In this work, a theory of temperature dependence of electrical resistivity is developed, with a particular emphasis on dilute magnetic semiconductors (DMSs). The approach is based on the equation of motion of the Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction and considers both spin and charge disorder. The formalism is applied to the specific case of Ga1−xMnxAs.Using the RKKY exchange interaction, the relaxation time τand the exchange interaction J are calculated. Then using spin-dependent relaxation time, electrical resistivity of the material is calculated. The electrical resistivity of Mn-doped III—V DMS is decreased with increasing temperature and magnetic impurity concentration.
- RKKY interactions
- electrical resistivity
- spin-dependent relaxation time
The motivation for this study has been to better understand semiconductors, which could be the building block for the next generation of computers. Semiconductors have been the subject of very extensive research over recent decades, because of ever more numerous and powerful applications. As conventional electronics, semiconductors are more appropriate than normal metals, since metal devices do not amplify current. Because of the complementary properties of semiconductor and ferromagnetic material systems, a growing effort is directed toward studies of semiconductor-magnetic nanostructures [1, 2].
Semiconductors and magnetic materials both play essential roles in modern electronic industry. Since the applications of semiconductors and magnetic have evolved independently, it appears logical to combine their properties for possible spin-electronic applications with increased functionalities . This is known as spintronics, which utilizes both charge and spin of the electrons to process and store data. This is to control the spin degree of freedom of electrons in semiconductors. The direct method to introduce spin degrees of freedom in semiconductors is to introduce magnetic ions into semiconductors. Such semiconductors are referred to as diluted magnetic semiconductors (DMSs) . The curie temperature of the DMS material and whether the ferromagnetism in the DMS material originates from free carrier mediation or purely from localized magnetic dopants limit DMS practical applications for devices .
For practical applications, increasing beyond room temperature is a necessity. Thus, exploring various roots of raising the value of is a great fundamental and practical significance. Carrier-mediated ferromagnetism in p-type II-VI DMS hetero-structures occurs only at very low temperatures typically below . Therefore, very significant strides have been made in developing ferromagnetic-based semiconductors containing which remaining ferromagnetic to much higher [3, 6]. A discovery of ferromagnetic ordering in III–V DMSs with critical temperatures of annealed could exceed 110 k; reaching 191 K has renewed and greatly intensified an interest in those materials, but still too low for actual applications. This was, at least partially, related to expectations that their Curie temperatures can be relatively easily brought to room temperature range through a clearly delineated path [7, 8, 9].
For the ferromagnetic transition to occur in , the concentration of Mn impurities should be relatively high, x = 0.01–0.05. Such high concentration of impurities makes the study of the DMS more challenging. Moreover, the issue of solubility of the magnetic ions into the regular semiconductor was a major challenge to fabricate high-quality DMSs. In recent years, Molecular Beam Epitaxy techniques have led to the successful growth of various DMS including. In order to prevent phase separation order, magnetic should be grown at low temperatures (T; 200–300°C). These problems are not specific to , in fact, all presently known DMS materials suffer from the same problems. As a result, theoretical study of DMS electrical properties such as resistivity is very difficult. Since these properties are influenced by the exchange interaction between the carriers and the localized moments, it must be taken into account non-perturbatively [10, 11]. Due to these influences, spin fluctuation scattering contributes to the resistivity. The experimentally measured dc resistivity in the DMS materials shows interesting behavior strongly depending on the concentration of the magnetic impurity and temperature. In (Ga,Mn)As, there is a metal–insulator transition between insulating samples with small manganese concentration and metallic samples with larger concentration. In and with low concentration (x < 0.03), only an insulating behavior has been observed in transport measurements. Insulating behavior is here characterized by a diverging resistivity for T → 0, indicating localization of carriers. However, near-optimal doping x = 0.05, where the highest value of is reported, the nonmonotonic behavior of insulator–metal–insulator of the resistivity as a function of temperature is observed. A resistivity peak appears near the critical temperature, and the resistivity shows metallic behavior below and insulating behavior at higher temperatures. In metallic samples, the resistivity decreases and eventually saturates for T → 0. The peak has been understood as the critical scattering effects of spin fluctuations [11, 12, 13]. In this paper, a Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction was used for the calculation of the resistivity. This paper focuses on the group ferromagnetic semiconductors, more specifically . Even though we focus on III–V compound based DMS such as , the results presented in this paper are general for all DMS materials.
2. Theoretical formalism
2.1 RKKY interaction approaches to p-d exchange interaction
The carrier-mediated spin–spin coupling is usually described in terms of the Ruderman-Kittel-Kasuya-Yosida (RKKY) model, which provides the energy of exchange coupling. When magnetic impurity is introduced into semiconductors, spin flip processes become possible. The spin of the scattered electron and the spin of magnetic impurities are flipped simultaneously while the component of the total spin along the quantization axis is conserved. The interaction of the impurity with conduction electron is described as p-d or s-d interaction. The spin density of electrons determines this interaction at the localized moment. Like in the Heisenberg model, the scalar product of two spins is taking with the exchange coupling [14, 15]. Writing the spin density of electrons in terms of the field operators, the interaction between the spin at and the conduction electron is
Where is the field operator states expanded in terms of the block states. Using the field operators, the Fourier transform of number density operator becomes
Where and are the annihilation and creation operators of electrons, respectively. The integral in Eq. (2) is vanished unless due to the lattice periodicity of the function s . Where G is the reciprocal lattice vectors. The coupling strength depends on state the state in which the electron of wave vector is scattered. If we express Eq. (2) in terms of the spin density and the system contains localized spins, the interaction with conduction electrons is determined by
The interaction of localized spin ions with electron gives rise 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
Where is the state of the scattering of 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 distribution function of spin-up and spin-down electrons is
Where is the transition probability, and are the distribution functions of the initial states and the scattered states, respectively. The transition probability between two states and is given by
Where is the eigen state of the initial system, and is the corresponding eigen values, is the final eigen states, and is the corresponding eigen values, respectively. If the spin is flipped in the intermediate state, an electron of quantum numbers appears in the first process, and the impurity spin goes over from the initial state to. Where the electron–hole pair is composed of a spin-up electron and spin-down hole, the impurity spin has to flip down, to the state. Then by substituting state and and using the distribution function for the spin of the electron–hole pair, the total contribution is
Where is the state of the spins of the z-component of the intermediate states, and is the corresponding eigen values, is the state of the spins of the z-component of the scattered states, and is the corresponding eigen value, and is the state of the spins of initial state, and is the corresponding eigen value.
Eq. (7) can be considered as the matrix element of the operator, when we write the exchange energy between and which is written as
To determine the exchange coupling strength , we introduce where
Replacing the sum in Eq. (9) by an integral, using the quadratic dispersion relation valid for free electrons and the notations, the parabolic energy , in which is the effective mass, and trigonometric identity and the complex variable instead of, where is an infinitesimal quantity, the angular integral becomes
Then the effective exchange interaction or the coupling exchange interaction constant between magnetic impurities and delocalized charge carriers is given by
Where is the Fermi wave vector and the function , Eq. (11) gives the well-known RKKY interaction. As the value of the oscillatory function varies, the value of the exchange interaction is either positive or negative, and hence, the interaction in the system can be ferromagnetic or antiferromagnetic.
The scattering (transition) probability in Eq. (13) can be expressed in terms of the interaction Hamiltonian up to third order in the coupling constant. Then using the second-order correction to the matrix, the Fermi-Dirac distribution function, which is expressed by and if the z-component of the impurity spin is unchanged by the scattering, the combined contribution is
If the impurity spin goes over from the initial state to , the electron–hole pair is composed of a spin-up electron and a spin-down hole (i.e., the impurity spin has to flip down) to the state to, and on the account of conservation of energy, then the combined contribution of these processes gives
The first term in Eq. (15) is negligible since it is small; however, the second term is no longer small. The second term diverges logarithmically. Using this form, the transition probability Eq. (13) in by summing over the possible spin orientations is given by
Where is the concentration of magnetic impurities, which is where is the concentrations of the impurity added to the semiconductors, is lattice constant, and the singular function in the third order correction is
We shall investigate the dependence of Eq. (16) on the energy of the initial state , which is entirely involved in the function . At the absolute zero of temperature, can be replaced by a step function, which is unity when and zero when , where is the magnitude of the Fermi momentum.
By using Fermi-Dirac distribution, and for electron close to the Fermi surface at low temperature, the singular function becomes
Where is the number of conduction electrons per atom. Since at T ≠ 0, the average of for thermally excited electrons is proportional to T, even at this stage of calculation we can expect a term proportional to in the expression of the resistivity. It is true that diverges when approaches to . Eq. (18) makes the calculation easier to retain the definition of here and to carry out the summation after we get the expression for the resistivity.
Then substituting Eq. (16) into collision integral, the relaxation time is the inverse of the transition probability. Therefore, the spin-dependent relaxation time is
The Fermi energy is given by In semiconductors, it is found that the maximum and minimum energy locations of the Fermi level typically do not deviate by more than 1ev. In the specific case of , the conduction band is located at and the valence band at . Where is the Fermi-level stabilization energy, and it is found to be located at below the vacuum level and is the same for all III-V and II-VI SCs. since -doped is a p-type, we use Fermi level energy of the valence band. By using this value of Fermi energy and the values of other constant, we can calculate.
In the presence of external perturbation due to the electric field or temperature gradient, there is a variation in the function . Supposing the electric field lies along the z-direction, then using Eq. (19), the rate change of the probability with which the state is occupied due to the collision with the localized spins becomes
Then from standard theory, we can easily obtain the resistivity as
Where is defined by
Where V is the volume of the crystal. Then using some techniques, the total electrical resistivity becomes
Since substitutional ions in (i.e.,) are a p-type semiconductor, we use the hole concentration , instead of . The hole concentration would ideally to be given by where is the concentration of sites in and is the concentration of impurity . For , the hole effective mass , where is the electron mass .
3. Results and discussion
As mentioned in Section 2, the temperature dependence of electrical resistivity is calculated. The electrical resistivity of p-type -doped has been investigated theoretically. This theoretical calculation is done using RKKY interaction models. This theoretical calculation is somewhat similar to experimental work, which is done on this material. In this paper, we use RKKY to calculate the exchange interaction constant and spin-dependent relaxation time approximation. Using these exchange interaction constant and spin-dependent relaxation time, the electrical resistivity of this material is calculated
Electrical resistivity is the inverse of electrical conductivity of the material. The electrical resistivity of semiconductors decreases exponentially with increasing temperature in contrast to that of pure metals. The electrical resistivity of extrinsic semiconductor is decreased with increasing both the concentration of magnetic impurity and the temperature. From this concept the electrical resistivity of DMSs, in which magnetic impurities are incorporated into standard semiconductors, is decreased with temperature and concentration of magnetic ions. The temperature dependence of resistivity in for 0.08 is shown in Figure 1. From Eq. (23) the electrical resistivity is inversely proportional to the hole density or hole concentration (i.e., ). Since is an extrinsic semiconductor, its hole concentration is increased with adding the impurity. Then the electrical resistivity decreases with increasing concentration. From Figure 1 the electrical resistivity of for x = 0.035, x = 0.05, and x = 0.08 is drawn. From the graph as the magnetic impurity concentration is increased, its resistivity decreases. As the temperature and hole concentration increase, the electrical resistivity is decreased. At x = 0.08 or at high concentration, there is low resistivity and at x = 0.035, the resistivity is high.
In this paper, we studied the theoretical temperature dependence of electrical resistivity of DMS specifically has become the most understood and extensively studied ferromagnetic semiconductor. DMSs show interesting electrical properties, which strongly depend on the concentration of magnetic dopants and temperature. These properties of is affected by concentration and temperature. Since the electrical properties are dependent on the density of states, the impurity added to the semiconductor increases this density of state and decreases energy gap. The electrical resistivity of is exponentially decreased with an increase of temperature from and concentration of magnetic ions in the range of . In general, if of the DMSs can be increased, there is a possibility of utilizing the system under consideration for the spintronic purpose at room temperature.
First of all, I would like to thank the almighty God for keeping me safe and continually blessing me in all aspects of my life. I extend my heartful thanks to my parents for their encouragement and support both morally and psychologically.
Conflicts of interest
The authors declare no conflicts of interest.
No data were used to support this study.
Dietl T. Ferromagnetic semiconductors. Semiconductor Science and Technology. 2002; 17:377-392
Sapoval B, Hermann C. Physics of Semiconductors. New York: Springer; 1995. pp. 1-2
Dietl T, Awschalom DD, Kaminska M, Ohno H. Semiconductor and Semimetals. Vol. 82. Elseiver; 2008
Shinjo T. Nanomagnetism and Spintronics. 1st ed. Elsevier; 2009. pp. 20-22
Zheng Y. A prespective of recent progress in DMSs, Springer-Verlag Berlin Heidelberg. Applied Physics A. 2013; 112:241–254
Twardowski A. Diluted magnetic III-Vsemiconductors, international school of semiconducting compounds. 2000; 98
Kossut J, Gaj JA. Introduction to the Physics of Diluted Magnetic Semiconductors. Berlin Heidelberg: Springer-Verlag; 2010. p. 356
Macdonald AH, Schiffer P, Samarth N. Ferromagnetic Semiconductors. Nature Publishing Group; Nature Materials. 2005; 4
Chen L, Yang X, et al. Enhancing the curie temperature of ferromagnetic semiconductor (Ga,Mn)As to 200 K via nanostructure engineering. Nano Letters. 2011; 11:2584-2589
Kaminski A, Das Sarma S. Magnetic and transport percolation in diluted magneticSemiconductors. Physical Review B. 2003; 68:235210
Hwang EH, Das Sarma S. Transport properties of diluted magnetic semiconductors. Physical Review B. 2005; 72:035210
Matsukura F, Ohno H, Shen A, Sugawara Y. Physical Review B. 1998; 57:R2037
Timm C. Disorder effects in diluted magnetic semiconductors. Journal of Physics: Condensed Matter. 2003; 15:R1865
Dietl T. Ferromagnetic transition in diluted magnetic semiconductors. Condensed Matter Physics. 1999; 2(3(19)):495-508
Solyum J. Fundamentals of the Physics of Solids. Vol. II. Berlin Heidelberg: Springer-Verlag; 2009. pp. 213-218
Blakemore JS. Semiconducting and other major properties of gallium arsenide. Journal of Applied Physics. 1982; 53(10)