Magnetic, Electric and Optical Properties of Ion Doped CuCr₂O₄ Nanoparticles

Abstract

The magnetic, electric and optical properties of pure and ion doped CuCr${}_2$O${}_4$ - bulk and nanoparticles are investigated theoretically. The magnetization $M_s$ and the band gap $E_g$ decrease with increasing particle size. By Co ion doping $M_s$ and the polarization P show a maximum whereas by Pr ion doping they decrease with increasing the doping concentration. The dielectric constant decreases with enhancing Pr dopants. It is shown that the difference between the doping and host ions radii leads to appearing of a compressive or tensile strain and to different exchange interaction constants in the doped state. $E_g$ decreases by Co doping, whereas it increases by Pr doping.

1. Introduction

Spinel structured chromium oxides ACr${}_2$O${}_4$ (A = Cu, Mn, Fe, Co, and Ni) exhibit a wide range of electronic, magnetic, and optical properties [1]. Their physical properties depend upon the distribution of cations among the tetrahedral (A) and octahedral sites (B), and relative superexchange interactions via anions. They have a great research interest in the last years due to their multiferroic (MF) properties and applications in nanoscience and nanotechnology. Moreover, CuCr${}_2$O${}_4$ (CCO) structure has attracted also large attention due to thermal and chemical stability and various applications such as: photocatalytic degradation, photocatalytic H${}_2$ production, oxidation of carbon monoxide, and water treatment [2]. CCO is a normal spinel in which Cr${}^{3+}$ and Cu${}^{2+}$ ions are situated at the octahedral (B) and tetrahedral (A) sites, respectively [3]. The strongly negative $J_{BB}$ (Cr-O-Cr) and $J_{AB}$ (Cu-O-Cr) exchange interactions play an important role in controlling the magnetic properties of chromites and observing a ferrimagnetic phase. The magnetization data of the bulk CCO compounds indicate ferrimagnetic order below $T_C^{FM}$ = 122 K [3]. Enhanced magnetism and Curie temperature are observed in CCO thin films due to substrate effects [4,5]. Ali et al. [6] presented a detailed study of magnetic and magnetocaloric effect of the 3d-metal chromites ACr${}_2$O${}_4$ (where A = Mn, Fe, Co, Ni, Cu, and Zn) near $T_C$ and $T_N$. Enhanced magnetism and Curie temperature are observed in CCO thin films due to substrate effects [4,5]. Ye et al. [7] studied different properties of CCO single crystals. The effects of Co doping on the MF order in CCO is reported recently by Chatterjee et al. [8]. Structure and vibrational studies of La doped CCO are performed recently by Rajeswari et al. [9]. Jahn-Teller and geometric frustration effects on the structural and magnetic ground states of substituted spinels (Ni,A)Cr${}_2$O${}_4$, (A = Mn/Cu) are studied by Yadav et al. [10]. Multiferroicity is found in other chromites, too, for example FeCr${}_2$O${}_4$ and CoCr${}_2$O${}_4$ spinels [11].

The optical spectra of CCO nanoparticles (NPs) are reported by Habibi et al. [12], Beshkar et al. [13], Krause et al. [14], Lahmar et al. [15]. The band gap $E_g$ of CCO NPs (∼3.35 eV) as direct semiconductor [13] shows an increasing compared with that of the bulk compound (around 1.2–1.4 eV) [15,16,17]. The band gap energy in Mn doped CCO NPs is studied in [18].

As mentioned above the effects of ion doping on the MF and optical properties of CCO are not so intensive studied - experimentally and theoretically. In the present paper we will investigate and explain for the first time using a microscopic model and the Green’s function theory the appearance of MF and optical properties in pure and ion doped CCO NPs which can find different applications.

2. The Model

The s-d Hamiltonian which describes the CCO NP can be written as (for review see Refs. [19,20]:

$$H_{s-d}=H_{sp}+H_{el}+H_{sp-el}.$$

(1)

The Heisenberg Hamiltonian is corresponding for the ferrimagnetic properties of CCO:

$$H_{sp}=-\frac{1}{2}\sum _{i,j}{J}_{ij}{\mathbf{S}}_i·{\mathbf{S}}_j-\sum _i{D}_i{\left(S_i^z\right)}^2,$$

(2)

where ${\mathbf{S}}_{\mathit{i}}$ is the Heisenberg spin operator. The Cu-Cu, Cr-Cr and Cu-Cr exchange interactions are antiferromagnetically ($J<0$). D is single-ion anisotropy constant, $\mid D\mid <<\mid J\mid$. By doping with Co or Pr ions the Hamiltonian is modified as follows, there appear some additive terms with $\propto (1-x)J(Cu-Cu)$ and $\propto xJ(Co-Co)$ or $\propto (1-x)J(Cr-Cr)$ and $\propto xJ(Pr-Pr)$, respectively.

$H_{el}$ is the Hamiltonian of the conduction band electrons

$$H_{el}=\sum _{ij\sigma }{t}_{ij}{c}_{i\sigma }^{+}{c}_{j\sigma }+\frac{1}{2}\sum _{ijkl,\sigma {\sigma }^{\prime }}v\left(ijkl\right)c_{i\sigma }^{+}{c}_{j{\sigma }^{\prime }}^{+}{c}_{k{\sigma }^{\prime }}{c}_{l\sigma },$$

(3)

$t_{ij}$ is the hopping integral, v is the Coulomb interaction and $c_{i\sigma }^{+}$ and $c_{i\sigma }$ are Fermi-creation and -annihilation operators.

The operator $H_{sp-el}$ couples the spin and electron subsystems by an intra-atomic exchange interaction $I_i$:

$$H_{sp-el}=\sum _i{I}_i{\mathbf{S}}_i{\mathbf{s}}_i.$$

(4)

The spin operators $s_i$ of the conduction electrons at site i can be expressed as $s_i^{+}=c_{i+}^{+}{c}_{i-}$, $s_i^z=(c_{i+}^{+}{c}_{i+}-c_{i-}^{+}{c}_{i-})/2$.

$H_{sp-ph}$ describes the spin-phonon interactions which are important in CCO [21] and must be taken into account:

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

(5)

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 first and second order, respectively. h.c. represents the hermitian conjugate.

The total magnetization is $M=(M^A+M^B)$. The localized-spin magnetization $M^{A,B}$ for a given sublattice A or B and arbitrary spin S is calculated to

$$M^{A,B}=\frac{1}{N^2}\sum _{i,j}\left[(S^{A,B}+0.5)coth\left[(S^{A,B}+0.5)\beta E_{ij}^{A,B}\right]-0.5coth\left(0.5\beta E_{ij}^{A,B}\right)\right].$$

(6)

$\beta =1/k_{B}T$, $E_{ij}^{A,B}$ is the spin-wave excitation energy observed from the poles of the Green’s functions $G_{ij}^{A,B}=\langle \langle S_i^{-A,B};S_j^{+A,B}\rangle \rangle$.

For the approximate calculation of the Green’s function $G_{ij}$ we use a method proposed by Tserkovnikov [22]. After a formal integration of the equation of motion for $G_{ij}$, one obtains

$$G_{ij}\left(t\right)=-i\theta \left(t\right)\langle [S_i^{-};S_j^{+}]\rangle exp(-i{\omega }_{ij}\left(t\right)t)$$

(7)

where

$$\begin{array}{ccc}\hfill {\omega }_{ij}\left(t\right)={\omega }_{ij}& -& \frac{i}{t}{\int }_0^{t}d{t}^{\prime }{t}^{\prime }\left(\frac{\langle [j_i\left(t\right);j_j^{+}\left(t^{\prime }\right)]\rangle }{\langle [S_i^{-}\left(t\right);S_j^{+}\left(t^{\prime }\right)]\rangle }\right.\hfill \\ & -& \left.\frac{\langle [j_i\left(t\right);S_j^{+}\left(t^{\prime }\right)]\rangle \langle [S_i^{-}\left(t\right);j_j^{+}\left(t^{\prime }\right)]\rangle }{{\langle [S_i^{-}\left(t\right);S_j^{+}\left(t^{\prime }\right)]\rangle }^2}\right)\hfill \end{array}$$

(8)

with the notation $j_i\left(t\right)=\langle [S_i^{-},H_{interaction}]\rangle$. The time-independent term

$${\omega }_{ij}=\frac{\langle [[S_i^{-},H];S_j^{+}]\rangle }{\langle [S_i^{-};S_j^{+}]\rangle }$$

(9)

is the spin excitation energy in the generalized Hartree-Fock approximation. The time-dependent term in Equation (8) includes damping effects.

The spin induced electric polarization $P_{ij}$ produced between the two magnetic moments $S_i$ and $S_j$ is given by [23]

$$P=a{e}_{ij}\times (S_i\times S_j),$$

(10)

where $e_{ij}$ being the unit vector connecting the sites i and j, a is a proportional constant as determined by the spin exchange interaction and the spin-orbit interaction [24].

To obtain the dielectric function defined as [25]:

$$({(\Lambda /(ϵ\left(E\right)-1))}_{\alpha \beta }+\Lambda \frac{k_{\alpha }{k}_{\beta }}{k^2}){\tilde{G}}^{\alpha \beta }\left(E\right)={\delta }_{\alpha \gamma };\Lambda =4\pi Z^2/V,$$

(11)

where Z is the electron charge and V is the volume, we have calculated beyond the random phase approximation the longitudinal Green’s function

$${\tilde{G}}_{ij}^{zz}\left(E\right)=\frac{2(E^2-{\left(E_{fi}\right)}^2+2iE{\gamma }^{11})}{(E^2-{\left(E_{fi}\right)}^2+2iE{\gamma }^{11})(E+i{\gamma }^{33})-E{\left({ϵ}^{13}\right)}^2}.$$

(12)

$E_{fi}$ and ${\gamma }^{11}$ are the transverse pseudo-spin energy and its damping, whereas ${\gamma }^{33}$ is the longitudinal damping. ${ϵ}^{13}$ describes the coupling between the longitudinal and transverse modes. In order to determine ${ϵ}^{\prime }$ and ${ϵ}^{″}$ we must evaluate the real and imaginary part of the Green’s function (12).

Bulk CCO is a p-type semiconductor with a small band gap $E_g$ of 1.4 eV [3]. $E_g$ is defined by the difference between the valence and conduction bands [26]:

$$E_g={\omega }^{+}(k=0)-{\omega }^{-}(k=k_{\sigma })$$

(13)

with the electronic energies

$${\omega }^{\pm }\left(k\right)={ϵ}_k-\frac{\sigma }{2}IM+\sum _{k^{\prime }{\sigma }^{\prime }}[v\left(o\right)-v(k-k^{\prime }){\delta }_{\sigma {\sigma }^{\prime }}]\langle n_{k^{\prime }{\sigma }^{\prime }}\rangle$$

(14)

observed from the Green’s function $g(k,\sigma )=\ll c_{k,\sigma };c_{k\sigma }^{+}\gg$, $\sigma =\pm 1$. $\langle n_{k^{\prime }{\sigma }^{\prime }}\rangle$ is the occupation number distribution.

3. Numerical Results and Discussion

The following model parameters are used for the numerical calculations: $J_1$(Cu-Cu) = −0.26 meV, $J_2$(Cu−Cr) = −3.16 meV, $J_3$(Cr−Cr) = − 3.01 meV [27], D = −0.1 meV, I = 0.2 eV, $T_N$ = 122 K, $T_C$ = 170 K [8], F = 23 cm${}^{-1}$, R = −18 cm${}^{-1}$.

It should be noted that the magnetic exchange interactions are taken from Ref. [27]. The spin-phonon constants F and R are determined from the Raman spectra of CCO [28] at very low temperatures, taking two values at two different temperatures from the Raman phonon energy and solving the system of two equations with two unknown parameters (see for more [29]).

Let us emphasize that the exchange interaction $J_{ij}=J(r_i-r_j)$ depends on the distance between the spins, on the lattice parameters. It is different on the surface and the doped states, denoted as $J_s$ and $J_d$, respectively, than in the bulk and undoped states, $J_b$. This is valid for all interaction constants. Moreover, the spin-phonon interaction renormalizes J to $J^{eff}=J+2F^2/({\omega }_0-MR)$ which is now temperature dependent.

Firstly, the spontaneous magnetization $M_s$ for a CCO NP is calculated in dependence on size at room temperature for the following relation $J_s>J_b$. It can be seen from Figure 1 that $M_s$ increases with decreasing NP size. A similar behavior of $M_s$ is observed by Iwata et al. [5]. This increase of $M_s$ is due to surface and size effects, due to the changed number of next neighbors on the surface and the reduced symmetry. All these factors lead to the different exchange interaction constants on the surface and in the bulk. The properties are very sensitive to the surface value $J_s$ of all interaction constants. With increasing $J_s$ the increase of $M_s$ with decreasing N is stronger, the value of $M_s$ is larger (see Figure 1, curve 2.) Below a critical size of $N_{cr}$ = 3 the ferrimagnetism disappears, we have superparamagnetism which is not studied here. The magnetic transition temperature $T_N$ also decreases with increasing the NP size.

The magnetization could be changed by doping with different ions. Firstly, we will substitute Cu${}^{2+}$ ions with ferromagnetic transition metal Co${}^{2+}$ ions. Thus, we can compare our results with those of Ref. [8] and check our model. By doping with Co${}^{2+}$ ions (0.65 $\dot{A}$ [30]), which radius is smaller than that of the Cu${}^{2+}$ ion (0.73 $\dot{A}$ [31]), there appears a compressive strain. This means that the exchange interaction constant at the doped state $J_{d1}$ is larger than that in the undoped one $J_{b1}$, i.e., $J_{d1}>J_{b1}$. All other interaction constants in the doped states are assumed to be nearly the same as in the undoped ones. Thus the magnetization $M_s$ increases with increasing Co doping concentration up to x = 0.6. After x> 0.6, where appears a secondary phase we use the relation $J_{d1}<J_{b1}$ and the magnetization $M_s$ (see Figure 2, curve 1) begins to decrease, in accordance with the experimental data of Chatterjee et al. [8]. $T_N$ decreases with increasing the Co dopants. It must be mentioned that $T_N$ of bulk CCO is 122 K, whereas of CoCr${}_2$O${}_4$ it is 93 K. A similar behavior we expect to obtain also by doping with Ni or Mn ions at the Cu site due to the relation between the ionic radii of these ions.

As the next we will consider the case of rare earth doping in a CCO NP. It is shown that for example by Pr${}^{3+}$ ion doping at the Cr${}^{3+}$ site, we observe a different behavior compared to the Co${}^{2+}$ doping at the Cu${}^{2+}$ site. We were looking for a doping with larger-sized ions, such as the rare earth ions which substitution is commonly used. Pr was randomly chosen, it could have been any other rare earth ion, La etc., which ionic radius is larger than that of Cr, and the results would have been qualitatively the same. Unfortunately, there are not experimental data for rare earth doped CCO. The lattice parameters increase with the substitution of the Pr ion, i.e., there appears a tensile strain. This is due firstly to the larger ionic radius of the Pr${}^{3+}$ ion (1.013 $\dot{A}$) in comparison with that of the Cr${}^{3+}$ ion (0.755 $\dot{A}$). The strain is also produced in the crystal’s volume, due to this mismatch in ionic sizes. Moreover, the substitution of rare-earth ions into the copper-based spinel chromites creates a crystalline anisotropy. In addition, the system remains in stable form due to the balance of volume strain with crystalline anisotropy. Due to the tensile strain we have to use the following relation $J_{d3}<J_{b3}$. All other interaction constants in the doped states are assumed to be nearly the same as in the undoped ones. Then the spontaneous magnetization $M_s$ decreases with increasing the Pr${}^{3+}$ doping concentration x (see Figure 2, curve 2). The substitution of the Pr${}^{3+}$ ions at the Cr${}^{3+}$ ion sites causes partial disorder and weakens the Cr${}^{3+}$-O-Cr${}^{3+}$ super exchange interactions. Moreover, there appears changes of the valence of the Cr ion from Cr${}^{3+}$ with a high spin state to Cr${}^{2+}$ with a low spin state. This is due to the deviation from collinear to non-collinear order, leading to a decrease of $T_N$. This decrease in the magnetic phase transition temperature $T_N$ may be also due to the fact that Pr-Cr interactions on the B sites are smaller than Cr-Cr interactions, $J(Pr-Cr)<J(Cr-Cr)$. Similar results for the spontaneous magnetization $M_s$ and the magnetic phase transition temperature $T_N$ we would obtain in the case of La ion doped CCO.

It is observed also the dependence of the polarization P on the ion doping concentration for Co and Pr doping ion substitution at Cu and Cr sites, respectively. The results are shown in Figure 3, curves 1 and 2, respectively. The discussion for the ion doping dependence of the polarization P is similar to that of the spontaneous magnetization $M_s\left(x\right)$. Moreover, the doping of Cu${}^{2+}$ with Co${}^{2+}$ systematically tunes the ferroelectric $T_C$ as well as the magnetic ordering temperature $T_N$, which decrease with increasing x.

From Equations (11) and (12) the real ${ϵ}^{\prime }$ and imaginary ${ϵ}^{″}$ part of the dielectric constant $ϵ$ is calculated for small doping concentration x in Pr doped CCO NPs. The results are presented in Figure 4. It can be seen that both decrease with increasing x. The rare-earth ions substitute the ferric ions on the B-site due to their larger ionic radius which hinders the electrons exchange between the Cr${}^{2+}$ and Cr${}^{3+}$ ions. This leads to a decrease of the polarization P and the dielectric constant $ϵ$.

Finally, we will consider the band gap energy $E_g$ of pure and ion doped CCO NPs. We observe that $E_g$ increases strongly with decreasing NP size (see Figure 5), in agreement with the experimental results of Refs. [13,15]. By doping of a CCO NP with Co ions $E_g$ decreases whereas by Pr ions it increases (see Figure 6). These results coincide with those of Lopez et al. [32]. The authors have studied the band gap energy in CdS thin films and have shown that the smaller the lattice parameter the larger the band gap. Band gap changes in doped semiconductors could be due as a result of exchange and Coulomb interactions. The band gap energy $E_g$ increases with increasing the Coulomb interaction (not shown here). Unfortunately, there are not experimental data for $E_g$ in ion doped CCO. Let us emphasize that the increase of the energy gap for semiconductors leads to decrease the absorption region and the substance may be transparent and the absorption edge shifted toward low wavelengths. This may be useful for solar cell applications.

4. Conclusions

In the present paper we have investigated different properties of pure and ion doped CCO - bulk and NPs for the first time using a microscopic model. Our model is microscopic, because the exchange interaction constants are dependent on the lattice parameters, on the microstrain, on the crystal symmetry, on the number of nearest neighbors etc. They are renormalized through the spin-phonon interaction and are temperature dependent. So we can discuss the properties on microscopic level. They are connected with different strains which lead to different lattice parameters and different exchange interaction constants. Using the Green’s function theory it is observed a decrease of the magnetization $M_s$ and the band gap energy $E_g$ with increasing NP size. The spontaneous magnetization $M_s$ and the polarization P show a maximum value around x = 0.6 for Co dopants, whereas they decrease with increasing Pr doping concentration. The real and imaginary parts of the dielectric constant decrease with increasing Pr dopants. By the doping concentration of the band gap energy $E_g$ it is vice versa. $E_g$ decreases by Co doping and increases by Pr ion doping.

In summary, it is shown that in Co doped CCO NPs the MF properties, magnetization M and polarization P, as well as the coercivity are enhanced below x = 0.6, the phase transition temperatures $T_N$ and $T_C$ are reduced, so that this doping case is more appropriate for application in spintronics in comparison with the rare earth substitution, where M and P decrease with increasing Pr dopants. On the other hand, by Pr ion doping the band gap energy $E_g$ is enhanced which can be used for solar cell applications.

It should be noted that the most observed properties of ion doped CCO NPs, for example the band gap energy, the dielectric constant, the rare earth ion doping, are not reported experimentally till now. Thus we hope that our results will be confirmed in the future.

Moreover, more experimental and fundamental theoretical investigations need to be done to find new materials with high multiferroic properties which can be applied in nanoscience and nanotechnology. We hope to have contributed to this view with this article.