQMC) method, our non-perturbative calculations reveal that a ferromagnetic-like behaviour may be induced by a relatively large interaction at the zigzag edge in the graphene nanoribbons. The proposed edge magnetism is important for the realisation of graphenebased nanoelectronics [25, 26]. The sketch for the zigzag graphene nanoribbons has been shown in figure 1, in which the blue and white circles
The graphene-based system is an emerging topic of research which attracts impressive interest in various fields, including physics, chemistry, materials, and engineering [1–8]. Perfect graphene consists of a layer of carbon atoms arranged in a honeycomb lattice, and each unit cell of the hexagonal Bravais lattice contains two carbon atoms, which give rise to two sublattices [1]. Due to this bi-particle nature, graphene nanoribbons can have two different kinds of edge: zigzag or armchair; see figure 1, where a graphene nanoribbon with a zigzag edge is produced if we consider the system is infinite in the x-direction while posessing a finite length along the y-direction. The possible magnetism in graphene nanoribbons may hold the promise of many applications in the design of nanoscale magnetic and spintronics devices [9–11]. It has been shown that the zigzag graphene nanoribbons exhibit a ferromagnetic correlation along the edge at half-filling due to electron–electron correlation [12–15], and a room temperature magnetic order on zigzag edges of narrow graphene nanoribbons is proposed [16]. Armchair graphene nanoribbons, may have ferromagnetic fluctuations in a heavily doped region around the nearly flat band [17]; as well as the perfect graphene [18]. The ferromagneticlike behaviour is also predicted in the graphene quantum dot under strain [19, 20]. The already reported possible magnetism referenced above is waiting for experimental 0953-8984/16/086001+5$33.00
1
© 2016 IOP Publishing Ltd Printed in the UK
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
transverse width and also the edge topology [31–33]. The basic algorithm of the DQMC method is to supply the partition function as a high-dimensional integral over a set of random auxiliary fields, and then the integral is accomplished by Monte Carlo techniques, where 8000 sweeps were used to equilibrate the system, and an additional 30 000 sweeps were then made, each of which generated a measurement. These measurements were separated into ten bins which provide the basis of coarse-grain averages and errors were estimated based on standard deviations from the average. To explore the magnetic properties in zigzag graphene nanoribbons, we calculate the magnetic susceptibility in the z direction at zero frequency,
Figure 1. A piece of a honeycomb lattice displaying zigzag edges with Ly = 4 which defines the width of the ribbon in the transverse direction and Lx = 12 indicating the length in the longitudinal direction. The lattice size N = 2 × 4 × 12 = 96 where the blue and white circles indicate the A and B sublattices respectively, and the sites at the top edge (white circles, Ni = 73, 75, ..., 95) and the bottom edge (blue circles, Ni = 2, 4, ..., 24) are plotted with larger size.
χ=
respectively represent sublattices A and B, and the transverse integer index y = 1, 2, ..., Ly defines the width of the ribbon while x = 1, 2, ..., Lx at the zigzag edge defines the length. We construct the ribbon to be infinite in the x direction and finite in the y direction. Recently, the Peierls–Feynman–Bogoliubov variational principle shows that a generalised Hubbard model with non-local Coulomb interactions is mapped onto an effective Hubbard model with on-site effective interaction U only, which is a good starting point for graphene-based material [27]. Thus the physics of zigzag graphene nanoribbons can be described by the Hubbard model [8, 12, 18, 30, 33], ijσ
+µ ∑(naiσ + nbiσ )
dτ
∑ ∑ eiq⋅ (i − j ′ )〈mi (τ ) ⋅ m j ′ (0)〉 d
d , d ′ = a, b i , j
d
d
d
where mia(τ ) = e Hτ mia(0)e−Hτ with mia = a†i ↑ai ↑ − a†i ↓ai ↓ and mib = b†i ↑bi ↑ − b†i ↓bi ↓. We measure χ in units of 1/ | t|, and in the following we shall study not only the uniform magnetic susceptibility χu, but also the magnetic susceptibility at different edges, especially the edge magnetism at the zigzag edge. The uniform magnetic susceptibility χu refers to the bulk system, and the summation in equation (2) is made over every site in the simulations. For the edge magnetic susceptibility χe at the zigzag edge, the summation is made over the sites along a single zigzag edge firstly, and then an average over the results from both the top edge and the bottom edge is made. The top (white circles, Ni = 73, 75, ..., 95) and the down (blue circles, Ni = 2, 4, ..., 24) zigzag edges, have been marked with a larger size in figure 1. For comparison, the edge magnetism at the armchair edge, χa is simulated as a summation of sites at a single armchair edge; for example, the sites marked by 23, 48, 71, 96 at the right armchair edge. We also define the inter-edge magnetic susceptibility χin which refers to the summation in equation (2), indicating that it is made over every site except the sites at both the top and bottom zigzag edges. Firstly, we present the temperature-dependence of the edge magnetic susceptibility χe in figure 2 around half-filling with U = 3.0| t| for L x = 12, L y = 4. The hole doping level studied here δ = 1.00 − n is no larger than 5 percent which could be realised in experiments either by the electric gate and/or chemical doping methods [1, 2]. It is interesting to see that the χe increases as the temperature decreases, especially in the low temperature region, which shows that the ferromagnetic fluctuations may dominate in the behaviour of the edge magnetic susceptibility. Comparing results with different electron fillings, one may also see that the χe is suppressed as the system is doped away from half-filling. To qualitatively estimate the behaviour of the temperature dependence of the edge magn etic susceptibility, we plot the function y = 1/x based on the Curie–Weiss law χ = C /(T − Tc ) which can describe the magnetic susceptibility χ for a ferromagnetic material in the temperature region above the Curie temperature Tc. It is important to see that the χe at half-filling ( blue line with circles ) is beyond the function y = 1/x, which indicates that the χe tends to be divergent at some low temperatures, and a ferromagnetic behaviour could be expected.
H = −t ∑(a†iσbi + jσ + h.c.) + U ∑(nai ↑nai ↓ + nbi ↑nbi ↓)
∫0
β
i
(1)
iσ
where t is the nearest hopping integrals, μ is the chemical potential and U is the on-site repulsion. Here, aiσ (a†iσ) annihilates (creates) electrons at the site Ri with spin σ (σ =↑, ↓) on sublattice A, as well as biσ (b†iσ) acting on the electrons of sublattice B, naiσ = aiσ a†iσ and nbiσ = biσ b†iσ. In the simulations, we apply periodic boundary conditions at the x direction and open boundary conditions at the zigzag edge, and we assume that the hoping integral is the same for all atoms in our model. This assumption is a reasonable approximation because the providing conditions are at the low-energy region so that the disturbance of the atoms has little influence on other additional atoms [28, 29]. On the honeycomb network for graphene nanoribbons, the nearest-neighbour hopping energy t reported in the literature ranges from 2.5 to 2.8 eV, and the value of the onsite repulsion U can be taken from the estimation from 6.0 to 16.93 eV in polyacetylene [30]. From the Peierls–Feynman– Bogoliubov variational results [27] and following the latter reference we study the model Hamiltonian in the range of U / | t | = 1.0 ∽ 5.0, which puts graphene-based material in the region of a moderate electronic correlation system. Thus, we treat the electronic interactions in such a system by using the DQMC method, which is very suitable for simulating and investigating the nature of magnetic correlation in the presence of moderate Coulomb interactions; especially when treating the band structure changes with respect to the finite 2
S Li et al
J. Phys.: Condens. Matter 28 (2016) 086001
Figure 3. The temperature-dependent edge magnetic susceptibility χe at n = 1.00 with various on-site repulsion.
Figure 2. The temperature-dependent edge magnetic susceptibility
for different electron fillings at U = 3.0 | t|.
The ferromagnetic-like behaviour in the edge magnetic susceptibility at half-filling is exciting as it is important for realising the graphene-based materials into the application of spintronics [34]. We further our story with different interaction, and the temperature-dependent edge magnetic susceptibility with different U at half-filling have been shown in figure 3. The edge magnetic susceptibility is enhanced as the interaction increases, which leads to a possible divergence of edge magnetic susceptibility at higher temperature. And we fit the DQMC data of U = 3.0| t| with a formula of χe(T ) = a /(T − Tc ) + b (2)
as shown (dashed lines) in figure 3, which allows us to estimate the transition temperature Tc. From equation (2), one may see that at T = Tc, the χe(T ) tends to diverge, and we may estimate the Tc from the fitting data where the χe may diverge at some temperatures. For the data shown in figure 3 for U = 3.0| t| and n = 1.00, Tc is estimated about ∼0.013t (roughly ∼ 325 K) according to our fitting. One can notice significant error bars on the susceptibility, related to the Monte Carlo sampling in lower temperatures. To estimate the error bar of the Tc, we use the standard rule for estimating errors of indirect measurement, and, we use the susceptibility at the lowest temperature Tlowest to estimate δTc = aδχe(Tlowest )/χ2e (Tlowest ) 0.004 | t|, which indicates that the value of Tc should be statistically distinguishable from zero. In graphene-based material, one of its important properties is that its chemical potential can be tuned and hence it is possible to induce electron or hole doping in graphene; this opens the door for carbon-based electronics [18]. From the results shown in figure 2, it is likely that the edge magnetic susceptibility at n = 0.98 has no potential to divergence at U = 3.0| t|. A system with mobile electrons or holes shows that a ferromagnetic behaviour may also be important. We study the edge magnetic susceptibility χe at n = 0.98 with different interactions U in figure 4, in which the χe reveals that it is also possible to show a ferromagnetic behaviour as the interaction is large enough, for example, U ⩾ 4.0| t|.
Figure 4. The temperature-dependent edge magnetic susceptibility χe at n = 0.98 with various on-site repulsion.
The temperature-dependent uniform magnetic susceptibility is shown in figure 5, in which results for n = 0.80 (dark line with circle), n = 0.75 (red line with down triangle) and n = 0.70 (green line with up triangle) are shown. The ferromagnetic fluctuations may dominate in the heavily doped region, where the electron filling is below n = 0.80. This result is similar to that of the armchair graphene nanoribbons [17] or perfect graphene [18], which indicates that uniform magnetism is difficult to realise in graphene-based material as the doping level is far from current experimental ability. Comparing results present in figures 2–5, we argue here that the edge magnetism is more easily achieved experimentally [35, 36]. Finally, we compare the uniform magnetic susceptibility χu, the magnetic susceptibility χe at the zigzag edge, the magnetic susceptibility χa at the armchair edge, and also the inter-edge magnetic susceptibility χin in figure 6 for different fillings and lattice widths. Clearly, the χe (solid lines with circles) at the zigzag edge is much larger than the uniform magnetic susceptibility χu (dashed red lines with square) for the bulk system, 3
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needed for nonvolatile transistors, many different characteristics are considered including high transition temperature, high carrier mobility, and intrinsic insulation conditions for carrier doping [37]. The proposed possible ferromagnetic-like behaviour here should be important for such kinds of searching, and hence should be useful for future nanodevices. Acknowledgments T Ma thanks Chinese Academy of Engineering Physics (CAEP) for partial financial support. This work is supported by National Natural Science Foundation of China (NSFC)s (Grant. Nos. 11374034 and 11334012), and the Fundamental Research Funds for the Central Universities (No. 2014KJJCB26). This research work is supported by a Tianhe-2JK computing time award at the Beijing Computational Science Research Center (CSRC).
Figure 5. The temperature-dependent uniform magnetic susceptibility χu for several values of n at U = 3.0 | t|.
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Figure 6. The temperature-dependent χu, χe, χin, and χa for several
values of n at U = 3.0 | t| on different lattice widths.
and also larger than the χa (dark lines with solid circle) at the armchair edge. Compared with different lattice width, Ly = 4 (blue lines) and Ly = 6 (pink lines), the χe is slightly enhanced as the width increases. It is, however, difficult to estimate the possible difference between divergence depends on different width, due to the notorious sign problem in DQMC methods, which prevent us from having accurate results in the lower temperature region and with larger system sizes. The interedge magnetic susceptibility χin for n = 0.98 (dotted dark lines) and n = 0.95 (dotted green lines) are shown for a lattice with 2 × 4 × 12 sites. The χin is much lower than χe , and it increases slightly as the electron filling decreases. In summary, we have studied the ferromagnetic properties in doped zigzag graphene nanoribbons by using the DQMC method. Our unbiased numerical results show that a ferromagn etic-like behaviour is induced by relative large interactions at the zigzag edge in the graphene nanoribbons. In nanoelectronics, microelectronic circuits that retain their logic state when the power is off would permit entirely new kinds of computers, and ferromagnetic semiconductors might make this technology possible. When searching for the ferromagnetic semiconductors 4
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