Spin effects in transport through single-molecule magnets in the sequential and cotunneling regimes

We analyze the stationary spin-dependent transport through a single-molecule magnet weakly coupled to external ferromagnetic leads. Using the real-time diagrammatic technique, we calculate the sequential and cotunneling contributions to current, tunnel magnetoresistance, and Fano factor in both linear and nonlinear response regimes. We show that the effects of cotunneling are predominantly visible in the blockade regime and lead to enhancement of tunnel magnetoresistance (TMR) above the Julliere value, which is accompanied with super-Poissonian shot noise due to bunching of inelastic cotunneling processes through different virtual spin states of the molecule. The effects of external magnetic field and the role of type and strength of exchange interaction between the LUMO level and the molecule’s spin are also considered. When the exchange coupling is ferromagnetic, we find an enhanced TMR, while in the case of antiferromagnetic coupling we predict a large negative TMR effect.

Figure 1

(Color online) Schematic representation of the system under consideration. The system consists of a SMM weakly coupled to two ferromagnetic electrodes with the collinear configuration of their magnetic moments, i.e., either parallel or antiparallel. Due to symmetrically applied bias voltage $V=\left({\mu }_L-{\mu }_R\right)/e$, where ${\mu }_{L\left(R\right)}$ denotes the electrochemical potential of the left (right) lead, the LUMO level is independent of $V$. Position of the LUMO level, however, can be tuned by the gate voltage $V_g$.

Figure 2

(Color online) The total (first plus second order) differential conductance in the parallel (a) and antiparallel (b) configurations for the parameters: $S=2$, $J=0.2\text{meV}$, $D=0.05\text{meV}$, $D_1=-0.005\text{meV}$, $D_2=0.002\text{meV}$, $U=1\text{meV}$, $k_{B}T=0.04\text{meV}$, $P_L=P_R=0.5$, and $\Gamma =0.002\text{meV}$.

Figure 3

(Color online) Density plot of the total (first plus second order) TMR (a) and TMR calculated in the sequential tunneling approximation (b) plotted in the same scale and for the same parameters as in Fig. 2. The sequential TMR is smaller than the total TMR. The dashed lines are only a guide for eyes and they represent positions of the main conductance peaks, Fig. 2, separating thus regions corresponding to different occupation states of the LUMO level.

Figure 4

(Color online) The average value of the $z\text{th}$ component of the molecule’s total spin $⟨S_t^z⟩$ for the parallel (a) and antiparallel (b) magnetic configurations. All parameters as in Fig. 2.

Figure 5

(Color online) Energy spectrum of the molecule under consideration (relevant parameters given in the caption of Fig. 2) for $\epsilon =-0.5\text{meV}$ (a,c) and $\epsilon =0.75\text{meV}$ (b,d) in the case of ferromagnetic [(a) and (b)] and antiferromagnetic [(c) and (d)] coupling between the SMM’s core spin and the spin of electrons in the LUMO level. The dashed line represents the Fermi level of the leads when no external voltage bias is applied $\left(V=0\right)$. Different sets of molecular states correspond to different values of the SMM’s total spin $S_t$, and/or the occupation number of the LUMO level: $|2;0,m⟩$ (●), $|5/2;1,m⟩$ (◆), $|3/2;1,m⟩$ (▲), and $|2;2,m⟩$ (◼). Note that in (a) and (c) the degeneracy between states $|2;0,m⟩$ and $|2;2,m⟩$ takes place only for $m=0$.

Figure 6

(Color online) (a) TMR in the linear-response regime for the parameters as in Fig. 2 (solid line). The dot-dashed line shows the TMR calculated in the first-order approximation. (b) Average value of the $z\text{th}$ component of the molecule’s total spin in the (P) parallel (solid line) and (AP) antiparallel (dashed line) magnetic configurations. The dotted lines in (a) and (b) correspond to the case of $D_1=D_2=0$.

Figure 7

(Color online) [(a) and (b)] Schematic representation of the elastic and inelastic electron cotunneling processes. The two bottom panels show examples of inelastic cotunneling processes leading to increase in the $z\text{th}$ component of the SMM’s spin in the situation when the LUMO level is (c) empty $\left(Q=0\right)$ and (d) doubly occupied $\left(Q=2\right)$. Note that $V\to 0$ in the linear-response regime, so we assume ${\mu }_L={\mu }_R+0^{+}$. When in the ground state the molecule is occupied by a single electron, $Q=1$, inelastic processes can generally occur via two virtual states associated with empty and doubly occupied LUMO levels.

Figure 8

(Color online) Bias dependence of the current (a,e), differential conductance (b,f), Fano factor (d,h) in the parallel (solid lines) and antiparallel (dashed lines) configurations and TMR (c,g) for $\epsilon =-0.5\text{meV}$ [(a)–(d)] and $\epsilon =0.75\text{meV}$ [(e)–(h)]. The parameters are the same as in Fig. 2 and $I_0=e\Gamma /\hslash \approx 0.5\text{nA}$. The dotted lines show the results obtained taking into account only first-order tunneling processes. The effect of cotunneling is most pronounced in the TMR and Fano factor.

Figure 9

(Color online) Density plot of TMR for $J=0.05\text{meV}$ (a) and $J=0.4\text{meV}$ (b). The bottom panels display cross sections of (a) and (b) for the constant energy of the molecule’s LUMO level $\epsilon =-0.5\text{meV}$ (c) and the constant bias voltage $V=0\text{mV}$ (d). Other parameters are the same as in Fig. 2.

Figure 10

(Color online) (a) Density plot of TMR in the case when the external magnetic field $H_z=0.216\text{T}$ $\left(g{\mu }_B{H}_z=0.025\text{meV}\right)$ is applied along the easy axis of the molecule. (b) TMR in the linear-response regime (solid line). For comparison, TMR in the absence of external magnetic field [dotted line in (b)] is also shown. The other parameters are the same as in Fig. 2.

Figure 11

(Color online) Average value of the $z\text{th}$ component of the total molecule’s spin in the parallel (a) and antiparallel (b) magnetic configurations, when an external field $H_z=0.216\text{T}$ is applied along the $z$ axis. The other parameters are the same as in Fig. 2.

Figure 12

(Color online) The current (a,e), differential conductance (b,f), Fano factor (d,h) in the parallel (solid lines) and antiparallel (dashed lines) configurations, and the TMR (c,g) for $\epsilon =-0.5\text{meV}$ [(a)–(d)] and $\epsilon =0.75\text{meV}$ [(e)–(h)] as a function of the bias voltage. An external magnetic field $H_z=0.216\text{T}$ is applied along the $z$ axis, while the other parameters are the same as in Fig. 2.

Figure 13

(Color online) (a) The total tunnel magnetoresistance in the case of antiferromagnetic coupling between the SMM’s core spin and the spin in the LUMO level calculated for $J=-0.2\text{meV}$ and other parameters as in Fig. 2. (b) Representative cross sections of the density plot in (a) for several values of the LUMO level energy $\epsilon$.

Figure 14

(Color online) Tunnel magnetoresistance (a) and the $z\text{th}$ component of the molecule’s total spin (b) calculated in the linear-response regime for the antiferromagnetic coupling of the SMM’s core spin with the spin of the LUMO level ($J=-0.2\text{meV}$ and other parameters as in Fig. 2). Dotted lines show the results obtained for the case of ferromagnetic exchange coupling, see Fig. 6—in (b) the dotted line corresponds to the parallel configuration.