Mapping Spin Interactions from Conductance Peak Splitting in Coulomb Blockade

TL;DR

This work addresses how to extract complete spin-Hamiltonian information from electronic transport in a Coulomb-blockaded quantum dot coupled to a spin dimer. By modeling a three-terminal device and deriving a generalized master equation that includes both populations and coherences, the authors relate differential conductance peak structure to four key spin parameters: magnetic anisotropy $D$, inter-spin exchange $J_{23}$, dot–spin exchange $J_{1i}$, and $g$-factors. They demonstrate that field-dependent energy shifts and bias–gate dependences yield distinct peak patterns, enabling a staged protocol to map each parameter step-by-step, even in the presence of decoherence. The approach provides a practical route to characterize nanoscale magnetic systems using only differential conductance measurements, with potential implications for molecular magnets and quantum-dot qubits.

Abstract

We investigate the transport properties of a quantum dot coupled to leads interacting with a multi-spin system using the generalized master equation within the Coulomb blockade regime. We find that if two states for each scattering region electron manifold are included, several signatures of the interacting spin system appear in steady-state transport properties. We provide a theoretical mapping of differential conductance peak signatures and all spin Hamiltonian parameters related to the inclusion of excited state transitions between uncharged and charged electron manifolds. Our predictions describe a scheme of only using a quantum dot and differential conductance to measure magnetic anisotropy, inter-spin exchange coupling, exchange coupling between the spin system and itinerant electron, and applied magnetic field response.

Figures (11)

• Figure 1: Schematic of the system consisting of a central scattering region, containing an $S_{2,3}=1$ spin dimer complex interacting through an exchange interaction $J_{23}$, coupled to polarized leads at temperature $T$. The central region’s eigenenergy levels are tuned via gate voltage $V_{G}$ so that the charged ground state energy $E_{N+1}^{(0)}$ with $N+1$ electrons is aligned with the unbiased leads, i.e., $E_{N+1}^{(0)}=\mu_{L}=\mu_{R}=0$ eV. Applying a symmetric bias voltage as shown enables transport of a single electron (particle 1) through the $N + 1$ electron states. Once the electron has transported into the central region, and prior to leaving the central region, additional exchange interactions $J_{1i}$ couples the electron’s spin to the dimer.
• Figure 2: First four energy levels of the $N$ (black) and $N+1$ (blue) central region electron manifolds. Energy differences $\Delta E^{(0,0)}_{N+1,N}$ (green) and $\Delta E^{(1,1)}_{N+1,N}$ (red) are also plotted. (a) Only $\mathcal{H}_{23}$ and $\mathcal{H}_{E}$ interactions are turned on, with $J_{23}=0.6\;\text{cm}^{-1}$ and $E_{C}=1\;\text{meV}$. (b) The zero-field splitting term $\mathcal{H}_{A}$ is turned on with $D=-0.6\;\text{cm}^{-1}$. (c) The applied magnetic field term $\mathcal{H}_{Z}$ is turned on with a sufficiently high field, $B_{x}=0.5\;\text{T}$, resulting in degenerate energy differences. (d) Finally, the exchange interaction interaction $\mathcal{H}_{eS}$ is turned on, $J_{1i}=-0.8\;\text{cm}^{-1}$, breaking the degeneracy.
• Figure 3: Energy levels $E$ of the $N$ (black) and $N+1$ (blue) central region electron manifold. The ground (solid), first excited (dashed), and higher-order (light solid) states are plotted using the parameters given in the text, as a function of applied transverse magnetic field $B_{trans}$.
• Figure 4: Energy difference $\Delta E$ values for the $N\rightarrow N+1$ electron manifold transitions. Energy differences are plotted by their transition type: $\Delta E^{(0,0)}_{N+1,N}$ (green solid), $\Delta E^{(0,1)}_{N+1,N}$ (green dashed), $\Delta E^{(1,0)}_{N+1,N}$ (red dashed), $\Delta E^{(1,1)}_{N+1,N}$ (red solid), and the subset of differences involving the second excited state of both manifolds (gray).
• Figure 5: $S^{2}$ projections of the first two states in the uncharged and charged manifolds.