Role of coherence in transport through engineered atomic spin devices

We give a further step in the quantum mechanical description of engineered atomic spin structures by deriving a master equation of the Redfield type that governs the dynamics of the atomic spin density matrix. By generalizing this approach to charge-specific density matrices, we are able to describe magnetic transport quantities, such as the average inelastic current and the shot noise, accessible by tunneling spectroscopy. Our method suitably describes moderate lead-atom coupling regimes where quantum coherence effects cannot be disregarded. We contrast our approach with the existing descriptions in terms of rate equations and show examples where coherence effects are crucial to understand the physics of spin-polarized tunnel current through spin structures.

Figure 1

(a) Schematics of a typical EASD. Magnetic atoms are deposited on an insulating layer coating a metallic substrate. Upon applying a voltage difference between the metallic STM tip and the substrate, a charge current ensues. (b) Sketch of the model. The substrate is modeled by a set of independent reservoirs sharing the same chemical potential ${\mu }_S$. The tip is modeled by an additional reservoir with ${\mu }_T={\mu }_S-eV$. All reservoirs are assumed to be wideband metals.

Figure 2

Steady state characteristics for a single spin $S=1/2$ in a perpendicular geometry ($\mathbf{B}$ along $z$ axis, $\mathbf{P}$ along $x$ axis) for different values of the polarization parameter $p$ and for $g=2$, $B=5$ T, ${\gamma }_T={\gamma }_S=0.8$, ${\left(\beta k_B\right)}^{-1}=1$ K. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_x\rangle$, (d) average spin component $\langle S_y\rangle$, (e) average spin component $\langle S_z\rangle$, (f) entropy. In this geometry RE and ME approaches are not equivalent for $p\ne 0$, as the coherences determined by $\langle S_x\rangle$ and $\langle S_y\rangle$ do not vanish and give contribution to the results. While ME gives different $dI/dV$ and $dS/dV$ curves for different $p$, the results obtained with REs are independent of $p$ and coincide with the ME results for $p=0$.

Figure 3

Dependence of (a) average current and (b) shot noise on the coupling strength for a single spin $S=1/2$ in a perpendicular geometry ($\mathbf{B}$ along $z$ axis, $\mathbf{P}$ along $x$ axis). $dI/dV$ and $dS/dV$ curves are computed with REs and ME for different values of ${\gamma }_T={\gamma }_S=\gamma$ and for $g=2$, $B=5$ T, ${\left(\beta k_B\right)}^{-1}=2$ K, $p=1$. In the limit of weak coupling $dI/dV$ and $dS/dV$ curves obtained with ME coincide with the results obtained with REs.

Figure 4

Steady state characteristics for a single spin $S=5/2$ in a perpendicular geometry ($z$ is the easy axis of the crystal and $\mathbf{P}$ is along $x$ axis) for different values of the polarization parameter $p$ and for $D=-0.04$ meV, ${\gamma }_T={\gamma }_S=0.6$, ${\left(\beta k_B\right)}^{-1}=0.5$ K. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_x\rangle$, (d) entropy. In this geometry RE and ME approaches are not equivalent for $p\ne 0$, as the coherences determined by $\langle S_x\rangle$ do not vanish and give contribution to the results. Other components of the spin vanish, i.e., $\langle S_y\rangle =\langle S_z\rangle =0$. While ME gives different $dI/dV$ and $dS/dV$ curves for different $p$, the results obtained with REs are independent of $p$ and coincides with the ME results for $p=0$.

Figure 5

Steady state characteristics for a chain of 4 spins $S=1/2$ in zero magnetic field for different values of the polarization parameter $p$ and for ${\left(\beta k_B\right)}^{-1}=1$ K, ${\gamma }_T={\gamma }_S=0.8$, $J=0.3$ meV. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_z\rangle$, (d) entropy. Other components of the spin vanish, i.e., $\langle S_x\rangle =\langle S_y\rangle =0$.

Figure 6

Steady state characteristics for a chain of 4 spins $S=1/2$ in a parallel geometry (both $\mathbf{B}$ and $\mathbf{P}$ along $z$ axis) for different values of the polarization parameter $p$ and for $g=2$, $B=5$ T, ${\left(\beta k_B\right)}^{-1}=1$ K, ${\gamma }_T={\gamma }_S=0.8$, $J=0.3$ meV. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_z\rangle$, (d) entropy. Other components of the spin vanish, i.e., $\langle S_y\rangle =\langle S_z\rangle =0$.

Figure 7

Steady state characteristics for a chain of 4 spins $S=1/2$ in a perpendicular geometry ($\mathbf{B}$ along $z$ axis, $\mathbf{P}$ along $x$ axis) for different values of the polarization parameter $p$ and for $g=2$, $B=5$ T, ${\left(\beta k_B\right)}^{-1}=1$ K, ${\gamma }_T={\gamma }_S=0.8$, $J=0.3$ meV. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_x\rangle$, (d) average spin component $\langle S_y\rangle$, (e) average spin component $\langle S_z\rangle$, (f) entropy.

Figure 8

Steady state characteristics for a single spin $S=1/2$ in a parallel geometry (both $\mathbf{B}$ and $\mathbf{P}$ along $z$ axis) for different values of the polarization parameter $p$ and for $g=2$, $B=5$ T, ${\gamma }_T={\gamma }_S=0.8$, ${\left(\beta k_B\right)}^{-1}=1$ K. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_z\rangle$, (d) entropy. In this geometry ME and RE approaches are equivalent, as the coherences vanish due to $\langle S_x\rangle =\langle S_y\rangle =0$.

Figure 9

Steady state characterization for a single spin $S=5/2$ in a parallel geometry ($\mathbf{P}$ along the easy axis $z$ of the crystal) for different values of the polarization parameter $p$ and for $D=-0.04$ meV, ${\gamma }_T={\gamma }_S=0.6$, ${\left(\beta k_B\right)}^{-1}=0.5$ K. The quantities presented as functions of voltage are (a) differential conductance, (b) differential shot noise, (c) average spin component $\langle S_z\rangle$, (d) entropy. Other components of the spin vanish, i.e., $\langle S_x\rangle =\langle S_y\rangle =0$. As in the case of spin $S=1/2$, ME and RE approaches are equivalent in this geometry.