Independent and coherent transitions between antiferromagnetic states of few-molecule systems

Spin-electronic devices are poised to become part of mainstream microelectronic technology. Downsizing them led to the field of molecular spintronics. Here, we provide proof-of-concept data that allow expanding this area from its traditional focus on single-molecule magnets to molecules in which spin centers are antiferromagnetically (AFM) coupled to result in a singlet ground state. In this context, and in contrast to all previous work on molecular spintronics, we develop a detection scheme of the spin state of the molecule that does not rely on a magnetic moment. Instead, we use quantum dot devices consisting of an isolated, contacted single-wall carbon nanotube covalently bound to a limited number of molecular AFMs, for which we chose representative coordination complexes incorporating four Mn(II) or Co(II) ions. Time-dependent quantum transport measurements along the functionalized nanotube show steplike transitions between several distinct current levels that we attribute to transitions between different AFM states of individual molecular complexes grafted on the nanotube. A statistical analysis of the switching events using factorial cumulants indicates that the cobalt complexes switch independently from each other, whereas a coherent superposition of the AFM spin states of the molecules along the nanotube is observed for the manganese complexes. The long coherence time of the superposition state (several seconds at 100 mK) is made possible by the absence of spin and orbital momentum in the relevant states of the manganese complex, while the cobalt complex includes a significant orbital momentum contribution due to the pseudo-octahedral coordination environment of the $d^7$ metal centers.

Figure 1

(a) General structure of the metal complexes (M: metal ion). H is omitted for clarity, the central (distorted) ${\mathrm{M}}_4{\mathrm{O}}_4$ cubane highlighted in yellow, all coordinative bonds in orange. All aromatic rings are made transparent to improve spatial representation. Ligand backbones differentiate between chelate and acetate ligands. Spin density $({\rho }_{\uparrow }-{\rho }_{\downarrow })$ on (b) the carbon nanotube $\left\{{\mathrm{Mn}}_4\right\}$-(CNT) and (c) the $\left\{{\mathrm{Co}}_4\right\}$-CNT hybrid system computed from first principles in an open-system setup [7, 8, 9]. Light blue and yellow indicate positive and negative values of the isosurfaces normalized to the maximum spin density.

Figure 2

(a) Stability diagram of the quantum dot formed on the $\left\{{\mathrm{Mn}}_4\right\}$-functionalized carbon nanotube (CNT), measured at a temperature of 4 K $\left(B=0\mathrm{T}\right)$, exhibiting a very clean and regular pattern of Coulomb diamonds with a quantum dot size comparable with the distance between the electrodes [39]. (b) Coulomb diamonds of a CNT quantum dot functionalized with $\left\{{\mathrm{Co}}_4\right\}$ complexes at 100 mK $\left(B=0\mathrm{T}\right)$. The width of the current steps as well as that of the Coulomb peaks yield an electron temperature of 600 mK $\left(\sim 50\mu \mathrm{eV}\right)$.

Figure 3

(a) Short section of the time trace of the current at the edge of a Coulomb diamond of the $\left\{{\mathrm{Mn}}_4\right\}$-functionalized carbon nanotube (CNT) quantum dot and probability density function (PDF) of the entire 3600 s signal displayed with 11 Gaussian peaks indicating the current levels (we omit the three highest levels for better visibility). (b) Histogram of the derivative of the entire signal (note the logarithmic scale) fitted with five Gaussian peaks leading to a standard deviation of the direct signal ${\sigma }_{\mathrm{direct}}={\sigma }_{\mathrm{deriv}}/\sqrt{2}=40\left(1\right)\mathrm{pA}$, respectively.

Figure 4

Short section of the current trace at the edge of a Coulomb diamond of the $\left\{{\mathrm{Co}}_4\right\}$-functionalized carbon nanotube (CNT) quantum dot and corresponding probability density function (PDF) of the entire signal (1300 s) with three Gaussian peaks fitted at $B=0\mathrm{T}$.

Figure 5

(a) Part of the stability diagram of a carbon nanotube (CNT) quantum dot functionalized with $\sim 70 \left\{{\mathrm{Co}}_4\right\}$ complexes at $B=8\mathrm{T}$ (perpendicular to CNT axis). White dots indicate the positions where the current traces shown in (b) were measured (gate and bias voltage of the lower point the same as in Fig. 4). (b) Parts of the time traces of the current taken at the indicated positions in a (top, in blue, for $V_{\mathrm{sd}}=3.9\mathrm{mV}$, bottom, in black, for $V_{\mathrm{sd}}=2.0\mathrm{mV}$) with the respective probability density function (PDF) of the entire direct current traces after background correction on the right. Discrete current levels are indicated by dashed lines.

Figure 6

Sketch of the different states of the system. Molecules can be in their ground state (open circles), the first excited state (full circles), or a superposition thereof (striped circles). The higher the current level, the more molecules are in an excited state. Here, only three of the attached molecules are shown with the possible transitions. In the case of $\left\{{\mathrm{Co}}_4\right\}$, the different transitions are independent. For $\left\{{\mathrm{Mn}}_4\right\}$, the states exist in a superposition, and thus, the transitions are coherent.

Figure 7

(a) Occupation probability of the different levels found in the $\left\{{\mathrm{Mn}}_4\right\}$-device on a logarithmic scale for the direct current data (black squares) and for models assuming 50 independent two-level fluctuators (blue triangles) and 50 coherently coupled two-level fluctuators (red crosses). The dashed line is a fit of the data to an exponential distribution. (b) First factorial cumulant $\left(C_{\mathrm{F},1}\right)$ as extracted from the measurements on the carbon nanotube (CNT) quantum dots functionalized with $\left\{{\mathrm{Mn}}_4\right\}$ (red) and $\left\{{\mathrm{Co}}_4\right\}$ (blue) complexes. Fits (dashed lines) yield the transition rates used for modeling, in (c), the second $\left(C_{\mathrm{F},2}\right)$ and, in (d), the third $\left(C_{\mathrm{F},3}\right)$ factorial cumulant. The solid lines in (b)–(d) are the respective cumulants extracted directly from the digitized measurement signal of the two devices ($\left\{{\mathrm{Mn}}_4\right\}$: red, $\left\{{\mathrm{Co}}_4\right\}$: blue). The positive second cumulant in (c) is related to a super-Poissonian Fano factor by $F=\left(C_{\mathrm{F},2}/C_{\mathrm{F},1}\right)+1$. All lines of the $\left\{{\mathrm{Co}}_4\right\}$ sample are multiplied by a factor of 20 for better visibility.