Spin-orbit interaction induces charge beatings in a lightwave-STM – single molecule junction

Recent lightwave-STM experiments have shown space and time resolution of single molecule vibrations directly on their intrinsic length and timescales. We address here theoretically the electronic dynamics of a copper-phthalocyanine in a lightwave-STM explored within a pump-probe cycle scheme. The spin-orbit interaction in the metallic center induces beatings of the electric charge flowing through the molecule as a function of the delay time between the pump and the probe pulses. Interference between the quasidegenerate anionic states of the molecule and the intertwined dynamics of the associated spin and pseudospin degrees of freedom are the key aspects of such phenomenon. We study the dynamics directly in the time domain within a generalized master-equation approach.

Figure 1

Schematic representation of a lightwave-STM single molecule junction. The molecule is adsorbed on an insulator to hinder its hybridization to the underlying metallic substrate. The sharp tip allows one to image the molecule with atomic resolution. The bias across the junction is modulated by laser pulses, illustrated by the green (red) wave packet for the pump (probe) pulse.

Figure 2

Pump-probe scheme in lightwave-STM single-molecule junction. The THz pulse induces a bias pulse across the junction. Charge flows whenever this transient bias pulse overcomes a threshold. The first pulse brings CuPc, due to interference, into a superposition of many-body states. The intertwined spin-orbital dynamics on the molecule dominates the free-evolution during the delay time and thus influences the readout signal, i.e., the transferred charge into the tip averaged over several pump-probe cycles.

Figure 3

Isosurfaces of the four frontier orbitals: SOMO, HOMO, and the real-valued LUMOs. The color indicates the sign of the wave function at the isosurface. Green corresponds to a positive, purple to a negative sign.

Figure 4

Low-energy spectrum of CuPc. With only direct Coulomb interaction the spectrum is eightfold degenerate. Exchange coupling splits the triplet from the singlet states. The SOI further lifts the degeneracy of the triplet states. Adapted from Ref. [15].

Figure 5

Schematic representation of the lightwave-STM dynamics within the many-body spectrum of CuPc. With the choice of chemical potential $\mu =-4\mathrm{eV}$, the low-energy anionic states have a lower energy than the ground states of the contiguous neutral and doubly charged configurations. The bias pulses induce tunneling transitions to the doublet states. The red arrow represent the SOI-induced precession.

Figure 6

Blocking state in the high-bias limit. (a) Eigenvalues of the RDM plotted as a function of time. After a short excursion to the neutral state, the system gets blocked in the anionic state. A threefold (twofold) degeneracy for the eigenvalues associated with the triplets (doublets) follows from spin isotropy. (b) Time evolution of the pseudospin components under the same conditions as in panel (a). The blocking state has nonzero expectation values for the pseudospin.

Figure 7

Strong dependence of the dark-state preparation on the tip position. We plot the $x$ and $y$ component of the pseudospin as a function of the tip position. In panels (a) and (b) we neglect SOI and the Lamb shift. The SOI is introduced in panels (c) and (d). The black cross marks the tip position associated with the simulations presented in the rest of the paper. The SOI reduces the strength of the pseudospin obtained by the preparing pulse.

Figure 8

Dynamics of the $x$ components of the pseudospin in constant high bias. (a) Comparison between the total pseudospin $\langle {\tau }_x\rangle$ and its singlet $\langle {\tau }_x^{\mathrm{s}}\rangle$ and spin-resolved triplet components $\langle {\tau }_x^{+1}\rangle , \langle {\tau }_x^0\rangle$, and $\langle {\tau }_x^{-1}\rangle$. The SOI is responsible for the shoulder appearing in the time evolution of $\langle {\tau }_x\rangle$ around $t=3\mathrm{ps}$ and for the oscillations in the $S_z=+1$ and $S_z=-1$ components. (b) Trajectories corresponding to the first $3\mathrm{ps}$ evolution of the spin resolved components of the pseudospin. The vectors $\langle {\mathit{\tau }}^{\pm 1}\rangle$ undergo precession dynamics inverse to each other due to the SOI, superimposed to the pumping along the direction opposite to ${\mathit{P}}_{\tau }$. The $\langle {\mathit{\tau }}^0\rangle$ component is not influenced by the SOI and just grows along the direction opposite to ${\mathit{P}}_{\tau }$.

Figure 9

Dependence on the tip position of $\langle {\tau }_z\rangle$ after the pump pulse. (a) The combined action of tunneling, the SOI, and the Lamb shift during the pump pulse induces nonzero values of $\langle {\tau }_z\rangle$. (b) The $z$ component of the torque exerted by the Lamb shift on the pseudospin associated with the dark state normalized to its maximum value, plotted as a function of tip position.

Figure 10

Free evolution of the pseudospin components after the pump pulse. The purple area indicates the time lapse during which the bias pulse is applied. In panel (a) we show the dynamics only driven by the SOI with the analytical solution for the free evolution on top of the numerics. In panel (b) the Lamb-shift contribution is also included in the calculation. While $\langle {\tau }_x\rangle$ and $\langle {\tau }_y\rangle$ show qualitatively the same beating dynamics, $\langle {\tau }_z\rangle$ changes radically. The Lamb shift couples in fact more strongly the $\langle {\tau }_z\rangle$ dynamics to that of the $x$ and $y$ components, thus mixing the evolution timescales.

Figure 11

Dynamics of a mixed correlator. Also the mixed correlators, such as, for example, $\langle S_{z^2}^{\mathrm{t}}{\tau }_x^{\mathrm{t}}\rangle$, show beatings dynamics. Moreover, the comparison with product of the single spin and pseudospin expectation values reveals the correlation between these two degrees of freedom triggered by the SOI.

Figure 12

Transferred charge per pump probe cycle plotted as a function of the delay time between the two pulses. The blue and the orange solid lines represent analytical results calculated according to Eq. (56) and its improved version, Eq. (57), respectively.

Figure 13

Current flowing towards the tip during the probe pulse. The current decay during the probe pulse and the backflow from the tip to the molecule are essential for a quantitative reconstruction of the readout signal.