# Magnetic Properties of Gd-Doped Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
## 2 Model and [PERSON]'s Functions
This assumption is related to the fact that the antiferromagnetic nature of the interaction between octahedral and tetrahedral sites is preserved and makes possible the use of the Neel model for antiferromagnetism. For the magnetite, a strong-spin phonon interaction has been demonstrated [18], which, from the microscopic point of view, renormalizes the magnetic exchange interactions. This is taken into account with the following Hamiltonian:  
\[H_{sp}=-\frac{1}{2}\sum_{i,j,k}F(i,j,k)Q_{i}S_{j}^{Xaz}S_{k}^{Xaz}-\frac{1}{4} \sum_{i,j,k,l}R(i,j,k,l)Q_{i}Q_{j}S_{k}^{Xaz}S_{l}^{Xaz}+h.c. \tag{2}\]  
where \(F_{ijk}\) and \(R_{ijkl}\) are the spin-dependent lattice vibration constants, which are the first and second derivatives of the exchange interaction constants in relation to the phonon displacements, respectively. \(Q_{i}\) is the normal coordinate of the lattice mode. The spontaneous magnetization \(M_{s}\) of the Gd-doped Fe\({}_{3}\)O\({}_{4}\) is  
\[M_{s}=\mid M_{A}-M_{B}\mid. \tag{3}\]  
\(M_{A}\) and \(M_{B}\) are the magnetizations of the two sublattices:  
\[M_{A}=M^{FeA};\qquad M_{B}=(1-x)M^{FeB}+xM^{GdB}, \tag{4}\]  
where \(M^{Xa}=(1/N)\sum_{i}\langle(S_{i}^{Xa})^{z}\rangle\) with \(X\) = Fe, Gd and \(\alpha\) = A, B. \(\langle(S_{i}^{Xa})^{z}\rangle\) is determined from the following expression:  
\[\langle(S_{i}^{Xa})^{z}\rangle=\left[(S^{Xa}+0.5)\coth[(S^{Xa}+0.5)\beta E_{i} ^{Xa}]-0.5\coth(0.5\beta E_{i}^{Xa})\right], \tag{5}\]  
where \(\beta=\frac{1}{k_{B}T}\) and \(E_{i}^{Xa}=(1/N^{a})\sum_{j}E_{ij}^{Xa}\). \(E_{ij}\) is the excitation energy observed from the poles of the following Green's functions:  
\[\langle(S_{i}^{FeA+};S_{j}^{FeB-})\rangle;\langle(S_{i}^{FeB+};S_{j}^{FeA-}) \rangle;\langle(S_{i}^{GB+};S_{j}^{FeA-})\rangle;\langle(S_{i}^{FeA+};S_{j}^{ CdB-})\rangle;\]  
\[\langle(S_{i}^{FeB+};S_{j}^{GdB-})\rangle;\langle(S_{i}^{GB+};S_{j}^{FeB-}) \rangle;\langle(S_{i}^{FeA+};S_{j}^{FeA-})\rangle;\langle(S_{i}^{FeB+};S_{j}^{ FeB-})\rangle,\]  
where the operators \(S_{i}^{Xa}\) are determined from the following expression: \((S_{i}^{Xa})^{\pm}=(S_{i}^{Xa})^{x}\pm i(S_{i}^{Xa})^{y}\). The spin-phonon interaction is observed within the mean field approximation. This leads to a renormalization of the exchange interactions in the system making them temperature dependent \(J_{eff}=J+2F^{2}/(\omega_{0}-RM_{s})\) where \(\omega_{0}\) is the harmonic frequency of the polar lattice mode and \(J\) is a formal denotation for all exchange interaction constants. As mentioned in the Introduction, by doping the magnetite with RE ions, the band gap width increases. The band gap width in Fe\({}_{3}\)O\({}_{4}\) defines the latter as a semiconductor, which makes these oxide NPs suitable for optical sensors. The changes in the magnetic characteristics caused by the Gd doping will cause changes in the band structure and the width of the band gap. We will account for this influence by including in the Hamiltonian of the system the interaction \(I\) between the localized spins and conduction ones at the \(i\) site, i.e., the s-d model:  
\[- x\sum_{ij}J_{ij}^{FeA-GdB}\mathbf{S}_{i}^{FeA}\cdot\mathbf{S}_{ j}^{GdB}-x\sum_{ij}J_{ij}^{FeB-GdB}\mathbf{S}_{i}^{FeB}\cdot\mathbf{S}_{j}^{GdB}\]0  
where \(\mathbf{s}_{i}\) are the spin operators of the conduction electrons. Then, we can define the band gap energy \(E_{g}\) as the difference between the valence and conduction bands:  
\[- x\sum_{ij}J_{ij}^{FeA-GdB}\mathbf{S}_{i}^{FeA}\cdot\mathbf{S}_{ j}^{GdB}-x\sum_{ij}J_{ij}^{FeB-GdB}\mathbf{S}_{i}^{FeB}\cdot\mathbf{S}_{j}^{GdB}\]1  
with the electronic energies  
\[- x\sum_{ij}J_{ij}^{FeA-GdB}\mathbf{S}_{i}^{FeA}\cdot\mathbf{S}_{ j}^{GdB}-x\sum_{ij}J_{ij}^{FeB-GdB}\mathbf{S}_{i}^{FeB}\cdot\mathbf{S}_{j}^{GdB}\]2  
which are observed from the Green's functions [19]\(g(\mathbf{k}\sigma)=\ll c_{kv}\); \(c_{kv}^{+}\gg\), \(\sigma=\pm 1\) where \(c_{kv}^{+}\) and \(c_{iv}\) are Fermi-creation and -annihilation operators.