Current-induced switching of a single-molecule magnet with an arbitrary oriented easy axis *

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1 Materials Science-Poland, Vol. 5, No. 4, 007 Current-induced switching of a single-olecule agnet with an arbitrary oriented easy axis * M. MISIORNY 1 1, **, J. BARNAŚ 1 Departent of Physics, A. Mickiewicz University, Uultowska 85, Poznań, Poland Institute of Molecular Physics, Polish Acadey of Sciences, Soluchowskiego 17, Poznań, Poland The ain objective of this work was to investigate theoretically how tilting of an easy axis of a singleolecule agnet (SMM) fro the orientation collinear with agnetic oents of the leads affects the switching process induced by current flowing through the syste. We consider a odel syste that consists of a SMM ebedded in the nonagnetic barrier of a agnetic tunnel junction. The anisotropy axis of the SMM fors an arbitrary angle with agnetic oents of the leads (the latter ones are assued to be collinear). The reversal of the SMM s spin takes place due to exchange interaction between the olecule and electrons tunnelling through the barrier. The current flowing through the syste as well as the average z-coponent of the SMM's spin are calculated in the second-order perturbation description (Feri golden rule). Key words: single-olecular agnet; agnetic switching; spin-polarized current 1. Introduction Single-olecule agnets (SMMs) [1] draw attention as potential candidates for devices which can cobine conventional electronics with spintronics []. Characterized by a relatively large energy barrier the spin reversal of the olecule, an SMM can be used at low teperatures as a olecular eory cell [3]. For these reasons, transport through SMMs is of current interest [3 5]. It has been shown that the olecule spin can be reversed by a spin current (also in the absence of external agnetic field) [6, 7]. The phenoenon of current-induced spin switching is of great iportance for future applications. Furtherore, it is now possible to investigate experientally transport through a single olecule [8 10]. However, using present-day experiental techniues, one can hardly control the orientation of the olecule easy axis [11]. * Presented at the Conference of the Scientific Network New Materials for Magnetoelectronics MAG-EL-MAT 007, Będlewo near Poznań, 7 10 May 007. ** Corresponding autor, e-ail: barnas@au.edu.pl

2 136 M. MISIORNY, J. BARNAŚ The ain objective of this paper was to investigate theoretically how tilting of the easy axis of a SMM fro the orientation collinear with agnetic oents of the electrodes (leads) affects the switching process and current flowing through the syste.. Model The syste under consideration consists of a SMM ebedded in a nonagnetic barrier between two ferroagnetic electrodes. Electrons tunnelling through the barrier can interact via exchange coupling with the SMM, which ay result in agnetic switching of the olecule. Furtherore, we assue that the spin nuber S of the SMM does not change when current flows through the syste, i.e. the charge state of the olecule is fixed. In the case considered here, the anisotropy axis of the olecule (used as the global uantization axis z ) can for an arbitrary angle ϕ with agnetic oents of the leads. To siplify the following description, we neglect the influence of exchange interaction with the leads on the ground state of the olecule. Such an influence, however, can be included via an effective exchange field. The full Hailtonian of the syste reads: H = HSMM + HL + HR + HT The first ter describes the free SMM and takes the for HSMM = DSz, where S z is the z coponent of the spin operator, and D is the uniaxial anisotropy constant. The next two ters of the Hailtonian H correspond to the two ferroagnetic electrodes, H ε + = αa αa kα k k kα for = L (left) and = R (right). The electrodes are represented by a band of non-interacting electrons with the energy dispersion ε k α, where k denotes the wave vector, α is the electron spin index (+ for spin ajority and for spin inority + electrons), and ak α ( ak α) is the relevant creation (anihilation) operator. Finally, the last ter of the Hailtonian H stands for the tunnelling processes and is given by the Appelbau Hailtonian [1] rotated by the angle ϕ around the axis y = y (see Fig. 1) H T 1 J, = σαβ S NN, kk αβ ' ( k β ) ϕ ϕ Td L+ R + ak αak α + H.c. NN cos ak αak β sin ak αa sinϕ a αa a αa k β k k k β kk α (1) where σ = (σ x, σ y, σ z ) are the Pauli atrices, S is the SMM s spin operator, and α = α. The first ter in the above euation describes exchange interaction of the SMM and electrons in the leads, with J, denoting the relevant exchange paraeter.

3 Current-induced switching of a single-olecule agnet 137 Fig. 1. Scheatic picture of the syste under consideration for two collinear configurations of the leads agnetic oents: parallel (white arrows) and antiparallel (black arrows). The axis y (y ) is noral to the x z (x -z ) plane For the sake of siplicity, we consider only syetrical situation, where J L,L = J R,R = J L,R = J R,L J. The second ter in E. (1) represents direct tunnelling between the leads, with T d denoting the corresponding tunnelling paraeter. We also assue that J and T d are independent of energy and polarization of the leads. Finally, N ( = L, R) denotes the nuber of eleentary cells in the -th electrode. 3. Theoretical description The electric current I flowing through the syste is deterined with the use of the Feri golden rule [4, 6]. Up to the leading ters with respect to the coupling constants J and T d, the current can be expressed by the forula π L R L R I = e Td + J cos ϕ S z ( D D + D D ) V π L R 4ϕ L R 4ϕ + ej P Aη ( ) η DD cos + DD sin η=+, ϕ ϕ ζ η + + η η + ζ η + η L R 4 L R 4 ( D( 1) ev) D D sin D D cos ( D( 1) ev) sin ϕ L R L R + ( DD + DD) ζ ( D( η+ 1) + ηev) + ζ D( η+ 1) ηev 4 ( ) () where e is the electron charge (for siplicity assued e > 0, so the current is positive for electrons tunnelling fro the left to right). In the above euation D σ is the density of states (DOS) at the Feri level in the -th electrode for spin σ, and S = P, where P denotes the probability of finding the SMM in the spin z state. The voltage V is defined as the difference of the electrocheical potentials of the leads, ev = μ L μ R. Finally, we introduced: A ± () = S(S + 1) ( ± 1) and ζ(ε) = ε[1 exp( εβ)] 1 with β 1 = k B T.

4 138 M. MISIORNY, J. BARNAŚ To copute nuerically the current I fro E. (), one needs to know the probabilities P. To deterine the, the SMM s spin is assued to be saturated in the initial state S, and then voltage growing linearly in tie is applied [6]. Since the reversal process occurs through all the consecutive interediate spin states, the probabilities P can be found by solving the set of relevant aster euations, S + + { γ δ, + 1 γ δ, 1 ( γ γ ) δ, } dp c = + + dv l= S dp± S ± c = γ± S P± S + γ± S 1P± S 1 dv l l l l l l l l for S < < S, where c = V/t is the rate at which the voltage is increased. The paraeters γ ± describe the rates at which the spin z-coponent () is increased/decreased by one. These tunnelling rates have been calculated fro the Feri golden rule and have the for, P (3) ± π γ = J A± ( ) 4 ϕ L R L R cos ( ( 1) ) ( ( 1) ) DDζ D± + ± ev + DDζ D± + ev 4 ϕ L R L R + sin DDζ ( D( 1) ev) DD ζ ( D( 1) ev) ± + + ± + ± sin ϕ ± ± + 4ϕ 4ϕ L L R R + sin + cos ( DD + DD ) sin ϕ + 4 L R L R ( D D D D ) ζ ( D( 1) ev ) ζ ( D( 1) ev ) ± + L L R R ( D D D D ) ζ ( D( 1) ) (4) 4. Nuerical results and discussion Nuerical results have been obtained for the olecule Fe 8 [1, 13] corresponding to the total spin S = 10, whose anisotropy constant D is 0.9 K. Apart fro this, we assue J T d 100 ev. Calculations have been perfored for the teperature T = 0.01 K, which is below the olecule blocking teperature T B = 0.36 K, and for c = 10 kv/s. It has been also assued that both leads are ade of the sae etallic aterial characterized by the total DOS D= D + D 0.5 per electronvolt and per +

5 Current-induced switching of a single-olecule agnet 139 eleentary cell, where D + ( ) denotes the DOS of ajority (inority) electrons in the -th electrode. Furtherore, the -th electrode is described by the polarization paraeter P defined as P = ( D+ D )/( D+ + D ). The following discussion is liited to the case, where one electrode (the left one) is fully polarized, P L = 1, whereas the polarization factor of the second electrode can vary fro P R = 0 (nonagnetic) to P R = 1 (half-etallic ferroagnet). Fig.. The average value of the SMM spin S z (a) and the current I flowing through the syste (b) as functions of the voltage V, calculated for the parallel configuration of the agnetic oents of electrodes with P L = 1 and P R = 0.3 S z In Figure, we show the average value of the z-coponent of the SMM s spin, S = P, and the charge current I, calculated for several values of the angle = S ϕ and for parallel agnetic configuration of the leads. The case of ϕ = 0 (ϕ = π) corresponds to the situation when the initial SMM s spin is antiparallel (parallel) to the leads spin oents. One can note that the influence of current on the spin of a olecule gradually disappears as the angle ϕ approaches π. For ϕ < π/, the olecule s spin becoes switched fro the state S> to the state S>. The switching tie, however, becoes longer and longer as the angle ϕ approaches ϕ = π/. At ϕ = π/, which corresponds to the situation with the SMM s easy axis perpendicular to the leads agnetic oents, different olecular spin states > becoe eually probable with increasing voltage, and therefore S z 0. This is a conseuence of the fact that when the voltage exceeds the activation energy for the spin-flip process [6], the SMM undergoes transitions + to upper and lower spin states with eual rates γ = γ (see E. (4)). When ϕ > π/, the spin state of the olecule is only weakly odified by current, and reains strictly unchanged for ϕ = π. The absence of switching by positive current at large values of ϕ (for the assued paraeters) is consistent with the conclusion of

6 140 M. MISIORNY, J. BARNAŚ Ref. [6] where for collinear configurations and positive current only switching fro S> to S> states was allowed, whereas positive current had no influence on the state S>. The SMM s spin can be reversed due to exchange interaction with tunnelling electrons. The latter flip their spins and hence add to or subtract soe aount of angular oentu fro the olecule. As the angle ϕ grows, the spin orientation of tunnelling electrons seen by the olecule and conseuently also the transition rates given by E. (4) change as well. Figure b shows the current flowing in the syste as a function of the bias voltage. This current strongly depends on the orientation of the SMM s easy axis. This dependence is a conseuence of the fact that the doinant contribution to current is due to the exchange ter (first ter in E. (1)), which is sensitive to the orientation of the SMM s spin. The curves for ϕ = 0 and ϕ = π overlap (except for a sall voltage range where the switching for ϕ = 0 takes place not resolved in Fig. b). Fig. 3. The average value of the SMM s spin S z (a), and the current I flowing through the syste (b) as a function of the angle ϕ in the parallel (P) (a, b) and antiparallel (AP) (c, d) agnetic configurations for V = V and P L = 1 Figure 3 presents the spin z-coponent S z and the current I in both configurations of the leads agnetic oents, plotted as a function of the angle ϕ and calculated for V = V. For ϕ < π/, the spin switching takes place in both parallel and antiparallel agnetic configurations. Figure 3 also indicates that the current at ϕ = π/ is independent of the agnetic configuration as well as on the polarization paraeters of the leads. In conclusion, we have shown that tilting the easy axis of a SMM fro the collinear orientation relative to the leads agnetic oents has a significant influence on

7 Current-induced switching of a single-olecule agnet 141 the reversal process of the olecule s spin, as well as on current flowing through the syste. Acknowledgeents This work was supported by funds fro the Ministry of Science and Higher Education as a research project in One of us (M.M.) also acknowledges support fro the MAGELMAT network. References [1] GATTESCHI D., SESSOLI R., Angew. Che. Int. Ed., 4 (003), 68. [] JOACHIM C., GIMZEWSKI J.K., AVIRAM A., Nature, 408 (000), 541. [3] TIMM C., ELSTE F., Phys. Rev. B, 73 (006), [4] KIM G.-H., KIM T.-S., Phys. Rev. Lett., 9 (004), [5] ELSTE F., TIMM C., Phys. Rev. B, 73 (006), [6] MISIORNY M., BARNAŚ J., Phys. Rev. B, 75 (007), [7] ELSTE F., TIMM C., Phys. Rev. B, 75 (007), [8] HEERSCHE H.B., DE GROOT Z., FOLK J.A., VAN DER ZANT H.S.J., ROMEIKE C., WEGEWIJS M.R., ZOBBI L., BARRECA D., TONDELLO E., CORNIA A., Phys. Rev. Lett., 96 (006), [9] JO M.-H., GROSE J.E., BAHETI K., DESHMUKH M., SOKOL J.J., RUMBERGER E.M., HENDRICKSON D.N., LONG J.R., PARK H., RALPH D.C., Nano Lett., 6 (006), 014. [10] HENDERSON J.J., RAMSEY C.M., DEL BARCO E., MISHRA A., CHRISTOU G., J. Appl. Phys., 101 (007) 09E10. [11] TIMM C., Phys. Rev. B, 76 (007), [1] APPELBAUM J., Phys. Rev., 17 (1966), 91 [13] WERNSDORFER W., SESSOLI R., Science, 84 (1999), 133. Received 30 April 007

arxiv:cond-mat/ v1 [cond-mat.mes-hall] 17 Nov 2005

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