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Magnetic phase transitions in pure zigzag graphone nanoribbons
Accepted Manuscript Magnetic phase transitions in pure zigzag graphone nanoribbons L.B. Drissi, S. Zriouel
PII: DOI: Reference:
S0375-9601(14)01282-1 10.1016/j.physleta.2014.12.041 PLA 23025
To appear in:
Physics Letters A
Received date: 30 October 2014 Revised date: 19 December 2014 Accepted date: 22 December 2014
Please cite this article in press as: L.B. Drissi, S. Zriouel, Magnetic phase transitions in pure zigzag graphone nanoribbons, Phys. Lett. A (2015), http://dx.doi.org/10.1016/j.physleta.2014.12.041
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Highlights
• • • • •
We study pure graphone nanoribbons using MC calculations and mean field theory. We show the effects of the nanoribbon width on thermodynamic quantities. We study magnetic properties of GONR in presence of external magnetic field. We describe the influence of the system’s parameters on the hysteresis curves. This work offer promise for use of GONR in high-energy-storage-capacitor applications.
Magnetic phase transitions in pure zigzag graphone nanoribbons L. B. Drissi
1,2,
*, S. Zriouel1
1- LPHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco 2- CPM, Centre of Physics and Mathematics, Faculty of Science, Mohammed V University, Rabat, Morocco and *Corresponding author: [email protected]
Abstract Magnetic properties and hysteresis loops of pure graphone nanoribbons (GONR) are studied using both Monte Carlo calculations and mean field theory. This study is relevant for understanding the magnetic behavior of pure GONR that exhibits magnetism due to the localized electrons on the carbon atoms without hydrogens. Magnetization and its corresponding susceptibility are given for various ribbon widths 3 ≤ Nz ≤ 100 and external magnetic field 0 < h ≤ 10kOe. The critical temperature Tc is deduced. It is shown that temperature Tc reduces as a step function versus the ribbon widths Nz for low values of h up to 0.3kOe. The effect of temperature, low/strong h and Nz parameter on the hysteresis curves is also examined. The findings of this work offer considerable promise for use of GONR in various nanoelectronic devices especially for high-energystorage-capacitor applications that require square hysteresis loops behavior. Keywords: Graphone, Nanoribbons, Triangular lattice, Monte Carlo calculations, Mean Field Theory, Magnetic phases, Critical Temperature, Hysteresis cycles
1
I.
INTRODUCTION
Since its first isolation in 2004, graphene has attracted a lot of interest for its fundamental studies [1, 2] and its high potential applications [3, 4] . Graphene is a zero gap semiconductor which leads to the challenges of opening up and controlling the band gap to adapt this material to future high tech-electronic devices. In order to overcome this limitation several approaches are considered using various processes. Quantum confinement of electrons by forming nanoribbons [5, 6] is one way that not only modifies the electronic structure but it introduces also magnetism in this non magnetic material. Graphene nanoribbons (GNRs) with different widths can be made either by cutting exfoliated graphene sheet along a straight line [7, 8] or by epitaxial graphene pattern [9, 10]. The resulting one dimensional systems can have either armchair or zigzag edges. Zigzag graphene nanoribbon (ZGNR) is a semi-conductor at its ground state with band gap depending on its width W [11]. Due to the zigzag edge states, electron spin polarization appears spontaneously. Indeed, in each edge, localized states are ferromagnetically ordered [12] while the magnetic moments on the two edges interact antiferromagnetically since they have opposite spin orientation [13]. FM-AFM energy differences per unit cell is a few meV and could present metal state at finite temperature [14]. Different applications in a number of exceptional spintronic devices have been proposed for one-dimensional zigzag graphene nanoribbons [15, 16]. Assuming that graphene is a giant macromolecule allows the use of chemical reactions that create new derivatives such as graphane [17, 18]. The use of this strategy tunes the gap energy [19] and affects the magnetic properties of graphene. Graphane attain permanent magnetic moment through hydrogen vacancy domains [20] or by partial dehydrogenation giving rise to graphone [21]. Graphone is a semiconductor with a small indirect gap. This new material exhibits ferromagnetism ground state originating from interactions between 2p moments attributed to extended p-p interactions between the localized and unpaired electrons on the unhydrogenated carbon atoms [22]. The partial hydrogen coverage has a striking effect on physical properties of graphene that can be restored by annealing [23]. Removing half of the hydrogen atoms from graphone leads to non magnetic material as pz orbitals of two nearest unsaturated C atoms form π-bonding that quenches magnetism [21]. Although magnetic carbon nanostructures are of great interest as they are light, stable,
2
simple to treat, and cheaper to produce, they are not investigated extensively. The number of theoretical studies existing are focussing on the role of topological defects [24], carrier density [25] and edge states [26, 27] to induce magnetism in graphene nanoribbons. Recently, structural and electronic properties of long nanoribbons of graphone have been studied [28]. Unlike short samples with specific sizes that form carbon nanotubes [29], long graphone nanoribbons, due to their sp3 hybridization, spontaneously roll up to form spiral structures with interesting localization of frontier molecular orbitals. These rolled hydrocarbon structures are stable beyond room temperature up to at least T = 1000K. The feasibility of engineering magnetic graphene nanostructure devices is questioned by magnetic ordering at finite temperature. This opens a new field in the research of the critical magnetic phenomena at nanoscale. In [30] and [31], the influence of the disorder on the magnetic properties of nanoribbon are investigated. In [30], the effect of both the number and the positions of s=3/2 substituted magnetic atoms on the magnetic phase transitions of mono-, bi- and tri-doped graphone nanoribbons in different configurations are reported. It is shown that the critical temperature TC increases with the number of dopants but for configurations with fixed number of magnetic impurities, TC is more sensitive to edges. In [31], the magnetic behavior of a mixed s=1/2 core and S=1 shell nanoribbon with an anti-ferromagnetic interface coupling shows a number of characteristic behaviors such as the occurrence of compensation temperature and the existence of single and triple hysteresis loops. To understand more the fundamental behavior of the magnetic properties of graphone nanostructures, we study in the present work the GONR width dependence of Curie temperature in the absence and presence of external magnetic field. In particular, the influence of the changing width on the magnetic behavior of ferromagnetic nanoribbon. Graphone nanoribbons is modeled by triangular lattice with periodic boundary conditions in the X -direction and free boundary conditions in the Y-direction. In the literature, 2d Ising model phase transition on a triangular lattice is a long standing topic [32] investigated mainly for systems with periodic boundary conditions using different methods as MC calculations [33] and Wang-Landau simulation [34]. However, we are not aware of any work reporting the boundary effects on the magnetic behavior of triangular lattice with finite width such as nanoribbons even if these nanostructures are very much required in engineering magnetic devices such as graphene derivatives and other 2D materials. Using Monte Carlo (MC) 3
simulations and mean field theory (MFT), we show the effects of external magnetic field, as well as temperature and size on the relevant thermodynamic quantities such as magnetization, susceptibility, hysteresis loops and critical temperature of the system. A detailed description of the model and the formalism are given in Section 2. In Section 3 we present the obtained results and discuss, for weak and strong magnetic field, the effect of different cited parameters on the magnetic phases and hysteresis cycles that change from squares to loops to anhysteretic curves. We end with a conclusion.
II.
THE MODEL
In the present work, we consider infinitely long one-dimensional graphone nanoribbons with zigzag edges. The system is periodic only along X-direction and the unit cell used in our calculations is delimited by dashed lines as shown in Fig 1.
FIG. 1: Structural model of 8-Zigzag graphone nanoribbon. Transparent atoms represent hydrogenated C1 atoms, black atoms are C2 atoms, and small blue atoms denote atoms passivating the edges of graphone.
Following previous customary notation [13, 35], the finite width W of the ribbon is characterized by the number Nz of zigzag chains of carbons that run along X. In Fig 1 we plot the structure of Nz = 8 H-ZGONR-H with the edge carbon atoms all saturated 4
with non magnetic atoms, namely H atoms to avoid the dangling bond states. In this model, we have two kind of carbon atoms: C1 and C2 forming the hexagonal structure. ˚ In each The bond length dC1 −C2 , between the two first nearest neighbors (NN), is 1.495A. hexagon, carbon atoms C1 are decorated with hydrogen in sp3 hybridization while C2 atoms that remain unsaturated are sp2 hybridized. So only the unhydrogenated C2 atoms carry a magnetic moment of about 1μB as their p- electrons are localized and unpaired. Each C2 atom, except those near the edges, has three non magnetic first nearest neighbors C1 and six magnetic second nearest neighbors C2 . According to [21], the valence electrons in p-states are more delocalized than those in d or f-states. Therefore, the ferromagnetic (FM) ground state in graphone is due to interactions between 2p moments of localized electrons spins in C2 atoms. By ignoring non magnetic C1 atoms as they don’t contribute in our calculations, we depict the corresponding C2 nanoribbon in Fig 2. In this figure, ZGONR is modeled by triangular lattice with periodic boundary conditions in the X -direction and free boundary conditions in the Y-direction.
FIG. 2: (a) Representation of 8-Zigzag graphone nanoribbon (b) showing only magnetic nearest neighbors and ignoring non magnetic atoms.
The system is described by the hamiltonian:
H=
J0 Siz Sjz + h
Siz
(1)
i
i=j
where J0 is the coupling between magnetic next-nearest neighbors in the hexagonal structure of ZGONR, namely J0 is the interaction between the 2p S-moments at two different sites i 5
and j of C2 atoms and Szi is the spin of the magnetic C2 atom at site i. All the spins are set to be ±1/2 for pure ZGONR. h is the external magnetic field ranging as 0 ≤ h ≤ 10kOe.
III.
RESULTS AND DISCUSSION
We consider ZGONR having different widths W . We concentrate on the effect of the parameter W on the variation of Curie temperature for pure ZGONR on the basis of both Monte Carlo calculations and mean field theory. Two cases are investigated: ZGONRs in the absence and presence of external magnetic field.
A.
Pure ZGONRs
Monte Carlo calculations: By mean of MC simulations for the Ising model described above, we study rectangular hydrogen-terminated one-dimensional zigzag graphone nanoribbons of different widths Nz varying from 3 up to 100. The periodic boundary conditions are applied in X-direction and free boundary conditions are applied in Y -direction. The ground state electronic configuration of pure ZGONR is characterized by the ferromagnetic arrangement of spins. The MC steps are 5.105 steps per spin discarding the first 5.104 Monte Carlo simulations. We build a program using the Metropolis algorithm [36] to calculate the thermal average of magnetization M and energy E. We set the coupling J0 = 1 and the Boltzmann’s constant kB = 1. We calculate also the corresponding magnetic susceptibility χ=
1 2 M − M2 T
and the specific heat CV given by CV =
1 2 E − E2 . 2 T
Magnetization and susceptibility as functions of T for pure ZGONR with specific parameter width NZ = 8, 20 and 40 are plotted in Fig 3. From temperature dependence susceptibility we deduce the Curie temperature Tc = 3.67, 3.08 and 2.66 respectively for the three values of Nz mentioned before. ˚ up For 3 ≤ Nz ≤ 100, that corresponds to varying the width W in the range from 5A ˚ we collect data from temperature dependence susceptibility. The obtained result to 223A, 6
FIG. 3: Magnetization and susceptibility versus temperature for pure zigzag graphone nanoribbons for three representative widths Nz =8, 20 and 40.
is plotted in figure 4. We deduce that critical temperature TC decreases as a step function of the ribbon width Nz , to minimize and stabilize in its minimal value Tcmin for Nz ranging from 32 to 100. We expect recovering the limit of graphone infinite sheet for large width that corresponds to high values of Nz .
FIG. 4: Curie temperature versus ZGONR width parameter NZ ≤ 100 (The lines guide the eye).
Indeed, when Nz varies in the large interval [100, 4000], the width W reaches the value ˚ The obtained results reveal that the evolution of Tc has a different behavior of 8970A. compared to the previous case where Nz takes integer values in the interval [3, 100] . As plotted in figure 5, the transition temperature Tc increases with Nz until it stabilizes for 7
Nz ≥ 1300. The presence of two regions with different behaviors suggests a 1D → 2D dimensionality crossover for Nz around 100.
FIG. 5: ZGONR width parameter dependence of the Curie Temperature for high values of NZ ≥ 100(The lines guide the eye).
To reproduce results in agreement with literature, we plot in Fig. 6 the variation of the critical temperature as function of the ribbon width varying in the two intervals [3, 100] and [100, 4000] with large steps ΔNz . Notice that the step function found in Figs. 4 and 5 disappear leading to a decrease of the critical temperature in function of the size in the interval [3,100] and an increase of Tc in the interval [100,4000]. These results are in good agreement with [31, 37]. We deduce that the plateaus found in Figs. 4 and 5 are a direct consequence of the very small steps ΔNz used in our numerical simulation.
FIG. 6: Curie temperature versus ZGONR width parameter a) for NZ ≤ 100 and b) for NZ ≥ 100 for large steps (The lines guide the eye).
Mean field theory: In order to analyze the MC calculations of size effect on TC , we 8
have carried out an analytic calculations with mean field theory. For the Hamiltonian in eq(1), the Gibbs free energy per site [38] is given by F¯0 = F0 + H0 − H0 0
(2)
exp(−β.H0 ) denotes the average value performed over the effective Hamilwhere · · · 0 = tr ···exp(−β.H 0)
tonian of the system H0 , and F0 is its associated free energy. We have N i , H0 = − hi .S
F0 = with β =
i=1 − β1 log Z0
(3)
1 , kB T
kB is the Boltzmann constant and T is the temperature. −→ i = ± 1 , the partition function generated by the For the effectif field h parallel to OZ and S 2 above Hamiltonian is
N Z0 = 2. cosh β.h 2
where N is the total number of C2 atoms. Therefore, eqs (3) can be rewritten as
and
F0 = − Nβ log 2. cosh β.h 2
(4)
H0 = −J0 N1 + 32 .N2 tanh2 β.h 2
(5)
with N1 is the number of C2 atoms near the edge that have only f our C2 nearest neighbors while N2 is the number of C2 atoms inside ZGONR having six nearest neighbors. Replacing eqs (4) and (5) in eq (3), the minimization of the free energy (dF¯0 /dh) = 0 leads to: β.h 3 N.β.h = J0 .β. tanh . N1 + .N2 . (6) 4 2 2 Finally, for h −→ 0, we deduce from eq (6) that
2 Tc = J0 . 2 + (N1N+N 2) In the unit cell, when we vary the width parameter Nz , only the number N2 vary while N1 remains constant. As the mean field theory is an approximation that becomes exact only in the limit of infinite system size [39], we deduce that when N2 −→ ∞, the critical temperature of the system Tc −→ 3J0 which agrees well with previous MC result for large value of Nz having Tc = 3.47. This approach can not reproduce the 1D limit as the mean field type approximation fail in one dimensional system where the quantum fluctuations dominate and the behavior is very exotic and non-standard [40]. 9
B.
Magnetic field effect
In this section, we run our MC calculations to study the influence of external magnetic field h on hydrogen-terminated periodic one-dimensional graphone nanoribbons of different widths 3 ≤ Nz ≤ 100. We vary the magnetic field in steps of Δh = 0.01 when h ≤ 1kOe and Δh = 0.5 for strong h, equilibrating at each state for 5.105 Monte Carlo steps per spin. The results reveal that we have two main cases: (i) h takes very low values 0 ≤ h ≤ 0.3kOe and (ii) external magnetic field is strong. In fig 7, we display the magnetization M and the corresponding susceptibility χ versus
FIG. 7: Magnetization and susceptibility versus temperature of 3-zigzag graphone nanoribbon for various values of external magnetic field h .
temperature for different values of external magnetic field h = 1, ..., 10kOe for Nz = 3 ZGONR. All the magnetization curves decrease from their saturation value 0.5 and go to zero when T increases. As magnetic field increases, the slope of magnetization curves becomes less pronounced. For the temperature dependence of the susceptibility, the curves peak for different values of T that increases as h increases. These peaks that characterize the phase transition from ferromagnetic phase to paramagnetic one become less pronounced for high values of h. This is due to the strong effect of h on the alignment of spins. Notice that the same behavior for M and χ versus T is obtained for different ribbon widths 3 ≤ Nz ≤ 100. The same calculations are performed as a function of low magnetic field for width Nz up to 100. The temperature dependence of the susceptibility is reported in Fig 8 for the case Nz = 6. Compared to previous results for strong h, the behavior is the same as the curves peak for different values of T that increases with h. However, the maximum of χ is larger than that found for strong h and the transition temperature is much lower. 10
FIG. 8: Susceptibility versus temperature for 6-zigzag graphone nanoribbon at low external magnetic field h=0.03, 0.15, 0.2 and 0.28 with a zoom on susceptibility peaks.
FIG. 9: GONR width parameter dependence of the Curie temperature for external magnetic field with values h ≤ 0.3. (The lines guide the eye).
Notice that the most important influence on the critical temperature Tc seems to be the size [41], which indicates that the critical temperature becomes tunable for switches functioning in a designed temperature range [42]. To determine the evolution of the critical temperature in function of the width, we plot the estimated transition temperatures as
11
function of Nz ≤ 10 for low magnetic field in Figs 9 and 10 and for strong magnetic field in Fig 11. When 0 < h ≤ 0.3kOe, the critical temperature Tc decreases as a step function of ZGONR
FIG. 10: Transition temperature versus GONR width parameter Nz at some fixed values of 0.35 ≤h≤ 0.8.(The lines guide the eye).
width Nz . At fixed width Nz , Tc increases as h increases and stabilizes at its approximate value 4.08 for h = 0.3kOe as shown in Fig.9. When h vary in the range [0.35, 0.8] , the temperature Tc increases as a step function of the ribbon width Nz to maximize and saturate as depicted in Fig.10. For strong external magnetic field 1 ≤ h ≤ 4, the transition temperature increases slowly and similarly with the size Nz . When h ≥ 5, the variation becomes slow for Nz ≥ 6 and stabilizes for higher values of the width. It is worth noting that the ZGONs is insensitive to h when 0 < h ≤ 0.3kOe and the behavior of the critical temperature versus the width Nz is the same as the one obtained for zero magnetic field. In order to investigate the influence of the temperature on the hysteresis behavior of the Nz ferromagnetic GONR in the case of low external magnetic field, a series of hysteresis curves at various values of T is plotted in Fig 12 for the case of Nz = 6 GONR having Tc = 3.88. The curves show hysteretic behavior with loops becoming narrower as the temperature 12
FIG. 11: Transition temperature as function of GONR width parameter Nz for strong external magnetic field h
FIG. 12: Magnetic hysteresis cycle of NZ = 6 graphone nanoribbons in presence of low h at four different temperatures T=0.8, 1.6, 2 and 2.2.
increases. We can also see clearly that the area of the hysteresis loops disappears completely when the temperature approaches its critical value Tc . Moreover, all the magnetization curves are symmetric for both positive and negative values of the external magnetic field. This property comes directly from the symmetry of the Hamiltonian with respect to changes h → −h and M → −M . At T = 0.8, the hysteresis is almost square indicating that the normal orientation is an easy axis. In this case, the remanent magnetization is equal to the saturation magnetization that is an obvious consequence of the square hysteresis loop. The 13
square behavior of curves is due to more the temperature is low the closer we get to the ground state, therefore the system becomes ordered and encounters the thermal equilibrium. At higher T, the shapes of the ferromagnetic hysteresis loops change from square to loop. The slanted curves demagnetize and remagnetize quickly. It is worth noting that all curves saturate quickly and when increasing temperature, the remanent magnetization decreases while the saturation magnetization increases. For T 2.2, we find that the slanted curves disappear. This behavior can be explained by the fact that higher temperature disorders the system and namely, the state of the nanoribbon system is changing from ferromagnetic phase to paramagnetic phase. For strong external magnetic field, the curves are reversible for all T, and show linear behavior with absence of remanence due to the reduction of spontaneous magnetization by strong h. The anhysteretic magnetization curves are plotted in Fig 13 for Nz = 6 ZGONR at two different temperatures T = 0.8 and T = 1.6. At T = 0.8 the saturation is achieved at low magnetic field hs = 110 Oe but when T = 1.6 the curve saturates at hs = 950 Oe. The absence of hysteresis loop in nanoribbon of graphone can be attributed to the small
FIG. 13: Hysteresis loops for Nz =6 GONR at different temperature (a) T=0.8 and (b) T=1.6 calculated in weak and strong external magnetic field.(The lines guide the eye).
size of our system in agreement with the work [43] studying magnetic nanoparticles. Indeed, this result is confirmed for higher widths of ZGONR as showed in Fig 14 where we plot the effect of graphone nanoribbon width on hysteresis curves. For strong magnetic field at fixed temperature (see Fig 14 -a), the area of hysteresis curves increase as the width Nz increase. Moreover, the slope of the curves becomes more important with Nz and the remanence at zero external field equals the saturation magnetization due to the nearly square hysteresis loops. 14
For low external h, the results are plotted in Fig 14-b for small Nz = 3, 5, 9, 19 and large width Nz = 30, 42, 65. It is shown that at fixed temperature T = 1.1, the area of hysteresis curves increases as the width increases. Notice that the same behavior is found for higher values of nanoribbon widths up to 100. When Nz increases, the curves saturate more quickly with a square loop behavior indicating that the perpendicular external field is strong enough to keep ZGONR in a perpendicularly magnetized state.
FIG. 14: The dependence of magnetic hysteresis loops on the width Nz for a) strong magnetic field at T=0.8, b) for weak external magnetic field h at T=1.1. (The lines guide the eye).
With its square hysteresis loop behavior, ZGONR is a suitable material for high-energystorage-capacitor applications.
IV.
CONCLUSION
In this work, we have studied magnetic phases of graphone nanoribbons with width W ˚ to 223A ˚ in the presence and absence of external magnetic field. The Monte ranging from 5A Carlo simulations were performed to determine the corresponding critical temperature Tc . The obtained results agree well with Tc calculated with mean field theory. From this study we learned that Tc is a reduced step function versus the ribbon widths Nz for magnetic field 0 ≤ h ≤ 0.3kOe. However, Tc increases with Nz for higher values of h. It was also shown that the hysteresis curves change from squares to loops to anhysteretic curves for appropriate values of temperature and ribbons width in both low and strong external magnetic field. These results have direct implications for the control of the critical temperature in ZGONR using suitably modulated magnetic fields and ribbon widths which is very required for the practical realization of magnetic graphone nanoribbon-based devices in the nanotechnology industry. 15
Acknowledgement 1 L.B. Drissi would like to thank Prof E. H. Saidi for discussions. L.B. Drissi would like to thank ICTP (Trieste) for the junior associateship scheme and acknowledge financial support by Centre National de Recherche Scientifique (CNRS), Morocco under project URAC09.
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PII: DOI: Reference:
S0375-9601(14)01282-1 10.1016/j.physleta.2014.12.041 PLA 23025
To appear in:
Physics Letters A
Received date: 30 October 2014 Revised date: 19 December 2014 Accepted date: 22 December 2014
Please cite this article in press as: L.B. Drissi, S. Zriouel, Magnetic phase transitions in pure zigzag graphone nanoribbons, Phys. Lett. A (2015), http://dx.doi.org/10.1016/j.physleta.2014.12.041
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
• • • • •
We study pure graphone nanoribbons using MC calculations and mean field theory. We show the effects of the nanoribbon width on thermodynamic quantities. We study magnetic properties of GONR in presence of external magnetic field. We describe the influence of the system’s parameters on the hysteresis curves. This work offer promise for use of GONR in high-energy-storage-capacitor applications.
Magnetic phase transitions in pure zigzag graphone nanoribbons L. B. Drissi
1,2,
*, S. Zriouel1
1- LPHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco 2- CPM, Centre of Physics and Mathematics, Faculty of Science, Mohammed V University, Rabat, Morocco and *Corresponding author: [email protected]
Abstract Magnetic properties and hysteresis loops of pure graphone nanoribbons (GONR) are studied using both Monte Carlo calculations and mean field theory. This study is relevant for understanding the magnetic behavior of pure GONR that exhibits magnetism due to the localized electrons on the carbon atoms without hydrogens. Magnetization and its corresponding susceptibility are given for various ribbon widths 3 ≤ Nz ≤ 100 and external magnetic field 0 < h ≤ 10kOe. The critical temperature Tc is deduced. It is shown that temperature Tc reduces as a step function versus the ribbon widths Nz for low values of h up to 0.3kOe. The effect of temperature, low/strong h and Nz parameter on the hysteresis curves is also examined. The findings of this work offer considerable promise for use of GONR in various nanoelectronic devices especially for high-energystorage-capacitor applications that require square hysteresis loops behavior. Keywords: Graphone, Nanoribbons, Triangular lattice, Monte Carlo calculations, Mean Field Theory, Magnetic phases, Critical Temperature, Hysteresis cycles
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I.
INTRODUCTION
Since its first isolation in 2004, graphene has attracted a lot of interest for its fundamental studies [1, 2] and its high potential applications [3, 4] . Graphene is a zero gap semiconductor which leads to the challenges of opening up and controlling the band gap to adapt this material to future high tech-electronic devices. In order to overcome this limitation several approaches are considered using various processes. Quantum confinement of electrons by forming nanoribbons [5, 6] is one way that not only modifies the electronic structure but it introduces also magnetism in this non magnetic material. Graphene nanoribbons (GNRs) with different widths can be made either by cutting exfoliated graphene sheet along a straight line [7, 8] or by epitaxial graphene pattern [9, 10]. The resulting one dimensional systems can have either armchair or zigzag edges. Zigzag graphene nanoribbon (ZGNR) is a semi-conductor at its ground state with band gap depending on its width W [11]. Due to the zigzag edge states, electron spin polarization appears spontaneously. Indeed, in each edge, localized states are ferromagnetically ordered [12] while the magnetic moments on the two edges interact antiferromagnetically since they have opposite spin orientation [13]. FM-AFM energy differences per unit cell is a few meV and could present metal state at finite temperature [14]. Different applications in a number of exceptional spintronic devices have been proposed for one-dimensional zigzag graphene nanoribbons [15, 16]. Assuming that graphene is a giant macromolecule allows the use of chemical reactions that create new derivatives such as graphane [17, 18]. The use of this strategy tunes the gap energy [19] and affects the magnetic properties of graphene. Graphane attain permanent magnetic moment through hydrogen vacancy domains [20] or by partial dehydrogenation giving rise to graphone [21]. Graphone is a semiconductor with a small indirect gap. This new material exhibits ferromagnetism ground state originating from interactions between 2p moments attributed to extended p-p interactions between the localized and unpaired electrons on the unhydrogenated carbon atoms [22]. The partial hydrogen coverage has a striking effect on physical properties of graphene that can be restored by annealing [23]. Removing half of the hydrogen atoms from graphone leads to non magnetic material as pz orbitals of two nearest unsaturated C atoms form π-bonding that quenches magnetism [21]. Although magnetic carbon nanostructures are of great interest as they are light, stable,
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simple to treat, and cheaper to produce, they are not investigated extensively. The number of theoretical studies existing are focussing on the role of topological defects [24], carrier density [25] and edge states [26, 27] to induce magnetism in graphene nanoribbons. Recently, structural and electronic properties of long nanoribbons of graphone have been studied [28]. Unlike short samples with specific sizes that form carbon nanotubes [29], long graphone nanoribbons, due to their sp3 hybridization, spontaneously roll up to form spiral structures with interesting localization of frontier molecular orbitals. These rolled hydrocarbon structures are stable beyond room temperature up to at least T = 1000K. The feasibility of engineering magnetic graphene nanostructure devices is questioned by magnetic ordering at finite temperature. This opens a new field in the research of the critical magnetic phenomena at nanoscale. In [30] and [31], the influence of the disorder on the magnetic properties of nanoribbon are investigated. In [30], the effect of both the number and the positions of s=3/2 substituted magnetic atoms on the magnetic phase transitions of mono-, bi- and tri-doped graphone nanoribbons in different configurations are reported. It is shown that the critical temperature TC increases with the number of dopants but for configurations with fixed number of magnetic impurities, TC is more sensitive to edges. In [31], the magnetic behavior of a mixed s=1/2 core and S=1 shell nanoribbon with an anti-ferromagnetic interface coupling shows a number of characteristic behaviors such as the occurrence of compensation temperature and the existence of single and triple hysteresis loops. To understand more the fundamental behavior of the magnetic properties of graphone nanostructures, we study in the present work the GONR width dependence of Curie temperature in the absence and presence of external magnetic field. In particular, the influence of the changing width on the magnetic behavior of ferromagnetic nanoribbon. Graphone nanoribbons is modeled by triangular lattice with periodic boundary conditions in the X -direction and free boundary conditions in the Y-direction. In the literature, 2d Ising model phase transition on a triangular lattice is a long standing topic [32] investigated mainly for systems with periodic boundary conditions using different methods as MC calculations [33] and Wang-Landau simulation [34]. However, we are not aware of any work reporting the boundary effects on the magnetic behavior of triangular lattice with finite width such as nanoribbons even if these nanostructures are very much required in engineering magnetic devices such as graphene derivatives and other 2D materials. Using Monte Carlo (MC) 3
simulations and mean field theory (MFT), we show the effects of external magnetic field, as well as temperature and size on the relevant thermodynamic quantities such as magnetization, susceptibility, hysteresis loops and critical temperature of the system. A detailed description of the model and the formalism are given in Section 2. In Section 3 we present the obtained results and discuss, for weak and strong magnetic field, the effect of different cited parameters on the magnetic phases and hysteresis cycles that change from squares to loops to anhysteretic curves. We end with a conclusion.
II.
THE MODEL
In the present work, we consider infinitely long one-dimensional graphone nanoribbons with zigzag edges. The system is periodic only along X-direction and the unit cell used in our calculations is delimited by dashed lines as shown in Fig 1.
FIG. 1: Structural model of 8-Zigzag graphone nanoribbon. Transparent atoms represent hydrogenated C1 atoms, black atoms are C2 atoms, and small blue atoms denote atoms passivating the edges of graphone.
Following previous customary notation [13, 35], the finite width W of the ribbon is characterized by the number Nz of zigzag chains of carbons that run along X. In Fig 1 we plot the structure of Nz = 8 H-ZGONR-H with the edge carbon atoms all saturated 4
with non magnetic atoms, namely H atoms to avoid the dangling bond states. In this model, we have two kind of carbon atoms: C1 and C2 forming the hexagonal structure. ˚ In each The bond length dC1 −C2 , between the two first nearest neighbors (NN), is 1.495A. hexagon, carbon atoms C1 are decorated with hydrogen in sp3 hybridization while C2 atoms that remain unsaturated are sp2 hybridized. So only the unhydrogenated C2 atoms carry a magnetic moment of about 1μB as their p- electrons are localized and unpaired. Each C2 atom, except those near the edges, has three non magnetic first nearest neighbors C1 and six magnetic second nearest neighbors C2 . According to [21], the valence electrons in p-states are more delocalized than those in d or f-states. Therefore, the ferromagnetic (FM) ground state in graphone is due to interactions between 2p moments of localized electrons spins in C2 atoms. By ignoring non magnetic C1 atoms as they don’t contribute in our calculations, we depict the corresponding C2 nanoribbon in Fig 2. In this figure, ZGONR is modeled by triangular lattice with periodic boundary conditions in the X -direction and free boundary conditions in the Y-direction.
FIG. 2: (a) Representation of 8-Zigzag graphone nanoribbon (b) showing only magnetic nearest neighbors and ignoring non magnetic atoms.
The system is described by the hamiltonian:
H=
J0 Siz Sjz + h
Siz
(1)
i
i=j
where J0 is the coupling between magnetic next-nearest neighbors in the hexagonal structure of ZGONR, namely J0 is the interaction between the 2p S-moments at two different sites i 5
and j of C2 atoms and Szi is the spin of the magnetic C2 atom at site i. All the spins are set to be ±1/2 for pure ZGONR. h is the external magnetic field ranging as 0 ≤ h ≤ 10kOe.
III.
RESULTS AND DISCUSSION
We consider ZGONR having different widths W . We concentrate on the effect of the parameter W on the variation of Curie temperature for pure ZGONR on the basis of both Monte Carlo calculations and mean field theory. Two cases are investigated: ZGONRs in the absence and presence of external magnetic field.
A.
Pure ZGONRs
Monte Carlo calculations: By mean of MC simulations for the Ising model described above, we study rectangular hydrogen-terminated one-dimensional zigzag graphone nanoribbons of different widths Nz varying from 3 up to 100. The periodic boundary conditions are applied in X-direction and free boundary conditions are applied in Y -direction. The ground state electronic configuration of pure ZGONR is characterized by the ferromagnetic arrangement of spins. The MC steps are 5.105 steps per spin discarding the first 5.104 Monte Carlo simulations. We build a program using the Metropolis algorithm [36] to calculate the thermal average of magnetization M and energy E. We set the coupling J0 = 1 and the Boltzmann’s constant kB = 1. We calculate also the corresponding magnetic susceptibility χ=
1 2 M − M2 T
and the specific heat CV given by CV =
1 2 E − E2 . 2 T
Magnetization and susceptibility as functions of T for pure ZGONR with specific parameter width NZ = 8, 20 and 40 are plotted in Fig 3. From temperature dependence susceptibility we deduce the Curie temperature Tc = 3.67, 3.08 and 2.66 respectively for the three values of Nz mentioned before. ˚ up For 3 ≤ Nz ≤ 100, that corresponds to varying the width W in the range from 5A ˚ we collect data from temperature dependence susceptibility. The obtained result to 223A, 6
FIG. 3: Magnetization and susceptibility versus temperature for pure zigzag graphone nanoribbons for three representative widths Nz =8, 20 and 40.
is plotted in figure 4. We deduce that critical temperature TC decreases as a step function of the ribbon width Nz , to minimize and stabilize in its minimal value Tcmin for Nz ranging from 32 to 100. We expect recovering the limit of graphone infinite sheet for large width that corresponds to high values of Nz .
FIG. 4: Curie temperature versus ZGONR width parameter NZ ≤ 100 (The lines guide the eye).
Indeed, when Nz varies in the large interval [100, 4000], the width W reaches the value ˚ The obtained results reveal that the evolution of Tc has a different behavior of 8970A. compared to the previous case where Nz takes integer values in the interval [3, 100] . As plotted in figure 5, the transition temperature Tc increases with Nz until it stabilizes for 7
Nz ≥ 1300. The presence of two regions with different behaviors suggests a 1D → 2D dimensionality crossover for Nz around 100.
FIG. 5: ZGONR width parameter dependence of the Curie Temperature for high values of NZ ≥ 100(The lines guide the eye).
To reproduce results in agreement with literature, we plot in Fig. 6 the variation of the critical temperature as function of the ribbon width varying in the two intervals [3, 100] and [100, 4000] with large steps ΔNz . Notice that the step function found in Figs. 4 and 5 disappear leading to a decrease of the critical temperature in function of the size in the interval [3,100] and an increase of Tc in the interval [100,4000]. These results are in good agreement with [31, 37]. We deduce that the plateaus found in Figs. 4 and 5 are a direct consequence of the very small steps ΔNz used in our numerical simulation.
FIG. 6: Curie temperature versus ZGONR width parameter a) for NZ ≤ 100 and b) for NZ ≥ 100 for large steps (The lines guide the eye).
Mean field theory: In order to analyze the MC calculations of size effect on TC , we 8
have carried out an analytic calculations with mean field theory. For the Hamiltonian in eq(1), the Gibbs free energy per site [38] is given by F¯0 = F0 + H0 − H0 0
(2)
exp(−β.H0 ) denotes the average value performed over the effective Hamilwhere · · · 0 = tr ···exp(−β.H 0)
tonian of the system H0 , and F0 is its associated free energy. We have N i , H0 = − hi .S
F0 = with β =
i=1 − β1 log Z0
(3)
1 , kB T
kB is the Boltzmann constant and T is the temperature. −→ i = ± 1 , the partition function generated by the For the effectif field h parallel to OZ and S 2 above Hamiltonian is
N Z0 = 2. cosh β.h 2
where N is the total number of C2 atoms. Therefore, eqs (3) can be rewritten as
and
F0 = − Nβ log 2. cosh β.h 2
(4)
H0 = −J0 N1 + 32 .N2 tanh2 β.h 2
(5)
with N1 is the number of C2 atoms near the edge that have only f our C2 nearest neighbors while N2 is the number of C2 atoms inside ZGONR having six nearest neighbors. Replacing eqs (4) and (5) in eq (3), the minimization of the free energy (dF¯0 /dh) = 0 leads to: β.h 3 N.β.h = J0 .β. tanh . N1 + .N2 . (6) 4 2 2 Finally, for h −→ 0, we deduce from eq (6) that
2 Tc = J0 . 2 + (N1N+N 2) In the unit cell, when we vary the width parameter Nz , only the number N2 vary while N1 remains constant. As the mean field theory is an approximation that becomes exact only in the limit of infinite system size [39], we deduce that when N2 −→ ∞, the critical temperature of the system Tc −→ 3J0 which agrees well with previous MC result for large value of Nz having Tc = 3.47. This approach can not reproduce the 1D limit as the mean field type approximation fail in one dimensional system where the quantum fluctuations dominate and the behavior is very exotic and non-standard [40]. 9
B.
Magnetic field effect
In this section, we run our MC calculations to study the influence of external magnetic field h on hydrogen-terminated periodic one-dimensional graphone nanoribbons of different widths 3 ≤ Nz ≤ 100. We vary the magnetic field in steps of Δh = 0.01 when h ≤ 1kOe and Δh = 0.5 for strong h, equilibrating at each state for 5.105 Monte Carlo steps per spin. The results reveal that we have two main cases: (i) h takes very low values 0 ≤ h ≤ 0.3kOe and (ii) external magnetic field is strong. In fig 7, we display the magnetization M and the corresponding susceptibility χ versus
FIG. 7: Magnetization and susceptibility versus temperature of 3-zigzag graphone nanoribbon for various values of external magnetic field h .
temperature for different values of external magnetic field h = 1, ..., 10kOe for Nz = 3 ZGONR. All the magnetization curves decrease from their saturation value 0.5 and go to zero when T increases. As magnetic field increases, the slope of magnetization curves becomes less pronounced. For the temperature dependence of the susceptibility, the curves peak for different values of T that increases as h increases. These peaks that characterize the phase transition from ferromagnetic phase to paramagnetic one become less pronounced for high values of h. This is due to the strong effect of h on the alignment of spins. Notice that the same behavior for M and χ versus T is obtained for different ribbon widths 3 ≤ Nz ≤ 100. The same calculations are performed as a function of low magnetic field for width Nz up to 100. The temperature dependence of the susceptibility is reported in Fig 8 for the case Nz = 6. Compared to previous results for strong h, the behavior is the same as the curves peak for different values of T that increases with h. However, the maximum of χ is larger than that found for strong h and the transition temperature is much lower. 10
FIG. 8: Susceptibility versus temperature for 6-zigzag graphone nanoribbon at low external magnetic field h=0.03, 0.15, 0.2 and 0.28 with a zoom on susceptibility peaks.
FIG. 9: GONR width parameter dependence of the Curie temperature for external magnetic field with values h ≤ 0.3. (The lines guide the eye).
Notice that the most important influence on the critical temperature Tc seems to be the size [41], which indicates that the critical temperature becomes tunable for switches functioning in a designed temperature range [42]. To determine the evolution of the critical temperature in function of the width, we plot the estimated transition temperatures as
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function of Nz ≤ 10 for low magnetic field in Figs 9 and 10 and for strong magnetic field in Fig 11. When 0 < h ≤ 0.3kOe, the critical temperature Tc decreases as a step function of ZGONR
FIG. 10: Transition temperature versus GONR width parameter Nz at some fixed values of 0.35 ≤h≤ 0.8.(The lines guide the eye).
width Nz . At fixed width Nz , Tc increases as h increases and stabilizes at its approximate value 4.08 for h = 0.3kOe as shown in Fig.9. When h vary in the range [0.35, 0.8] , the temperature Tc increases as a step function of the ribbon width Nz to maximize and saturate as depicted in Fig.10. For strong external magnetic field 1 ≤ h ≤ 4, the transition temperature increases slowly and similarly with the size Nz . When h ≥ 5, the variation becomes slow for Nz ≥ 6 and stabilizes for higher values of the width. It is worth noting that the ZGONs is insensitive to h when 0 < h ≤ 0.3kOe and the behavior of the critical temperature versus the width Nz is the same as the one obtained for zero magnetic field. In order to investigate the influence of the temperature on the hysteresis behavior of the Nz ferromagnetic GONR in the case of low external magnetic field, a series of hysteresis curves at various values of T is plotted in Fig 12 for the case of Nz = 6 GONR having Tc = 3.88. The curves show hysteretic behavior with loops becoming narrower as the temperature 12
FIG. 11: Transition temperature as function of GONR width parameter Nz for strong external magnetic field h
FIG. 12: Magnetic hysteresis cycle of NZ = 6 graphone nanoribbons in presence of low h at four different temperatures T=0.8, 1.6, 2 and 2.2.
increases. We can also see clearly that the area of the hysteresis loops disappears completely when the temperature approaches its critical value Tc . Moreover, all the magnetization curves are symmetric for both positive and negative values of the external magnetic field. This property comes directly from the symmetry of the Hamiltonian with respect to changes h → −h and M → −M . At T = 0.8, the hysteresis is almost square indicating that the normal orientation is an easy axis. In this case, the remanent magnetization is equal to the saturation magnetization that is an obvious consequence of the square hysteresis loop. The 13
square behavior of curves is due to more the temperature is low the closer we get to the ground state, therefore the system becomes ordered and encounters the thermal equilibrium. At higher T, the shapes of the ferromagnetic hysteresis loops change from square to loop. The slanted curves demagnetize and remagnetize quickly. It is worth noting that all curves saturate quickly and when increasing temperature, the remanent magnetization decreases while the saturation magnetization increases. For T 2.2, we find that the slanted curves disappear. This behavior can be explained by the fact that higher temperature disorders the system and namely, the state of the nanoribbon system is changing from ferromagnetic phase to paramagnetic phase. For strong external magnetic field, the curves are reversible for all T, and show linear behavior with absence of remanence due to the reduction of spontaneous magnetization by strong h. The anhysteretic magnetization curves are plotted in Fig 13 for Nz = 6 ZGONR at two different temperatures T = 0.8 and T = 1.6. At T = 0.8 the saturation is achieved at low magnetic field hs = 110 Oe but when T = 1.6 the curve saturates at hs = 950 Oe. The absence of hysteresis loop in nanoribbon of graphone can be attributed to the small
FIG. 13: Hysteresis loops for Nz =6 GONR at different temperature (a) T=0.8 and (b) T=1.6 calculated in weak and strong external magnetic field.(The lines guide the eye).
size of our system in agreement with the work [43] studying magnetic nanoparticles. Indeed, this result is confirmed for higher widths of ZGONR as showed in Fig 14 where we plot the effect of graphone nanoribbon width on hysteresis curves. For strong magnetic field at fixed temperature (see Fig 14 -a), the area of hysteresis curves increase as the width Nz increase. Moreover, the slope of the curves becomes more important with Nz and the remanence at zero external field equals the saturation magnetization due to the nearly square hysteresis loops. 14
For low external h, the results are plotted in Fig 14-b for small Nz = 3, 5, 9, 19 and large width Nz = 30, 42, 65. It is shown that at fixed temperature T = 1.1, the area of hysteresis curves increases as the width increases. Notice that the same behavior is found for higher values of nanoribbon widths up to 100. When Nz increases, the curves saturate more quickly with a square loop behavior indicating that the perpendicular external field is strong enough to keep ZGONR in a perpendicularly magnetized state.
FIG. 14: The dependence of magnetic hysteresis loops on the width Nz for a) strong magnetic field at T=0.8, b) for weak external magnetic field h at T=1.1. (The lines guide the eye).
With its square hysteresis loop behavior, ZGONR is a suitable material for high-energystorage-capacitor applications.
IV.
CONCLUSION
In this work, we have studied magnetic phases of graphone nanoribbons with width W ˚ to 223A ˚ in the presence and absence of external magnetic field. The Monte ranging from 5A Carlo simulations were performed to determine the corresponding critical temperature Tc . The obtained results agree well with Tc calculated with mean field theory. From this study we learned that Tc is a reduced step function versus the ribbon widths Nz for magnetic field 0 ≤ h ≤ 0.3kOe. However, Tc increases with Nz for higher values of h. It was also shown that the hysteresis curves change from squares to loops to anhysteretic curves for appropriate values of temperature and ribbons width in both low and strong external magnetic field. These results have direct implications for the control of the critical temperature in ZGONR using suitably modulated magnetic fields and ribbon widths which is very required for the practical realization of magnetic graphone nanoribbon-based devices in the nanotechnology industry. 15
Acknowledgement 1 L.B. Drissi would like to thank Prof E. H. Saidi for discussions. L.B. Drissi would like to thank ICTP (Trieste) for the junior associateship scheme and acknowledge financial support by Centre National de Recherche Scientifique (CNRS), Morocco under project URAC09.
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