# Impact of Bismuth Incorporation into (Ga,Mn)As Dilute Ferromagnetic Semiconductor on Its Magnetic Properties and Magnetoresistance

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{C}. Such negative MR at around T

_{C}has usually been understood as the reduction of spin-disorder scattering of charge carriers caused by the ordering of localized Mn spins in an external magnetic field, a mechanism well known in ferromagnetic metals [21]. Instead, at low temperatures, when the Mn spins in (Ga,Mn)As are fully ferromagnetically ordered, the field-induced destruction of quantum interference contribution to the resistivity caused by the effect of weak localization has been proposed to account for the negative MR [11,22,23,24].

## 2. Materials and Methods

_{xx}and Hall resistance R

_{xy}of the Hall-bars have been measured, using a dc ±10 μA sensing current, in a helium cryostat with superconducting electromagnet at temperatures down to 1.5 K and perpendicular magnetic field up to ±13.5 T.

^{2}specimens by means of mechanical polishing. Such a metallic contamination exerts a magnetic moment m of a magnitude that may well exceed that of 10 nm thin (Ga,Mn)As and exhibits a ferromagnetic-like magnetization curve m(H) [28]. On the other hand, the superconductivity of In would mar and obscure the response of (Ga,Mn)As below some 4 K, rendering the low temperature magnetic studies impossible. All the magnetic measurements have been carried out according to the well-established protocols to eliminate experimental artifacts [29].

## 3. Results and Discussion

#### 3.1. Electrical Characterization

_{C}. They result from the spin-disorder scattering of current carriers by magnetic fluctuations while entering the paramagnetic-to-ferromagnetic phase transition [30]. These maxima at about 83 K and 92 K for the (Ga,Mn)(Bi,As) and (Ga,Mn)As layers, respectively, correspond pretty well to the T

_{C}values determined from our SQUID magnetometry results, shown in the next section. Novák et al. [31] have shown that in (Ga,Mn)As DFS layers with rather high Curie temperatures, their values can be better estimated from maxima of temperature derivatives of resistance vs. temperature dependences, which are also shown in Figure 1 for the presently investigated layers. However, the latter maxima at about 69 K and 71 K, respectively, evidently underestimate the T

_{C}values. Such behavior is characteristic of the (Ga,Mn)As layers with Curie temperatures of about 100 K and below [17,32]. The increase in the Hall bar resistances, observed while lowering the temperature below about 30 K, indicates that the WL correction to the Drude–Boltzmann conductivity may become dominating at low temperatures in both the investigated layers.

_{⊥}measured for the Hall bars of two investigated layers at low temperatures, 1.6 K and 4.2 K, is presented in Figure 2. For magnetic materials the Hall resistivity can be described by the relation [12]:

_{xy}= R

_{H}B

_{⊥}+ R

_{s}M

_{⊥},

_{⊥}and dominates at low magnetic fields. Similar hole concentrations p ≅ 2 × 10

^{20}cm

^{−3}have been determined for both the layers from the high-field (above about 1 T) results at T = 4.2 K, presented in Figure 2, where the variation of anomalous Hall effect with a magnetic field is sufficiently small.

#### 3.2. Magnetic Properties

_{0}H = 0.1 T to the base temperature T = 2 K, where the field is quenched and the thermoremnant magnetization, TRM, for a given orientation is collected on warming. The warming continues until above the magnetic moment vanishes. The temperature when TRM drops to zero marks the Curie temperature, T

_{C}, for the given sample. The procedure is repeated for [100], [−110] and [110] in-plane orientations, yielding the same values of T

_{C}= 96 K for the (Ga,Mn)As layer and T

_{C}= 83 K for the (Ga,Bi)(Mn,As) one, regardless of the layer orientation. This visibly lower magnitude of T

_{C}for (Ga,Bi)(Mn,As) is the first direct evidence of the influence of the enhanced SOC strength on the magnetism of (Ga,Mn)As [33].

_{C}. At about 15 K below T

_{C}, a clear hump develops on TRM [110] and TRM along [100] starts to exhibit the same magnitudes as that of [−110]. This is the signature of the beginning of the spin reorientation transition (SRT) process of the π/4 in-plane rotation of the [−110] uniaxial anisotropy [43]. This remarkable feature of (Ga,Mn)As has been observed in layers with the hole concentration exceeding p ≅ 6 × 10

^{20}cm

^{−3}and with T

_{C}in excess of about 120 K. Since in our layers p ≅ 2 × 10

^{20}cm

^{−3}we observe only the precursory behavior of this SRT (the hump mentioned above) and its further development is hampered by relatively low magnitudes of T

_{C}. That is why instead of exchanging their intensities as observed previously in layers with much higher Curie temperatures, all three magnitudes of TRM are quenched to zero much earlier at T

_{C}. Accordingly, the quenching of TRM is sharper in (Ga,Mn)(Bi,As), which T

_{C}is smaller than that of (Ga,Mn)As.

_{0}H

_{A}≅ 0.4 T, a value which exceeds by far the shape anisotropy in this dilute ferromagnetic material μ

_{0}H

_{D}= μ

_{0}M

_{S}≅ 0.04 T. This discrepancy in favor of exchange effects is the direct manifestation that the magnetism in DFS, as in (Ga,Mn)As and its derivatives, is predominantly determined by the anisotropy of the carrier-mediated exchange interaction reflecting the anisotropic properties of the top of the valence band [33]. On the other hand, the magnetization hysteresis loops recorded at the magnetic field along the main in-plane crystallographic directions, shown in Figure 5, evidence the same magneto-crystalline anisotropy at low temperatures for both the investigated layers. However, the enhanced SOC strength in the Bi-contained layer results in a distinct increase in the layer coercive fields by a factor of about 1.5 for all three main in-plane crystallographic directions.

#### 3.3. Magnetoresistance and Weak Localization

_{⊥}| < 0.4 T, the dependences display a positive magnetoresistance. This positive MR is caused by the reorientation of the layer magnetization vector from its original in-plane direction at zero magnetic field to the perpendicular one at B

_{⊥}corresponding to the perpendicular anisotropy field, which is just 0.4 T for both the studied layers, as determined from the SQUID magnetometry results shown in the previous section. We interpret this positive MR as resulting from the effect of anisotropic magnetoresistance (AMR) occurring in conducting ferromagnetic materials, which depends on the angle between the magnetization vector and the electric current direction and reaches the maximum value at the magnetization vector perpendicular to the current, cf. [17,37,46].

_{⊥}= 13 T. It reflects a significant role of both the chemical disorder and SOC that are expected to be introduced by Bi incorporation.

_{i}are defined (assuming they are independent of the spin sense) by the scattering times τ

_{i}as B

_{i}= ħ/(4eDτ

_{i}), where D is the diffusion coefficient, and the index i stands for the following scattering processes: p—elastic, φ—inelastic, so—spin–orbit. The reason for applying the 2D correction for the investigated here 3D thin layers is that the phase coherence length, L

_{φ}= (Dτ

_{φ})

^{1/2}, at low temperatures is expected to be of the order of 100 nm [47,48], which is much larger than the layer thickness, d = 10 nm. Additionally, the magnetic length, L

_{B}= (ħ/(eB))

^{1/2}, while comparing with d, implies the dimensional crossover from 3D to 2D quantum correction case, as L

_{B}> d for B < 6.6 T, but even at the maximum field used, 13.5 T, L

_{B}is still of the order of d—about 7 nm. Taking into account Equation (2), we examine the following formula to fit the total 3D resistivity in a perpendicular magnetic field:

_{c}is the semi-classical Boltzmann resistivity, and F

_{σ}is the scaling factor introduced by us in order to adapt the 2D WL correction to the 3D samples. Equation (3) is derived from the inversion of conductivity tensor for weak fields approximation, i.e., µB

_{⊥}<< 1 (µ denotes mobility of free charge carriers—holes in our case), which is fulfilled in the whole range of fields used.

_{⊥}| > 0.4 T, where the negative MR appears, the fitted curves in Figure 7 are plotted in the whole range of fields to show that the theory accounts for the negative MR around B

_{⊥}= 0 as well. The fitting procedure includes three fitting parameters: ρ

_{c}, F

_{σ}, and the spin–orbit scattering length L

_{so}= (Dτ

_{so})

^{1/2}. Further, there are four fixed-value parameters: the technological layer thickness d = 10 nm, reduced effective mass of holes, set to be 0.7 [23], the hole concentration, p, set to be 2 × 10

^{20}cm

^{−3}, determined from the Hall effect results, and L

_{φ}(T). We set L

_{φ}as fixed, because when fitting both L

_{φ}and L

_{so}, the fitting procedure is not convergent. We set L

_{φ}(T = 1.6 K) = 100 nm based on our previous estimation for (Ga,Mn)As nanoconstriction [47], and consequently L

_{φ}(T = 4.2 K) is taken equal to 50 nm according to the L

_{φ}(T)~T

^{−3/4}temperature dependence that is expected for the 3D disordered systems, cf. [49].

_{c}are lower than the corresponding experimental ρ

_{xx}(B = 0) ones that are due to the quantum correction Δσ

_{xx}at the B = 0 limit, which takes the non-zero negative value. Note that both, Δσ

_{xx}and ρ

_{c}, are related to each other by the momentum relaxation time τ

_{p}. The obtained ρ

_{c}values correctly reproduce the increase of ρ

_{xx}(B = 0) with decreasing temperature and their larger values for the Bi-contained layer. On the other hand, the fitted spin–orbit scattering length, L

_{so}, reveals distinctly lower values for the Bi-contained layer, as seen in Table 1, in agreement with the enhanced SOC strength.

_{so}= ∞). The visible difference between the corresponding curves implies that the spin–orbit interaction leads to antilocalization at zero magnetic field and to a positive contribution to the total MR at low field range. This positive contribution (WAL phenomenon) manifests itself more significantly for the Bi-contained layer and while lowering temperature. However, the effect is relatively weak and is dominated and obscured by the negative contribution of WL.

_{c}values and measured hole concentrations in the layers. The product of the Fermi wave vector, k

_{F}, and mean free path, l, that is a measure of disorder, is seen to be smaller than one, indicating the charge transport regime in studied samples is actually a border between the WL and Anderson–Mott localization. We assign this conclusion as a reason for not perfect agreement between the fitted and experimental curves (compare curvatures in Figure 7), and partly for the fitted values of factor F

_{σ}(see Table 1), which are smaller than one. It turns out that fitting the experimental MR needs about 10 times smaller magnitude (F

_{σ}~ 0.1) of WL correction than that expected by the 2D theory.

^{2}/(Vs) is typical for (Ga,Mn)As layers and results from heavy doping with Mn ions. Since the values in Table 2 are determined based on the classical contribution (ρ

_{c}) to the total resistivity, thus the obtained mobility values are expected to reflect the classical momentum scattering. It is clearly seen that Bi incorporation into the layer reduces the hole mobility, confirming the increased disorder. On the other hand, the influence of low temperatures on the mobility seems to be rather weak in this classical approach.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Temperature dependences of longitudinal resistance for the Hall bars of (Ga,Mn)(Bi,As) and (Ga,Mn)As layers. Right inset presents temperature derivatives of the resistance, in the temperature range around their maxima, where their intersections with zero value correspond to the maxima in the main figure. Microscopic image of the Hall bar and its geometry is shown in the left inset. Here, the darker contrast corresponds to non-conducting areas etched to the substrate.

**Figure 2.**Hall resistance measured for the Hall bars of (Ga,Mn)(Bi,As) and (Ga,Mn)As layers, at temperatures of 1.6 K and 4.2 K, as a function of an external magnetic field perpendicular to the layer plane.

**Figure 3.**Temperature dependent magnetization M of (

**a**) (Ga,Mn)As and (

**b**) (Ga,Mn)(Bi,As) layers. On both panels open diamonds represent M collected during field cooling (FC) in μ

_{0}H = 0.1 T, with magnetic field H applied along the [100] in-plane direction. Solid symbols mark the thermoremnant magnetization, TRM, measured upon warming the samples in the absence of H, right after FC, along the in-plane crystallographic directions: [100]—red diamonds, [−110]—green bullets, and [110]—blue squares. The magnitudes of the Curie temperatures, T

_{C}, are indicated by arrows.

**Figure 4.**Magnetic field H dependence of the in-plane magnetization, represented here by the [100] one (red symbols), and of the perpendicular one (black symbols) measured at temperature T = 2 K for (

**a**) (Ga,Mn)As and (

**b**) (Ga,Mn)(Bi,As) layers studied here.

**Figure 5.**In-plane magnetic hystereses of (

**a**) (Ga,Mn)As and (

**b**) (Ga,Mn)(As,Bi) layers at temperature T = 4 K. Magnetic field H has been applied along three main in-plane directions, as described in the panels.

**Figure 6.**Relative longitudinal resistance measured for the Hall bars of (Ga,Mn)(Bi,As) and (Ga,Mn)As layers at temperatures of 1.6 K and 4.2 K, while sweeping an external magnetic field perpendicular to the layer plane in opposite directions.

**Figure 7.**Longitudinal resistivity vs. perpendicular magnetic field measured for the Hall bars of (Ga,Mn)(Bi,As) and (Ga,Mn)As layers at temperatures of 1.6 K and 4.2 K (open circles), compared with fitting curves (red lines) within the weak localization theory for the 2D ferromagnet (see text for details). The visible set of experimental points, that are chosen to fit, is limited to the |B

_{⊥}| > 0.4 T range. The results are shown in the absolute resistivity scale (

**a**) and their changes with respect to their values at the maximum field, Δρ

_{xx}, (vertically offset for clarity) (

**b**) to compare the changes.

**Figure 8.**The fitted curves, the same as in Figure 7b, (red) compared with the curves calculated by Equation (3) (blue) with the spin–orbit interaction switched off, i.e., L

_{so}set as infinite (B

_{so}= 0), and with the rest of parameters unchanged comparing the red curves. The curves are vertically offset for clarity.

**Table 1.**The best fit values of fitting parameters: ρ

_{c}, F

_{σ}, and L

_{so}, corresponding to the fitted curves in Figure 7.

Layer | T (K) | ρ_{c} (10^{−2} Ωcm) | F_{σ} | L_{so} (nm) |
---|---|---|---|---|

(Ga,Mn)(Bi,As) | 1.6 | 3.0 | 0.13 | 44 |

4.2 | 2.5 | 0.15 | 70 | |

(Ga,Mn)As | 1.6 | 2.1 | 0.12 | 52 |

4.2 | 1.8 | 0.11 | 140 |

**Table 2.**Hole mobilities, µ, and disorder parameter, k

_{F}l, derived from the fitted ρ

_{c}and measured p values.

Layer | T (K) | µ (cm^{2}/(Vs)) | k_{F}l |
---|---|---|---|

(Ga,Mn)(Bi,As) | 1.6 | 1.0 | 0.23 |

4.2 | 1.2 | 0.27 | |

(Ga,Mn)As | 1.6 | 1.5 | 0.32 |

4.2 | 1.7 | 0.37 |

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## Share and Cite

**MDPI and ACS Style**

Andrearczyk, T.; Levchenko, K.; Sadowski, J.; Gas, K.; Avdonin, A.; Wróbel, J.; Figielski, T.; Sawicki, M.; Wosinski, T.
Impact of Bismuth Incorporation into (Ga,Mn)As Dilute Ferromagnetic Semiconductor on Its Magnetic Properties and Magnetoresistance. *Materials* **2023**, *16*, 788.
https://doi.org/10.3390/ma16020788

**AMA Style**

Andrearczyk T, Levchenko K, Sadowski J, Gas K, Avdonin A, Wróbel J, Figielski T, Sawicki M, Wosinski T.
Impact of Bismuth Incorporation into (Ga,Mn)As Dilute Ferromagnetic Semiconductor on Its Magnetic Properties and Magnetoresistance. *Materials*. 2023; 16(2):788.
https://doi.org/10.3390/ma16020788

**Chicago/Turabian Style**

Andrearczyk, Tomasz, Khrystyna Levchenko, Janusz Sadowski, Katarzyna Gas, Andrei Avdonin, Jerzy Wróbel, Tadeusz Figielski, Maciej Sawicki, and Tadeusz Wosinski.
2023. "Impact of Bismuth Incorporation into (Ga,Mn)As Dilute Ferromagnetic Semiconductor on Its Magnetic Properties and Magnetoresistance" *Materials* 16, no. 2: 788.
https://doi.org/10.3390/ma16020788