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Temperature Dependence of Electrical Resistivity of (III, Mn)V Diluted Magnetic Semiconductors

Written By

Edosa Tasisa Jira

Submitted: December 11th, 2021Reviewed: February 4th, 2022Published: March 31st, 2022

DOI: 10.5772/intechopen.103046

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New Advances in SemiconductorsEdited by Alberto Adriano Cavalheiro

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New Advances in Semiconductors [Working Title]

Dr. Alberto Adriano Cavalheiro

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Abstract

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.

Keywords

  • RKKY interactions
  • electrical resistivity
  • spin-dependent relaxation time

1. Introduction

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 [3]. 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) [4]. 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 [5].

For practical applications, increasing TCbeyond room temperature is a necessity. Thus, exploring various roots of raising the value of TCis a great fundamental and practical significance. Carrier-mediated ferromagnetism in p-type II-VI DMS hetero-structures occurs only at very low temperatures TCtypically below 2.0K. Therefore, very significant strides have been made in developing ferromagnetic-based IIIVsemiconductors containing Mn,which remaining ferromagnetic to much higher TC[3, 6]. A discovery of ferromagnetic ordering in III–V DMSs with critical temperatures of annealed Ga1xMnxAscould 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 Ga1xMnxAs, 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 includingGaMnAs. In order to prevent phase separation order, magnetic Ga1xMnxAsshould be grown at low temperatures (T; 200–300°C). These problems are not specific to Ga1xMnxAs, 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 IIIMnVferromagnetic semiconductors, more specifically Ga1xMnxAs. Even though we focus on III–V compound based DMS such as Ga1xMnxAs, the results presented in this paper are general for all DMS materials.

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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 Jijof exchange coupling. When magnetic impurity Mnis 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 J[14, 15]. Writing the spin density of electrons in terms of the field operators, the interaction between the spin Sjat rjand the conduction electron is

Hpd=2JSj.Sirj=JαβSj.σαβψ̂+αrjψ̂βrjE1

Where ψ̂is the field operator states expanded in terms of the block states. Using the field operators, the Fourier transform of number density operator nrbecomes

nq=nreq.rdr=1Vkkσeik+qk.rukrukrc+ckσdrE2

Where cand c+are the annihilation and creation operators of electrons, respectively. The integral in Eq. (2) is vanished unless k=k+q+Gdue to the lattice periodicity of the function s ukr. Where G is the reciprocal lattice vectors. The coupling strength depends on state k,the state in which the electron of wave vector kis scattered. If we express Eq. (2) in terms of the spin density and the system contains Nilocalized spins, the interaction with conduction electrons is determined by

Hpd=JVl=1Nik,kk,keikk.rSl+ck+ck+Slck+ck+Slzck+ck+ck+ckE3

The interaction of localized spin Mnions with electron gives rise to the creation or annihilation of an electron–hole pair. By considering elastic transitions and the common energy Ek=Ekdenoted byE0, the Hamiltonian is written as

kHeffk=jkλHeffjjλHeffkEjE0E4

Where kis ground state,kis the state of the scattering of electron, λ is the coupling constant, and Heffis the effective Hamiltonian, which is equal toHpd. 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

fktcoll=kWk,kfk1fkfk1fk+kWk,kfk1fkfk1fkE5

Where Wk,kis the transition probability, fkand fkare the distribution functions of the initial states and the scattered states, respectively. The transition probability between two states αand βis given by

Wαβ=2πћαHpdβ2δEαEβE6

Where αis the eigen state of the initial system, and Eαis the corresponding eigen values, βis the final eigen states, and Eβis the corresponding eigen values, respectively. If the spin is flipped in the intermediate state, an electron of quantum numbers Mlappears in the first process, and the impurity spin goes over from the initial state Sz=MtoSz=M+1. 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 stateSz=M1. Then by substituting state αand βand using the distribution function for the spin of the electron–hole pair, the total contribution is

fHeffi=JN2ijf0Ek1f0EkEkEkeikk.rjri{2MlSizSjz+12Si+Sj+SiSj+MlE7

Where Mlis the state of the spins of the z-component of the intermediate states, and Mlis the corresponding eigen values, Mlis the state of the spins of the z-component of the scattered states, and Mlis the corresponding eigen value, and Mlis the state of the spins of initial state, and Mlis the corresponding eigen value.

Eq. (7) can be considered as the matrix element of the operatorH, when we write the exchange energy between Siand Sj,which is written as

H=ijJrirjSi.SjE8

To determine the exchange coupling strength Jrirj, we introduce r=rirjwhere ris the displacement vector between the ions i and j, and using the following integral notation Ias

I=1/V2kkf0Ek1f0EkEkEkeikk.rE9

Replacing the sum in Eq. (9) by an integral, using the quadratic dispersion relation valid for free electrons and the notationsκ=kr,κ=kr, the parabolic energy Ek=ћ2k22m, in which mis the effective mass, and trigonometric identity sinκ=eiκeiκ2iand the complex variable κ+instead ofκ, where ηis an infinitesimal quantity, the angular integral becomes

I=mekFJ24ћ2π3sin2kFr2kFrcos2kFr2kFr4E10

Then the effective exchange interaction or the coupling exchange interaction constant between magnetic impurities and delocalized charge carriers is given by

Jr=meJ2kF4ћ2π3F2kFrE11

Where kFis the Fermi wave vector and the function Fx=xcosxsinxx, x=2kFr.Eq. (11) gives the well-known RKKY interaction. As the value of the oscillatory function F2kFrvaries, the value of the exchange interaction is either positive or negative, and hence, the interaction in the system can be ferromagnetic or antiferromagnetic.

By using Eq. (11), we can determine the relaxation time τof the impurity spin of electrons. For spin conserving scattering, the matrix element of Eqs. (7) and (8) is

kHeffk=JVSzE12

A better approximation can be obtained for the transition probability by replacing the matrix element of the scattering matrix Tin Eq. (6). Then Eq. (6) is rewritten as

Wαβ=2πћαTβ2δEαEβE13

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 Tmatrix, the Fermi-Dirac distribution function, which is expressed byf0k=ck+ckand if the z-component of the impurity spin is unchanged by the scattering, the combined contribution is

JV2kSz21f0kEkEkf0kEkEk=JN2Sz2k1EkEkE14

If the impurity spin goes over from the initial state Sz=Mto Sz=M+1, 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 Sz=MtoSz=M1, and on the account of conservation of energyEk=Ek, then the combined contribution of these processes gives

JV2kSS+1M21EkEkM12f0kEkEkE15

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 Jby summing over the possible spin orientations is given by

Wkk=Wkk=Ni2πJ2SS+13ћ1+4JgEkδEkEkE16

Where Niis the concentration of magnetic impurities, which is Ni=4x/alc3where xis the concentrations of the impurity added to the semiconductors, alcis lattice constant, and the singular function in the third order correction is

gEk=1Vkf0kEkEkE17

We shall investigate the dependence of Eq. (16) on the energy of the initial state EK, which is entirely involved in the function gEk. At the absolute zero of temperature, f0kcan be replaced by a step function, which is unity when k<k0and zero when k>k0, where k0is the magnitude of the Fermi momentum.

By using Fermi-Dirac distribution, Ek=ћ2k22mand for electron close to the Fermi surface at low temperature, the singular function becomes

gEk=3z2EF{1+k/2k0logkk0/kk0}E18

Where zis the number of conduction electrons per atom. Since at T ≠ 0, the average of kk0for thermally excited electrons is proportional to T, even at this stage of calculation we can expect a term proportional to logTin the expression of the resistivity. It is true that gEkdiverges when Ekapproaches to EF. Eq. (18) makes the calculation easier to retain the definition of gEkhere and to carry out the summation after we get the expression for the resistivity.

Then substituting Eq. (16) into collision integral, the relaxation time τis the inverse of the transition probability. Therefore, the spin-dependent relaxation time is

1τ=Ni2πJ2SS+13ћEF1+4JgEkE19

The Fermi energyEFis given by EF=ћ2k22m=ћ22m3π2NV23.In IIIVsemiconductors, 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 ofGaAs, the conduction band is located at EFS+0.9eVand the valence band at EFS0.5eV[3]. WhereEFSis the Fermi-level stabilization energy, and it is found to be located at 4.9eVbelow the vacuum level and is the same for all III-V and II-VI SCs. since Mn-doped GaAsis 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 calculateτ0.

In the presence of external perturbation due to the electric field or temperature gradient, there is a variation in the function fk±. Supposing the electric field lies along the z-direction, then using Eq. (19), the rate change of the probability fk±with which the state K±is occupied due to the collision with the localized spins becomes

fk±tcoll=fk±fk0τE20

Then from standard theory, we can easily obtain the resistivity as

ρspin=xρ01(2jπmhk0gEkdf0dEkd3k}E21

Where ρ0is defined by

ρ0=3πmhJ2SS+1V/N/2e2EFE22

Where V is the volume of the crystal. Then using some techniques, the total electrical resistivity becomes

ρspin=xρ013zJ/EFlogTE23

Since substitutionalMnions in GaAs(i.e.,Ga1xMnxAs) are a p-type semiconductor, we use the hole concentration p=NV, instead of n. The hole concentration pwould ideally to be given by p=xN0where N0=2.2x1022cm3is the concentration of Gasites in GaAsand xis the concentration of impurity [3]. For GaMnAs, the hole effective mass mh=0.5m0, where m0=meis the electron mass [16].

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3. Results and discussion

As mentioned in Section 2, the temperature dependence of electrical resistivity is calculated. The electrical resistivity of p-type Mn-doped GaAshas 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 Ga1xMnxAsfor 0.08x0.01is shown in Figure 1. From Eq. (23) the electrical resistivity is inversely proportional to the hole density or hole concentration (i.e., ρ1p). Since Ga1xMnxAsis 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 Ga1xMnxAsfor 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.

Figure 1.

Electrical resistivity ofGa0.92Mn0.08Asversus temperature.

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4. Conclusion

In this paper, we studied the theoretical temperature dependence of electrical resistivity of DMS specificallyGa1xMnxAs.Ga1xMnxAshas become the most understood and extensively studied III1xMnxVferromagnetic semiconductor. DMSs show interesting electrical properties, which strongly depend on the concentration of magnetic dopants and temperature. These properties of Ga1xMnxAsis 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 Ga1xMnxAsis exponentially decreased with an increase of temperature from 0K300Kand concentration of magnetic ions in the range of 0.08x0.01. In general, if TCof the DMSs can be increased, there is a possibility of utilizing the system under consideration for the spintronic purpose at room temperature.

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Acknowledgments

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.

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Conflicts of interest

The authors declare no conflicts of interest.

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Data availability

No data were used to support this study.

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Written By

Edosa Tasisa Jira

Submitted: December 11th, 2021Reviewed: February 4th, 2022Published: March 31st, 2022