Theoretical investigation on the magnetization enhancement of Fe3O4-reduced graphene oxide nanoparticle system

We present a theoretical study on the enhancement of magnetization of Fe3O4 nanoparticle system upon addition of reduced graphene oxide (rGO). Experimental data have shown that the magnetization of Fe3O4-rGO nanoparticle system increases with increasing rGO content up to about 5 wt%, but decreases back as the rGO content increases further. We propose that the enhancement is due to spin-flipping of Fe ions at the tetrahedral sites assisted by oxygen vacancies at the Fe3O4 particle boundaries. These oxygen vacancies are induced by the presence of rGO flakes that adsorb oxygen atoms from Fe3O4 particles around them. To understand the enhancement of the magnetization, we construct a tight-binding based model Hamiltonian for the Fe3O4 nanoparticle system with the concentration of oxygen vacancies being controlled by the rGO content. We calculate the magnetization as a function of the applied magnetic field for various values of rGO wt%. We use the method of dynamical mean-field theory and perform the calculations for a room temperature. Our result for rGO wt% dependence of the saturated magnetization shows a very good agreement with the existing experimental data of the Fe3O4-rGO nanoparticle system. This result may confirm that our model already carries the most essential idea needed to explain the above phenomenon of magnetization enhancement.


Introduction
Fe 3 O 4 , informally known as magnetite, is a magnetic material with ferrimagnetic ordering. Due to its half-metallic characteristic with 100% spin polarization at the Fermi level [1,2] and high Curie temperature (∼ 851 K), magnetite becomes a prospective material for spintronic devices [3][4][5]. To some extent, Fe 3 O 4 nanoparticles still preserve the magnetic properties as those of bulk Fe 3 O 4 . Due to high Curie temperature and biocompatibility [6], Fe 3 O 4 nanoparticle system also becomes promising for use in photo-catalysis and biomedical applications [7,8].
A recent study by Taufik et al. suggests that the magnetization of Fe 3 O 4 nanoparticle system enhances when mixed with reduced graphene oxide (rGO) [9]. The mechanism behind this magnetization enhancement has still not been fully understood. However, a hint to explanation of this phenomenon may be given by another recent experimental study by Herng et al., suggesting that the magnetization enhancement occurs due to spin flipping of Fe ions at tetrahedrally coordinated sites, which is suspected to be induced by oxygen vacancies. These spin flippings lead the system to transform from ferrimagnetic to ferromagnetic ordering [10]. Motivated by that experimental hint, we develop the idea and propose a theoretical study to explain the magnetization enhancement phenomenon of the Fe 3 O 4 -rGO nanoparticle system. We hypothesize that the magnetization enhancement of Fe 3 O 4 -rGO nanoparticle system is induced by oxygen vacancies formed on the surface of Fe 3 O 4 nanoparticle clusters due to the presence of rGO flakes nearby the Fe 3 O 4 clusters. This argument is based on a previous experimental study by Santoso et al. [11], supported by a theoretical study by Majidi et al. [12], indicating that graphene can easily adsorb oxygens. Our main goal is to calculate magnetization of Fe 3 O 4 nanoparticle system with and without oxygen vacancies, and use this to mimic the experimental data of magnetization of the combined Fe 3 O 4 -rGO nanoparticle system by connecting the concentration of oxygen vacancies with the rGO content.

Model of Fe 3 O 4 -rGO nanoparticle system
The experimental data of reference [9] shows that the magnetization of Fe 3 O 4 -rGO nanoparticle system increases with increasing rGO content up to about 5 wt%, but decreases back as the rGO content increases further. We propose a model to explain this situation as illustrated in figure 1. When there is no rGO content (figure 1(a)), all the Fe 3 O 4 clusters are assumed to have no oxygen vacancies. As the system is added with rGO content less than 5 wt% (figure 1(b)), rGO flakes start to fill spaces in between neighboring Fe 3 O 4 clusters. At this low concentration of rGO, we suppose that each space between two neighboring Fe 3 O 4 clusters can only be filled by at maximum of one layer of rGO flake. The insertion of an rGO flake induces the extraction of oxygen atoms from the nearby Fe 3 O 4 clusters, which then get adsorbed onto the rGO surface. As a result, oxygen vacancies are formed on the surfaces of Fe 3 O 4 clusters, inducing spin flippings of the Fe ions at tetrahedral sites in those clusters, thus increasing the total magnetization of the whole system by some amount. We suppose that when the rGO content is ∼ 5 wt% (figure 1(c)), each space between neighboring Fe 3 O 4 clusters has been filled with one rGO layer. At this rGO concentration, the adsorption of oxygen atoms by the rGO flakes is optimum, hence the enhancement of magnetization of the system becomes maximum. When the rGO content is exceeding 5 wt% (figure 1(d)), there are some spaces between Fe 3 O 4 clusters that are filled with rGO flakes of more than one layer. The increase of number of layers of an rGO flake does not increase its capability of adsorbing oxygen atoms since the area of its open surfaces remains the same. However, the addition of more rGO content without more formation of oxygen vacancies in Fe 3 O 4 clusters causes the total magnetization per unit mass of the system to decrease. Therefore, adding rGO flakes of more than 5 wt% decreases the total magnetization per unit mass of the system monotonically.
The rGO content added to Fe 3 O 4 nanoparticles is experimentally measured using weight percent unit (wt%). This weight percentage can be understood as ratio of the total mass of rGO flakes to the total where N F and N G , respectively, are the number of Fe 3 O 4 clusters and rGO layers in the system, M F and M G are the averaged mass of Fe 3 O 4 clusters and the averaged mass of rGO layers, respectively, while x is wt% of rGO layers added to the Fe 3 O 4 nanoparticles. Now, let the mass ratio of Fe 3 O 4 clusters to rGO flakes be α. As an Fe 3 O 4 cluster is likely heavier than an rGO layer, α should be larger than unity. In our model, a space between neighboring Fe 3 O 4 clusters can be filled by an rGO flake or one or more layers. How rGO layers fill such spaces affects the magnetization enhancement of the system. Hence, we include this into our model. Let n be the maximum number of rGO layers filling a space between two Fe 3 O 4 clusters while the system still alows a vacant space. This means, no space will get filled with an rGO flake of more than n layers as long as there is still a vacant space. Hence, ratio of the total number of rGO layers in the system to the maximum number of rGO layers that can be accomodated by the system with no vacant spaces can be formulated as Based on the definition of n stated above, the probability of finding a space between two Fe 3 O 4 clusters with no rGO flake is Lastly, let γ be the portion of unit cells in an Fe 3 O 4 cluster that have oxygen vacancies. Thus, we may construct the relation between the total magnetization M , that depends on the rGO content x, with the quantities just described as with M 0 = µ 0 M F . By dividing both the numerator and the denominator with N F M F , the equation can be rearranged to be To use equation (5), we first specify the n value corresponding to the model we choose. Next, for every x value we take β according to equation (2), and then P 0 according to equation (3). Along with n, γ and α can be adjusted to obtain the best fit to the experimental data. As for M 0 , we take the empirical value of the saturation magnetization from the experimental data, that is ∼ 80 emu/g. Whereas, M ν is the theoretical value of the saturation magnetization if spin flipping occurs in all unit cells in the 3. Model to calculate the magnetization as a function of temperature and external magnetic field To calculate the magnetization of the system as a function of temperature we model the Fe 3 O 4 nanoparticle cluster as if it is a bulk system. The model has been proposed in our recent study [13]. Following reference [13], we use (001) Fe 3 O 4 surface and simplify the crystal structure by taking only the bottom-most two-layers of the actual unit cell. The half metallic property along the (001) Fe 3 O 4 surface has been confirmed by first-principle calculation [14]. We further simplify the unit cell by restricting only the square block containing one unit of Fe 3 O 4 as shown in figure 2. With this model unit cell, for the system without oxygen vacancies (referring to figure 2(a)), we choose 11 basis atomic orbitals to construct our Hilbert space, that we order as follows: |FeA − e g , |FeB 1 − e g and |FeB 2 − e g . The 11 basis orbitals along with the two spin orientations make up a set of 22 basis states. Whereas, for the system with oxygen vacancies, we remove one oxygen atom, i.e. O B , hence removing also |O B − p x and |O B − p y orbitals, leaving only 9 basis orbitals or 18 basis states remaining. Using the above sets of basis states, we propose our model Hamiltonian as The first term in the Hamiltonian is the kinetic term which is constructed using tight-binding approximation. Here, [a k ] is a column vector containing the corresponding fermion annihilation operators, while [a k ] † is its hermitian conjugate, with k index denotes the k-points in the Brillouin zone. [H 0 (k)] is a 22 × 22 (18 × 18) matrix in momentum space for the system without (with) oxygen vacancies, constructed using tight-binding approximation, which consists of four 11 × 11 (9 × 9) blocks corresponding to their spin directions as See reference [13] for the detailed content of [H 0 (k)] for both systems without and with oxygen vacancies. The second term represents the magnetic exchange interaction between the collective spin 5

1234567890
International S of the t 2g electrons and the spin s of an e g electron in each Fe ion, with i denoting the site of the Fe ion, and J H being Hund's coupling constant. The last term represents the Coulomb repulsive interaction between two electrons occupying the same Fe-e g orbital, with n iσ being the electron occupation number operator at site i with spin component σ ,and U the Hubbard parameter. To simplify our calculations, we replaced this term with the mean-field approximated version, that is with n iσ to be calculated self-consistently.

Calculation Methods
We use dynamical mean-field theory (DMFT) algorithm to calculate magnetization of the system. We begin the algorithm by defining the Green function matrix for our system as where z is the complex frequency variable defined as z = ω + i0 + for the real-frequency domain and z = iω n + µ for the Matsubara-frequency domain, with ω n = (2n + 1)πT being fermion Matsubara frequency, T temperature and µ chemical potential. Note that throughout this paper, we seth = k B = 1.
Here, the self-energy matrix [Σ(z)], along with n iσ , are taken from initial guess. We then average the Green function matrix over all k-points in the Brillouin zone as where N is the number of k-points in the Brillouin zone. Next, we extract the mean-field Green function matrix through We use this mean-field Green function matrix to calculate the local interacting Green function matrix that depends on the orientation of the three Fe spins where [Σ(θ 1 , θ 2 , θ 3 )] loc is the local self-energy matrix manifesting the dynamics of the Coulomb repulsive interactions and magnetic exchange interactions at a local unit cell, containing the dependence on the angles θ 1 , θ 2 and θ 3 , of the three Fe spins with respect to the direction of net magnetization (taken as the direction of the positive z axis), and the averaged occupancies of electrons at Fe-e g orbitals, n iσ . The datailed form of [Σ(θ 1 , θ 2 , θ 3 )] loc can be seen in reference [13]. Now [G(z, θ 1 , θ 2 , θ 3 )] loc must be averaged over all possible orientations of the three Fe spins in a unit cell. The averaging process is implemented using the Boltzmann weight defined as where is the effective action, which is calculated only in Matsubara-frequency domain and International serves as the partition function. The term hS(cos θ 1 + cos θ 2 + cos θ 3 )/T is added to accomodate the effect of external magnetic field h (defined in unit of energy) at a temperature T . Thus, the averaged interacting Green function matrix is obtained through This averaging process is also used to calculate the magnetization through The new self-energy matrix can then be extracted as Along with the self-energy matrix, we also need to update the chemical potential, µ, by imposing a fixed electron filling per unit cell, that is n filling = 20 for the system without oxygen vanancies, such that where is the electron density of states, and f (µ, ω, T ) = (e (ω−µ)/T + 1) −1 is the Fermi-Dirac distribution function. The calculated chemical potential is then used to update the averaged electron occupancy of each e g orbital of Fe ion at site i in a unit cell with spin component σ through with PDOS iσ (ω) = −(1/π) Im G ave iσ (ω + i0 + ) being the corresponding projected density of states. Finally, we iterate the algorithm using the updated self energy matrix, chemical potential and averaged electron occupancy, and the self-consistency process is repeated until convergence is achieved.

Results and discussion
Since in this paper we use DMFT to obtain mainly the magnetization as a function of temperature and magnetic field, we restrict our calculations only for the system with oxygen vacancies. While, for the system with oxygen vacancies, we take for granted the result of reference [13] from which the system with oxygen vacancies has a ferromagnetic ordering with magnetization being three times greater than that of the system without oxygen vacancies that has a ferrimagnetic ordering. Further, we normalize the magnetization such that the saturation magnetization of the pure Fe 3 O 4 nanoparticle system is ∼ 80 emu/g, as obtained from the experiment. For our calculations we use the following parameters:  Figure 3 shows our result of spin-resolved density of states (DOS) for a room temperature (∼ 300 K). The density of states shows that our model reveals its half-metallic characteristic, where in the majority spin channel, the chemical potential (or the Fermi level) lies inside of the band, while in the minority spin channel, the chemical potential lies outside the band. This result demonstrates that our model captures   [2]. Having this result, we argue that our model is adequate to be used for further calculations.
In calculating the saturation magnetization as a function of rGO wt% (x), we tune the parameters α, γ and n, and find that the best fit to experimental data is obtained using α = 20, γ = 0.574 and n = 1. Figure  4 shows our results of the saturation magnetization for n = 1, 2, and 3, as compared to the experimental data. By varying n, we can track how the distribution of number of layers of the rGO flakes influences the result. For n = 1, the result shows the linear increase of the saturation magnetization with increasing x up to 5 wt%, after which the additional rGO flakes make the saturation magnetization decreases also almost linearly. A little discrepancy is shown by our n = 1 result, that is, the saturation magnetization values above x = 5% are obtained to be slightly higher than the experimental data. For higher values of n, the maximum saturation magnetization shifts to the right. That is because there is still a probability of finding spaces unoccupied by rGO flakes when some other spaces are already filled by rGO flakes with two (for n = 2) or three (for n = 3) layers. This mechanism also implies that the peak of saturation magnetization for n = 2 and n = 3 become broader. Meanwhile, the fact that our saturation magnetization calculation result is in agreement with experimental data for γ = 0.574 indicates that not all Fe 3 O 4 unit cells have oxygen vacancies. This is because the capability of the rGO flakes to adsorb more oxygens from the Fe 3 O 4 clusters are limited by the oxygen adatoms that are already precedingly attached on the graphene surface. As for the value of α which is about 20, it shows us that the Fe 3 O 4 nanoparticle cluster is much heavier than the rGO layer which is consistent with our presumption.
The results of applied magnetic field dependent magnetization for all rGO wt% variation are shown in figure 5. The magnetization of the model for Fe 3 O 4 nanoparticle system with no additional rGO flakes is represented by the black curve. While, the red curve represents the magnetization of the system with the optimum rGO content (x = 5%), leading to highest saturation magnetization. For the other rGO wt% values (x = 0.10, 0.15 and 0.20) the calculations are performed using equation 5. The overall trend shows by our calculation results is in agreement with the experimental data, in which the highest magnetization occurs at 5 wt% of rGO, and the magnetization subsequently decreases until the content of rGO flakes achieving 20 wt%, which is still higher than that of the pure Fe 3 O 4 nanoparticle system.
In contrast to the saturation magnetization results, our calculated remanent magnetization values do not really fit with the experimental data, instead they are significantly higher than experimental data, and are rather closer to the saturation magnetization values. This is related to the fact that the Curie temperature (T C ) for our model of  the experimental value of 858 Kelvin. This may partly arise as a consequent of the over-simplification of our model unit cell. Another possible source of discrepancy may come from the mean-field treatment of the Hubbard term in our model Hamiltonian, which commonly overestimates physical quantities such as T C . Nevertheless, overall, our model is able to qualitatively confirm the magnetization enhancement of Fe 3 O 4 nanoparticle system with addition of rGO flakes.

Conclusions
In conclusion, our present study supports our previous hypothesis that the enhancement of magnetization of Fe 3 O 4 nanoparticle system upon addition of reduced graphene oxide (rGO) is due to spin-flipping of the Fe ions in the tetrahedral sites and that oxygen vacancies at the Fe 3 O 4 nanoparticle clusters assist the occurrence of spin-flipping. The presence of rGO flakes nearby Fe 3 O 4 clusters induces the formation of oxygen vacancies in the Fe 3 O 4 clusters, leading to the occurrence of spin-flipping and thus enhancing the magnetization of the system. Our model for Fe 3 O 4 nanoparticle cluster can describe the saturation magnetization enhancement of Fe 3 O 4 nanoparticle system upon addition of reduced graphene oxide (rGO). However, our calculation method overestimates the Curie temperature and the remanent magnetization of the Fe 3 O 4 nanoparticle system. This may be caused by the over-simplification of our model unit cell and the mean-field approximation used in our method.