Level splittings in exchange-biased spin tunneling

The level splittings in a dimer with the antiferromagnetic coupling between two single-molecule magnets are calculated perturbatively for arbitrary spin. It is found that the exchange interaction between two single-molecule magnets plays an important role in the level splitting. The results are discussed in comparison with the recent experiment. 75.45.+j, 75.50.Xx, 75.50.Tt Typeset using REVTEX The quantum properties of single-molecule magnets have generated considerable interest over the past decade in connection with macroscopic quantum phenomena [1]. High-spin molecules with spin-10, Mn12 and Fe8 have been such good candidates because all the clusters are identical with no dispersion on the size of the clusters and the number of interaction spins, and the spin ground state and the magnetic anisotropy are known with great accuracy. These molecules display particularly interesting phenomena such as quantum resonant tunneling [2,3] and quantum phase interference [4]. Such phenomena have received much attention, both theoretically and experimentally in view of macroscopic realization of quantum tunneling, and also because of some potential application to quantum computing [5]. Many efforts have been made to understand their mechanisms by considering a giant spin Hamiltonian with a single-molecule magnet [2,3,5,6]. Most of the study have neglected exchange interactions that depend on the distance and the non-magnetic atoms in the exchange pathway. Recently, however, it has been reported that a supramolecular single-molecule magnet dimer with antiferromagnetic coupling exhibits quantum behavior different from that of the individual single-molecule magnets [7]. This result implies that exchange interaction between two single-molecule magnets can have a large influence on the quantum properties of single-molecule magnets. It is therefore important to understand the effect of the exchange interaction on magnetization tunneling. The issue of spin tunneling with the exchange interaction has been raised by several groups [8]. In their studies exchange interaction is enhanced to magnetic anisotropy for studying tunneling of the Néel vector in antiferromagnetic particles. Using the instanton technique based on spin coherent state path integral, they calculated the tunneling rate of the Néel vector in uniaxial or biaxial antiferromagnetic particles. However, the previous works applicable in the limit S ≫ 1 have been confined to the spin tunneling of the ground state in an antiferromagnetic particle having two collinear ferromagnetic sublattices. In this paper, we will study magnetic tunneling in a system of identical, antiferromagnetically coupled dimer. By employing a perturbative approach [9], we obtain the level splitting of the states degenerate pairwise for arbitrary spin in some typical cases and show that even 2 weak exchange interaction plays a crucial role in inducing spin tunneling. The spin Hamiltonian of the dimer system can be written in the form H = H1 +H2 + JŜ1 · Ŝ2, (1) where Hi (i = 1, 2) is the Hamiltonian of each single-molecule magnet which can be modeled as a giant spin of Si. The corresponding Hamiltonian is given by Hi = −DŜ 2 zi +H trans i −HzŜzi, (2) where D is the anisotropy constant and H i includes the transverse anisotropy or field. Also, H stands for gμBH where g is the electronic g-factor and μB is the Bohr magneton. Henceforth, we will usually drop the combination gμB for better readability of the formula. Since the dimer consists of two single-molecule magnets with antiferromagnetic coupling, we take J > 0 much less than the anisotropy constant D. The system has (2S1 + 1)(2S2 + 1) degenerate energy levels which in the absence of the transverse terms of Eq. (1) are labeled by the spin projectionM1 andM2 on the z-axis and given by EM1,M2 = −D(M 2 1+M 2 2 )+JM1M2. It can be easily checked that for the longitudinal field Hz satisfying Hz = D(M 1 +M 2 2 −M ′ 1 2 −M ′ 2 ) + J(M ′ 1M ′ 2 −M1M2) M ′ 1 +M ′ 2 −M1 −M2 , (3) the energy levels are degenerate: EM ′ 1 ,M ′ 2 = EM1,M2. (4) Tunneling among the (2S1 + 1)(2S2 + 1) energy states is allowed by the transverse terms containing Ŝxi and Ŝyi. In the case of small transverse terms which is relevant for the dimer, the level splittings can be calculated in a more direct and simple way using the high-order perturbation theory. In such cases, the level splitting of the degenerate level pair (M ′ 1,M ′ 2) and (M1,M2) is represented as the shortest chain of matrix elements and energy denomenators connecting the states |M ′ 1,M ′ 2〉 and |M1,M2〉 for the typical situations which will be considered. 3 Let us consider as model I the level splitting induced by the transverse terms in the exchange interaction: H = −DŜ z1 −DŜ 2 z2 + JŜ1 · Ŝ2. (5) Noting that Ŝ1 ·Ŝ2 = Ŝ1zŜ2z+ 1 2 (Ŝ1+Ŝ2−+Ŝ1−Ŝ2+) and considering Ŝ1−Ŝ2+, the level splitting of the degenerate pair (M ′ 1,M ′ 2), (M1,M2) appears only in the chain of matrix elements with connecting the states |M ′ 1 + k,M ′ 2 − k〉 and |M ′ 1 + k + 1,M ′ 2 − k − 1〉 where M ′ 1 = −M1, M ′ 2 = M1 > 0, M2 = −M1, and k is an integer with 0 ≤ k ≤ M1 −1−M ′ 1. It corresponds to the level splitting of the degenerate pair (−M1,M1) → (M1,−M1). In this case the magnetic field does not contribute to the level splitting and thereby the longitudianl field (3) is not taken into consideration. Then, the level splitting of the degenerate pair becomes ∆EM ′ 1 M ′ 2 ,M1M2 = 2VM ′ 1M ′ 2,M ′ 1+1,M ′ 2−1 1 EM ′ 1 +1,M ′ 2 −1 − EM ′ 1 M ′ 2 VM ′ 1 +1,M ′ 2 −1,M ′ 1 +2,M ′ 2 −2 × 1 EM ′ 1 +2,M ′ 2 −2 − EM ′ 1 M ′ 2 ...VM1−1,M2+1,M1M2 , (6)