Spin-polarized tunneling from a Kondo insulator: equilibrium surface magnetism versus magneto-electric response in SmB$_6$

Technical Summary: Spin-polarized tunneling from a Kondo insulator: equilibrium surface magnetism versus magneto-electric response in SmB$_6$

Problem Statement
Recent spin-polarized scanning tunneling microscopy (SP-STM) experiments on SmB$_6$ tips coupled to a bi-collinear antiferromagnetic Fe$_{1+x}$Te substrate revealed a surprising magnetic contrast that reverses sign when the bias voltage polarity is reversed. This contrast vanishes above approximately 10 K, a temperature well above the bulk Curie temperature of surface ferromagnetism. Two competing interpretations have been proposed for this bias reversal:

1. Magneto-electric (ME) mechanism: The electric field of the tunnel junction induces a magnetic moment in the tip via a surface magneto-electric effect, where the induced moment changes sign with the bias polarity (Banerjee et al.).
2. Equilibrium mechanism: The tip surface possesses a time-reversal symmetry breaking (TRSB) equilibrium order that gaps the surface Dirac cones. This equilibrium order creates a spin-resolved spectral function that naturally changes sign near zero bias, causing the contrast reversal without requiring a field-induced moment (Dzero and Andersen).

The central problem addressed is determining which mechanism dominates the magnetic contrast within the experimental energy window ($|eV| \lesssim 10$ meV) and whether the field-induced contribution is significant enough to explain the observed phenomena.

Methodology
The authors employ a microscopic theoretical framework based on the eight-orbital model of SmB$_6$ with a (001) termination, previously developed to describe the TRSB surface order. The methodology involves:

• Hamiltonian Construction: A mean-field Hamiltonian is constructed for a semi-infinite slab of $N$ layers, including terms for the surface order parameters (pairing $\Delta_j$ and magnetic order $h_j$).
• Perturbation Theory: The tunneling junction electric field is treated as a perturbation. The external Hamiltonian includes electrostatic gating ($e\phi_j$) and an on-site $d$-$f$ electric dipole coupling ($\lambda_j$).
• Linear Response Calculation: The authors compute the change in the spin-resolved spectral function $\rho_{spin}(\omega, V)$ to linear order in the bias voltage $V$. This involves calculating the field-induced corrections to the order parameters ($\Delta^{(1)}_j, h^{(1)}_j$) using self-consistency equations and susceptibility functions ($\chi_{\alpha\beta}$).
• Decomposition of Conductance: The differential tunneling conductance is decomposed into an equilibrium part ($g_s(V)$) derived from the static order and a field-induced part ($G(V) + eV X(eV)$) arising from the bias-dependent electric field.
• Numerical Evaluation: The authors numerically evaluate the field-induced contribution $X_\Delta(\omega)$ (from the order parameter channel) and compare it against the equilibrium spectrum $\rho_s(\omega)$ at temperatures approaching the critical temperature $T_c \approx 9.8$ K.

Key Contributions and Results

1. Indistinguishability at Fixed Distance: The authors demonstrate that a tunneling spectrum taken at a fixed tip-substrate distance cannot separate the equilibrium contribution from the field-induced contribution. They show that the ratio of the zero-bias intercept to the response extracted via the inversion procedure used in previous work equals $-2V^*$ (where $V^*$ is the bias at which contrast changes sign), regardless of the relative strength of the two mechanisms. Thus, this ratio measures the position of the sign change, not the strength of the magneto-electric effect.
2. Dominance of Equilibrium Mechanism: Within the experimental energy window ($|eV| \lesssim 10$ meV), the field-induced contribution is negligible compared to the equilibrium contribution.
   • Dipole and Gating Channels: The direct coupling of the electric field to the single-particle spectrum (via $d$-$f$ dipole and electrostatic gating) changes the spectral function by at most a few percent per millivolt.
   • Order Parameter Channel: The field-induced change of the order parameter ($X_\Delta$) is enhanced near $T_c$ but remains three to four orders of magnitude smaller than the equilibrium spectrum per millivolt of bias. Even at $T = 8.5$ K (close to $T_c$), the field-induced correction to the magnetic contrast does not exceed 1% of the equilibrium part.
3. Bias Dependence: The bias-dependent field-induced contribution becomes comparable to the equilibrium contribution only at biases of order 100 meV or higher, far outside the experimental window.
4. Distance Dependence: The ratio of the field-induced to the equilibrium contribution is inversely proportional to the tip-substrate distance $d$. Consequently, the bias $V^*$ at which the contrast changes sign should be independent of the set-point current (and thus distance) if the equilibrium mechanism dominates. A measurable shift in $V^*$ with set-point current would indicate a dominant field-induced moment.

Significance and Claims
The paper claims that the observed bias reversal of magnetic contrast in SmB$_6$ tips is primarily determined by the equilibrium spin-resolved spectrum of the apex, which arises from a time-reversal symmetry breaking surface order. The authors conclude that:

• The field-induced magneto-electric response is too small (at most a few percent) to account for the observed contrast reversal within the experimental energy range.
• Previous interpretations attributing the reversal solely to a field-induced moment are insufficient because they ignore the strong energy dependence of the equilibrium spectral function.
• The observation of magnetic contrast implies the existence of TRSB on the SmB$_6$ surface, which gaps the Dirac cones.
• A direct experimental test to distinguish the mechanisms involves measuring the bias $V^*$ at various set-point currents; if $V^*$ is constant, the equilibrium mechanism is confirmed.

The authors emphasize that their conclusions rely on the microscopic calculation of the field-induced corrections and do not require the Fermi level to lie strictly inside the gap of a surface Dirac cone, provided the surface remains metallic due to the specific band structure of the ordered state.