RIXS calculations for an atomic model

Here we show how to compute RIXS for a single site atomic model with crystal field and electron-electron interactions. We take the case of Sr₂YIrO₆ from Ref. [1] as the material in question. The aim of this example is to illustrate the procedure and to provide what we hope is useful advice. What is written is not meant to be a replacement for reading the docstrings of the functions, which can always be accessed on the edrixs website or by executing functions with ?? in IPython.

import edrixs
import numpy as np
import matplotlib.pyplot as plt

Specify active core and valence orbitals

Sr₂YIrO₆ has a \(5d^4\) electronic configuration and we want to calculate the \(L_3\) edge spectrum i.e. resonating with a \(2p_{3/2}\) core hole. We will start by including only the \(t_{2g}\) valence orbitals.

shell_name = ('t2g', 'p32')
v_noccu = 4

Slater parameters

Here we want to use Hund’s interaction \(J_H\) and spin orbit coupling \(\lambda\) as adjustable parameters to match experiment. We will take the core hole interaction parameter from the Hartree Fock numbers EDRIXS has in its database. These need to be converted and arranged into the order required by EDRIXS.

Ud = 2
JH = 0.25
lam = 0.42
F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH)
info = edrixs.utils.get_atom_data('Ir', '5d', v_noccu, edge='L3')
G1_dp = info['slater_n'][5][1]
G3_dp = info['slater_n'][6][1]
F0_dp = edrixs.get_F0('dp', G1_dp, G3_dp)
F2_dp = info['slater_n'][4][1]

slater_i = [F0_d, F2_d, F4_d]   # Fk for d
slater_n = [
    F0_d, F2_d, F4_d,   # Fk for d
    F0_dp, F2_dp,        # Fk for dp
    G1_dp, G3_dp,        # Gk for dp
    0.0, 0.0           # Fk for p
]
slater = [slater_i, slater_n]

Diagonalization

The EDRIXS interface separates the physical model from the methods used to compute XAS or RIXS. model_1v1c() returns one-body matrices (emat_i and emat_n), Coulomb tensors (umat_i and umat_n), compact Fock-basis specifications (basis_i and basis_n), and Cartesian dipole matrices (trans_mat). The impurity one-body matrix imp_mat contains the valence spin–orbit coupling and, below, the crystal field. i quantities describe the initial and final states without a core hole, while the n quantities describe the intermediate state with a core hole. Both orbital-space Hamiltonians include the core, and basis_i explicitly fixes its occupancy to a full shell. The v_noccu argument counts only valence electrons.

get_ops() converts these backend-independent ingredients into many-body initial/final and intermediate Hamiltonians, plus dipole operators that map the initial Fock space to the intermediate one. We choose the SciPy backend, which represents these many-body operators as sparse matrices. ed() then obtains the retained low-energy eigenpairs used by xas() and rixs(). Note that the calculation does not know the core hole energy, so we need to adjust the energy at which the resonance will appear by hand. We know empirically that the resonance is at 11215 eV, and there is a further few-eV shift from the Coulomb energy of adding the valence electrons that is not captured by the raw calculation. We fold both into a single offset off and tune the small correction (here \(\approx6\) eV) so the computed spectrum lines up with experiment. In this case we are assuming a perfectly cubic crystal field, which we have already implemented by specifying the use of the \(t_{2g}\) subshell only, so we do not need to pass an additional v_cfmat matrix.

backend = 'scipy'
off = 11215 - 6
imp_mat = edrixs.atom_hsoc(shell_name[0], lam)
out = edrixs.model_1v1c(
    shell_name, v_noccu=v_noccu, shell_level=(0, -off),
    v_othermat=imp_mat, slater=slater,
)
emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, shift = out

hmat_i, hmat_n, trans_ops = edrixs.get_ops(
    emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat,
    backend=backend,
)


eval_i, evec_i = edrixs.ed(hmat_i, shift=shift, num_evals=2, backend=backend)

Compute XAS

To calculate XAS we need to correctly specify the orientation of the x-rays with respect to the sample. By default, the \(x, y, z\) coordinates of the sample’s crystal field, will be aligned with our lab frame, passing loc_axis to model_1v1c can be used to specify a different convention. The experimental geometry is specified following the angles shown in Figure 1 of Y. Wang et al., Computer Physics Communications 243, 151-165 (2019). The default setting has x-rays along \(z\) for \(\theta=\pi/2\) rad and the x-ray beam along \(-x\) for \(\theta=\phi=0\). Parameter scatter_axis can be passed to xas to specify a different geometry if desired.

Variable pol_type specifies a list of different x-ray polarizations to calculate. Here we will use so-called \(\pi\)-polarization where the x-rays are parallel to the plane spanned by the incident beam and the sample \(z\)-axis.

EDRIXS weights the retained low-energy eigenstates by their Boltzmann factors. The spectral broadening is dominated by the inverse core-hole lifetime gamma_c, the Lorentzian half width at half maximum.

ominc = np.linspace(11200, 11230, 50)
temperature = 300  # in K

thin = 30*np.pi/180
phi = 0
pol_type = [('linear', 0)]
gamma_c = info['gamma_c'][0]

xas = edrixs.xas(
    eval_i, evec_i, hmat_n, trans_ops, ominc,
    gamma_c=gamma_c, thin=thin, phi=phi, pol_type=pol_type,
    temperature=temperature, backend=backend,
)

Compute RIXS

Calculating RIXS is overall similar to XAS, but with a few additional considerations. The spectral width in the energy loss axis of RIXS it not set by the core hole lifetime, but by either the final state lifetime or the experimental resolution and is parameterized by gamma_f – the Lorentzian half width at half maximum.

The angle and polarization of the emitted beam must also be specified, so we pass pol_type_rixs to the function. Each entry is a 4-tuple that specifies both the incoming and the outgoing x-ray polarization. If, as is common in experiments, the emitted polarization is not resolved, one needs to add both outgoing polarization channels, which is what we do here.

eloss = np.linspace(-.5, 6, 400)
pol_type_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2)]

thout = 60*np.pi/180
gamma_f = 0.02

rixs = edrixs.rixs(
    eval_i, evec_i, hmat_i, hmat_n, trans_ops, ominc, eloss,
    gamma_c=gamma_c, gamma_f=gamma_f,
    thin=thin, thout=thout, phi=phi,
    pol_type=pol_type_rixs,
    temperature=temperature, backend=backend,
)

The array xas will have shape (len(ominc), len(pol_type))

Plot XAS and RIXS

Let’s plot everything. We will use a function so we can reuse the code later. Note that the rixs array rixs has shape (len(ominc), len(eloss), len(pol_type)). We will use some numpy tricks to sum over the two different emitted polarizations.

fig, axs = plt.subplots(2, 2, figsize=(10, 10))


def plot_it(axs, ominc, xas, eloss, rixscut, rixsmap=None, label=None):
    axs[0].plot(ominc, xas[:, 0], label=label)
    axs[0].set_xlabel('Energy (eV)')
    axs[0].set_ylabel('Intensity')
    axs[0].set_title('XAS')

    axs[1].plot(eloss, rixscut, label=f"{label}")
    axs[1].set_xlabel('Energy loss (eV)')
    axs[1].set_ylabel('Intensity')
    axs[1].set_title(f'RIXS at resonance')

    if rixsmap is not None:
        art = axs[2].pcolormesh(ominc, eloss, rixsmap.T, shading='auto')
        plt.colorbar(art, ax=axs[2], label='Intensity')
        axs[2].set_xlabel('Incident energy (eV)')
        axs[2].set_ylabel('Energy loss')
        axs[2].set_title('RIXS map')


rixs_pol_sum = rixs.sum(-1)
cut_index = np.argmax(rixs_pol_sum[:, eloss < 2].sum(1))
rixscut = rixs_pol_sum[cut_index]

plot_it(axs.ravel(), ominc, xas, eloss, rixscut, rixsmap=rixs_pol_sum)
axs[0, 1].set_xlim(right=3)
axs[1, 0].set_ylim(top=3)
axs[1, 1].remove()

plt.show()

Full d shell calculation

Some researchers have questioned the appropriateness of only including the \(t_{2g}\) subshell for iridates [2]. Let’s test this. We specify that the full \(d\) shell should be used and apply cubic crystal field matrix v_cfmat. We shift the energy offset by \(\frac{2}{5}10D_q\), which is the amount the crystal field moves the \(t_{2g}\) subshell.

ten_dq = 3.5
v_cfmat = edrixs.cf_cubic_d(ten_dq)
off = 11215 - 6 + ten_dq*2/5
imp_mat = edrixs.atom_hsoc('d', lam) + v_cfmat
out = edrixs.model_1v1c(
    ('d', 'p32'), v_noccu=v_noccu, shell_level=(0, -off),
    v_othermat=imp_mat, slater=slater,
)
emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, shift = out
hmat_i, hmat_n, trans_ops = edrixs.get_ops(
    emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat,
    backend=backend,
)
eval_i, evec_i = edrixs.ed(hmat_i, shift=shift, num_evals=2, backend=backend)

xas_full_d_shell = edrixs.xas(
    eval_i, evec_i, hmat_n, trans_ops, ominc,
    gamma_c=gamma_c, thin=thin, phi=phi, pol_type=pol_type,
    temperature=temperature, backend=backend,
)

rixs_full_d_shell = edrixs.rixs(
    eval_i, evec_i, hmat_i, hmat_n, trans_ops, np.array([11215]), eloss,
    gamma_c=gamma_c, gamma_f=gamma_f,
    thin=thin, thout=thout, phi=phi, pol_type=pol_type_rixs,
    temperature=temperature, backend=backend,
)

fig, axs = plt.subplots(1, 2, figsize=(10, 4))
plot_it(axs, ominc, xas, eloss, rixscut, label='$t_{2g}$ subshell')
rixscut = rixs_full_d_shell.sum((0, -1))
plot_it(axs, ominc, xas_full_d_shell, eloss, rixscut, label='$d$ shell')

axs[0].legend()
axs[1].legend()
plt.show()

As expected, we see the appearance of excitations on the energy scale of \(10D_q\) in the XAS and RIXS. The low energy manifold is qualitatively, but not quantitatively similar. This makes it clear that the parameterization of Sr₂YIrO₆ is dependent on the model.

Footnotes

[1]

Bo Yuan et al., Phys. Rev. B 95, 235114 (2017).

[2]

Georgios L. Stamokostas and Gregory A. Fiete Phys. Rev. B 97, 085150 (2018).