Jupyter notebook of the superexchange + double exchange model of INS intensity spectra
Neutron scattering spectrum for a superexchange and double exchange model
J. K. Glasbrenner
August 3, 2017
Table of contents
Exchange model Hamiltonian
We adopt the following ansatz as our Hamiltonian for a two spin system,
where the first term corresponds to superexchange as defined in Ref. 1 and the second term to double exchange as defined in Ref. 2.
We are modeling the exchange interaction of a pair of manganese atoms, which in the atomic limit — and ignoring spin-orbit coupling — can be idealized as a system of two identical spin-5/2 particles. It is convenient to work with Pauli matrices when taking a computational approach. The Pauli matrices for a spin-5/2 particle 3 are:
The total spin operator is a three-vector as follows:
Let \(\textbf{S}_1\) be the spin operator for particle 1 and \(\textbf{S}_1\) be the spin operator for particle 2. These operators are tensor products as follows:
where I is the identity matrix. For shorthand, we define:
We then write the rest of the relevant two-spin operators, which are
We implement the above operators in Python as the following SpinOperators
class. This class also solves the eigenvalue problem for the two-spin operators \(\textbf{S}_1^2\), \(\textbf{S}_2^2\), \(\textbf{S}^2\), \(\textbf{S}_z\), and \(2 \textbf{S}_1 \cdot \textbf{S}_2\).
Our goal is to model the inelastic neutron scattering (INS) spectra in the figure below
The main features we would like to explain using a model and our DFT calculations are
The peak around 13.3 meV for Ba(Zn0.85Mn0.15)2As2.
The broadened peak around 13.3 meV for Ba0.8K0.2(Zn0.85Mn0.15)2As2.
The outline for the rest of this notebook is as follows. First, we focus on the superexchange term and derive the energy differences for the \(\Delta{}S_z = \pm 1\) transitions, which are what the INS experiment detects. Next, we focus on the double exchange term, show that we can eliminate the infinite sum for a system of two spin-5/2 particles, and obtain the energy for different eigenvalues of \(\textbf{S}z\). Finally, we combine the results together by assuming that the superexchange and double exchange terms are independent such that the energy contributions from each can be simply summed together. We then use our DFT calculations to fit the superexchange \(J^{SE}\) and double exchange \(t\) parameters. We then assume that the superexchange parameter \(J^{SE}\) remains constant under doping and that we should multiply \(J^{SE}\) and \(t\) by 0.548 and 0.529 respectively to take into account magnetic moment softness. This allows us to get an approximate estimate of how double exchange and moment softness modifies the \(\Delta{}S{z} = \pm 1\) spectra. The notebook concludes with an interactive model of the INS spectra, assuming that Gaussian peaks, whose widths and amplitudes are tunable empirical parameters, are centered around the model's predicted \(\Delta{}S_{z} = \pm 1\) spectra.
Superexchange term
It's well-known that the eigenvalues for the superexchange term \(2 J^{SE} \textbf{S}_1 \cdot \textbf{S}_2\) are given by the expression \(J^{SE} \left[ S_z \left( S_z + 1 \right) - S_1 \left( S_1 + 1 \right) - S_2 \left( S_2 + 1 \right) \right]\). Even though the result is known, there is pedagogical worth in carrying out the analysis and reproducing the result, so let's do that.
The eigenvalue problem is already solved in the SpinOperators
class from the previous section, so the reader is referred to the code for the details. The eigenvalues for the different two-particle operators are as follows
\(\textbf{S}_1^2\) eigenvalues
Eigenvalue | Degeneracy |
---|---|
35/4 | 36 |
\(\textbf{S}_2^2\) eigenvalues
Eigenvalue | Degeneracy |
---|---|
35/4 | 36 |
\(\textbf{S}^2\) eigenvalues
Eigenvalue | Degeneracy |
---|---|
0 | 1 |
2 | 3 |
6 | 5 |
12 | 7 |
20 | 9 |
30 | 11 |
\(\textbf{S}_z\) eigenvalues
Eigenvalue | Degeneracy |
---|---|
-5 | 1 |
-4 | 2 |
-3 | 3 |
-2 | 4 |
-1 | 5 |
0 | 6 |
1 | 5 |
2 | 4 |
3 | 3 |
4 | 2 |
5 | 1 |
\(2 \textbf{S}_1 \cdot \textbf{S}_2\) eigenvalues
Eigenvalue | Degeneracy |
---|---|
-35/2 | 1 |
-31/2 | 3 |
-23/2 | 5 |
-11/2 | 7 |
5/2 | 9 |
25/2 | 11 |
Using the above tables, you can quickly confirm that the \(2 \textbf{S}_1 \cdot \textbf{S}_2\) eigenvalues match the predictions of the analytic expression.
The two-spin operators \(\textbf{S}^2\) and \(\textbf{S}_z\) commute with \(\textbf{S}_1 \cdot \textbf{S}_2\), and so we know that the eigenvectors for \(\textbf{S}_1 \cdot \textbf{S}_2\) will diagonalize \(\textbf{S}^2\) and \(\textbf{S}_z\). We can use this to associate the ferromagnetic and antiferromagnetic configurations from our DFT calculations with two eigenvectors from this model.
First, we claim that the following eigenvector can be associated with the antiferromagnetic configuration
This eigenvector has the following eigenvalue for \(2 \textbf{S}_1 \cdot \textbf{S}_2\)
the following eigenvalue for \(\textbf{S}^2\)
and the following eigenvalue for \(\textbf{S}_z\)
This is in line with what we would expect with an antiferromagnetic state. This state is a singlet, it has the lowest energy, and the eigenvalues for \(\textbf{S}^2\) and \(\textbf{S}_z\) are consistent with a state where half the spins point up and the other half point down.
Next, we claim that the following eigenvector can be associated with the ferromagnetic configuration
This eigenvector has the following eigenvalue for \(2 \textbf{S}_1 \cdot \textbf{S}_2\)
the following eigenvalue for \(\textbf{S}^2\)
and the following eigenvalue for \(\textbf{S}_z\)
This is in line with what we would expect with a ferromagnetic state. It has the highest energy and the eigenvalues for \(\textbf{S}^2\) and \(\textbf{S}_z\) are consistent with a state where all the spins are pointing in the same direction. During subsequent analysis, we will refer to the state with \(2 S_1 S_2 = -35/2 \), \(S^2 = 0\), and \(S_z = 0\) as the antiferromagnetic state and the state with \(2 S_1 S_2 = 25/2 \), \(S^2 = 30\), and \(S_z = 5\) as the ferromagnetic state.
Now that we have the eigenvalues, we want to find the change in energy for all \(\Delta{}S_z = \pm 1\) transitions. The analytic expression for calculating the transition energy is \(J^{SE} \left[ S_z' \left( S_z' + 1 \right) - S_z \left( S_z + 1 \right) \right]\). This results in the following transition energies
Sz → S'z | ΔE |
---|---|
-5 → -4 | -8JSE |
-4 → -3 | -6JSE |
-3 → -2 | -4JSE |
-2 → -1 | -2JSE |
-1 → 0 | 0 |
0 → 1 | 2JSE |
1 → 2 | 4JSE |
2 → 3 | 6JSE |
3 → 4 | 8JSE |
4 → 5 | 10JSE |
Double exchange term
We start with the two spin double exchange Hamiltonian ansatz defined in equation (5.213) in Ref. 2,
In order to show that this is a valid way to model doube exchange, we need to complete two tasks:
The infinite summation can be exactly replaced with a six-term sum for a system of two spin-5/2 particles.
There is a solution to the six-term reformulated model that yields the double exchange energy \(E^{DE} = -t \dfrac{\left\lvert S_z \right\rvert}{S_1 + S_2 + 1}\), where the eigenvalues \(S_1 = S_2 = 5/2\).
For task 1, we follow the procedure on pages 223-226 of Nolting and Ramakanth's book [2]. For convenience, we define the eigenvalue \(S_1 = S_2 \equiv S_{1,2}\). We start by calculating the values of \(\dfrac{1}{\hbar{}^{2}} \left( \textbf{S}1 \cdot \textbf{S}2 \right) = \dfrac{1}{2} S_z \left(S_z + 1\right) - S{1,2} \left( S{1,2} + 1 \right) \) for all values of \(S\) using the following helper functions,
We then run the calculation to get the following table
S | ℏ-2 (S1 ⋅ S2) |
---|---|
0 | -35/4 |
1 | -31/4 |
2 | -23/4 |
3 | -11/4 |
4 | 5/4 |
5 | 25/4 |
Following the procedure of Nolting and Ramakanth, we write down a second ansatz
If a set of coefficients \(\left( \alpha, \beta, \gamma, \delta, \epsilon, \zeta \right)\) exists for a system of two spin-5/2 particles, then all terms with powers higher than \( \left( \textbf{S}{1} \cdot \textbf{S}{2} \right)^5 \) can be re-expressed as products of the lower-order terms, resulting in a six-term Hamiltonian. To show this, we implement the second ansatz and the method of solving for the coefficients in the functions below:
As a sanity check, let's test out the above functions against the \(S_{1,2} = 1\) case, where we know that the coefficients should be
coeff | value |
---|---|
α | 2 |
β | 1 |
γ | -2 |
The coefficients match as expected, so now we apply the same procedure to our ansatz of interest:
The solution exists, which means that we can satisfy the second ansatz as follows
Thus, the series for \(H^{DE}\) with \(S_{1,2} = \dfrac{5}{2}\) terminates after the quintic term.
For task 2, we continue to follow the lead of Nolting and Ramakanth, writing down the following new six-term ansatz
In task 2, we specified that the eigenvalues for the double exchange interaction need to be
which means our Hamiltonian should yield the following eigenvalues when \(S_z\) equals the following values:
We follow a similar procedure as before. This time, we will solve a series of equations and obtain the constants \(J_0(5/2)\) through \(J_5(5/2)\). The functions below construct and solve this series of equations:
As a sanity check, let's again test out the above functions against the \(S_{1,2} = 1\) case, where we know that the constants should be
coeff | value |
---|---|
J0(1) | 5/9 |
J1(1) | ℏ-2/6 |
J2(1) | - ℏ-4/18 |
The coefficients match as expected, so now we apply the same procedure to our ansatz of interest:
The coefficients are therefore
coeff | value |
---|---|
J0(5/2) | 1267325/1990656 |
J1(5/2) | 601247ℏ-2/17418240 |
J2(5/2) | -86357ℏ-4/10886400 |
J3(5/2) | -1633ℏ-6/2721600 |
J4(5/2) | 43ℏ-8/272160 |
J5(5/2) | ℏ-10/48600 |
and the double exchange Hamiltonian for a pair of spin \(\textbf{S} = \frac{5}{2}\) particles is
We implement the double exchange Hamiltonian into the function below
How double exchange affects the superexchange transition spectra
We now combine the superexchange term and the double exchange term and analyze how this affects the \(\Delta{}S_z = \pm 1\) transition spectra.
First, we will proceed by assuming that we can treat the superexchange and double exchange terms as independent. We've already solved both systems separately in the previous sections, so all we need to do is add together the energies. We construct a table showing how the energy spectra changes with \(S_z\):
Sz | ESE | EDE | ESE + EDE |
---|---|---|---|
-5 | 5JSE/2 | -5t/6 | \( \dfrac{5J^{SE}}{2} - \dfrac{5t}{6} \) |
-4 | -11JSE/2 | -2t/3 | \( -\dfrac{11J^{SE}}{2} - \dfrac{2t}{3} \) |
-3 | -23JSE/2 | -t/2 | \( -\dfrac{23J^{SE}}{2} - \dfrac{t}{2} \) |
-2 | -31JSE/2 | -t/3 | \( -\dfrac{31J^{SE}}{2} - \dfrac{t}{3} \) |
-1 | -35JSE/2 | -t/6 | \( -\dfrac{35J^{SE}}{2} - \dfrac{t}{6} \) |
0 | -35JSE/2 | 0 | \( -\dfrac{35J^{SE}}{2} \) |
1 | -31JSE/2 | -t/6 | \( -\dfrac{31J^{SE}}{2} - \dfrac{t}{6} \) |
2 | -23JSE/2 | -t/3 | \( -\dfrac{23J^{SE}}{2} - \dfrac{t}{3} \) |
3 | -11JSE/2 | -t/2 | \( -\dfrac{11J^{SE}}{2} - \dfrac{t}{2} \) |
4 | 5JSE/2 | -2t/3 | \( \dfrac{5J^{SE}}{2} - \dfrac{2t}{3} \) |
5 | 25JSE/2 | -5t/6 | \( \dfrac{25J^{SE}}{2} - \dfrac{5t}{6} \) |
In the superexchange section, we showed that the \(2 S_1 S_2 = -35/2 \), \(S^2 = 0\), and \(S_z = 0\) state can be associated with the antiferromagnetic configuration and the state \(2 S_1 S_2 = 25/2 \), \(S^2 = 30\), and \(S_z = 5\) with the ferromagnetic one. This means that the energy cost for a full spin is:
We apply this model to (Ba1-xKx)(Zn, Mn)2As2 and consider the case for a nearest-neighbor pair of Mn atoms for doping levels of x = 0, x = 0.2, and x = 0.4. The VASP calculated spin flip energies are
x | E(AFM) - E(FM) (meV) |
---|---|
0.0 | 407 |
0.2 | 207 |
0.4 | 148 |
For x = 0, there is no double exchange and thus t = 0. We can calculate the superexchange constant \(J^{SE}\) for \(0 \to 5\):
This yields a superexchange parameter of \(J^{SE} = 13.6\) meV. Let's assume that \(J^{SE}\) is relatively unchanged by doping. This allows us to solve for \(t\) for the x = 0.2 and x = 0.4 cases.
Now that we have \(J^{SE}\) and \(t\), we can predict the \(\Delta S_z = \pm 1\) transition energies. The function below implements this calculation:
We run the calculation using parameters from above, and get
For x = 0.0, J = 13.6, t = 0
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -108.8 | -108.8 |
-4 | -3 | -81.6 | -81.6 |
-3 | -2 | -54.4 | -54.4 |
-2 | -1 | -27.2 | -27.2 |
-1 | 0 | 0 | 0 |
0 | 1 | 27.2 | 27.2 |
1 | 2 | 54.4 | 54.4 |
2 | 3 | 81.6 | 81.6 |
3 | 4 | 108.8 | 108.8 |
4 | 5 | 136 | 136 |
For x = 0.2, J = 13.6, t = 241
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -68.6 | -108.8 |
-4 | -3 | -41.4 | -81.6 |
-3 | -2 | -14.2 | -54.4 |
-2 | -1 | 13 | -27.2 |
-1 | 0 | 40.2 | 0 |
0 | 1 | -13 | 27.2 |
1 | 2 | 14.2 | 54.4 |
2 | 3 | 41.4 | 81.6 |
3 | 4 | 68.6 | 108.8 |
4 | 5 | 95.8 | 136 |
For x = 0.4, J = 13.6, t = 312
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -56.8 | -108.8 |
-4 | -3 | -29.6 | -81.6 |
-3 | -2 | -2.4 | -54.4 |
-2 | -1 | 24.8 | -27.2 |
-1 | 0 | 52 | 0 |
0 | 1 | -24.8 | 27.2 |
1 | 2 | 2.4 | 54.4 |
2 | 3 | 29.6 | 81.6 |
3 | 4 | 56.8 | 108.8 |
4 | 5 | 84 | 136 |
Quantitatively, these don't align all that well with the neutron scattering peaks, especially for x = 0.0. In that case, we're off by a factor of 2. However, an interesting outcome of this is that including double exchange lifts the degeneracy between \(S_z = -1\) and \(S_z = 0\). And, since INS picks up both \(\Delta{}S_z = 1\) and \(\Delta{}S_z = -1\), both negative and positive energies will be detected. We already see that, in the x = 0.2 case, two different peaks can be centered nearby each other, leading to overlap. Overlapping peaks can be a potential source of the strong peak broadening we see in the INS data.
We should note that the previous analysis did not take into account magnetic moment softening. The DFT local Mn moments in the undoped case are approximately 3.7 \(\mu_B\) each, which is significantly reduced from the ideal 5 \(\mu_B\) case. We can make a back of the envelope-style estimate for how much this will reduce the eigenvalues of the two-spin operators:
What this means is, if we assume that the basic physics of the two-spin model with spin-5/2 particles is correct, but that hybridization reduces the effective exchange coupling strength, then a direct fit of this model to the DFT energies will yield parameters \(J^{SE}\) and \(t\) that are too large by a factor of nearly two. A quick fix then is to reduce the superexchange parameter \(J\) and the hopping parameter \(t\) using the above ratios:
For x = 0.0, J = 7.452800000000001, t = 0
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -59.6 | -59.6224 |
-4 | -3 | -44.7 | -44.7168 |
-3 | -2 | -29.8 | -29.8112 |
-2 | -1 | -14.9 | -14.9056 |
-1 | 0 | 0 | 0 |
0 | 1 | 14.9 | 14.9056 |
1 | 2 | 29.8 | 29.8112 |
2 | 3 | 44.7 | 44.7168 |
3 | 4 | 59.6 | 59.6224 |
4 | 5 | 74.5 | 74.528 |
For x = 0.2, J = 7.452800000000001, t = 127.489
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -38.4 | -59.6224 |
-4 | -3 | -23.5 | -44.7168 |
-3 | -2 | -8.6 | -29.8112 |
-2 | -1 | 6.3 | -14.9056 |
-1 | 0 | 21.2 | 0 |
0 | 1 | -6.3 | 14.9056 |
1 | 2 | 8.6 | 29.8112 |
2 | 3 | 23.5 | 44.7168 |
3 | 4 | 38.4 | 59.6224 |
4 | 5 | 53.3 | 74.528 |
For x = 0.4, J = 7.452800000000001, t = 165.048
Sz | S'z | ΔESE+DE (meV) | ΔESE (meV) |
---|---|---|---|
-5 | -4 | -32.1 | -59.6224 |
-4 | -3 | -17.2 | -44.7168 |
-3 | -2 | -2.3 | -29.8112 |
-2 | -1 | 12.6 | -14.9056 |
-1 | 0 | 27.5 | 0 |
0 | 1 | -12.6 | 14.9056 |
1 | 2 | 2.3 | 29.8112 |
2 | 3 | 17.2 | 44.7168 |
3 | 4 | 32.1 | 59.6224 |
4 | 5 | 47 | 74.528 |
The spectra seem more plausible now for x = 0.0. And the trend for x = 0.2 of overlapping peaks becomes even more plausible with these renormalized energies. It's worth modeling the INS peaks as Gaussians and using the above energies as their respective means. The widths and amplitudes are left as empirical parameters to be tuned. The goal is to show that this explanation of overlapping peaks is a plausible explanation for the INS data. We implement this visualization in the next section.
Interactive model of INS spectra
We implement an interactive model visualization for the INS spectra, using our above results for input. This visualization will plot two main curves, the predicted peaks if only the superexchange term is used, and the predicted peaks if both superexchange and double exchange are used. Some things to note about this visual model:
We take the absolute value of all transition energies, as we assume that both \(\Delta{}S_z = 1\) and \(\Delta{}S_z = -1\) transitions will be detected.
The width of the Gaussian peaks is an empirical parameter that we can tune. Based on the INS data, peak widths seem to be in the general neighborhood of 4 to 6 meV.
The amplitudes of the Gaussian peaks are an empirical parameter that we can tune. Based on the INS data, peak amplitudes seem to be in the general neighborhood of 3 to 4 units.
The default widths assume that the doped spectra have a slightly larger Gaussian width than the undoped spectra.
A major assumption is that the amplitude for transitions between states with larger \(\left\lvert S_z \right\rvert\) eigenvalues is reduced compared to the smaller ones, for example the transition \(0 \to 1\) has a larger amplitude than \(1 \to 2\).
For this model, we reduce the amplitude of each higher transition — such as going from \(0 \to 1\) to \(1 \to 2\) — by a factor of 2.
The code below implements the interactive dashboard for the INS spectra plot.
The following figure illustrates the "best case" scenario for the model, yielding the qualitative features of the INS spectra and approximate quantitative agreement.
References
E. C. Svensson, M. Harvey, W. J. L. Buyers, and T. M. Holden, "Excitations of isolated clusters of magnetic ions", J. Appl. Phys. 49, 2150 (1978)
W. Nolting and A. Ramakanth, Quantum Theory of Magnetism (Springer, Heidelberg ; New York, 2009)
Stefan Stoll and Arthur Schweiger, Spin operators and matrices, WWW Document, http://easyspin.org/documentation/spinoperators.html