Unexpected dependence of the anomalous Hall angle on the Hall conductivity in amorphous transition metal thin films

J. Karel, D. S. Bouma, C. Fuchs, S. Bennett, P. Corbae, S. B. Song, B. H. Zhang, R. Q. Wu, and F. Hellman

Phys. Rev. Materials 4, 114405 – Published 4 November 2020

Abstract

The anomalous Hall effect (AHE), and magnetic and electronic transport properties were investigated in a series of amorphous transition metal thin films— ${Fe}_{x} {Si}_{1 - x}, {Fe}_{x} {Ge}_{1 - x}, {Co}_{x} {Ge}_{1 - x}, {Co}_{x} {Si}_{1 - x}$ , and ${Fe}_{1 - y} {Co}_{y} Si$ . The experimental results are compared with density functional theory calculations of the density of Berry curvature and intrinsic anomalous Hall conductivity. In all samples, the longitudinal conductivity ( $σ_{x x}$ ), magnetization ( $M$ ), and Hall resistivity ( $ρ_{x y}$ ) increase with increasing transition metal concentration; due to the structural disorder $σ_{x x}$ is lower in all samples than a typical crystalline metal. In the systems with Fe as the transition metal (including ${Fe}_{1 - y} {Co}_{y} Si$ ), the magnetization and AHE are large and in some cases greater than the crystalline analog. In all samples, the AHE is dominated by the intrinsic mechanism, arising from a nonzero, locally derived Berry curvature. The anomalous Hall angle (AHA) $(= σ_{x y} / σ_{x x})$ is as large as 5% at low temperature. These results are compared with the AHAs reported in a broad range of crystalline and amorphous materials. Previous work has shown that in a typical crystalline ferromagnet the Hall conductivity ( $σ_{x y}$ ) and $σ_{x x}$ are correlated and are usually either both large or both small, resulting in an AHA that decreases with increasing $σ_{x y}$ . By contrast, the AHA increases linearly with increasing $σ_{x y}$ in the amorphous systems. This trend is attributed to a generally low $σ_{x x}$ , while $σ_{x y}$ varies and can be large. In the amorphous systems, $σ_{x x}$ and $σ_{x y}$ are not coupled, and there may thus exist the potential to further increase the AHA by increasing $σ_{x y}$ .

Received 12 August 2020
Accepted 13 October 2020

DOI:https://doi.org/10.1103/PhysRevMaterials.4.114405

Physics Subject Headings (PhySH)

Electrical conductivity Electrical generation of spin carriers Ferromagnetism Magnetotransport Spin current

Metallic glasses

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Karel^1,2,*, D. S. Bouma^3,4, C. Fuchs ^4,5, S. Bennett¹, P. Corbae ⁶, S. B. Song⁷, B. H. Zhang⁷, R. Q. Wu⁸, and F. Hellman ^3,4

¹Department of Materials Science and Engineering, Monash University, Clayton, VIC 3800, Australia
²ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Clayton, VIC 3800, Australia
³Materials Science Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
⁴Physics Department, University of California Berkeley, Berkeley, California 94720, USA
⁵Department of Physics and Astronomy, Julius-Maximilians-Universität, Würzburg 97074, Germany
⁶Department of Materials Science and Engineering, University of California Berkeley, Berkeley, California 94720, USA
⁷State Key Laboratory of Surface Physics, Key Laboratory of Computational Physical Sciences, and Department of Physics, Fudan University, Shanghai 200433, China
⁸Department of Physics and Astronomy, University of California, Irvine, California 92697, USA

^*julie.karel@monash.edu

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Vol. 4, Iss. 11 — November 2020

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Images

Figure 1
(a) Longitudinal resistivity ( $ρ_{x x}$ ) as a function of temperature for a series of amorphous ${Fe}_{y} {Co}_{1 - y} Si$ thin films. The compositions listed are experimentally determined from RBS and correspond from top to bottom to the nominal compositions $y = 0$ , 0.1, 0.2, 0.3. (b) Longitudinal conductivity ( $σ_{x x}$ ) at low temperature (2–4.2 K) as a function of transition metal concentration for all samples, as indicated. The line is a guide to the eye, and representative error bars are shown. Amorphous ${Fe}_{x} {Si}_{1 - x}$ and ${Fe}_{x} {Ge}_{1 - x}$ data adapted from Refs. [5, 6], respectively. Each material system in (b) is represented by a different symbol: ${Fe}_{x} {Si}_{1 - x}$ (blue circle), ${Fe}_{x} {Ge}_{1 - x}$ (purple diamond), ${Co}_{x} {Ge}_{1 - x}$ (gold left-pointing triangle), ${Co}_{x} {Si}_{1 - x}$ (red down-pointing triangle), and ${Fe}_{1 - y} {Co}_{y} Si$ (green square).
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Figure 2
Saturation magnetization ( $M_{s}$ ) at 2 K as a function of transition metal content in a series of amorphous thin films. $M_{s}$ is the value of $M$ at high $H$ (well above the coercive field) extrapolated to $H = 0$ , as shown in the inset. The amorphous ${Fe}_{x} {Si}_{1 - x}$ and ${Fe}_{x} {Ge}_{1 - x}$ data were reported in Refs. [5, 6]. The two lines, one for the Fe-based compounds and the other for the Co-based systems, are a guide to the eye. Representative error bars are shown. Each material system is represented by a different symbol: ${Fe}_{x} {Si}_{1 - x}$ (blue circle), ${Fe}_{x} {Ge}_{1 - x}$ (purple diamond), ${Co}_{x} {Ge}_{1 - x}$ (gold left-pointing triangle), ${Co}_{x} {Si}_{1 - x}$ (red down-pointing triangle) and ${Fe}_{1 - y} {Co}_{y} Si$ (green square). The inset shows a representative $M$ ( $H$ ) curve for ${Fe}_{0.44} {Co}_{0.18} {Si}_{0.38}$ with $H$ applied in the plane of the film for 2 and 300 K, as indicated.
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Figure 3
(a) A representative curve of $ρ_{x y}$ versus $H$ for an $a − {Fe}_{0.42} {Co}_{0.20} {Si}_{0.38}$ thin film at various temperatures, as indicated. The inset shows normalized $ρ_{x y}$ and out-of-plane magnetization ( $M_{z}$ ) versus $H$ for the same sample, indicating the Hall resistivity scales with the magnetization. (b) Transverse resistivity ( $ρ_{x y}$ ) versus transition metal content for amorphous thin films at low temperature (2– 4.2 K). For all samples, these data are determined by extrapolating the high-field slope of $ρ_{x y}$ versus H [as shown in (a)] to $H = 0$ . Data from amorphous ${Fe}_{x} {Si}_{1 - x}$ and ${Fe}_{x} {Ge}_{1 - x}$ are from Refs. [5, 6]. The two lines, one for the Fe-based compounds and the other for the Co-based systems, are a guide to the eye. Representative error bars are shown in (b). Each material system is represented by a different symbol: ${Fe}_{x} {Si}_{1 - x}$ (blue circle), ${Fe}_{x} {Ge}_{1 - x}$ (purple diamond), ${Co}_{x} {Ge}_{1 - x}$ (gold left-pointing triangle), ${Co}_{x} {Si}_{1 - x}$ (red down-pointing triangle), and ${Fe}_{1 - y} {Co}_{y} Si$ (green square).
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Figure 4
Hall conductivity ( $σ_{x y}$ ) normalized by $M_{z}$ and $n_{h}^{2 / 3}$ versus longitudinal conductivity ( $σ_{x x}$ ) for the amorphous samples studied in this work at low temperature (2–4.2 K). $a − {Co}_{x} {Ge}_{1 - x}$ is not shown since a meaningful carrier concentration could not be determined for $x = 0.60$ , which was the only sample in the series exhibiting an AHE. Data from amorphous ${Fe}_{x} {Si}_{1 - x}$ and ${Fe}_{x} {Ge}_{1 - x}$ are from Refs. [5, 6]. Each material system is represented by a different symbol: ${Fe}_{x} {Si}_{1 - x}$ (blue circles), ${Fe}_{x} {Ge}_{1 - x}$ (purple diamonds), ${Co}_{x} {Si}_{1 - x}$ (red inverted triangles), and ${Fe}_{1 - y} {Co}_{y} Si$ (green squares). Representative error bars are shown on the ${Fe}_{1 - y} {Co}_{y} Si$ data. The inset shows the composition dependence of the carrier concentration for $a − {Fe}_{x} {Si}_{1 - x}$ and $a − {Fe}_{x} {Ge}_{1 - x}$ . For samples with $x > 0.60, δ ρ_{x y} / δ H$ is too small to measure. Thus, for $a − {Co}_{x} {Si}_{1 - x}, a − {Fe}_{1 - y} {Co}_{y} Si$ , and $a − {Fe}_{x} {Si}_{1 - x}$ ( $x > 0.60$ ), the carrier concentration was approximated based on the composition dependence shown in the inset.
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Figure 5
AHA versus Hall conductivity for (a) crystalline materials and (b) amorphous materials; and (c) AHA versus longitudinal conductivity for the amorphous materials in this work. The data and gray dashed line (guide to the eye) in (a) are based on Ref. [4]. The solid line in (b) is a linear fit to the data. In (b), closed symbols are data from this work (at 2–4.2 K), and open symbols are literature values (at 50–77 K). The gray lines in (c) are a guide to the eye. Data points were taken from the following references: ${Co}_{3} {Sn}_{2} S_{2}$ [4], MnGa [33], $L 1_{0} − FePt$ [34], TbCo [35], SmFe [36], GaMnAs [37], $SrRu O_{3}$ [38], Fe [39], Gd [39], LaSrCoO [39], CuZnCrSe [39], ${Mn}_{3} Ge$ [2], ${Mn}_{3} Sn$ [3], ${Mn}_{5} {Ge}_{3}$ [40], ${Fe}_{3} {Sn}_{2}$ [14], ${Mn}_{2} {Ru}_{x} Ga$ [41], ${Fe}_{0.28} Ta S_{2}$ [42], ${Co}_{2} FeSi$ [43], ${Co}_{2} FeAl$ [43], MnSi [28], $a$ -FePC [44], $a − YF e_{2.5}$ [44], $a − YC o_{2}$ [44], $a − {Co}_{0.5} {Au}_{0.5}$ [44], $a$ -CoFeB [45], $a$ -Co-Gd-Mo [44], $a$ -Fe-Gd [44], $a$ -Co-Gd [44], $a$ -Gd-Au [44], $a$ -Co-Gd-Au [44], and GdPtBi [14].
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