[Retracted] Quadrupole Excitation in Tunnel Splitting Oscillation in Nanoparticle 𝐌𝐧₁₂

Abstract

We analyze the interference between tunneling paths that occur for a spin system with special Hamiltonian both for dipole and quadrupole excitations. Using an instanton approach, we find that as the strength of the second-order transverse anisotropy is increased, the tunnel splitting for both excitations is modulated, with zeros occurring periodically and the number of quenching points for quadrupole excitation decreasing. This effect results from the interference of four tunneling paths connecting easy-axis spin orientations and occurs in the absence of any magnetic field.

1. Introduction

Berry phase effects play an important role in spin dynamics. Tunneling of a spin (or magnetic particle) between degenerate orientations can be calculated by such Berry phase effects via the interference between tunneling paths, with quenching of the tunnel splitting occurring when tunneling paths destructively interfere [1, 2].

The first conclusive experimental evidence of tunneling in molecular magnet Mn₁₂O₁₂(CH₃COO)₁₆(H₂O)₄ called Mn₁₂ was provided by Friedman [3]. He applies a magnetic field along the easyaxis of the crystal at a variety of temperatures and found a curve punctuated by sharp steps at regular intervals in the parameter 𝜆. Also this observation that is confirmed by Hernandez et al. [4] and Thomas et al. [5] shows a series of steps in the hysteresis loops in Mn₁₂ in low temperature. The steps observed in the hysteresis loops at nearly equal intervals of magnetic field are due to enhanced relaxation of the magnetization at the resonant fields when levels on opposite side of the anisotropy barrier coincide in energy (see Figure 1).

(a)

(b)

Figure 1 

The first curve was obtained from numerical calculation of the tunnel splitting for Mn12 and the second curve is the magnetization plot of this magnetic molecule in low temperature.

The common feature of molecular magnets responsible for their interesting behavior is the strong spin-spin coupling between metallic ions in the core of the molecule. In this magnet molecule, there are 4 Mn⁴⁺ atoms (spin 3/2) surrounded by 8 Mn³⁺ atoms (spin 2), and they retain a relative orientation such as to give the whole molecule a spin of 10 (= 8(2)−4(3/2)) [3].

The Mn₁₂ single-molecule magnet has a fourfold transverse magnetic anisotropy and displays resonant tunneling between two easy-axis orientations [3]. Here we consider the interference that occurs between tunneling paths in such a system and how that interference can be modified by the presence of a second-order transverse anisotropy perturbation and multipole excitations. Such a perturbation could potentially be induced by the application of uniaxial pressure to a sample of Mn₁₂. We find that for both excitations the tunnel splitting is periodically quenched as a function of the strength of the perturbation. This interference effect takes place in the absence of any magnetic field. In the present study, the fourth-order anisotropy term is the primary transverse anisotropy in the problem, producing four interfering paths, but the interference is modulated by the strength of the second-order anisotropy, leading to periodic quenches as that term is varied. In particular, we consider a spin governed by the Hamiltonian𝐻=𝐽²̂−𝐽²_𝑧̂+𝑘𝐽⁴₊̂+𝐽⁴₋̂+𝜆𝐽²_𝑥̂−𝐽²_𝑦̂, (1) where the 𝐽𝑠̂ are standard spin operators. We restrict the values of 𝑘 and 𝜆 such that the z-axis is the spin easy axis. In the above Hamiltonian, the leading term −𝐽²_𝑧̂ established the double well potential and the easy axis and the term fourth-order transverse anisotropy 𝑘(𝐽⁴₊̂+𝐽⁴₋̂) are responsible for 4-fold rotational symmetries. The third term is a second-order transverse anisotropy that is present in many low-symmetry SMMs.

In this paper, firstly we calculated periodically depended of spin tunneling for Mn₁₂ in SU(2) group, in other word, we considered only dipole excitation in Hamiltonian. Due to the symmetry of spin operators in Hamiltonian, for being more accurate, we have to consider other multipole excitation. Then, we consider both dipole and quadrupole excitations.

2. Theory and Calculations

The instanton method is an efficient way of calculating tunnel splitting, both for particles [6] and for spin [7]. It is based on evaluating the path integral for a certain propagator in the steepest-descent approximation and is designed to be asymptotically correct in the semiclassical limit (𝐽→∞ or ℏ→0). Instantons are classical paths that run between degenerate classical minima of the energy. By classical we mean that the path obeys the principle of least action and satisfies energy conservation. However, a path along which energy is conserved can not run between two minima and still have real coordinates and momenta. Hence, one must enlarge the notion of a classical path and allow the coordinates and/or momenta to become complex.

In the continuous approach the tunnel splitting can be computed as the functional integral [8]exp(−𝑆(𝑥̇,𝑥,𝑡))𝑑𝜇(𝑥), (2) where 𝑑𝜇(𝑥)=∏^𝑁_𝑖=1𝑑𝑥_𝑖. In Su(2) group 𝑁=2 and in Su(3) group 𝑁=4, and 𝑁 is the number of degree of freedoms, and the action is given by𝑆(𝑥̇,𝑥,𝑡)=𝑆_𝐾+𝑆_𝐷+𝑆_𝐵. (3)

The first term, 𝑆_𝐾, is kinetic term which has the properties of a Berry phase, which can give rise to interference between different trajectories. The second term, 𝑆_𝐷, is dynamical term and the final term is boundary term this term depends explicitly on the boundary values of the path.

When the functional action is specified, the instanton recipe for calculating the tunnel splitting is as follows. Let there be a number of instantons, that is, least action paths, labeled by k, and let the actions for these various paths be 𝑆_𝑘^cl. The tunneling amplitude is given by Δ=_k𝐷_𝑘exp−𝑆^cl_k. (4)

The prefactor 𝐷_𝑘 results from integrating the Gaussian fluctuations around the kth instanton.

For nanoparticle Mn₁₂, the tunneling amplitude calculated and obtained the following relation [3]:Δ≈4𝑒^−𝑆𝑅cos(𝐽𝜋)cos𝐽𝑆_𝐼, (5) where𝑆_𝐼=Re^3𝜋/2_𝜋/2𝑃𝜙𝑑𝜙, (6) and 𝑃(𝜙) is solution of classical enrgy equation. 𝑆_𝑅 is the real part of the Euclidean action and the same for all instantons. The factor cos(𝐽𝜋) ensures that the tunnel splitting is zero for half-integer spin.

The classical energy is (the x-axis is chosen as the azimuth because the-path will not encounter it and 𝜙 is measured from the z axis)𝐸_cl𝜃,𝜙=𝐽²1−sin²𝜃sin²𝜙+𝜆𝐽²cos²𝜃−sin²𝜃cos²𝜙+2𝑘𝐽⁴cos⁴𝜃+sin⁴𝜃cos⁴𝜙−6cos²𝜃sin²𝜃cos²𝜙. (7)

The minimum of energy obtained at𝜋𝜃=2𝜋𝜙=2,3𝜋2. (8)

The minima of enregy are𝐸_cl𝜋2,𝜋2𝐸=0_cl𝜋2,3𝜋2=0. (9)

Making the substitution 𝑃=cos𝜃, the instanton must satisfy the constraint𝐽²1−1−𝑃²sin²𝜙+𝜆𝐽²𝑃²−1−𝑃²cos²𝜙+2𝑘𝐽⁴𝑃⁴+1−𝑃²²cos⁴𝜙−6𝑃²1−𝑃²cos²𝜙=0. (10)

The four solutions for the above equation are:𝑃𝜙=±𝐴𝜙±𝑖𝐵𝜙𝐶𝜙, (11) where𝐴𝜙=2𝑎cos²𝜙7+cos2𝜙−𝜆1+cos²−sin²𝜙𝐵𝜙=𝑎cos²𝜙×−𝜆1+𝑘−𝜆+𝑘cos2𝜙35+28cos2𝜙+cos4𝜙1+cos²𝜙−2𝑎cos²𝜙7+cos2𝜙+sin²𝜙²𝐶𝜙=4𝑎cos⁴𝜙+6cos²𝜙+1, (12) and 𝑎=𝐽²𝑘.

Solutions (11), despite conserving energy, fail to meet the boundary conditions 𝑃(𝜋/2)=𝑃(3𝜋/2)=0. These solutions gives rise to what Garg called boundary jump instantons [9]. As a final note, we consider only the following value of 𝑃(𝜙):𝑃𝜙=𝐴𝜙+𝑖𝐵𝜙𝐶𝜙. (13)

Substituting above relation in (6) and using from (5), splitting of level energy calculated. Up this section, we have just considered the dipole excitation of spin system. Because we will analyze the periodical dependence of spin tunneling, by relation (5), plotted function Cos(𝐽𝑆_𝐼) versus 𝜆.

If quadrupole excitation is considered, the classical energy is changed in the following form:𝐸_cl𝜃,𝜙=𝐽²1−sin²𝜃cos²2𝑔sin²𝜙+𝜆𝐽²cos²2𝑔cos²𝜃−sin²𝜃cos²𝜙+2𝑘𝐽⁴cos⁴2𝑔cos⁴𝜃+sin⁴𝜃cos²𝜙−6cos²𝜃sin²𝜃cos²𝜙. (14)

The minimum of energy obtained at𝜋𝜃=2,𝜋𝑔=4,𝜋𝜙=2,3𝜋2. (15)

Also minima of enrgy are𝐸_cl𝜋2,𝜋2,𝜋4𝐸=0_cl𝜋2,3𝜋2,𝜋4=0. (16)

Making the substitution 𝑃=Cos𝜃, the instanton must satisfy the constraint:𝐽²1−1−𝑃²cos²𝑔sin²𝜙+𝜆𝐽²cos²2𝑔𝑃²−1−𝑃²cos²𝜙+2𝑘𝐽⁴cos⁴2𝑔𝑃⁴+1−𝑃²²cos⁴𝜙−6𝑃²1−𝑃²cos²𝜙=0. (17)

Similar to previous section, solution that has the following form is considered:𝑃𝜙,𝑔=𝐴𝜙,𝑔+𝑖𝐵𝜙,𝑔𝐶,𝜙,𝑔 (18) where 𝐴𝜙,𝑔=2𝑎cos²𝜙sec²−12𝑔7+cos2𝜙21+3𝜆+(𝜆−1)cos2𝜙sec⁴𝐵2𝑔𝜙,𝑔=8𝑎1+6cos²𝜙+cos⁴𝜙×1+2𝑘cos⁴2𝑔cos⁴𝜙−cos²2𝑔𝜆cos²𝜙+sin²𝜙−𝜆+sin²𝜙+cos²𝜙𝜆−2𝑎cos²2𝑔7+cos2𝜙²𝐶𝜙,𝑔=4𝑎1+6cos²𝜙+cos⁴𝜙cos²2𝑔. (19)

Substituting relation (19) in (6) and using (5), splitting of energy level was calculated. In this section, we consider both dipole and quadrupole excitations of spin system. Like the previous section, plotted function cos(𝐽𝑆_𝐼) versus 𝜆. In this plot, 𝑎=10⁻⁴ and 𝑘=10⁻⁶ are considered.

As seen from Figures 2 and 3, when we added the quadrapole excitation to Hamilton, the number of quenching point reduced from 5 to 4, also by using Figure 1 we see that the number of steps is 4; that is equal to the number of quenching points and then these steps in magnetization graphic are quenching point is in spin tunneling phenomena.

Figure 2 

Red curve is Re(𝑃)/𝜋 and blue curve is cos(𝐽𝑆𝐼) versus 𝜆 obtained by using (11). In this graph, only dipole excitation was considered.

Figure 3 

Red curve is Re(𝑃)/𝜋 and blue curve is cos(𝐽𝑆𝐼) versus 𝜆 obtained using (16). In this graph both dipole and quadrupole excitations are considered.

As mentioned in the introduction, there are step changes in the experimental hysteresis loop of this crystal that have been explained considering only dipolar interaction between molecules. For a more accurate description of the experimental results, we found that we have to consider both dipolar and quadrupole excitations.

3. Discussion

In this paper, we consider the spin tunneling phenomena to parameter 𝜆 in both groups SU(2) and SU(3). Graph of function Re(𝑃)versus 𝜆 in SU(2) group is a straight line started from 5 and with increasing 𝜆 up to 0.04, this straight line decreases and in point 𝜆=0.04 the value is zero and remains in this value. But above function graph in SU(3) group, started less than 4, is not straight line and in point 𝜆=0.04 the value is zero and remains in this value. The number of quenching points in SU(2) group (dipole excitation) is 5, these points decreases to 4, when quadrupole excitation is considered.