1 Introduction

In recent years, dilute magnetic semiconductors (DMSs) have been of much interest from the industrial viewpoint, because of their potentiality as a new functional material, which will open a way to introduce the freedom of spin into semiconductor devices [1–3]. For practical applications, it is required to fabricate a high Curie temperature (T C ) ferromagnet with a homogeneous distribution of the magnetic dopants. Doping—the intentional introduction of impurities into a material—is a common approach for tuning the electronic, optical, mechanical, and magnetic properties of materials. Since the original report of ferromagnetism in Mn-doped InAs and GaAs above 100 K [4, 5], II–VI-based semiconductors have been intensively studied for they are a typical semiconductor in which the carrier control technique is well established and they display ferromagnetism when doped with most of the transition-metal (TM) ions, like Mn +2 and Fe +2. Partly, substitution of the transition-metal atoms into the cationic site of the host semiconductor results unique physical and magnetic properties [6]. The 3d state of transition metal is responsible for the unusual behavior of these materials [7]. Therefore, the investigation of 3d state of TM-doped semiconductors is of fundamental physical interest and technological importance [8].

CdSe is a semiconductor with an energy band gap of 1.74 eV at room temperature [9]. It is expected that TM alloys like Fe-doped and Co-doped CdSe can offer half-metallic (HM) ferromagnetic characteristic, so they can be useful in spintronics and other applications. Recently, much effort has been made to improve the properties of CdSe materials by doping Fe or Co. Experimentally, Fe or Co can be doped up to 30 % concentration in the CdSe crystal. Hanif et al. [10] doped Cd1-x Co x Se at different concentration of Co (0 ≤ x ≤ 30). They confirmed linear lattice compression with the increasing Co concentration. Orlowski et al. [11] studied the electronic structure of Cd1-x M x Se (M = Mn, Fe, Co) at 12 and 3 % by ultraviolet photoemission technique. The introduction of transition-metal atoms (Mn, Fe, Co) instead of rare-earth metals in II–VI semiconductors clearly induces magnetic and crystalline disordering in the crystal. Further, they found that the electronic and magnetic properties largely depend on the impurity 3d-electrons and their hybridization with the s-p valence band.

The first-principles pseudo-potential approach has been widely used in material simulations [12–14]. Recent first-principles calculation by Ahmadian et al. [15] indicated that different concentration of V in CdSe could contribute to the magnetic moment in CdSe-based DMSs. However, there is seldom theoretical study on the magnetic and optical properties for Fe-doped and Co-doped CdSe. The aim of this paper is to provide a more accurate and complementary study on the structural, electronic, magnetic, and optical properties of Fe-doped CdSe and Co-doped CdSe with ZB phase using the first-principles technique.

2 Model and Calculation Methods

The lattice constants a of pure ZB phase CdSe is a=6.04 Å. The supercell employed contains 64 atoms which correspond to a 2×2×2 supercell of CdSe. In the present theoretical studies, two Cd atoms are replaced by Fe/Co in the CdSe crystal to make ordered alloys in the ZB phase. To make our simulation closest to the dilute doping limit, we put the two Fe/Co atoms on positions not adjacent. The doped structures were then optimized with respect to both the lattice constants and the atomic positions.

The optimization of geometrical structures and calculations of the magnetic and electric properties of the ZB compounds were performed using “castep” [16] based on density functional theory. Ultra-soft pseudo-potentials [17] are chosen to represent the interactions between ionic core and valence electrons. The generalized gradient approximation (GGA) and PBEsol [18] are used to describe the exchange and correlation potential. PBEsol is a revised Perdew-Burke-Ernzerhof GGA that improves equilibrium properties of densely packed solids and their surfaces [19, 20]. For the Brillouin-zone sampling, the Monkhorst-pack scheme [21] was adopted. To confirm the convergence of our calculations, we carefully investigate the dependences of the total energy on the cutoff energy from 250 to 850 eV and the k-point set mesh according to the Monkhorst-Pack grid from 3×3×3 to 12×12×12, respectively. First-principles electronic structure calculations allow an accurate evaluation of the total energy of a system as a function of the cutoff energy and k-points, and it may be performed with sufficient accuracy to resolve energy differences as small as a few milli electron volts per atom. The test results show that, when the k-point set mesh is fixed in 7×7×7, the change in total energy is less than 1 meV when the cutoff energy is higher than 550 eV; on the other hand, when the cutoff energy is fixed in 550 eV, the convergence in total energy is very well when the k-point set mesh is beyond 7×7×7. In consideration of computational accuracy, we choose the cutoff energy to be in 550 eV, and the Brillouin-zone sampling mesh parameters for the k-point set are not lower than 7×7×7 in this study for CdSe with the ZB structure. The crystal structure and the atomic coordinates are fully relaxed without any restriction using the Broyden-Fletcher-Goldfarb-Shanno method [22] until the force on each atom converges to less than 0.01 eV/Å. The total energy is converged to lower than 5×10−6 eV/atom. The maximum stress of crystal is less than 0.02 GPa. The maximum displacement of atom is less than 5×10−4 Å. Then, the electronic band structures and magnetic properties are calculated on the optimized supercells. The valence electron configurations considered in this study included Cd: 4 d 105s 2, Se: 4 s 24p 4, Fe: 3 d 64s 2, and Co: 3 d 54s 2.

3 Results and Discussion

3.1 Structural Properties

The equilibrium lattice constants, bulk modulus, and direct band-gap energies of the pure CdSe, Fe-doped CdSe, and Co-doped CdSe in the ZB phase are calculated and presented in Table 1. The calculated lattices constant for pure CdSe is 6.077 Å, which is in good agreement with the experimental and other simulated results [23–26]. Recently, Arif et al. [24] have reported the optimized lattice constant of 6.04 Å by the full potential linearized augmented plane wave (FPLAPW) method in 2012. The constant of the ZB CdSe 6.055 Å has been reported by Benkabou et al. [23] from the molecular dynamics method. Using the most popular hybrid functional like B3LYP, Xiao et al. [26] calculated the constant of 6.217 Å in 2011. The experimental lattice constant reported by Passler et al. [25] is 6.077 Å. Our result is in accord with the experimental and other calculated values. This fact will also support the validity of our theoretical predictions of other propertied of CdSe and doped CdSe.

Table 1 Calculated lattice constants, bulk modulus and direct band-gap energies of the pure CdSe, Fe-doped CdSe and Co-doped CdSe in the ZB phase
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When two Cd atoms in the 2×2×2 CdSe supercell are replaced by Fe/Co, there are some decrease in the lattice constants to 0.147 Å for Fe doping and 0.167 Å for Co doping after structural optimization, which can be attributed to the fact that the ionic radius of Cd is larger than that of Fe/Co. The results are also in agreement with the experimental results of ref. [10] and the calculated results of ref. [24].

Table 1 also list another structural parameter (bulk modulus) from the GGA + U method for pure CdSe and doped CdSe, respectively. Other theoretical results are also presented for comparison. The theoretical ZB bulk modulus 51.2 GPa is in good accordance with the FPLAPW calculations of Arif et al. [24] and with the molecular dynamics calculations of Benkabou et al. [23]. The calculated bulk modulus of Fe-doped CdSe and Co-doped CdSe are 56.6325 and 56.6381 GPa. The bulk modulus of the material increases with the decrease in the lattice constant.

3.2 Electronic and Magnetic Properties

In order to understand the ferromagnetic properties from the microscopic point of view, we inspect the band structures of both pure CdSe and doped CdSe. The plots of the spin-polarized band structures for Fe-doped CdSe and Co-doped CdSe are presented in Fig. 1. Figure 1a, c is for the majority spin-channels, while Fig. 1b, d is for the minority spin-channels. The top of the valance bands and bottom of the conduction bands are located in the brillouin-zone G symmetry point. After doping, Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) are still direct band-gap semiconductors. It shows a half-metallic behavior with the majority spin being semiconductor and the minority spin being metallic with sufficient unfilled states above the Fermi level. These unfilled states behave like free holes, although slightly localized. The 100 % polarization of conduction carriers, which is required in spin injection where highly polarized spin current is desired, suggests that Fe-doped and Co-doped CdSe can be used for spin injection. The calculated band gaps for the pure CdSe and the majority spin-channels are presented in Table 1. Our result is in good agreement with the experimental and other simulated results.

Fig. 1
figure 1

The spin-polarized band structure of Cd1-x Fe x Se for majority spin state (a) and minority spin state (b), Cd1-x Co x Se for majority spin state (c) and minority spin state (d)

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To analyze the detailed electronic structure and the origin of the impurity energy levels of the Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %), the partial density of states of these systems was calculated, as shown in Fig. 2. It is seen that when Fe- or Co-dopants occupy cation host site Cd, it creates shallow acceptor states, partially fitted, in the top of the valance band. As shown in Fig. 2, the impurity states are mainly dominated by Fe-3d states and Co-3d states, which is in accordance with previous study [24]. From the point of view of the electrical behavior, these impurity states could be considered as the primary origin of the electrical conductivity in the doped CdSe systems.

Fig. 2
figure 2

Partial density of states of Cd1-x Fe x Se (a) and Cd1-x Co x Se (b). Fermi level is set to zero

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The total magnetic moment, the atomic magnetic moments, and the magnetization energy were calculated to determine the stable ferromagnetic phase in Fe/Co-doped CdSe systems (see Table 2). The magnetization energy is defined as the difference between the calculated ferromagnetic and anti-ferromagnetic energies (ΔE = E AFM−E FM). A positive value corresponds to stable ferromagnetism. From Table 2, it can be seen that the ferromagnetic state is more stable than the anti-ferromagnetic state, which is also reported in refs. [24] and [28]. It can be clearly seen that Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) showed stable ferromagnetic phase with permanent moments of 4 μ B and 3 μ B per supercell, respectively. One can see that the total magnetic moment of Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) mainly originate from Fe/Co atom with small contributions from Cd and Se atoms. It also reveals that Fe-doped CdSe exhibited more stable ferromagnetism than Co-doped CdSe. This is due to the fact that Fe atom has larger magnetic moment than Co atom because of the extra-unpaired electrons in the 3d state of Fe.

Table 2 The difference between the ferromagnetic and anti-ferromagnetic energies (meV), total and local magnetic moments μ B (in Bohr magneton) for Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %)
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The Curie temperature (T c ) can be estimated with the classical Heisenberg model in mean-field theory [18]:

$$ k_{B} T_{c} = 2{\Delta} E_{\text{FM}} /3x $$
(1)

where x is the concentration of doping atoms and k B is Boltzmann’s constant. By (1), the Curie temperature (T c ) of Fe-doped and Co-doped CdSe are obtained. It is about T c =344 K for Cd1-x Fe x Se and T c =210 K for Cd1-x Co x Se (x=6.25 %) and that of Cd1-x Fe x Se is higher than room temperature. These results illustrate that Fe-doped and Co-doped CdSe are more promising spintronics materials.

3.3 Optical Properties

The optical properties are studied by dielectric function ε = ε 1 + i ε 2; the imaginary part of the dielectric constant (ε 2) can be calculated theoretically based on DFT (see (2)), and then by applying the Kramers-Krönig relation [29], we can evaluate the real part (ε 1) at the same time. The peaks appearing in the ε 2 part of dielectric function are directly related to different intra-band or inter-band transitions in the first irreducible Brillouin zone. The refractive and extinction indices, optical conductivity, and reflectivity were also calculated using (3)–(6) [30–32].

$$\begin{array}{@{}rcl@{}} \varepsilon_{2} (w)&=&\frac{Ve^{2}}{2\pi \hslash m^{2}w^{2}}\int d^{3}k\sum\limits_{nn^{\prime}} \vert <kn\vert p\vert kn^{\prime}>\vert^{2}f(kn)\\ &&\times (1-f(kn^{\prime})\delta (E_{kn} -E_{kn^{\prime}} -\hslash w) \end{array} $$
(2)
$$ n(w)=\left( \frac{[{\varepsilon_{1}^{2}} (w)+{\varepsilon_{2}^{2}} (w)]^{1/2}+\varepsilon_{1} (w)}{2}\right)^{1/2} $$
(3)
$$ k(w)=\left( \frac{[{\varepsilon_{1}^{2}} (w)+{\varepsilon_{2}^{2}} (w)]^{1/2}-\varepsilon_{1} (w)}{2}\right)^{1/2} $$
(4)
$$ \sigma (w)=\frac{w}{4\pi} \varepsilon_{2} (w) $$
(5)
$$ R(w)=\frac{\left[ {n(w)-1} \right]^{2}+k^{2}(w)}{\left[ {n(w)+1} \right]^{2}+k^{2}(w)} $$
(6)

The real and imaginary parts of the dielectric functions were calculated in the range from 0 to 20 eV and plotted in Fig. 3a, b, which demonstrate that the curve shapes of ε 1 and ε 2 are similar for Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %). The real part of the dielectric function governs the propagation behavior of electromagnetic field in a material. The static dielectric constants ε 1 (0) obtained for CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %) were 5.969, 6.523, and 8.311 respectively, and reach their peaks of 7.006 at 1.06 eV, 7.788 at 1.34 eV, and 8.311 at 0 eV for CdSe, Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %). The imaginary part of the dielectric function represents the light absorption in the crystal. From Fig. 3b, the onsets of the spectra are at 8.27, 10.15, and 10.50 eV for CdSe, Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %), respectively. According to Burstein-Moss effect [33, 34], these values are connected with the optical vertical transition from the initial valence band states to the unoccupied states around the Fermi level.

Fig. 3
figure 3

Real part (a) and imaginary part (b) of the dielectric function, refractive (c) and extinction (d) indices, optical conductivity (e), and reflectivity (f) as functions of photon energy of the pure CdSe (black line), Cd1-x Fe x Se (red line) and Cd1-x Co x Se (green line)

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Figure 3c–f shows the refractive and extinction indices, the optical conductivity, and reflectivity of the CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %). From Fig. 3c, the refractive indices in the low region are nearly 2.44, 2.55, and 2.88 for CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %), respectively. In the high region, the refractive indices are all nearly 0.07. The trends of the refractive indices coincide with the real part of the dielectric functions. Figure 3c also shows the intensity of the refractive indices decrease with energy increase, which implies that with the increase of the incident photon energy, the refractive index decreases gradually. As displayed in Fig. 3d, there are three main peaks correspond to that of the imaginary part of the dielectric functions. The intensity of these three main peaks enhanced and reached the maximum at the third peak. The optical conductivity of CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %) are displayed in Fig. 3e. One can see that when doped with Fe/Co, the peaks of optical conductivity enhanced from 3.76 to 5.35. Figure 3f shows the reflectivity of CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %). It can be seen that the value of the peak did not increase but moved toward high region after doping.

The absorption coefficient α(ω) can be calculated from the real and imaginary parts of the dielectric function by using the following expressions [35]:

$$ \alpha (\omega )=\frac{4\pi} {\lambda} k(\omega ) $$
(7)

where λ,ω, and k(ω) represent the wavelength, frequency of the incident light, and extinction indices, respectively. The calculated absorption coefficients of CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se (x=6.25 %) varying with the wavelength are plotted in Fig. 4. As displayed in Fig. 4, Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) have stronger absorption than CdSe, the peaks of absorption coefficients enhanced from 18.2×104 to 22.5×104 cm −1 after doping. Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) have similar absorption coefficient curves because of their similar calculated band structures. As shown in Fig. 4, the absorption peaks in the wavelength region from 50 to 150 nm and from 150 to 300 nm correspond to the band-to-band transitions. The peaks originate from the transition of the inter-band transitions from the valence band maximum (VBM) to the conduction band minimum (CBM). After doping Fe/Co, the band gap is about 0.4/0.5 eV and is much smaller than 1.5 eV of pure CdSe. As a result, the transition of an electron from valence band to conduction band will need less energy, which may induce intense absorption, improving the photocatalytic property. The inter-band transition was attributed to the Fe/Co 3d and Se 4p orbital at VBM to the Se 4s and Cd 5s orbital at CBM. This transition would be more intense compared with that of pure CdSe because of the hybridization between the Fe/Co 3d and Se 4p orbital at VBM. Figure 4 shows that Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) have strong absorption in the UV-light range and visible region. The result indicates that doping with Fe/Co into CdSe is a potential method to make the systems to meet the requirement as the absorber of solar cells.

Fig. 4
figure 4

Calculated absorption coefficients of the pure CdSe, Cd1-x Fe x Se, and Cd1-x Co x Se

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4 Conclusion

The structural, electronic, magnetic, and optical properties have been studied by GGA and PBEsol. The calculated spin-polarized band structures and magnetic properties reveal that Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) are half-metallic ferromagnets. Moreover, the doped systems of Cd1-x Fe x Se have perfect Curie temperature higher than room temperature. The dielectric functions and absorption coefficients of Cd1-x Fe x Se and Cd1-x Co x Se (x=6.25 %) are studied from the electric ground states. The calculated results show that the VBM of the doped systems are attributed primarily to the hybridization of the dopant element Fe/Co 3d with Se 4p orbital. The impurity states of Fe- and Co-dopant induced strong absorption in the UV-light range and visible region. In summary, the study of magnetic and optical properties indicates that Fe- and Co-doped CdSe are promising spintronic devices and are potential materials for optical devices, such as absorber of solar cells.