Electrical Readout of Individual Nuclear Spin Trajectories in a Single-Molecule Magnet Spin Transistor

We present the electrical readout of time trajectories obtained from an isolated nuclear spin. The device, a ${\mathrm{TbPc}}_2$ single-molecule magnet spin transistor, detects the four different nuclear spin states of the ${\mathrm{Tb}}^{3+}$ ion with fidelities better than 69%, allowing us to measure individual relaxation times ($T_1$) of several tens of seconds. A good agreement with quantum Monte Carlo simulations suggests that the relaxation times are limited by the current tunneling through the transistor, which opens up the possibility to tune $T_1$ electrically by means of bias and gate voltages.

Figure 1

(a) Artist view of the molecular spin transistor, consisting of a ${\mathrm{TbPc}}_2$ molecular magnet, connected to source and drain gold electrodes, and a back gate underneath. The Pc ligands (white cloud) are acting as a readout quantum dot. The terbium ion (pink sphere in the center) possesses an electronic spin with $J=6$ (orange upward arrow) and a nuclear spin with $I=3/2$ (green sideward arrow). The uniaxial anisotropy axis of the Tb ion is perpendicular to the Pc plane. (b) Zeeman diagram of the energy levels of the ${\mathrm{TbPc}}_2$ molecule with the magnetic field parallel to the easy axis of magnetization ($H_{||}$). Because of a strong hyperfine interaction with the Tb nuclear spin, the electron’s up and down states are each split into four energy levels. Off diagonal terms in the spin Hamiltonian lead to avoided energy level crossings (shaded rectangles), enabling tunneling of the electronic spin. Note that for each QTM transition, the nuclear spin is preserved and therefore, their positions reveal the nuclear spin states.

Figure 2

Scheme of the measurement procedure. (a) The magnetic field $H_{||}$ is swept up and down as a function of time $t$ over the four avoided level crossings and (b) the conductance $g$ through the readout dot is simultaneously measured. Whenever the electronic spin undergoes a QTM transition, a conductance jump is observed (indicated by dashed lines) revealing the nuclear spin state.

Figure 3

Nuclear spin trajectory. (a) By continuously ramping the magnetic field up and down, the conductance jumps reveal the nuclear spin states (gray dots) as a function of time, yielding the nuclear spin trajectory (red curve). We found that the nuclear spin quantum number changes only by $\Delta m_I=\pm 1$ (see Supplemental Material [25]). (b) Histograms (gray) of about 40 000 conductance jumps, showing four nonoverlapping Gaussian-like distributions (dashed lines). The shaded bars, obtained by counting all events lying within three times the full width half maximum of the fitted Gaussians, contain $(95\pm 2)\%$ of the data points and represent the time-average population $P$ of each nuclear spin state.

Figure 4

(a)–(d) Measured expectation value and (e)–(h) QMC simulations of each nuclear spin state versus time. The given $T_1$ values were obtained by fitting the data to an exponential function $\exp (-t/T_1)$ (red dashed lines).

Figure 5

(a) The transition rates ${\Gamma }_{i,j}$, derived by fitting the results of QMC simulations to experimental data, exhibit a quadratic dependence on the nuclear spin level spacing ${\omega }_{i,j}$. This behavior is expected from a Weger relaxation process in which the nuclear spin is coupled via virtual spin waves to conduction electrons. (b) The decrease of lifetime with increasing current is probably due to an increase of electrons tunneling through the readout dot in addition to an increase of temperature.