Theoretical study on the origin of anomalous temperature-dependent electric resistivity of ferromagnetic semiconductor

Ferromagnetic semiconductors (FMSs), which embody both semiconductor and ferromagnetic properties, have garnered significant attention as potential materials for next-generation semiconductor spintronics device applications.^1–5 Historically, Mn-based FMSs such as (Ga,Mn) As and (In,Mn) were widely studied, establishing the foundation for this field.^6–10 Practical applications necessitate a Curie temperature (T_C) exceeding room temperature; consequently, the discovery of high-temperature ferromagnetic materials like (Zn,Cr)Te has invigorated the field of semiconductor spintronics.¹¹ Recent advancements have unveiled novel properties in Fe-based FMSs. For instance, both p- and n-type FMSs can be fabricated using (In,Fe)As; (Ga,Fe)Sb exhibits T_C values above room temperature, and gate voltage modulation of magnetism is achievable in (In,Fe)Sb.^12–17 This evolution of FMSs underscores the importance of exploring diverse material systems. The development of FMSs can be furthered by employing theory-driven computational materials design, given the numerous combinations of magnetic impurities and host semiconductors to consider.^18–20 To identify materials appropriate to practical applications, it is crucial to optimize not only T_C but also the electric transport properties at the device’s operating temperature. This is attributed to the fact that FMSs modulate magnetism through semiconductor characteristics. For example, despite its high T_C, (Zn,Cr)Te exhibited prohibitively high electric resistivity (ρ ∼ 10² Ω cm), rendering it unsuitable for spintronics device electrode applications.²¹ Therefore, first-principles calculation-based searches for new FMSs should also predict ρ at finite temperatures.

Despite its utility, the first-principles method currently faces limitations as it primarily addresses ground states at absolute zero temperature, necessitating innovative approaches to incorporate temperature effects. By contrast, conventional transport property estimations rely on the semi-classical Boltzmann method, which is ill-suited for systems containing transition metals frequently observed in spintronics materials. In transition-metal systems, large d states induce heavy effective mass around the Fermi level, leading to reduced electronic velocities. To account for the complex electronic structure in transport calculations, it is essential to consider energy-dependent relaxation times. Additionally, discussions of transport properties must acknowledge that localized d states can serve as impurity scattering centers for d-resonance scattering. Addressing these challenges, we previously assessed the temperature-dependent electronic structures and transport properties of the Heusler alloy Co₂MnSi.^22,23 In the mentioned study, the coherent potential approximation (CPA) was employed to incorporate electron–phonon/magnon scattering as a temperature effect. It was demonstrated that the spin polarization decreases with increasing temperature due to the disappearance of the half-metallic electronic structure, where one spin state is metallic and the other is a semiconductor. Furthermore, linear response theory, which accounts for quantum effects and incorporates impurity and temperature effect-related scattering processes, was utilized to compute transport properties. Consequently, the experimental temperature-dependent ρ was successfully reproduced quantitatively, even in transition-metal systems. Pioneering studies employing the CPA scheme have successfully determined the residual resistivity (ρ) at 0 K for FMSs. Turek et al.,²⁴ Lowitzer et al.,²⁵ and Ogura and Akai²⁶ calculated the residual resistivity of (Ga,Mn)As and achieved consistent results that aligned with experimental observations, ranging from 10⁻³ to 10⁻² Ω cm.^27–31 To enable effective computational materials' design, it is crucial to evaluate the temperature-dependent behavior of ρ through first-principles transport calculations. However, the quantitative assessment of temperature dependence is challenging for certain FMSs owing to their complex nature. A straightforward example is (Zn,Cr)Te, for which ρ monotonically decreases as the temperature increases.²¹ In contrast, (Ga,Mn)As exhibits anomalous behavior concerning temperature, with ρ initially increasing up to T_C and subsequently decreasing beyond T_C.^27–31 This distinct behavior can be attributed to the ferromagnetic–paramagnetic transition, resulting in a temperature dependence of ρ that differs from that observed in metallic systems. Accurate prediction of the intricate temperature-dependent behavior of ρ in FMSs through first-principles calculations would significantly advance theory-driven material design and remarkably enhance search efficiency. In this study, we focused on investigating (Ga,Mn)As a well-established FMS and analyzed its electrical transport properties at finite temperatures using the aforementioned computational techniques. There are several compelling reasons for focusing on (Ga,Mn)As in this study. (Ga,Mn)As stands out as the most extensively studied FMS, and its low ρ makes it potentially suitable for spintronic device applications like transistor electrodes and magnetoresistive random access memory devices. It is worth noting that recent studies have demonstrated new phenomena in (Ga,Mn)As, such as spin-orbit torque (SOT) switching in a single layer^32,33 and the ability to modulate the symmetry of magnetic anisotropy using the quantization size effect of a quantum well.³⁴ These features hold promise for improving device performance and reducing power consumption in the field of spintronics. Moreover, (Ga,Mn)As could serve as a material for cryogenic complementary metal-oxide-semiconductor (CMOS) applications, which are crucial for the realization of quantum computers. While the low T_C of (Ga,Mn)As has hindered practical applications thus far, it becomes inconsequential when operating at cryogenic CMOS temperatures. Consequently, (Ga,Mn)As has regained global attention, although the fundamental understanding of its physical properties remains a subject of controversy. In this study, we examined the impact of electronic structures, native defects, and magnetic properties on the electrical transport properties of (Ga,Mn)As from a micro-level perspective.

In this research, we employed density functional theory (DFT) calculations^35,36 using the AkaiKKR code,³⁷ which is based on the Korringa–Kohn–Rostoker (KKR) Green’s function method.^38,39 During the calculations, the randomness of the substitutional Mn atoms and co-doped impurities in (Ga,Mn)As was addressed using the CPA.^40,41 The CPA method can accurately account for configurational disorders by substituting multiple scattering effects with an effective medium potential. In this work, we utilized the Moruzzi–Janak–Williams (MJW) type of local density approximation (LDA).⁴² A total of 1661 and 11921 k-sampling points in the first irreducible Brillouin zone were employed for self-consistent calculations and the prediction of electrical transport properties, respectively. The relativistic effect was considered within the scalar relativistic approximation.⁴³ The cutoff angular momentum was set to l_max = 3.

The effect of the temperature was examined using the CPA method. In this approach, we considered both electron–phonon and electron–magnon scattering as temperature-dependent effects. For electron–phonon scattering, atoms that are slightly displaced from their equilibrium positions due to increasing temperature are treated as CPA impurities. Although the single-site t-matrix varies with the atomic displacement, it can be determined self-consistently. We implemented a method that expands the single-site t-matrix at the displaced atomic position during this self-consistent calculation. The displaced single-site t-matrix was then re-expanded to include partial waves at the equilibrium position, and the potential scattering matrix was adjusted with respect to the equilibrium position. In this case, the backscattering term of Green’s function was obtained with respect to the equilibrium position.

On the other hand, in addressing electron–magnon scattering within the CPA framework, we employed the local moment disorder (LMD) model.⁴⁶ In the LMD model, the tilt of the spins with increasing temperature is represented as a linear combination of the spin-up and spin-down magnetic moments. For (Ga,Mn)As, we assumed that the Mn atoms consist of two magnetic moments: spin-up (Mn^↑) and spin-down (Mn^↓), denoted as (Ga,Mn$1−x/2↑$⁠,Mn$x/2↓$⁠)As. For instance, the LMD model is (Ga,Mn$1↑$⁠,Mn$0↓$⁠)As at 0 K and (Ga, Mn$1/2↑$⁠,Mn$1/2↓$⁠)As at T_C. In previous studies, the LMD model successfully estimated the residual resistivity of (Ga,Mn)As.^24,26 It is important to note that the spin-flip x at each temperature was determined by referencing experimental magnetization data (T_C ∼ 80 K).³¹ 

Furthermore, we examined four types of impurities present in the crystal: substitutional Mn atoms (Mn_Ga), arsenic-antisite defects (As_Ga), interstitial Mn atoms at tetrahedral sites [Mn_i(T)], and interstitial Mn atoms at octahedral sites [Mn_i(O)]. The concentrations of these impurities were determined based on experimental data referenced in Ref. 49.

Figure 1(a) presents the calculated temperature-dependent ρ of (Ga,Mn)As, including intrinsic defects As_Ga, Mn_i(T), and Mn_i(O). The red, blue, and green stars represent the results accounting for temperature effects due to electron–phonon scattering, electron–magnon scattering, and both of them, respectively. The concentration of each impurity is summarized in Table I; this system corresponds to the right column [(Ga,Mn)As + As_Ga + Mn_i]. Figure 1(b) presents the temperature-dependent ρ results obtained by the pioneering calculations and experiments. The black points indicate the residual resistivity calculated by Turek et al.²⁴ and Ogura and Akai.²⁶ Our results reproduce these pioneering works well, indicating that the calculations are performed accurately. Furthermore, the black lines are the experimental results observed by Edmonds et al.²⁷ The solid and dashed lines indicate ρ of as-grown and annealed 6% Mn-doped GaAs samples, respectively. In the as-grown case, not only the defects assumed in this study but also Ga and As vacancies, Ga and As interstitial defects, and Ga antisite defects at As sites should be present. The annealing process reduces the crystal defects, resulting in the suppression of impurity scattering and ρ decreases. There are several experiments that observed ρ at the same Mn concentration as in this present study; the experimental ρ values vary depending on the sample quality, ranging from 10⁻³ to 10⁻² Ω cm.^27,31 As shown in Fig. 1(a), by considering both phonon and magnon effects simultaneously, we successfully reproduced behavior similar to experimental observations, where ρ has a peak at T_C. The electrical resistivity ρ increases, primarily reflecting magnon scattering up to T_C, and decreases due to the contribution of phonons above T_C. Accordingly, the obtained calculation results can be considered reasonable.

The following discussion examines the origin of this ρ behavior. Initially, our investigation centered on the effect of specific impurities on electronic structures and transport properties. Figure 2 presents the calculated temperature-dependent ρ for each type of impurity system: (a) pure (Ga,Mn)As, (b) As_Ga-doped (Ga,Mn)As [(Ga,Mn)As + As_Ga], (c) Mn_i-doped (Ga,Mn)As [(Ga,Mn)As + Mn_i], and (d) both As_Ga- and Mn_i-doped (Ga,Mn)As [(Ga,Mn)As + As_Ga + Mn_i]. Table I summarizes the corresponding impurity concentrations of Mn_Ga, As_Ga, Mn_i(T), and Mn_i(O) for each system. In our calculation, T_C is assumed to be independent of defect concentration in order to investigate systematic behavior. It is important to note that in experiments, T_C varies with Mn_i and As_Ga concentrations.^50,51 The contributions of electron–phonon scattering (illustrated by red circles) and electron–magnon scattering (indicated by blue squares) are also presented. Notably, Fig. 2(d) reflects the same results as those in Fig. 1. The electrical resistivity ρ demonstrates significantly different behaviors depending on the type of doped impurities, particularly evident in the case of magnon scattering. For (a) (Ga,Mn)As and (b) (Ga,Mn)As + As_Ga, ρ decreases as the temperature increases up to T_C. Conversely, in (c) (Ga,Mn)As + Mn_i and (d) (Ga,Mn)As + As_Ga + Mn_i, ρ displays an increase. As illustrated in both Figs. 1 and 2, the contributions of magnons to ρ remain steady at temperatures exceeding T_C. In our calculations, the LMD model was employed, which resulted in a fixed 50%:50% ratio for spin-up to spin-down local magnetic moments at temperatures exceeding T_C. The results corresponding to T_C are presented herein. It is important to note that in experimental scenarios, magnetic ordering does not completely vanish at temperatures above T_C, and ρ gradually decreases as temperature rises. From these observations, we deduce that Mn_i plays a significant role in modifying the transport properties of (Ga,Mn)As systems. We further discuss this mechanism from the perspective of electronic structures.

Figure 3 shows the impact of electron–magnon scattering on (a) the partial density of states (PDOSs) and (b) σ in (Ga,Mn)As. Figure 3(a) shows the Mn-3d  and As-4p PDOSs, represented by blue and red lines, respectively. The gradient of the color illustrates the variation in temperature, with darker lines indicating higher temperatures. The As-4p PDOS for spin-up electrons serves as the primary component around the Fermi level and is significantly influenced by increasing temperatures. As the As-4p states’ dispersion serves as a conduction channel in (Ga,Mn)As, the electrical conductivity σ also changes to reflect these electronic structures. In Fig. 3(b), the red and blue lines depict σ for spin-up and spin-down electrons, respectively. For the spin-up state, the As-4p PDOSs at the Fermi level decrease with increasing temperature, resulting in a reduction of σ. Conversely, for the spin-down state, σ increases as the PDOSs appearing around the Fermi level at finite temperatures become conduction channels. Figure 4(a) displays the PDOSs of Mn_i-doped (Ga,Mn)As. The blue, red, yellow, and green lines represent the PDOSs of the Mn_Ga-3d, As-4p, Mn_i(T)-3d, and Mn_i(O)-3d states, respectively. Upon doping donors into (Ga,Mn)As by adding Mn_i, the Fermi level shifts to a higher energy range. Consequently, the conduction channel comprising spin-up As-4p PDOSs is decreased at the Fermi level, resulting in lower conductivity (σ) than in (Ga,Mn)As even at 0 K, as depicted in Fig. 4(b). With increasing temperature, the As-4p PDOSs at the Fermi level further decrease, causing a reduction in σ. In the case of spin-down states, large Mn_i(T)-3d and Mn_i(O)-3d states are localized around the Fermi level. These localized d states become impurity scattering centers for the d resonance state, thus inhibiting conduction.^22,23 The impact of these localized states is substantial, rendering σ nearly zero irrespective of temperature, as shown in Fig. 4(b). Therefore, the presence of Mn_i significantly influences the electronic structure of (Ga,Mn)As resulting in altered transport properties.

In this analysis, we have omitted the discussion of phonon contributions because the change in σ due to phonon scattering is negligible compared to that caused by magnon scattering. The impact of phonon contributions on electronic structures and transport properties can be found in the supplementary material.

The origin of ferromagnetism in (Ga,Mn)As is still a topic of debate, with several proposed mechanisms, such as the impurity model (double exchange mechanism) and the valence band model (s/p-d exchange mechanism) (see details in Ref. 52). In the impurity model, electrons hop between magnetic ions while aligning their spins, leading to overall ferromagnetism in the system. By contrast, the valence band model posits that carriers in the valence s or p state give rise to intermediate ferromagnetism. In these models, the d states of magnetic atoms form localized states between the bandgap and the energetically deeper region in the valence band, respectively. In our calculations using the CPA and LDA methods, the electronic structure obtained was more consistent with the latter model. However, if the local environmental effect becomes significant owing to the emergence of a short-range magnetic order, such as a cluster, a different ferromagnetic mechanism might be observed. Moreover, as demonstrated in Figs. 3 and 4, the electronic structures significantly vary depending on the doped impurities. This suggests that the ferromagnetic mechanism may change depending on crystal quality and fabrication methods. Here, the single-site CPA cannot perfectly consider the localization effect due to the randomness of the impurity potentials. Furthermore, the present calculations employ the LDA for the exchange-correlation functional and do not take into account the strong Coulomb repulsion between electrons. Therefore, it should be noted that our calculations might have predicted more conductive transport properties. Readers can find further discussion of the impact of these effects on transport phenomena in Ref. 53. In Figs. 2(a)–2(d), which depict the temperature dependence of ρ, the vertical axes of the four panels have the same scale, but their ranges differ. Notably, the ρ range is higher for (b) (Ga,Mn)As + As_Ga and (c) (Ga,Mn)As + Mn_i. It can be inferred that impurity scattering, due to the presence of As_Ga or Mn_i, increases ρ. However, in the case of (d) (Ga,Mn)As + As_Ga + Mn_i, which should have the largest number of impurities, ρ is lower than that in the (b) (Ga,Mn)As + As_Ga and (c) (Ga,Mn)As + Mn_i cases. This can be explained from the perspective of magnetism. Figure 5 presents the calculated magnetic interactions at 0 K in (a) (Ga,Mn)As, (b) (Ga,Mn)As + As_Ga, (c) (Ga,Mn)As + Mn_i, and (d) (Ga,Mn)As + As_Ga + Mn_i systems. The bars indicate the magnetic interaction J₀₁ between the nearest neighbor Mn atoms: Mn_Ga–Mn_Ga (red), Mn_Ga–Mn_i(T) (orange), Mn_Ga–Mn_i(O) (yellow), Mn_i(T)–Mn_i(T) (green), Mn_i(O)–Mn_i(O) (light blue), and Mn_i(T)–Mn_i(O) (dark blue). Positive/negative values indicate ferromagnetic/antiferromagnetic interactions. It is important to note that J₀₁ depends on the Mn concentration, and because the Mn concentration varies in each system, the absolute values cannot be directly compared in this calculation. As a result, ferromagnetic interactions occur between the Mn_Ga atoms in (a) (Ga,Mn)As, while many antiferromagnetic pairs are present in (b) (Ga,Mn)As + As_Ga and (c) (Ga,Mn)As + Mn_i. In a system where antiferromagnetic interaction predominates, spin scattering becomes prominent, leading to high ρ. By contrast, in the (d) (Ga,Mn)As + As_Ga + Mn_i case, the number of antiferromagnetic Mn pairs is relatively small compared to the (b) (Ga,Mn)As + As_Ga and (c) (Ga,Mn)As + Mn_i cases, and it is predicted that ρ is not high. Therefore, it has been demonstrated that transport properties are influenced not only by the electronic structure but also by magnetic interactions.

In this study, we utilized first-principles calculations to estimate the temperature-dependent electrical resistivity of (Ga,Mn), a typical FMS. To incorporate the effects of impurity scattering, electron–phonon interaction, and electron–magnon interaction, we employed the CPA. Additionally, the transport properties were predicted using the linear response theory. As a result, we successfully reproduced the anomalous temperature-dependent behavior of the electrical resistivity around the Curie temperature, which can be attributed to the ferromagnetic–paramagnetic transition. We found that the co-doping of As_Ga and Mn_i played a crucial role in this behavior. In the pure (Ga,Mn)As system, the primary conduction channels in the ground state were identified as the spin-up As-4p states located around the Fermi level. However, as the temperature increased and magnons were excited, spin-down As-4p states emerged at the Fermi level, becoming additional conduction channels. Consequently, this led to a decrease in electrical resistivity. By contrast, in the case of Mn_i-doped (Ga,Mn)As, the conduction channel of spin-up As-4p states shifted significantly away from the Fermi level. Furthermore, Mn_i introduced a localized 3d state around the Fermi level on the spin-down side, acting as an impurity scattering center that hindered electrical conduction. Moreover, we discovered that the magnetic interaction between nearest-neighbor Mn pairs had an impact on the transport properties. In systems doped with As_Ga or Mn_i, the presence of a large number of antiferromagnetic Mn pairs resulted in prominent spin scattering, leading to high ρ. Conversely, in pure (Ga,Mn)As and in cases with co-doped As_Ga and Mn_i, the number of ferromagnetic Mn pairs increased, causing ρ to be lower than in the previous scenarios. Thus, we demonstrated that the transport calculations based on the KKR Green’s function method can effectively evaluate the anomalous transport properties of FMSs. By offering a micro-level perspective on the mechanism behind these transport properties, this approach becomes a valuable theory-driven tool for material design across various fields.

The calculated partial density of states and Bloch spectral functions are given in the supplementary material.

The authors gratefully acknowledge financial support from various organizations, including the Japan Science and Technology Agency (JST) CREST program (Grant Nos. JPMJCR1777, JPMJCR18I2, and JPMJCR17J5), the Ministry of Education, Culture, Sports, Science and Technology (MEXT) KAKENHI program (Grant Nos. 22K14285, 23H03802, and 23H03805), the “Program for Promoting Researches on the Supercomputer Fugaku” (Grant No. JPMXP1020230325), and the “Data Creation and Utilization-Type Material Research and Development Project (Digital Transformation Initiative Center for Magnetic Materials)” (Grant No. JPMXP1122715503). H.S. would like to express gratitude to H. Akai, M. Tanaka, L. D. Anh, and M. Kobayashi for their valuable discussions, which contributed to this study. Additionally, the authors would like to acknowledge Editage (www.editage.com) for providing English language editing services to improve the clarity and readability of the manuscript.

The authors have no conflicts to disclose.

Hikari Shinya: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (equal); Investigation (lead); Project administration (equal); Writing – original draft (lead). Tetsuya Fukushima: Conceptualization (supporting); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Kazunori Sato: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Shinobu Ohya: Supervision (equal); Validation (equal); Writing – review & editing (equal). Hiroshi Katayama-Yoshida: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.