Linear magneto-conductivity as a DC probe of time-reversal symmetry breaking

Abstract

Several optical experiments have shown that in magnetic materials the principal axes of response tensors can rotate in a magnetic field. Here we offer a microscopic explanation of this effect, and propose a closely related DC transport phenomenon -- an off-diagonal \emph{symmetric} conductivity linear in a magnetic field, which we refer to as linear magneto-conductivity (LMC). Although LMC has the same functional dependence on magnetic field as the Hall effect, its origin is fundamentally different: LMC requires time-reversal symmetry to be broken even before a magnetic field is applied, and is therefore a sensitive probe of magnetism. We demonstrate LMC in three different ways: via a tight-binding toy model, density functional theory calculations on MnPSe$_3$, and a semiclassical calculation. The third approach additionally identifies two distinct mechanisms yielding LMC: momentum-dependent band magnetization and Berry curvature. Finally, we propose an experimental geometry suitable for detecting LMC, and demonstrate its applicability using Landauer-Büttiker simulations. Our results emphasize the importance of measuring the full conductivity tensor in magnetic materials, and introduce LMC as a new transport probe of symmetry.

Figures (7)

• Figure 1: (a) The model considers a one electron $f\rightarrow d$ transition, with a degenerate initial state manifold and a final state manifold split by the three terms in Eq. \ref{['eq:Hamiltonian']}: $\Delta_{CF}$ separates the $d$ manifold into states of $e_g$ and $t_{2g}$ symmetry. (b) The ratio of off-diagonal symmetric conductivity to the average diagonal conductivity as a function of $H_z/H_y$, calculated for the different final states in Fig. \ref{['fig:Model']}a. (c) The orbital wave functions corresponding to $\left<L_{\alpha}^{\text{eff}}\right>=1$, for $\alpha=y, z, (z+y)/\sqrt{2}$.
• Figure 2: Rotation of Fermi surfaces and change of conductivity in a tight binding model as a function of magnetization. (a) Top view of the triangular lattice of magnetic ions with moment ${\bf M}=(0,M_y,M_z)$ in the $y-z$ plane. The panels (b-d) study quantities when ${\bf M}$ points along $-45^\circ$, $0^\circ$, or $+45^\circ$ from the $y$ axis, represented by red, grey, and blue arrows. (b) The band dispersion along high symmetry path in the Brillouin zone (BZ) for the three orientations of ${\bf M}$. $t$ is the overall hopping energy scale and $E_F$ is the Fermi energy. (c) FS at the Fermi energy $E = E_F$ for the three orientations of ${\bf M}$. As ${\bf M}$ is oriented from $-45^\circ$ to $+45^\circ$, the FS rotates accordingly in a clockwise way. (d) The conductivity angle $2\sigma^s_{xy}/(\sigma_{xx}+\sigma_{yy})$ as a function of $M_z/M_y$. For (c) and (d) we assumed the filling at $E=E_F$.
• Figure 3: (a) L-shape device geometry. The scattering region (leads) is denoted by the black (red) lattice. Leads 1 and 10 are the source and drain electrodes, while leads 2-8 are voltage probes. (b) Conductivity angle for the symmetric (red) and Hall (green) conductivity as a function of $M_z/M_y$. The calculations are done based on the tight binding model (Eq. \ref{['eq:TBHam']}, Fig. \ref{['fig:TightBinding']}), with a Fermi level of $-1.95$. Solid line is the average value performed over 5 disorder realizations (shown in dashed, paler color).
• Figure 4: (a) Top view and side view of the MnPSe$_3$ crystal structure. The red and blue arrows indicate the two magnetization directions used in the calculations. (b) Conductivity angle as a function of energy for magnetization orientation along $\pm45\degree$ with respect to the $z$ axis in the $yz$ plane.
• Figure S1: Longitudinal conductivity extracted from the two-terminal conductance for two strengths of Anderson disorder and for a device oriented along $x$ (black) and along $y$ (red). 10 disorder averages have been carrid out. The diffusive regime exists for lengths in which the conductivity remains constant. The vertical dashed (dotted) line denotes the chosen inter-lead distance (total length) of the L-shaped device. Importantly, lengths within these two vertical lines fall within diffusive transport.