Probing transverse magnetic anisotropy by electronic transport through a single-molecule magnet

By means of electronic transport, we study the transverse magnetic anisotropy of an individual ${\mathrm{Fe}}_4$ single-molecule magnet (SMM) embedded in a three-terminal junction. In particular, we determine in situ the transverse anisotropy of the molecule from the pronounced intensity modulations of the linear conductance, which are observed as a function of applied magnetic field. The proposed technique works at temperatures exceeding the energy scale of the tunnel splittings of the SMM. We deduce that the transverse anisotropy for a single ${\mathrm{Fe}}_4$ molecule captured in a junction is substantially larger than the bulk value.

Figure 1

(a) Schematic depiction of a molecular three-terminal transistor with a single ${\mathrm{Fe}}_4$ SMM bridging the junction. (b) Spatial orientation of an external magnetic field with respect to the principal axes set by the magnetic anisotropy of an SMM. (c) Differential-conductance map, $dI/d{V}_{\text{b}}$, measured as a function of gate $V_{\text{g}}$ and bias $V_{\text{b}}$ voltages showing two charge states $N$ (neutral) and $N+1$ (charged) for sample A. (d) Representative Coulomb peaks [corresponding to linear conductance $G\equiv dI/d{V}_{\text{b}}{|}_{V_{\text{b}}=0}$, e.g., marked by dashed line in (c)] measured at different values of the external magnetic field $B$. The bold arrowed lines and color dots serve as a guide for eyes to indicate the nonmonotonic change in the Coulomb peak height.

Figure 2

Signatures of transverse magnetic anisotropy in electronic transport at $T=1.8$ K. (a) Dependence of the Coulomb peak (CP) height $G_{\text{max}}$ [i.e., the maximal value of $G$; cf. Fig. 1 on magnetic field $B$ shown for two different samples where the orientation of the magnetic field lies in the hard plane $(\theta ={90}^{\circ })$. (b) Analogous to (a) for a single sample, except that now $\theta$ is varied and $\varphi$ is unknown. Note that the evolution of the CP position in magnetic field, and not $G_{\text{max}}$, was previously analyzed in Ref. [24] for samples A and C. (Bottom panels) Theoretical predictions for evolution of the CP height with magnetic field $B$ kept in the hard plane (c) for indicated values of $E/D$ and $\varphi =0^{\circ }$, whereas in (d) for several angles $\varphi$ and the fixed value of $E/D$ estimated from (a). Bold dashed lines represent the case of $E/D=0$ for $\varphi =0^{\circ }$ (c) and $\varphi ={90}^{\circ }$ (d). Notice that the shape of $G_{\text{max}}$ for $E/D=0$ is independent of $\varphi$ due to the rotational symmetry around the molecule's easy axis.

Figure 3

Theoretical analysis of transport for fixed $D=56\mu \mathrm{eV}$ and $E/D=0.17$ and $\mathit{B}$ along the hard axis ($\theta ={90}^{\circ }$ and $\varphi =0^{\circ })$. (a) Conductance $G_{\text{max}}\left(B\right)$ traces for various temperatures over the range $1.2\text{K–}2.4$ K at intervals of 0.2 K. (b) Occupation probabilities for several lowest-energy states in the spin multiplets for $N$ and $N+1$ at $T=1.8$ K. Here $k^{\prime }$ ($k)$ labels the states in order of increasing energy for $N (N+1)$, with $k^{\prime }=0 (k=0)$ denoting the ground state. (c) Relevant transition energies ${\varepsilon }_{N+1}^k-{\varepsilon }_N^{k^{\prime }}$ for $k,k^{\prime }⩽4$ determining the SET processes at the Coulomb resonance (note that ${\varepsilon }_{N+1}^0={\varepsilon }_N^0$ is restored for each $B$ by tuning $V_{\text{g}})$. Different colors of lines are used to distinguish groups of transitions with respect to possible combinations of indices $k$ and $k^{\prime }$ (see the main text). For the association of these lines with specific transitions as well as the energies of individual levels see Fig. 4. (d) Evolution of the current vs magnetic field at $T=1.8$ K calculated by including a restricted number of states per spin multiplet up to $r$, where $r=k_{\text{max}}^{\prime }+1=k_{\text{max}}+1$, showing that for small $r$ significant deviations are found compared to the calculation involving all the states (dashed line), used in all other plots. For a precise definition of the current $I_r$ see Appendix pp2.

Figure 4

Panel (a) is identical to Fig.  3, but now for each transition-energy line we specify the initial and final states, with respective energies ${\varepsilon }_N^{k^{\prime }}$ and ${\varepsilon }_{N+1}^k$, between which the transition occurs. Recall that $k$ is an index which numbers states in a given spin multiplet with respect to energy, with $k=0$ denoting the ground state. Moreover, by labeling the lines with $(k,k^{\prime })$ we mean that $k$ refers to the final state of a charged SMM $(N+1)$, whereas $k^{\prime }$ represents the initial state of a neutral SMM $(N)$. We note that information shown in (a) cannot be readily seen from energies ${\varepsilon }_n^k (n=N,N+1)$ of the individual levels, which for the completeness of the present discussion are plotted in (b). Observe that since energies in (b) are calculated at the Coulomb resonance, the curves for $k=0$ overlap.

Figure 5

How to determine the transverse magnetic anisotropy constant $E$ of an individual SMM from its transport characteristics. The position (a),(b) and amplitude (c)–(f) of the CP are shown for different values of the parameters $D$ and $E$ of the SMM model with $S_N=5$ and $S_{N+1}=9/2$ for $T=1.8$ K. Note that we employ the assumption for the ${\mathrm{Fe}}_4$ molecule from the main text, that is, $D=D_N=D_{N+1}/1.2$ and $E=E_N$, with $E_N/E_{N+1}=D_N/D_{N+1}$, and a relatively large value of $E/D$ (red lines) is used for clear illustration of the effects under discussion. In panels (a),(c),(e) the external magnetic field $\mathit{B}$ is oriented along the SMM's hard axis $x$ [see inset in (c)], whereas in panels (b),(d),(f) the field is parallel to the intermediate axis $y$ [see inset in (d)]. In panel (g) we present how temperature affects the occurrence of characteristic peaks associated with the presence of transverse magnetic anisotropy for $\mathit{B}$ along the hard axis $x$; for further details, see Fig. 16. To make the discussion complete, in panel (h) we show analogous dependencies but in the case when the field lies along the intermediate axis $y$. Finally, the frame at the bottom contains a schematic summary of the procedure leading to estimation of $E$: (i) Using the analysis of the CP position, find $D_n$ and adjust the magnetic field $\mathit{B}$ so that it is contained in the hard plane, i.e., the plane perpendicular to the easy axis $z$. (ii) Rotating systematically the magnetic field $\mathit{B}$ in the hard plane, analyze the CP amplitude to find the direction of the molecule's hard axis. This will be characterized by the occurrence of additional peaks in the amplitude, whose field position allows for estimating $E_n$. (iii) If no local maxima in the amplitude can be seen, adjust (try increasing) the temperature.

Figure 6

Details of the ${\mathrm{Fe}}_4$ SMM. (a) Sketch of the magnetic core of the ${\mathrm{Fe}}_4$ SMM. (b) Ground-state spin multiplet $(S_N=5)$ of the ${\mathrm{Fe}}_4$ SMM in a neutral charge state $N$; for further explanation see Appendix pp2. (c) Depiction of the ${\mathrm{Fe}}_4$ SMM illustrating the orientation of the phenyl rings [omitted in (a)] that terminate the molecule. Note that in both (a) and (c) hydrogen atoms are disregarded for clarity.

Figure 7

Three-terminal-junction fabrication. (a) Schematics of the three-terminal-device fabrication process. (b) Scanning electron microscope (SEM) image of a real three-terminal device before electromigration.

Figure 8

Coulomb peak position gate-voltage spectroscopy. The shift of the CP position due to magnetic field for samples A, B, and C. The solid lines are fits to ${\varepsilon }_{N+1}^0-{\varepsilon }_N^0$, calculated from the giant-spin Hamiltonian, Eqs. (1) and (2). From the fit we get the following values: for sample A in (a), $D_{N+1}=61\mu \mathrm{eV},{\theta }_N={87}^{\circ }$, and ${\theta }_{N+1}={86}^{\circ }$; for sample B in (b), $D_{N+1}=65\mu \mathrm{eV},{\theta }_N={86}^{\circ }$, and ${\theta }_{N+1}={84}^{\circ }$; for sample C, in (c) ${\theta }_N={87}^{\circ }$ and ${\theta }_{N+1}={85}^{\circ }$, whereas in (d) ${\theta }_N={63}^{\circ }$ and ${\theta }_{N+1}={62}^{\circ }$, with $D_{N+1}=68\mu \mathrm{eV}$ in both cases. We note that the evolution of the CP position in magnetic field, and not $G_{\text{max}}$, for samples A and C was previously analyzed in Ref. [24]. Also note that in the fitting for sample A we included $E/D=0.2$ and $\varphi =0^{\circ }$ obtained in Fig. 2.

Figure 9

Statistics. Differential-conductance maps, $dI/d{V}_{\text{b}}$, shown as a function of gate $V_{\text{g}}$ and bias $V_{\text{b}}$ voltages together with corresponding dependencies of the CP amplitude $G_{\text{max}}$ on magnetic field $B$ for six different ${\mathrm{Fe}}_4$ molecular junctions. (Top) [(a)–(c)] Junctions for which gate-voltage spectroscopy fits of the CP position (not shown) indicate $\theta <{60}^{\circ }$. (Bottom) [(d)–(f)] Junctions where $\theta \approx {90}^{\circ }$ is found. The shape of the field modulation of $G_{\text{max}}$ implies that for (d) and (e) the field is close to the intermediate axis $(\varphi \approx {90}^{\circ })$, whereas for (f) it is most likely in an intermediate $\varphi$ angle in the hard plane.

Figure 10

The effect of the reversed magnetic field polarity on $G_{\text{max}}$. Dependence of the scaled CP height $G_{\text{max}}/G_{\text{max}}(B=0)$ on magnetic field $B$ for the samples discussed in the main text, cf. Figs. 2 and 2, showing that the curves are symmetric upon reversal of the field polarity.

Figure 11

Cotunneling background. Differential conductance, $dI/d{V}_{\text{b}}$, measured as a function of magnetic field $B$ at two different points: (a) $V_{\text{g}}=-1.71$ V and $V_{\text{b}}=-12$ mV and (b) $V_{\text{g}}=-1.71$ V and $V_{\text{b}}=-10.5$ mV, which correspond to the cotunneling background in the left-hand charge state of sample A; cf. Fig. 1.

Figure 12

Effect of magnetic anisotropy on the energy spectrum of SMM. (Top)/(Bottom) [(a,c,e)/(b,d,f)] The case of a integer/half-integer value of a molecular spin. In particular, we use the values of spin known for a ${\mathrm{Fe}}_4$ SMM, $S_N=5$ for a neutral molecule and $S_{N+1}=9/2$ for a charged one [5]. (a),(b) In the presence of exclusively uniaxial magnetic anisotropy $D>0$ (and without magnetic field, $B=0)$ an energy barrier protecting the molecule's spin against reversal between two opposite, energetically degenerate, orientations arises. The excitation between the ground-state doublet and the first excited doublet is then commonly referred to as the ZFS. (c),(d) If additionally the transverse component of magnetic anisotropy occurs, it allows for mixing of pure $S_z$ states. Each new eigenstate is then formed from $S_z$ states belonging to one of two uncoupled, time-reversed sets, as schematically marked by two different colors. As follows from the Kramers theorem, for $S_N=5$ the transverse magnetic anisotropy introduces tunnel splittings $\Delta$, whereas for $S_{N+1}=9/2$ all states remain doubly degenerate. (e),(f) A characteristic feature of such anisotropic, large spins is that when an external magnetic field $B$ is applied along the system's hard axis, one observes periodic changes of the tunnel splittings [1, 14]. Other parameters assumed in the calculations: $D_N=56\mu \mathrm{eV},D_{N+1}=68\mu \mathrm{eV},$ and $E_N/D_N=E_{N+1}/D_{N+1}=0.3$.

Figure 13

Signatures of the transverse magnetic anisotropy in electronic transport (magnetic field along the hard axis, $\theta ={90}^{\circ }$ and $\varphi =0^{\circ })$. Analogous to Figs. 3–3 with each column corresponding now to a different value of $E/D$: (a)–(d) occupation probabilities for several lowest-in-energy states in the spin multiplets for $N$ and $N+1$ at $T=1.8$ K; (e)–(h) transition energies ${\varepsilon }_{N+1}^k-{\varepsilon }_N^{k^{\prime }}$ relevant for the SET processes at the Coulomb resonance (i.e., ${\varepsilon }_{N+1}^0={\varepsilon }_N^0$ is restored for each $B$ by tuning $V_{\text{g}})$ for $k,k^{\prime }⩽4$. Different colors of lines are used to distinguish groups of transitions with respect to possible combinations of indices $k$ and $k^{\prime }$ [see the discussion regarding Figs. 3 and 4; (i)–(l) energies ${\varepsilon }_n^k$ for $n=N,N+1$ at the Coulomb resonance (observe that the curves for $k=0$ overlap); (m)–(p) Dependence of the current on the number of spin-multiplet states $r$ included from each charge state. The left (right) most column represents the case of absent (significant) transverse magnetic anisotropy. Importantly, each column shows a detailed analysis of selected conductance curves from Fig. 2. We note that transition-energy lines in (e)–(h) can be easily identified with the use of Fig. 4. It can be seen that increasing $E/D$ results in shifting the minima of the transition-energy curves in (e)–(h) towards higher values of the field. Such a behavior, in turn, affects the occupation probabilities (a)–(d), so that the probability of finding an SMM in either the ground $(k=0)$ or the first excited $(k=1)$ state for both charge states $N$ and $N+1$ remain equal for a larger magnetic-field range (compare the outermost columns). Recall that the position of the CP is fixed mostly by $D$; see Fig. 8.

Figure 14

Signatures of the transverse magnetic anisotropy in electronic transport (magnetic field along the intermediate axis, $\theta ={90}^{\circ }$ and $\varphi ={90}^{\circ })$. Generally, this figure is analogous to Fig. 13, except that now the external magnetic field is rotated to align with the molecule's intermediate $\left(y\right)$ axis. To begin with, we note that the results shown in the leftmost column (i.e., for $E/D=0)$ are identical to those in the leftmost column of Fig. 13, which is the manifestation of the molecule's rotational symmetry about the easy $\left(z\right)$ axis in the absence of transverse component of magnetic anisotropy. Unlike for the case of $\varphi =0^{\circ }$, the consequence of the increase of $E/D$ is the displacement of the transition-energy curves minima (e)–(h) towards smaller values of the field. Interestingly enough, in the situation under discussion one thus observes a more abrupt decrease of the current [see dashed lines in (m)–(p)] for larger $E/D$ occurring at smaller values of $B$.

Figure 15

Dependence of transport signatures of the transverse anisotropy on the orientation of magnetic field in the hard plane $(\theta ={90}^{\circ })$ for $E/D=0.17$. Analogous to Figs. 3–3, with each column corresponding now to a different value of $\varphi$. Note that the case of $\varphi =0^{\circ }$ is presented in Figs. 3–3. Furthermore, here each column shows a detailed analysis of selected conductance curves from Fig. 2.

Figure 16

Evolution of the CP amplitude in the absence of transverse magnetic anisotropy $(E=0)$. This figure serves to illustrate the fact that even if the transverse magnetic anisotropy is absent, by making the uniaxial magnetic anisotropy parameter $D$ smaller (keeping a fixed temperature), one can eventually also produce a maximum as for $E\ne 0$. However, this maximum occurs at a completely different (smaller) value of magnetic field. Moreover, the shape of $G_{\text{max}}\left(B\right)$ remains invariant under rotation of the field in the hard plane; this is when the angle $\varphi$ is varied. None of these are the case in the experiment under discussion. (a),(b) Dependence of $G_{\text{max}}\left(B\right)$ on the value of the uniaxial magnetic anisotropy parameter $D\equiv D_N$ (and $D_{N+1}=1.2D)$ for an external magnetic field applied along the molecule's hard axis $(\theta ={90}^{\circ }$ and $\varphi =0^{\circ })$. A detailed analysis of selected curves from (a),(b) is carried out in (c)–(s), with each column corresponding to the indicated value of $D$.