Size and Ion-Doping Effects on Magnetic, Optical, and Phonon Properties of CuAlO₂ Nanoparticles

Abstract

The magnetic, optical, and phonon properties of ion-doped CuAlO${}_2$ nanoparticles on the Cu or Al site are theoretically investigated. The room temperature ferromagnetism in CuAlO${}_2$ nanoparticles can be due to the surface, size, and doping effects. The magnetization increases with the decreasing nanoparticle size. The different radii of the transition metal ion and the host Cu ion lead to compressive strain, to the enhancment of the exchange interaction constants, and to increased magnetization $M_s$ and Curie temperature $T_C$. By substitution with Mn or Cr on the Al site, tensile strain, a decrease in $M_s$, and an increase in dopants are observed. The size and ion-doping influence on the band-gap energy is also discussed. The phonon energy $\omega$ decreases, whereas the phonon damping $\gamma$ increases with increasing temperature and decreasing NP size. They show a kink around $T_C$ ∼ 400 K. The behavior of $\omega$ and $\gamma$ for different ion dopings is observed.

1. Introduction

Diluted magnetic semiconductors (DMS) play an important role in interdisciplinary materials science and future spintronics. Ferromagnetic DMS have been studied from first principles within mean field approximation [1,2]. Ferromagnetism in 3d transition metal (TM)-doped II–VI and III–V based DMS is predicted [3]. Kizaki et al. [4] have investigated the magnetism in a new DMS, namely CuAlO${}_2$ (CAO), by the Korringa–Kohn–Rostoker method. Using density functional theory (DFT) calculations, Iordanidou et al. [5] have examined the hole doping effect on the magnetic and electronic properties of CAO. It is known that pure CAO is a p-type width band gap semiconductor with $E_g$ of about 3.5 eV [6]. It has the hexagonal delafossite structure and the space group of R$\overline{3}$m. CAO can be applied as a DMS when doped with TM ions. Kizaki et al. [4,7] have investigated the electronic and magnetic properties under carrier doping treatment in TM-doped CAO on the Cu site (TM = Fe, Co, Mn, and Ni). The Curie temperature $T_C$ of the doped examples increases with increasing dopant concentration. Moreover, Fe or Co doping on the Al site of CAO bulk and thin films increases the spontaneous magnetization [8,9,10,11,12].

The origin of ferromagnetism in DMS is not yet clear, even if various methods have been proposed. First-principal calculations showed that (Cu,Fe)AlO${}_2$ can be a candidate of ferromagnetic DMS [13]. From the calculated density of states, it seems that the double exchange interaction is the dominant exchange mechanism in (Cu,Fe)AlO${}_2$ [13]. It was recently found that the weak magnetism of these materials may derive from the polarised unpaired electrons around impurities [14]. The magnetic properties of Mn-doped CAO, Cu(Al,Mn)O${}_2$ have been reported by Zhang et al. [15]. The magnetization decreases with increasing Mn concentration. The influence of Cd impurity at Cu and Al sites on the electronic properties of CAO from first-principles calculations is discussed in Ref. [16].

The TM doping on the Al site also influences the optical properties of CAO thin films and nanofibers [17,18,19,20,21]. Raman spectra of pure and ion-doped CAO bulk and NPs are studied in [18,22,23,24,25,26]. So, carrier concentration and ion doping together significantly affect the oxide-based DMS.

The aim of the present paper is to investigate the magnetic, optical, and phonon properties of ion-doped CAO nanoparticles (NPs) for the first time using a microscopic model and Greens’s function theory in order to clarify the origin of room temperature ferromagnetism in these systems on microscopic level.

It should be noted that the most theoretical papers consider the magnetic properties of CAO NPs using DFT. The DFT is a very powerful tool in investigation of many body problems. However, DFT is mostly concerned with ground-state properties at zero temperature. In our approach, we are able to cover the whole temperature regime. It is a finite temperature analysis including the entire excitation spectrum. In particular, the method allows us to study the total phase diagram, which is based on the different excitation energies realized in the system. The disadvantage of our approach consists of the consideration of collective properties from the beginning. Our basic quantities are not the “naked” electrons but effective spins of the underlying quasi-particles. Whereas within DFT all of the parameters of the system can be—at least in principle—calculated, we are forced to use additional models to find out those parameters. We are convinced that both approaches, DFT and Green’s function method, are appropriate and, to a certain extent, are an alternative in describing many body systems.

2. The Model and Green’s Functions

The room temperature ferromagnetism in bulk CAO can be caused by ion doping and can be described by the Heisenberg hamiltonian $H_d$:

$$H_d=\sum _{i,j}x{J}_{dij}({\mathbf{S}}_j·{\mathbf{S}}_j)-\sum _i{D}_i{\left(S_i^z\right)}^2-g{\mu }_{B}h\sum _i{S}_i^z.$$

(1)

${\mathbf{S}}_i$ is the Heisenberg spin-operator of the TM at site i. $J_d$ is the exchange interaction between the TM ions. Kizaki et al. [4] found that $J_d$ between the magnetic ions in the same Cu-plane are ferromagnetic, whereas those between the Cu-planes can be neglected. D is the single-ion anisotropy, h is an external magnetic field, and x is the doping concentration.

In the case of a CAO NP, there also appears a ferromagnetism from surface and size effects due to the uncompensated Al spins on the surface described by $H_s$ as well as due to ion-doping effects (see $H_d$):

$$H_s=-\sum _{i,j}(1-x)J_{sij}({\mathbf{S}}_j·{\mathbf{S}}_j).$$

(2)

From the spin Green’s function defined as $G_{ij}=\ll S_i^{+};S_j^{-}\gg$ the magnetization M for arbitrary spin S is calculated:

$$M=\frac{1}{N^2}\sum _k\left[(S+0.5)coth\left[(S+0.5)\beta E_m\left(\mathbf{k}\right)\right]-0.5coth(0.5\beta E_m\left(\mathbf{k}\right)\right],$$

(3)

where $\beta$ = $1/k_{B}T$, $E_m\left(\mathbf{k}\right)$ is the spin-wave energy.

The spin–phonon and phonon–phonon interactions are described by:

$$H_{sp-ph}=\frac{1}{2}\sum _{i,j,k}F(i,j,k)Q_i{S}_j^z{S}_k^z-\frac{1}{4}\sum _{i,j,r,s}R(i,j,r,s)Q_i{Q}_j{S}_r^z{S}_s^z+h.c.$$

(4)

The normal coordinate $Q_i$ can be expressed in terms of phonon creation $a^{+}$ and annihilation a operators, $Q_i={\left(2{\omega }_{0i}\right)}^{-1/2}(a_i+a_i^{+})$. F and R are the spin–phonon coupling constants in the first and second order, respectively.

$$\begin{array}{ccc}\hfill H_{ph-ph}& =& \frac{1}{2!}\sum _i{\omega }_{0i}{a}_i{a}_i^{+}+\frac{1}{3!}\sum _{i,j,r}B(i,j,r)Q_i{Q}_j{Q}_r\hfill \\ & +& \frac{1}{4!}\sum _{i,j,r,s}A(i,j,r,s)Q_i{Q}_j{Q}_r{Q}_s,\hfill \end{array}$$

(5)

where ${\omega }_{0i}$ is the frequency of the lattice mode.

From the poles of the phonon Green’s function ${\tilde{G}}_{ij}\left(t\right)=\langle \langle a_i\left(t\right);a_j^{+}\rangle \rangle$ the phonon energies are observed:

$$\begin{array}{ccc}\hfill {\omega }_{ij}^2& =& {\omega }_0^2-2{\omega }_0(M_i{M}_j{R}_{ij}{\delta }_{ij}\hfill \\ & -& \frac{1}{2N^{\prime }}\sum _r{A}_{ijr}(2{\overline{N}}_r+1)-B_{ij}\langle Q_{ij}\rangle {\delta }_{ij}),\hfill \end{array}$$

(6)

$$\langle Q_{ij}\rangle =\frac{M_i{M}_j{F}_{ij}{\delta }_{ij}-\frac{1}{N^{\prime }}{\sum }_r{B}_{ijr}(2{\overline{N}}_r+1)}{{\omega }_0-M_i{M}_j{R}_{ij}{\delta }_{ij}+\frac{1}{N^{\prime }}{\sum }_r{A}_{ijr}(2{\overline{N}}_r+1)}.$$

(7)

The phonon correlation function ${\overline{N}}_r=\langle a_r^{+}{a}_r\rangle$ is obtained via the spectral theorem. The phonon damping $\gamma ={\gamma }_{sp-ph}+{\gamma }_{ph-ph}$ is also calculated taking into account anharmonic spin–phonon and phonon–phonon interactions.

The band-gap energy $E_g$ of a CAO NP is defined by the difference between the valence and conduction bands:

$$E_g={\omega }^{+}(\mathbf{k}=0)-{\omega }^{-}(\mathbf{k}={\mathbf{k}}_{\sigma }).$$

(8)

The electronic energies

$${\omega }^{\pm }\left(k\right)={ϵ}_k-\frac{\sigma }{2}I\langle S^z\rangle$$

(9)

are observed from the Green’s function $g(k,\sigma )=\ll c_{k,\sigma };c_{k\sigma }^{+}\gg$, $\sigma =\pm 1$, $c_{i\sigma }^{+}$ and $c_{i\sigma }$ are Fermi operators, and I-s-d is the interaction constant [27].

3. Numerical Results and Discussion

A CAO NP with a cubo-octahedral shape is defined by fixing the origin at a certain Al spin in the center of the particle and including all other spins within the particle into shells, which are numbered by $n=1,\dots ,N$ from the central to the surface shell. For the numerical calculations, we use the following model parameters: $J_d$(Fe-Fe) = 195 meV, $J_d$(Co-Co) = 117 meV, $J_d$(Mn-Mn) = 78 meV, $J_d$(Ni-Ni) = 97.5 meV [4], $J_s$ = 20 meV, D = −0.068 meV, F = 23 cm${}^{-1}$, R = −18 cm${}^{-1}$, A = 6.61 cm${}^{-1}$, and B = −2.94 cm${}^{-1}$. The exchange interaction $J_{ij}=J(r_i-r_j)$ depends on the distance between the spins, i.e., on the lattice parameters. It is inverse proportional to the lattice parameters. Therefore, the exchange interaction constant on the surface $J_s$ and of the doped states $J_d$ can be changed or can increase or decrease in dependence on the strain in the lattice. So, we take into account the direct connection between the microstructure and magnetic behavior.

Firstly, we will consider the spontaneous magnetization $M_s$ in a pure CAO NP. It must be noted that bulk CAO is a p-type wide band gap semiconductor, and it is a nonmagnetic compound. However, due to the uncompensated surface spins of the Al${}^{3+}$ ions on the surface, a spontaneous magnetization $M_s$ appears in the CAO NP, contrary to the bulk case, where $M_s$ = 0. We obtain that $J_s$ increases and the spontaneous magnetization $M_s$ of a CAO NP increases with decreasing NP size. The result is presented in Figure 1. A finite magnetization $M_s$ is also observed experimentally in pure CAO NPs [12,28] and in pure CAO thin films [29], due to the existence of point defects. It must be noted that we obtain similar behavior for the size dependence of the phase-transition temperature $T_C\left(N\right)$, $T_C$, which increases with decreasing NP size.

The origin of ferromagnetism in DMS is not yet clear. In order to investigate the observed room temperature ferromagnetism (RTFM) in ion-doped CAO-bulk and NPs, we have replaced some Cu-ions with Fe-ions, i.e., we consider Cu${}_{1-x}$Fe${}_x$AlO${}_2$ NPs for x = 0–0.25. There are experimental data that show that in this concentration interval the samples are single-phase, whereas for larger x values, the second phase appears [4]. It must be noted that by the doping of TM on the Al site, the limit value of the ion doping concentration is smaller; it is around 0.05–0.1 [8,10,15,28,30], which is different for the different doping ions. The exchange interaction constant J depends on the distance between the spins, i.e., on the lattice parameter, on the different strain, on the lattice symmetry, and on the number of next neighbors. J is an inverse square function of the distance between two neighboring spins, i.e., $J\propto 1/{(r_i-r_j)}^2$. From the structural analysis, Cu${}^{1+}$ is replaced by Fe${}^{3+}$ or Fe${}^{2+}$, which leads to cation vacancies increasing because of charge variation [4,7,9]. Through the different radii of the dopants, lattice defects can be introduced or intrinsic host-lattice defects can be activated when different ions, such as Fe, occupied the Cu sites. Chen et al. [9] have shown that Fe element exists in the ion form, with the valence of +2 and/or +3. Since the ionic radius of Fe${}^{2+}$ (0.92 $\dot{A}$) or Fe${}^{3+}$ (0.65 $\dot{A}$) is always smaller than Cu${}^{1+}$ (0.95 $\dot{A}$), the lattice parameters decrease with increasing the Fe dopants as supported by Chen et al. [9]. The reduction of the lattice constants in Fe-doped CAO nanostructures leads to a compressive strain, i.e., in our microscopic model, it leads to an increase in the exchange interaction constant of the doped states $J_d$ with an increase in the Fe ion concentration because it is inverse proportional to the lattice parameters. So, we obtain an increase in the spontaneous magnetization $M_s$ and the Curie temperature $T_C$. The ferromagnetic coupling between the doping Fe-ions contributes additively to this increase. Fe${}^{3+}$-ion (Fe${}^{2+}$-ion) has 5d (6d) electrons with the total spin of S = 5/2 (S = 2), whereas the Cu${}^{1+}$-ion has S = 0. Thus, Fe${}^{3+}$ or Fe${}^{2+}$ substitution into Cu${}^{1+}$ induces an extra magnetic moment. The spontaneous magnetization $M_s$ and the Curie temperature $T_C$ in dependence on the Fe${}^{3+}$ doping concentration are calculated. We observe that both increase with increasing Fe doping concentration x. The result for $T_C\left(x\right)$ is demonstrated in Figure 2, curve 1. We would obtain a similar increase in the phase-transition temperature $T_C$, for example, for Ni-, Mn-, or Co-doped CAO, (Cu,TM)AlO${}_2$ (Figure 2, curves 2–4), where the doping ions have a smaller ionic radius in comparison with that of the Cu ion. This behavior of the doping dependence of $T_C\left(x\right)$ is in good qualitative agreement with the experimental data [8,9,10,11,12].

It must be noted that for example, in Mn- or Cr-doped CAO on the Al site, where the ionic radius of Mn${}^{3+}$(0.72 $\dot{A}$) or Cr${}^{3+}$(0.63 $\dot{A}$) is larger than that of Al${}^{3+}$(0.51 $\dot{A}$), the lattice parameters increase in agreement with the experimental data of Zhang et al. [15]. A tensile strain appears, i.e., $J_d$ decreases, which leads to a decrease in $M_s$ with increasing Mn or Cr ion doping concentration (see Figure 3 and Figure 4, respectively). Similar behavior is reported for Cu(Al,Mn)O${}_2$ by Zhang et al. [15]. Unfortunately, there are not experimental data for $M\left(x\right)$ of Cr-doped CAO. A decrease in the magnetization is observed additively with increasing Mn content x due to the antiferromagnetic coupling among the Mn spins (contrary to the case of Mn doping at the Cu site). By the substitution of the Al ion with a rare earth ion (RE) whose radius is larger than that of Al, tensile strain and a reduction of the spontaneous magnetization $M_s$ with an increase in the RE ion doping concentration appears again. Unfortunately, there are not experimental data for RE doped CAO.

From Equation (8), we have calculated the band-gap energy $E_g$ for pure and ion-doped CAO NPs (see Figure 5). It must be noted that there is a broadening of the band gap $E_g$ of pure CAO NPs with a decrease in the NP size. We now consider the case of a Cr${}^{3+}$-doped CAO NP, CuAl${}_{1-x}$Cr${}_x$O${}_2$. The lattice parameters increase with the increase in the Cr-doping concentration [17] because the ionic radius of the doping Cr ion (0.63 $\dot{A}$) is larger than that of the host Al ion (0.51 $\dot{A}$). A tensile strain appears, i.e., $J_d$ decreases with increasing x, which leads to a decrease in the magnetization (see Figure 4) and an increase in $E_g$ (curve 1). The observed behavior is in agreement with the experimental data of Jiang et al. [17]. We would also obtain a similar enhanced $E_g$ within our model by doping with Sb${}^{3+}$ (0.76 $\dot{A}$) ions or Y${}^{3+}$ (1.04 $\dot{A}$) ions, which also causes a tensile strain, i.e., an increase in the lattice parameters as well as a decrease in the exchange interaction constant $J_d$ and therefore reduced magnetization $M_s$ and enhanced $E_g$, reported in [18,31]. Otherwise, by doping with the Mg ion, we observe the contrary result: a decrease in the band-gap energy (curve 2), in agreement with [28,30,32]. The optical absorption of the CAO increases due to the reduction in $E_g$ by Mg doping. A smaller $E_g$ is also reported by Bi and Eu doping at the Al site in CAO [21,33].

Finally, we will discuss the temperature, size, and ion-doping dependence of the phonon energy $\omega$ and damping $\gamma$ in CAO NPs. The temperature dependence for the A${}_{1g}$ mode ${\omega }_0$ = 767 cm${}^{-1}$ of the phonon energy $\omega$ and phonon damping $\gamma$ is shown in Figure 6. We take into account the anharmonic spin–phonon interaction $R<0$, as well as the three-phonon $A>0$ and four-phonon $B<0$ anharmonic interactions. Let us emphasize that in order to obtain the experimentally observed softening of $\omega$, we have to chose $R<0$. For $R>0$, we would obtain a hardening, an increase in $\omega$ with T [34]. The damping $\gamma$ is proportional to $R^2$ and so independent of the sign of R. $\gamma$ corresponds to the full width at half-maximum of the Raman lines and is indirectly proportional to the phonon life time. The damping is a direct consequence of the anharmonic coupling. The phonon energy decreases, whereas the phonon damping increases with increasing temperature, which is in agreement with [22,26]. It can be seen from Figure 6 that around $T_c$ ∼ 400 K, we observe a kink due to the spin–phonon interaction R. For R = 0, the kink disappears. Moreover, the kink is size-dependent; it shifts to smaller T values with increasing NP size. In Figure 1, we have shown that with decreasing NP size spontaneous magnetization $M_s$ with the corresponding $T_C$ value appears. A similar anomaly in the phonon energy and damping of CAO thin films is observed experimentally between 423 and 573 K by Singh et al. [22] and around T ∼ 300 K by Pellicer-Porres et al. [26]. The authors have proposed that these changes may be related to a negative thermal expansion (NTE) in the delafossite structure along the O-Cu-O linkage. A similar NTE is also reported in other AMO${}_2$ (A = Cu or Ag; M = Al, Sc, In, or La) [35,36,37]. NTE means a volume contraction, i.e., there is a change in the structure, for example, by transitions between ferro- and paramagnetic phases.

The size dependence of the phonon energy $\omega$ and the phonon damping $\gamma$ in a CAO NP is presented in Figure 7. It can be seen that the phonon energy decreases whereas the damping increases with decreasing nanoparticle size, which is in agreement with the experimental data of Yassin et al. [23].

The ion-doping also influences the phonon properties of CAO NPs. It may cause lattice distortions and unrelaxed strains. To show this, we will consider the Fe${}^{3+}$ doping at the Al${}^{3+}$ site. The ionic radius of the Fe ion (0.65 $\dot{A}$) is larger than that of the Al ion (0.51 $\dot{A}$), i.e., we observe tensile strain, an increase in the lattice parameters that leads to a decrease in $J_d$ and $R_d$ and to a decrease in the phonon energy with increasing Fe ions, see Figure 8, curve 1. This corresponds to a red-shift in the peak positions. The contribution to the damping of the ion doping is in addition to those of the bulk, surface, spin–phonon, and phonon–phonon interactions, so that $\gamma$ increases with increasing Fe concentration (see Figure 8, curve 2). This means that the full-weight of the half maximum increases, i.e., the Raman lines are broadener. This is in agreement with the experimental results of Fe-doped CAO [38]. We obtain a similar decrease in $\omega$ and an increase in $\gamma$ for Eu-doped CAO NP as reported by Liu et al. [21].

4. Conclusions

To conclude, we have studied the size and ion-doping effects on the magnetic, optical, and phonon properties of transition metal (TM = Fe, Co, Mn, and Ni) ion-doped CAO NPs on the Cu and Al site using a microscopic model and Green’s function theory. The RTFM in bulk CAO can be due to ion doping, whereas CAO NPs make an additive contribution in terms of surface and size effects. Due to surface defects, a magnetization is obtained that increases with decreasing NP size. The difference in the radii of the TM${}^{3+}$ and the host Cu${}^{+}$ ions leads to a compressive strain, an increase in the exchange interaction constants, and increased spontaneous magnetization $M_s$ and Curie temperature $T_C$. The doping TM${}^{3+}$ ion has an additional spin value compared to the zero spin value of the Cu${}^{+}$ ion, which also enhances $M_s$. On the other hand, a tensile strain is observed by substitution with Mn${}^{3+}$ or Cr${}^{3+}$ on the Al${}^{3+}$ site, where the radius of the Mn or Cr ions is larger than that of the Al ion, which leads to a decrease in $M_s$ with an increase in the ion-doping concentration. The band-gap energy increases with decreasing NP size and by doping with Cr or Sb ions and decreases by Mg ion doping. The phonon energy $\omega$ decreases, whereas the phonon damping $\gamma$ increases, with increasing temperature. They show a kink around the phase-transition temperature $T_C$ ∼ 400 K, caused by the anharmonic spin–phonon interaction. Due to surface and size effects, $\omega$ decreases, whereas $\gamma$ increases, with decreasing NP size. The phonon energy decreases, whereas the damping increases, by Fe and Eu doping on the Al site, i.e., a red shift and broadening of the Raman peaks is obtained. The observed results are in good qualitative agreement with the existing experimental data.

Let us emphasize that in order to take into account the carrier doping treatment in transition-metal-doped CAO on the Cu site, which plays an additive role in the effect of the different ionic radii, we have to expand our model and consider the s-d model by also taking into account the electron–phonon interaction, which is important for CAO [39]. This could be done in the next paper.