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We discuss hole-induced magnetic solitons and metal-insulator transition of transport properties in diluted magnetic semiconductors Ga
_{1-x}Mn
_{x}As from the standpoint of a field theoretical formulation, and analyze experimental data of transport properties, using the supersymmetry sigma formula and the effective Lagrangian of diffusion model.

Diluted magnetic semiconductors (DMSs), which are formed by substitution of several percent of cation sites in a host semiconductor with magnetic impurities, are actively investigated both theoretically and experimentally, due to their potential applications in new generations of semiconductor spintronic devices [_{1}_{−}_{x}Mn_{x}As and In_{1}_{−}_{x}Mn_{x}As show severely limited chemical solubility due to the substitution of divalent Mn atoms for the trivalent Ga or In sites. In order to prevent phase separation, these materials should be grown at low temperature (T from 200˚C to 300˚C), which results in an abundance of different types of crystal defects. As a result, a theoretical study of DMSs is very difficult owing to two factors (strong disorder and exchange interaction), which must be taken into account nonperturbatively.

Understanding the mechanism behind the carrier-induced ferromagnetism is of significance for further development of semiconductor spintronic devices. Several theoretical models for carrier-induced ferromagnetism in (Ga, Mn)As have been proposed [

In this study, the anomalous transport properties in DMSs are discussed using a field-theoretical formulation. Then we analyze some conductivity data in DMSs (Ga, Mn)As, using the gauge-invariant effective Lagrangian density and quantized magnetic solitons.

According to the aggregation of hole-induced magnetic solitons, the non-monotonic temperature dependence of the transport properties of (Ga, Mn)As is qualitatively explained as being due to the hole localization around the Mn ions. It has been suggested that the ferromagnetic ordering might be due to a double-exchange-like interaction and the remarkable change of spin exchange interaction among Mn ions by the hole seems to be cooperative and non-linear (Yang Mills like). Kanazawa and coworkers [^{3} [

Then the Yang-Mills fields A μ a induced by the doped hole have a local SO(4) symmetry. Here we have thought that the SO(4) symmetry fields A μ a are spontaneously broken around the hole through the Anderson-Higgs mechanism, in the III-V-based diluted magnetic semiconductors with magnetic manganese ion-doping. Through the spontaneous symmetry breaking 〈 0 | ϕ a | 0 〉 = 〈 0,0,0, μ 〉 , the effective Lagrangian density has been introduced [

H = − J ∑ 〈 i ˜ , j ˜ 〉 cos ( θ i ˜ j ˜ / 2 ) O ( r i ˜ ) ⋅ O ( r j ˜ ) + 1 2 K ∑ i ˜ ≠ j ˜ O ( r i ˜ ) ⋅ O ( r j ˜ ) | r i ˜ − r j ˜ | . (1)

Here the first sum ∑ 〈 i ˜ , j ˜ 〉 is taken only over nearest neighbors (the distance between each magnetic soliton is ≤ 2 R c ), while the second sum is taken over all pair( i ˜ ≠ j ˜ means | r i ˜ − r j ˜ | > 2 R c ) [

− J = − g 2 2 4 π e − m 1 r r | r ∼ 1 m 1 ∼ − g 2 2 4 π m 1 e − 1 (2)

is the short-range attractive potential, which is derived from massive gauge fields A μ 1 , A μ 2 , and A μ 3 exchange interaction. When the magnetic soliton, O ( r i ˜ ) , with the effective spin N i ˜ is located at the nearest-neighbor site of the magnetic soliton, O ( r j ˜ ) , with the effective spin N j ˜ , holes are hopping between the two solitons O ( r i ˜ ) and O ( r j ˜ ) . If N i ˜ is parallel to N j ˜ , the p-d exchange interaction induces large reduction of the kinetic energy. The hopping term between the nearest neighbors of hedgehog-like solitons (clusters) leads to an additional term in the σ-model describing a coupling of the supermatrices, Q i ˜ , corresponding to different magnetic solitons (clusters) [

F ˜ ( Q ) = s t r ( − ∑ 〈 i ˜ , j ˜ 〉 J i ˜ j ˜ Q i ˜ Q j ˜ + i 4 ( ω + i δ ) ∑ i ˜ Δ i ˜ − 1 Λ Q ) , (3)

where J i ˜ j ˜ = J cos ( θ i ˜ j ˜ / 2 ) 1 Δ i ˜ Δ j ˜ . Then Δ i ˜ is the mean energy level spacing at

the hedgehog-like soliton (cluster) O ( r i ˜ ) and J > 0 . The diffusion coefficient D 0 is introduced as follows,

D 0 ∼ 4 Δ π ∑ j ˜ J i ˜ j ˜ ( r i ˜ − r j ˜ ) 2 ∼ 4 Δ π ∑ j ˜ J cos ( θ i ˜ j ˜ / 2 ) 1 Δ 2 ( r i ˜ − r j ˜ ) 2 ∼ 4 π Δ ∑ j ˜ J cos ( θ i ˜ j ˜ / 2 ) ( r i ˜ − r j ˜ ) 2 (4)

Here Δ = 1 ν π R c 2 and ν is the density of states of the carriers at the Fermi

surface. In the case of the low frequency limit of ω , the localization length L l o c is shown as follows,

L l o c ∝ π 2 ν R c 2 D 0 ∼ π 2 ν R c 2 4 π Δ ∑ j ˜ J cos ( θ i ˜ j ˜ / 2 ) ( r i ˜ − r j ˜ ) 2 (5)

We shall consider the variable range hopping conductivity and the system length L ≫ L l o c as follows,

σ ∝ exp [ − ( A / T ) 1 / ( d + 1 ) ] (6)

where d is the dimensionality of the system.

A ∝ ( 1 L l o c ) d / ( d + 1 ) ∼ ( 1 π 2 ν R c 2 D 0 ) d / ( d + 1 ) (7)

_{0.95}Mn_{0.05}As. The annealing is performed at 310˚C for 15 mins.

The as-grown sample and the sample annealed at 310˚C show insulating behavior above ~30 K and ~50 K, respectively. The annealing at 310˚C increases the conductivity. Annealing might reduce concentration of As antisites and interstitial Mn. As the conductivity σ increases, the high-temperature structure moves to higher temperatures, which means T c (Curie temperature) increases. Thus the concentration ρ of mobile holes and T c are enhanced by the annealing. The experimental data are fitted well with Equations (6) and (7), as shown with solid lines in

The hole-induced magnetic solitons and metal-insulating transition of transport properties in DMSs have been discussed based on a field theoretical formulation. We have analyzed experimental data on the transport properties of GaMnAs by using the effective Lagrangian of diffusion model.

The authors declare no conflicts of interest regarding the publication of this paper.

Kanazawa, I., Nakamura, S. and Maeda, R. (2018) Carrier-Induced Magnetic Solitons and Metal-Insulator Transition in Diluted Magnetic Semiconductors Ga_{1−x}Mn_{x}As. Journal of Modern Physics, 9, 2437-2442. https://doi.org/10.4236/jmp.2018.914156