Abstract
We investigated the transport properties of diluted magnetic semiconductors (DMSs) theoretically by using the Heisenberg and Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction models by considering both spin and charge disorder. The formalism is applied to the specific case of . Using the Heisenberg model and the Green function formalism the total thermal excitation of the magnon is calculated. The magnetization and Curie temperature of -doped is calculated. The theoretical calculation of of at x = 0.08 has a good agreement with the experimental calculation at x = 0.08 (i.e., 162 k). The exchange interaction constant and spin-dependent relaxation time is calculated by using RKKY interaction. The electrical conductivity and hole mobility are calculated by using the Boltzmann transport equation and the spin-dependent relaxation time. The electrical conductivity of -doped III–V DMS is exponentially increased with temperature and magnetic impurity concentration. Hole mobility of -doped III–V diluted magnetic semiconductor is increased with the magnetic impurity concentration.
1. Introduction
The modern electronic industry heavily relies on diluted magnetic semiconductors (DMSs). In DMSs, semiconducting and magnetic properties coexist. The semiconductor property processes information using the charge of electrons and holes, while the magnetic property stores information using the spin of magnetic ions/atoms. Since magnetic materials and semiconductor applications have developed independently, it makes sense to integrate their capabilities to create spintronics, which have greater functionality [1, 2]. The fundamental goal is to regulate the electrons’ degree of spin freedom in specific semiconducting materials. The most practical way to add spin degrees of freedom to semiconductors is to incorporate magnetic ions into them. These semiconductors can be defined as semimagnetic or DMSs [3, pp. 20–22].
According to the current studies on DMSs, the presence of magnetic ions significantly alters the electron transport phenomenon [3, pp. 20–22]. Beginning in the 1970s, it was realized that one material (DMS) might have both magnetic and semiconducting properties. The antiferromagnetic exchange between the magnetic ions () spins regulates the magnetic interaction in II–VI-based DMSs, leading to paramagnetic, spin glass, and finally long-range antiferromagnetic behavior. Nowadays, ferromagnetic properties are found in p-type II–VI DMS heterostructures, but they only exist at Curie temperatures (), which are typically below 2.0 K. As a result, installing ferromagnetic-based III–V semiconductors with magnetic ion, , which stay ferromagnetic to significantly higher , has made very substantial progress [1]. The recent discovery that ferromagnetic ordering in III–V DMSs with phase transition temperatures of annealed DMS could exceed 110 K; roughly approaching 170 K has greatly increased interest in those materials [4, p. 356; 5]. Although encouraging, the greatest temperature (173 K) in Mn-doped GaAs for obtained utilizing low-temperature annealing procedures is still too low for the practical applications [6].
A lot of theoretical and experimental work has been done on raising . Currently, the highest reported value is just about 190 K. There are numerous ways to calculate this temperature. According to Vašek et al. [7], measuring magnetization as a function of temperature is the most suitable approach. It is much easier to measure conductivity (resistivity) than magnetization, and can also be found by looking at how resistance changes with temperature. Studying these transport characteristics theoretically is fairly difficult. Since the exchange interaction between the carriers and the localized moments, which results in the ferromagnetic transition, affects these features, it is necessary to take this into nonperturbative consideration [6, 8]. The magnetic impurity concentration and temperature have a significant impact on the experimentally determined dc resistivity in the DMS materials, which exhibits interesting characteristics. According to Hwang and Sarma [8], there is a metal–insulator transition in the material (Ga, Mn)As between insulating samples with low-manganese concentrations and metallic samples with higher concentrations. In this study, the theoretical computation of the DMS transport properties was carried out using the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, the Heisenberg model, and the Green Function formalism.
2. Theoretical Models
2.1. RKKY Model Interaction Approaches for Spin Dependent Relaxation Time
In this case, the RKKY model, characterized by the energy of exchange coupling, is typically used to describe spin–spin interactions [9]. This interaction at the localized moment is determined by the spin density of the electrons [10, pp. 213–218]. Formulation of the spin density of electrons in terms of the given field operators, the coupling between the spin at and the conduction electron iswhere is the field operator states expanded in terms of the block states. By using the Fourier transform of field operators, the number density operator is expressed bywhere and are the annihilation and creation operators of electrons, respectively. If we translate Equation (2) into terms of the spin density, the overall spin is conserved in the interaction, although conduction electrons and localized spins may exchange positions. The electron of wave vector k is scattered in state k’, and this state determines the coupling strength. Assume that the k dependance may be ignored. The interaction with conduction electrons is determined by
The interaction of localized spin ions with electrons rises to the creation or annihilation of an electron–hole pair. By considering elastic transitions and the common energy , denoted by , the Hamiltonian is written as follows:where is the ground state, the state of the scattering of the electron, λ is the coupling constant, and is the effective Hamiltonian which is equal to . Since the flip of the impurity spin is accompanied by the flip of an electron spin, the collision term of the distribution function of spin-up and spin-down electrons iswhere is the transition probability, and are the distribution function of the initial states and the scattered states, respectively. Then by substituting and using the distribution function for the spin of the electron–hole pair, the matrix element of the operator is With the help of some integral notation, the quadratic dispersion relation that holds true for free electrons, and the notations, we can calculate the exchange coupling strength between magnetic impurities and delocalized charge carriers. This exchange interaction constant, also known as the effective exchange interaction, is given bywhere is the Fermi wave vector, and , The well-known RKKY interaction is provided by Equation (6). The interaction in the system can be either ferromagnetic or antiferromagnetic depending on the value of the oscillation function . is used to calculate different temperature properties such as Curie temperature, spin-dependent relaxation time, spin stiffness constants, and magnon energies. Then, by adding up all possible spin orientations, the transition probability in J is given by:where is the concentration of magnetic impurities and is the singular function in the third-order correction. By using Fermi–Dirac distribution and for electron close to the Fermi surface at low temperature (i.e., ) and substituting this into collision integral, the spin-dependent relaxation time is the inverse of the transition probability which is written as follows:where is constant, is the energy gap between the impurity band and valence band. It is also known as activation energy. The thermal excitation of holes from the impurity band to the significantly more conductive valence band is used to compute this energy. The maximum and minimum energy positions of the Fermi level are observed to commonly differ by no more than 1 EV in III–V semiconductors. In the case of specifically, the conduction band is found at and the valence band at [1].
2.2. Heisenberg Model Approaches for Thermal Excitation of Magnon
The energy difference is the central parameter for estimating magnetic transition temperature in the Heisenberg model [2]. If the Heisenberg exchange interaction term describes the interaction between the atomic spins at lattice locations and and the exchange coupling, which depends on spin separation, may be used to describe the strength of the interaction, then the Heisenberg Hamiltonian is . Atomic spins tend to align themselves in a preferred direction and take on an ordered structure at low temperatures, where the exchange interaction typically does not leave them independent [11]. The simplest of them is ferromagnetic order, where J is a positive sign in the exchange interaction. Heisenberg Hamiltonian has the following notation if the external field is applied in the z-direction:where is the lande -factor of the localized moment, is the Bohr magneton, and is the applied magnetic field. If we denote the ground state by , an excited state is obtained by the series of raising and lowering operators to the ground states. Then the eigenstate of the Hamiltonian possessing translational symmetry may be characterized by a wave vector , we shall seek the proper eigenstate in the form .Where N is the number of spins and S is the spins of localized magnetic ions. To transform spin operators to the deviation of creation and annihilation operators and , respectively, the wave functions of the spin–wave are created and annihilated bywhere and are spin-raising and lowering operators, respectively. If the operators and were boson operators, the operators in the lattice representation would also behave as boson operators. At low temperatures, the z-components of the spins can be well approximated by . By using the lattice representation of bosons inverse Fourier transforms, the spin operators are written as follows:where and are creation and annihilation operators of and atom on sublattice and , respectively. It is convenient to make a transition from the atomic, and to the magnon variables and for two sublattices, respectively. These magnon variables, for two sublattices are defined as follows:where is the spin of the atoms, and are the position vectors of atoms i and j. Then using an inverse Fourier transform of these new magnon variables is calculated as follows:
Then by substituting Equation (13) into Equation (9), and using the Dirac delta function, the Hamiltonian of excited state becomes
By considering exchange interactions of strength between nearest neighbors only and taking the average of this equation over and substituting into Equation (14) we obtain:where is the concentrations of the impurity added to the semiconductors, z is the number of nearest neighbors, and . According to Solyam [12], the leading correction of in the long wavelength limit is of the order which puts it extremely close to the unity. When using the simple cubic lattice with the lattice constant α and small values of the wave numbers , along with two new creation and annihilation operators that combine the operators of the two sublattices, along with the inverse transform of these two operators in Equation (15), the excitation energy for ferromagnetic materials is
The ground state of a simple ferromagnetic has all spins parallel. At long wavelengths, <<1 so that, the frequency of magnon is proportional to , in the same limit the frequency of a phonon is proportional to k. The second term in Equation (16) is small, for , the excitation energy for ferromagnetic material is expressed as follows:
2.3. The Green’s Function Formalism for the Heisenberg Model
To answer the question of how a quantum mechanical system reacts to an external perturbation, such as an electrical or magnetic field, Green’s functions naturally appear as response functions; the corresponding response functions would then describe the electrical conductivity of a system. For the quasiparticle spectrum of the system described by the Hamiltonian in Equation (15), we consider the following retarded Green’s function, at
By using the momentum space creation and annihilation operators and differentiating Equation (18) with respect to t and by taking its Fourier transform we get
Therefore, the equation of motion for fermions and bosons for finite temperature () iswhere and is the frequency of excitation energy. The correlation function is related to the analytic property of Green’s function by
By using Schrodinger picture, taking the Fourier transformation of Equation (24), and for and Equation (21) becomes . Again by considering this equation and taking equal time correlation gives the number operator which is
By using Dirac delta function in Equation (22), the total number of magnons in all modes excited at temperature T can be calculated as follows:
Then by using the integral function, the relation between the volume of the primitive cell and its wave function which is , the long magnon dispersion in ferromagnetic materials which is , and by assuming , the total excitation number of magnon is
To solve Equation (24) let’s take, , , and gives
Equation (25) is the total excitation number of magnon which is obtained by introducing zeta and gamma function where , . Then the total excitation energy of magnons becomes
To solve Equation (26) let us introduce variable , . Then by substituting these values in Equation (26), the total excitation energy is
By introducing zeta and gamma function , then the total excitation energy (activation energy) of magnon becomeswhere and a is the lattice constant which is given by Vegard’s law, [13]. Equation (28) indicates the relation between temperature, concentration, and energy gap of the DMSs. As the concentration of magnetic impurity, x, increases the total excitation energy decreases. That means the energy gap is decreased and the charge carriers are moved simply. The electrical properties of the semiconductors are changed. This total excitation energy, Equation (28), is used to calculate the transport properties of DMSs.
2.3.1. Magnetization
Several methods have been developed to precisely determine , the temperature at which a material changes from a ferromagnetic state to a paramagnetic one. Measuring of magnetization as a function of temperature is the most effective technique. To calculate the magnetization of ferromagnetic material and its alignment in the z-direction at a temperature T, we use the relation between magnetic moment and magnetization as follows:where , , and are the total number of particles, the volume, the Lande g-Factor and the Bohr magneton, respectively. Then by using and Equation (25), Equation (29) becomeswhere is constant, is the magnetization at absolute zero where all spins are parallel and V is the volume of the unit cell. The ratio of magnetization at temperature T and magnetization at absolute zero is called reduced magnetization. Equation (30) indicates that the total magnetization decreases by one unit of spin with the excitation (or creation) of a magnon and the temperature dependance of decrease in magnetization. -doped has an FCC lattice structure with with a lattice constant of about and spin of .
2.3.2. Critical Temperature
By the limit of approaches zero, then the critical temperature for ferromagnetic material is
The critical temperature of ferromagnetic increases as the concentration of magnetic ions impurity increases. Equation (31) indicates the relation between the transition temperatures and the concentration of magnetic ions in ferromagnetic materials.
2.3.3. Electrical Conductivity
For -doped , the scattering mechanism is the ionized magnetic impurity scattering. The Boltzmann transport theory is used to determine the electrical conductivity of the carriers when the predominant scattering mechanism is the scattering by ionized impurities because it is effective in treating along-range disorder [14]. According to Shinjo [3], the density of spin-up and spin-down in ferromagnetic materials is almost similar (i.e., ). Supposing the electric field lies along the z-direction and the magnetic field is zero, then using Equation (8), the rate of change of the probability , Equation (5), with which the state is occupied due to the collision with the localized spins becomes . Then using the hole density of particles with spin , the Fermi Dirac distribution function and parabolic energy, the current density is given by
If Fermi energy is located in the valence band, the term has a large value near the Fermi energy only; it may be approximated by Dirac delta function . Then the total current density becomes . Where is the relaxation time at . Then the electrical conductivity of spin-up and spin-down electrons is expressed as follows:
Since -doped has a zinc blende crystal structure similar to , the optical thresh hold at directly gives the band gap [11]. Where is the thermal energy excitation of a single magnon. Then the total thermal energy excitation is related to the energy gap by the relation . Where in our case by assuming , , which is the total excitation energy of magnon. As the concentration of magnetic impurity increases the energy gap is reduced. By substituting the spin-dependent relaxation time Equations (8) and (28) in Equation (33), the temperature dependance electrical conductivity becomes
Where , . is a p-type semiconductor, thus we utilize the hole concentration rather than . According to Dietl et al. [1], the optimal formula for the hole concentration, , is , where is the concentration of Ga sites in GaAs, which is equal to , and x is the concentration of the impurity. According to Blakemore [15], for (Ga,Mn)As, the hole effective mass is , where is the electron mass.
2.3.4. Hole Mobility
Since -doped is a p-type semiconductor, the mobility we calculated is hole mobility. By using the relation between electrical conductivity and mobility, the mobility of impurity is:where . From the relation of spin-dependent relaxation time, we have Where is the impurity density, is the p–d exchange interaction constant and s is the spin of magnetic impurity. The hole mobility is exponentially proportional to .
3. Results and Discussion
Different transport properties of DMSs are computed in this work. The transport properties of p-type -doped have been investigated theoretically. The theoretical calculation is done using different methods and models. Throughout this study, for better numerical estimations, we use the exchange integral of local magnetic moments, at different sites separated approximately by a distance of lattice constant (), , the spin of the atom S = 5/2, the Boltzmann constant , reduced Plank’s constant ћ = 1, hole effective mass , electron charge , and mass of an electron . The Curie temperature for this article, as determined theoretically, is around 162 K. Using this value, the transport properties of ferromagnetic are discussed as follows:
At x = 0.08, the magnetization of the the decreased with the temperature of , as seen in Equation (30) and Figure 1. This demonstrates that the magnetization of this material reduces to zero as the temperature rises. When there is no applied external field, the spontaneous magnetization of ferromagnetic occurs in the ground state. This indicates that the spins are positioned parallel to one another. The orientations of these spins are altered in the presence of an external field. At high temperatures, this disruption in spin orientation results in a reduction in spontaneous magnetization. As a result, the spontaneous magnetization is zero above the Curie temperature. There is no spontaneous magnetization at high temperatures.
In Equation (31), we determine the concentration dependance of the Curie temperature for -doped DMS. As shown in Figure 2, the transition temperature rises as magnetic ion concentration does. We determine the value for . The estimated curie temperature in accord well when compared to experimental and theoretical expectations. While our calculation places the Curie temperature of at x = 0.08 around 162 k, the experimental result for this value is 173 k [6]. Within a reasonable concentration range (about 10%), high is very challenging to achieve for homogeneous DMS systems. Therefore, as the content x increases, the concentration of occupying the sites () increases, leading to higher for . By increasing the Mn content and/or the free hole concentration in the alloy, the varied values of in might be enhanced. It has been demonstrated experimentally that in rises with increasing hole concentration. Due to the fact that goes to zero when holes are compensated and that of increases with the increasing hole concentrations.
Using Equation (34), Figure 3 illustrates the electrical conductivity of for x = 0.035, x = 0.05, and x = 0.08. A doped or extrinsic semiconductor’s electrical conductivity increases with increased temperatures and magnetic impurity concentration.
According to this idea, the electrical conductivity of DMSs, which are semiconductors with magnetic impurities added into them, increases exponentially with both temperature and magnetic ion concentration [11]. A small addition of a magnetic impurity has a significant impact on the electrical conductivity, as shown in Figure 3. We can see from the graph that electrical conductivity increases as magnetic impurity concentration does. Due to the fact that modern electronics are made by adding magnetic impurities (introducing spin) to semiconductors, or “Spintronics,” the electrical conductivity of semiconductors is altered by the presence of magnetic ions. The conductivity is high at x = 0.08, which corresponds to a high concentration, and low at x = 0.035. Electrical conductivity increases along with the magnon’s thermal excitation.
Since is a p-type semiconductor we discuss the hole mobility. From the relation of mobility and relaxation time in Equation (35) we can find the hole mobility of for . The hole mobility is exponentially increased with temperature as shown in the Figure 4. This can be understood qualitatively by considering that with increasing temperature, electrons can travel faster and this makes it easier to escape the ionized impurities. Figure 4 shows the hole mobility at x = 0.05. Holes are more mobile when they experience less scattering, i.e., the time between collision (relaxation time) is larger and the effective mass is smaller. As the temperature increase the hole mobility is also increase.
4. Conclusion
In this study, we studied the theoretical transport properties of DMS with a focus on . The ferromagnetic semiconductor known as has undergone the most extensive study to date. The concentration of magnetic dopants and temperature have a significant impact on the interesting transport characteristics of DMSs. Since the transport properties are depend on the density of states, the impurity added to the semiconductor increases this density of state. Magnons are thermally excited when an electromagnetic field is applied. We investigated the Curie temperature, magnetization, electrical conductivity, and hole mobility of using this thermal excitation of a magnon.
By using these theoretical work of at x = 0.08 is 162 K. From experimental work, of at x = 0.08 is 173 K. Therefore, the theoretical calculation is more approximated to the experimental work. The Curie temperature of is increasing with increasing concentration. Since it is a ferromagnetic material, it has spontaneous magnetization at low temperatures. The magnetization of this material is reduced to zero as . The electrical conductivity of is exponentially increased with an increase in temperature from 0 to 300 K and concentration of magnetic ions in the range of . Since hole mobility is directly proportional to the relaxation time (i.e., ) it exponentially increases with increasing temperature.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are deeply indebted to Dilla University, School of Graduate Studies for the financial support.