Excursion I on the Effective Hamiltonian
IX. Single-Molecule Magnets II: Quantum Tunneling and Two-Level System
IX.2. c. The Giant-Spin Hamiltonian
Some properties of the giant-spin Hamiltonian were already discussed in chapters V.3.e. und VI. As a most important feature it was found that that the various terms should be distinguished according to their commutation with Sˆz.
, where + ⊥
=Hˆ Hˆ Hˆ
||
[
Hˆ||,Sˆz]
=0 but[
Hˆ ⊥,Sˆz]
≠0The reason is that the terms commuting with , which we also call the longitudinal terms, are the dominating energy terms (i.e. D) and, moreover, are trivially diagonalized by the spin functions
Sˆz
SM . The non-commuting terms, however, which we also call the transversal terms, do not affect the energy spectrum much but lead to a mixing of the SM states.
Effect of the Longitudinal Terms
Some statements on the longitudianl terms:
1)
( )
B z z3) the Hamilton-Matrix is diagonal in the SM basis
The D term and longitudinal Zeeman term are the most dominating terms in the Hamiltonian, and therefore set the general structure of the energy levels. For D < 0 it is
characterized by an energy barrier, with resonances which may be tuned by the magnetic field.
- 1 - Vorl. #19 (2. Feb. 2011) Molecular Nanomagnets ©Oliver Waldmann
Effect of the Transversal Terms
The typical transversal terms to be considered are
(
Sˆ Sˆ)
B Oˆ g(
Sˆ B Sˆ B)
...E Hˆ
y y x x B 4 4 4 4 2 y 2
x − + +μ + +
⊥ =
If not forbidden by the symmetry of the molecule, the E term is the dominant term in zero field, such as in the Fe8 molecule. In the Mn12 molecule the E term is (nominally) forbidden by the molecular 4-fold axis, hence the B44 term would be the dominant contribution in zero field. However, also the transversal field components of course contribute to the transversal part of the Hamiltonian.
As discussed already in chapter VI.2. the fact that the transversal terms do not commute with implies a number of consequences, which can equivalently be expressed in (at least) four ways:
Sˆz
[
Hˆ⊥,Sˆz]
≠0⇔ 1) the states SM are not eigenstates of the Hamiltonian
⇔ 2) the Hamilton matrix in the basis SM exhibits non-diagonal elements
⇔ 3) the states SM get mixed, i.e., the true eigenstates are superpositions of the SM
⇔ 4) a system which is initially in a state SM will not continue to stay in this state, i.e., it will change in time
The last formulation may make the connection of the transversal terms to quantum tunneling most clear.
Order of Magnitude of the Non-Diagonal Elements SMHˆ⊥ SM'
The effect of the transversal magnetic field is "difficult" and will be sketched in the last chapter. In zero-field, however, the effect of the transversal terms can be explained at least qualitatively by perturbation theory arguments.
As specific example we consider the E term,
(
2y) (
2 2)
2
x Sˆ ESˆ Sˆ
Sˆ
E − = + + − .
The E term can only connect states for which the magnetic quantum number M differs by two, i.e., for which M−M′=±2, that is
2 M M 2
2 Sˆ SM'
Sˆ
SM + + − ∝δ − ′=±
Hence, the E term cannot directly connect, e.g., the M = -10 and M' = +10 states involved in the level-crossing at zero field in Mn12. It can do so only through higher-order perturbation theory. In the present example of M = -10 and M' = +10, a non-zero effect would start to appear in 10th order, as it takes ten times a step by ΔM=2 to get from M = -10 to M' = +10:
10
As a result, since it emerges in 10th order, the effect will be very small. Furthermore, it can be seen, that it should become larger the smaller the difference ΔM=M−M′ is, since a non-zero effect now appears already at lower orders of perturbation theory. The increase of the effect is in fact quite drastic, since it goes exponentially.
IX.2.d. From the Giant-Spin Hamiltonian to the Two-Level Hamiltonian
At a level crossing apparently only two levels are relevant, which in the present case of a single-molecule magnet would be the levels SM and SM' . Hence, being for the moment only interested in what happens near a level crossing, we may introduce a further effective Hamiltonian which only works in the space of these two levels. This problem of two levels is encountered in many areas in quantum mechanics, and has become known as two-level system or the two-level Hamiltonian. Generalizing the discussion, we introduce the following quantities, with obvious meaning:
0 , 1 , ε0 = 0Hˆ 0 , ε1 = 1Hˆ 1 , b= 0Hˆ 1
In the middle part of the equation the two-level Hamiltonian was expressed in matrix form choosing the states 0 , 1 as basis, and on the right-hand-side in operator form, i.e., as an effective spin Hamiltonian since the Pauli matrices are trivially related to the spin operators of a spin S = 1/2 particle.
The physical meaning of the non-diagonal matrix element b ≠0 is to allow an excitation to move form "0" to "1", which depending on the particular system under consideration is called
"hopping", "tunneling", "delocalization", etc.. On a quantum mechanical level it leads to a mixing of the states 0 and 1 , which, as we will see in a moment is connected to an energy gap in the energy spectrum.
Important Comment
Since this effective Hamiltonian is derived from or related to a more "sophisticated" higher-level Hamiltionian, the quantities ε0, ε1, and b will in general depend on the parameters in the higher-level Hamiltonian (as it is the case for any effective Hamiltonian). The functional dependence may often be simple or trivial, however, it does not always have to be trivial. It in fact can be quite complicated and surprising, an example we will see below.
Comments
1) since the quantities ε0 and appear in the diagonal we can expect that they are governed by , while the non-diagonal element b should be related to
ε1
Hˆ||
Hˆ⊥
2) for our present case, these quantities will depend on the magnetic field and we expect the field dependencies t be of the form ε0 =ε0
( )
B|| , ε1 =ε1( )
B|| , and b=b( )
B⊥ .- 3 - Vorl. #19 (2. Feb. 2011) 3) for the field dependencies of ε0 and ε1 one obtains the typical Zeeman-like splitting with longitudinal magnetic field, ε0 =−gμBB|| +const and ε1 =+gμBB|| +const; for b the relation is less obvious