Impact of random impurities on the anomalous Hall effect in chiral superconductors

Hao-Tian Liu, Weipeng Chen, and Wen Huang

Phys. Rev. B 107, 224517 – Published 28 June 2023

Abstract

The anomalous Hall effect and the closely related polar Kerr effect are among the most direct evidence of chiral Cooper pairing in some superconductors. While it has been known that disorder or multiband pairing is typically needed for these effects to manifest, there is a lack of direct real-space investigation with regard to how disorder impacts the Hall response in both single-band and multiband chiral superconductors. On the basis of chiral superconducting models often adopted for ${Sr}_{2} {RuO}_{4}$ , we study in this work the anomalous Hall effect in the presence of random nonmagnetic impurities on real-space lattices. The single-band chiral $p$ -wave $(p_{x} + i p_{y})$ calculation qualitatively reproduces the Hall conductivity obtained in previous skew-scattering-type diagrammatic analyses, along with some quantitative difference originating primarily from contributions involving impurity-induced in-gap states. The non- $p$ -wave chiral states, such as $d_{x^{2} - y^{2}} + i d_{x y}$ , generically exhibit finite Hall response in the presence of random impurities, in contrast to a conclusion drawn from the aforementioned diagrammatic study. In particular, while pointlike impurities appear to induce minuscule Hall conductivity in non-self-consistent calculations, self-consistency and finite-range impurity potentials can both lead to substantial Hall conductivity. However, the intrinsic Hall conductivity in multiband chiral superconductors, which is related to interband transitions, decreases parametrically as disorder suppresses the superconducting order parameter. In addition, we check that random impurities do not induce anomalous Hall effect in nonchiral but time-reversal symmetry breaking superconducting states the likes of $s + i d_{x^{2} - y^{2}}$ and $d_{x^{2} - y^{2}} + i g_{x y (x^{2} - y^{2})}$ . We briefly remark on the implications of our results for Kerr effect measurements.

Received 12 October 2022
Revised 16 June 2023
Accepted 16 June 2023

DOI:https://doi.org/10.1103/PhysRevB.107.224517

Physics Subject Headings (PhySH)

Anomalous Hall effect Impurities in superconductors Magneto-optical Kerr effect

Unconventional superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hao-Tian Liu , Weipeng Chen , and Wen Huang ^*

Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China; International Quantum Academy, Shenzhen 518048, China; and Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China

^*huangw3@sustech.edu.cn

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Vol. 107, Iss. 22 — 1 June 2023

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Images

Figure 1
The zero-temperature Hall conductivity and the respective density of states for various single-band superconductors in 2D square lattices with randomly distributed pointlike impurities with concentration $n_{imp} = 1.56 %$ . Through our calculations, $t = μ = 1$ , $u = 1000$ and $η = 10^{- 3}$ . In (a) and (b), $Δ_{0}$ is the quasiparticle excitation gap extracted respectively from the clean-limit DoS data in (c) and (d), with $Δ_{0} = 0.17 t$ and $Δ_{0} = 0.18 t$ respectively. The small upturn of $Im (σ_{H})$ near zero frequency in (a) is a numerical artifact that can be eliminated by choosing sufficiently small $η$ . The inset of (a) displays the imaginary part of the conductivity of the $p_{x} + i p_{y}$ state with different impurity concentrations. The set of data in (b) was obtained from an average over 90 random samples. The error bars reflect the scatter of data of different impurity configurations. For clarity, the error bars will be dropped in other figures.
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Figure 2
The imaginary part of the zero-temperature Hall conductivity of (a) $p_{x} + i p_{y}$ and (b) $d_{x y} + i d_{x^{2} - y^{2}}$ states at $n_{i} = 1.56 %$ and with different impurity potential profiles as determined by the potential $u^{'}$ assigned to the four nearest neighbors of each impurity site.
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Figure 3
The imaginary part of the zero-temperature Hall conductivity of (a) $p_{x} + i p_{y}$ and (b) $d_{x y} + i d_{x^{2} - y^{2}}$ states for pointlike impurities at the impurity concentration $n_{imp} ≃ 1.5 %$ after self-consistency. Parameters are the same as in Fig. 1. The dash curves are the same as the $Im [σ_{H}]$ in Figs. 1 and 1. The abbreviations in the legend NSC and SC denote non-self-consistent calculation and self-consistent calculation, respectively. The inset in panel (a) shows the drop of the spatial average of the pairing order parameter as a function of the impurity concentration. The error bars in the self-consistent data in panel (b) are to demonstrate the robustness of the lineshape.
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Figure 4
(a) The imaginary part of Hall conductivity in the two-band model. The results for finite impurity concentration scenarios are obtained after order parameter self-consistency. The inset shows the drop of the average order parameter potential with varying impurity concentration. (b) Zoom-in view of part of the high frequency Hall conductivity. The solid curves plot the clean-limit Hall conductivity evaluated using the pairing amplitude $Δ = Δ_{0}$ as indicated for each curve. The filled circles and squares depicted the conductivity of disordered samples in non-self-consistent (NSC) calculations with uniform input pairing amplitude $Δ_{0} = 0.2 t$ . The open circles and squares plot the conductivity obtained in self-consistent (SC) calculations, which use an interaction that produces a pairing amplitude of $Δ_{0} = 0.2 t$ in the clean limit. These calculations employed the parameter set $(t^{'}, μ) = (0.5, 1) t$ .
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Figure 5
The real part of longitudinal conductivity $σ_{x x}$ for the two-band $p_{x} + i p_{y}$ model in clean and disordered systems. The results for disordered systems are obtained from the same models as in Fig. 4. The clean-limit conductivity (blue dashed curve) was evaluated using the momentum-space formulation provided in the text. The Drude peak in $σ_{x x}$ is suppressed by superconductivity. Note that the oscillatory high-frequency conductivity is a numerical artifact of our finite-size modeling. It is more severe in smaller system size calculations and at lower impurity concentrations, where the energy distribution of the limited number of high-energy quasiparticle states is not yet sufficiently spread out in statistical sense.
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Figure 6
Comparison of the Hall conductivity obtained from calculations with respective system size of $40 \times 40$ and $80 \times 80$ in the presence of the same concentration of random pointlike impurities, for the single-band chiral $d$ -wave (upper panel) and $s + i d_{x^{2} - y^{2}}$ (lower panel) states. Except for the different system size, all other parameters are the same as in Fig. 1 for the chiral $d$ -wave model, and $Δ_{0} = 0.09 t$ for the $s + i d_{x^{2} - y^{2}}$ model. All data sets are obtained by averaging over 90 different impurity samples.
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