Abstract
We have developed the numerical software package chinook for the simulation of photoemission matrix elements. This quantity encodes a depth of information regarding the orbital structure of the underlying wavefunctions from which photoemission occurs. Extraction of this information is often nontrivial, owing to the influence of the experimental geometry and photoelectron interference, precluding straightforward solutions. The chinook code has been designed to simulate and predict the ARPES intensity measured for arbitrary experimental configuration, including photonenergy, polarization, and spinprojection, as well as consideration of both surfaceprojected slab and bulk models. This framework then facilitates an efficient interpretation of the ARPES, allowing for a deeper understanding of the electronic structure in addition to the design of new experiments which leverage the matrix element effects towards the objective of selective photoemission from states of particular interest.
Introduction
Angleresolved photoemission spectroscopy (ARPES) and its variants have developed in recent years to be established among the preeminent experimental methods in solidstate physics. With an intimate connection to the oneelectron removal spectral function, ARPES is unique among the suite of techniques available to condensed matter physicists in its direct correspondence to the electronic structure of crystalline materials, providing access to the one electron removal spectral function within its native momentum space.^{1,2,3}
In the framework of Fermi’s Golden Rule, the photoemission intensity is described as:
where \(\left {\left\langle {\psi _f} \right{\hat{\mathbf \Delta }}\left {\psi _i} \right\rangle } \right^2\) is the photoemission matrix element, and A_{f,i} is the oneelectron removal spectral function, given by:
which reflects the overlap between the initial Nparticle manybody wavefunction upon removal of an electron and the ensemble of (N − 1)particle final state wavefunctions, while preserving energy conservation. Written as the imaginary part of the retarded Green’s function, the spectral function becomes:
The spectral function is seen to carry details of both the underlying bare dispersion associated with the electronic structure of the material \(\epsilon _k^0\), as well as correlations via the self energy \(\Sigma (k,\omega ) = \Sigma {^\prime}(k,\omega ) + i\Sigma ^{\prime\prime} (k,\omega )\). In the opposing limits of vanishing and strong interactions, ARPES is described as an ideal probe of the bandstructure and correlation effects respectively.
In practice, the photoemission can be strongly modulated by the \(\left {\left\langle {\psi _f} \right{\hat{\mathbf \Delta }}\left {\psi _i} \right\rangle } \right^2\) term, altering the spectral intensity through the dependence of the initial and final states on energy, momentum, and band index. At worst, this suppresses all intensity from certain bands, precluding their study by ARPES entirely. From a different perspective however, this modulation can be viewed as an additional experimental signature in the ARPES intensity which encodes a description of the photoemitted electron’s wavefunction. This term can be simulated to allow for quantitative descriptions and insights regarding the experimental signal. While such an approach has been made at some level for a number of ARPES experiments, this requires substantial effort in developing a specific model for each study.^{4,5,6,7,8,9} The development of a standard numerical framework would allow for a much larger set of experiments to be analysed at this level, providing the opportunity to understand and leverage the matrix element effects in a broad class of materials. We have pursued this objective through the development of an open source software package, chinook, implemented in Python to enable a broad audience to perform quick and easy simulation of photoemissionrelated phenomena, thereby improving both the interpretation and analysis of experimental data.
In the following, we will outline the primary workflow of our numerical approach, and the various ways in which this package can be applied to the study of the electronic structure of solids via ARPES.
Results
Matrix element effects
In the design of an ARPES experiment, a cursory understanding or prediction of the matrix elements relevant to a given system can dramatically improve one’s ability to study aspects of the electronic structure of interest. Before proceeding to explicit description of our software, it is instructive to consider a motivating example, taken here to be the ironbased superconductor FeSe. In Fig. 1, we plot experimental data along the ΓM direction taken with two, orthogonal linear polarizations of light at hν = 37 eV. Near the Brillouin zone centre, three holebands disperse away from the Fermi level (c.f. Fig. 2). For light polarized along the momentum axis in Fig. 1a, only a single state is observed clearly, whereas the perpendicular polarization in Fig. 1b. illuminates this and several other states. The third hole band is almost imperceptible, for any choice of polarization. These observations can be explained through an understanding of the orbital structure of the underlying electronic states, indicated by the insets of Fig. 1, in combination with the orbitalmixing effects of spinorbit coupling (SOC), as will be discussed in more detail below.^{10,11} Recent experiments designed with these effects in mind have leveraged the dipole matrix elements to perform targeted spin and angle resolved photoemission from states of particular interest, extracting fundamental information pertaining to a broad variety of orbitalrelated phenomena.^{11,12,13,14,15,16}
In many cases, orbital symmetry can be extracted from the polarization dependence alone. The information available from these arguments is limited, in particular for highorbital (l > 1) states, where the dimension of the vector field provides insufficient means to identify all orbitals uniquely. This is further complicated in multiatom bases, where now relative phases between different sites can differ from symmetric combinations in a momentumdependent fashion. In these situations, the photoemission intensity pattern is found to depend sensitively on the relative phases within the initial state wavefunction, producing socalled photoelectron interference patterns. In this way however, the matrix elements encode further information regarding the initial state beyond orbital symmetry alone. These effects have been seen in for example graphene^{7,17} and topological insulators,^{8,9} demonstrating the full depth of information regarding the initial state wavefunction which is contained in the ARPES matrix element. To further leverage the information available from ARPES experiments, it is advantageous to be able to simulate the full ARPES experimental intensity, while maintaining physical transparency. By preserving access to the relevant model parameters, one can then establish a more fundamental, and conceptual understanding of the electronic structure.
Model Hamiltonian
There are various levels at which the ARPES matrix element can be modelled.^{1,4,6,18,19,20} While the most sophisticated approaches account for the possibility of scattering processes subsequent to the photoemission event such as those which make use of KorringaKohnRostoker final states,^{21,22,23} we make two important simplifying assumptions here. First, the final states are taken to be freeelectron plane waves:
At high photon energies, the assumption of the plane wave final state is particularly well justified, as the crystal potential can be treated as a perturbation and sensitivity to the momentum structure of the exact final states becomes negligible.^{3} The validity of this assumption is ultimately material dependent, however similar logic as that applied to the domain of suitability for the Born approximation can be made: such an assumption is reasonable when either the crystal potential V_{o} ≪ ħ^{2}/m_{e}a^{2} or in the highenergy limit, V_{o}/(ħ^{2}/m_{e}a^{2}) ≪ ka, where a is the range of the potential. At present it is possible within chinook to relax this assumption only in the restricted sense of ref. ^{18}, as one can include phase shifts to the final state expansion. While beyond the scope of chinook in its current form, it would be possible to write the final states in the form of more sophisticated scattering final states, where the radial and orbital components of the ket in Eq. (4) are modified appropriately to reflect the presence of a finite crystal potential. This can be done through modification of the radial integrals \(B_b^{l^\prime }\) defined below.
Secondly, we work within a tightbinding framework wherein the initial states can be described by localized atomiclike orbitals, centred on the sites of the lattice basis. In materials where the spin degree of freedom is relevant, the orbital basis can be doubled to define a complete spinor basis, represented here by χ_{±}. Formally, the tightbinding basis set is expressed typically as:
where a represents a basis index and n, l the principal and orbital quantum numbers respectively. \(R_{n,l}^a(r)\) is a radial wavefunction, \(K_l^a(\Omega )\) a cubic harmonic, and χ_{±} the spinor projection. Alternatives such as distorted and rotated basis states can also be accommodated, so long as a unitary transformation into the basis of spherical harmonics can be made for the purpose of photoemission calculations. While these simplifications are in some cases unable to capture the full structure of the experimental photoemission intensity, we trade this level of universality for the substantial gains in transparency and physical insight which can be extracted from this approach.
Regarding the definition of the tightbinding model, there are various formalisms which are found in the literature, including SlaterKoster,^{24} t_{ab}, and Wannier Hamiltonians: we have made an effort to accommodate all possible variations without loss of functionality. We require only that the Hamiltonian matrix elements can be written as a Fourier series of bilinear terms in the orbital Hilbert space:
In this expression r_{ab} denotes the full connecting vector between basis states ϕ_{a} and ϕ_{b}, as opposed to the equivalent form where one refers to the connecting lattice vector alone. In addition to H_{o}(k), any other bilinear functions of momentum can also be added to the full Hamiltonian at this stage, including spinorbit coupling, and orbital or spin order. Adherence to Bloch’s theorem can be further relaxed in the context of the lowenergy effective models which describe a narrow region of momentum space: in this scenario, a more general function of momentum which satisfies the pointgroup symmetry can be employed.^{25}
With the basis and Hamiltonian so defined, the eigenvalue problem can be solved, and the initial state wavefunctions then defined as a superposition of the basis states described by Eq. (5):
With this information, a full characterization of the model for a specific material can be performed, followed by subsequent simulation of ARPES matrix elements. This includes density of states, bandstructure, 3D Fermi surface, orbital projections of the eigenstates, as well as the expectation values of various operators of interest, such as for example, 〈L ⋅ S〉 as in Fig. 2 for tetragonal FeSe. By defining an N × N Hermitian matrix, the expectation value of any observable operator can be computed in this way.
Computation of matrix elements
The workflow of chinook is sketched in Fig. 3. Once a satisfactory material model is established, one can proceed to the simulation of ARPES intensity maps. A suitable region of interest in momentum and energy space must be defined, and the eigenvalue problem is then solved over this domain.
We model the matrix elements of the dipole operator as:
where we have made use of the commutation relations to express the dipole operator in the position representation.
In explicit evaluation of the ARPES matrix element, we expand both the initial and final states as prescribed by Eqs. (4) and (5), which allows us to express Eq. (8) as:
This sum over integrals can be expressed in compact form as:
where \(c_\alpha ^b(\vec k,\omega ) \equiv \left\langle {\phi _b\psi _i^\alpha } \right\rangle\), and \(\epsilon _\mu\) the components of the polarization vector. In the third line, we have absorbed an extinction factor \(e^{  z_b/2\xi }\) into η_{b}, where ξ represents the meanfree path of a photoemitted electron, and z_{b} the spatial extent of the basis orbital below the surface.^{8,26} The radial and angular integrals are contained in the following terms, respectively:
and:
Here, \(G_{l^\prime ,m^\prime }^{b,\mu }\) is equivalent to a small subset of Gaunt coefficients, allowing for efficient and exact evaluation of this term.
Meanwhile, the radial integrals cannot necessarily be expressed in an analytical form and must be computed numerically, as the radial wavefunction is loosely constrained in most tightbinding models.^{27} Whether a hydrogenic, Slater, or more complex object should be employed to describe the radial wavefunction is left to the discretion of the user, as the best choice is somewhat dependent on the nature of the material and states of interest. The user is given the opportunity to select from a variety of initial state wavefunctions in addition to importing their own functions or radial integrals at the start of the calculation. This could be the Wannier function grid as generated by for example Wannier90.^{28}
We note that elsewhere it is common to take advantage of the planewave final state to recast the matrix element as a polarization modulated Fourier transform of the initial state.^{19,29} Specifically, one can write
M_{FT} can be expanded as in Eq. (10), establishing some formal equivalence to Eq. (8). However, one commonly observes qualitative deviation from experiment within this description, due to the form of the radial integrals \(B_b^{l^\prime }\) introduced above. In the Fourier representation, these are written as:
One can contrast this with Eq. (11) which we employ in chinook. It is made explicit in Eq. (14) that the Fourier representation of the dipole operator imposes radial integrals which are independent of final state angular momentum. The implications for multiorbital systems, and for those where l > 0, are important as final state interference becomes relevant. This is ultimately why for a planewave final state, the position, rather than momentum representation of the dipole operator yields a better description of experiment. We emphasize that although the limited constraints of tightbinding imply that the integrals \(B_b^{l\prime }\) are to some extent parameters of the calculation, support for distinct final state angular momentum crosssections is essential to the success of the position representation used here. Furthermore, it offers a natural extension of our framework to scatteringfinal states, wherein the commutation relations required to establish Eq. (8) are more rigourously justified (Further discussion of these approximations can be found in the Supplementary Materials).
Returning to the calculations executed in chinook, the central object of importance is the coherent matrix element factor:
Evaluation of this object proceeds following Eq. (9). Each column corresponds to the projection of the polarization vector in the basis of spherical harmonics, (i.e. μ ≡ Δm = ±1, 0), and the rows indicate the spinor projections. By retaining the matrix element in this coherent form, the ARPES intensity for arbitrary polarization and spin projection can be recalculated at runtime with minimal computational overhead. For each band and kpoint in the region of interest, a spectral function as defined in Eq. (3) is added to the total intensity map, with its amplitude multiplied by:
The photoemission intensity is then computed as described in Eq. (1), with \(\left {\left\langle {\psi _f} \right{\hat{\mathbf \Delta }}\left {\psi _i} \right\rangle } \right^2 \to M_\alpha ^2\). Spin projection, polarization, resolution, temperature and selfenergy can all be updated with little overhead at runtime. With ARPES intensity maps then calculated for different experimental configurations, these results can be exported for further analysis, or combined to define quantities such as spinpolarization and circular/linear dichroism. In this sense, the output of the standard chinook calculation is a threedimensional array of intensity in coordinates of momentum and energy which can be explored and analyzed in the same way as an experimental ARPES measurement.
Discussion
Bulk electronic structure and orbital texture
Returning to the motivating case of the Febased superconductor FeSe, we can implement the model characterized above in Fig. 2 and compare the simulated ARPES intensity against the lowenergy region of Fig. 1. As with the experiments, the calculations were done at hν = 37 eV and T = 120 K. A Fermiliquid type selfenergy has been applied to the spectral features, resulting in an energydependent broadening of the photoemission linewidth. In the present case, the tightbinding model has already been renormalized to match the experimental spectra, such that the dispersion is more appropriately defined as \(\epsilon _k^\prime = \epsilon _k^0  \Sigma ^\prime (k,\omega )\). Consequently, the selfenergy used in the ARPES simulation is purely imaginary, \(\Sigma (\vec k,\omega ) = i\Sigma^{\prime\prime} (\omega ) =  i(0.005 + 1.0\omega ^2)\), which is plotted in Fig. 4e. As ARPES matrixelements can confound the evaluation of the spectral function and correlation effects in experimental data,^{30,31,32} the ability to model both components in the same environment can facilitate the disentanglement of these two objects of interest.
The simulation in Fig. 4 captures the relative intensity ratio between the three hole bands, with the heaviest (largest effective mass) band visible only through the SOCinduced hybridization gaps near E_{B} = 50 meV. This latter state, composed primarily of d_{xy} orbitals, has vanishing photoemission intensity along the normal emission (\(\lim _{k_{} \to 0}\)) direction due to the selection rules associated with its definition in terms of spherical harmonics \(Y_2^{ \pm 2}\): all possible final states have a node along the normal emission direction. While conventional interpretation of the remaining states assumes d_{xz/yz}like wavefunctions, SOC allows for finite intensity from both states, as observed both experimentally and in the simulation near k_{} = 0 Å^{−1}. This orbital character mixing of the initial states is supported by projection of the tightbinding eigenstates onto the basis of spherical harmonics, as done at select kpoints in Fig. 4a using builtin diagnostic tools from chinook.
A more direct measure of the influence of SOC can be achieved through combining circularly polarized light with spin resolution to gain explicit access to both spin and orbital degrees of freedom. One can define the polarization asymmetry as:
where subscripts indicate the helicity of light polarization, and superscripts the spinprojection of the photoelectrons. This quantity is closely related to the projection of \(\langle \vec{L}\cdot\vec{S}\rangle\) along the quantization axis of the experiment, allowing for a connection between Figs. 2c and 4d to be made. This technique has been applied to both ruthenates^{13,33} and Febased superconductors,^{11} and utilizes the dipole selection rules encoded within the matrix element factor to provide the most direct measure of spinorbital entanglement in solid state.
Supercell impurity model
Consideration of a supercell model illustrates the information encoded in the ARPES matrix element beyond orbital symmetry alone. Regarding the electronic structure of periodic systems, one can choose an arbitrarily large unit cell in exchange for a reduced Brillouin zone and additional backfolded bands. By contrast, impurities or other symmetrybreaking potentials (SBP) explicitly require such an expanded unit cell. While one can numerically perform an unfolding of these bands in an attempt to recover the spectrum within the extended Brillouin zone,^{34} such an unfolding is carried out naturally in the photoemission experiment. In the absence of the SBP, Bloch’s theorem would impose that the original band becomes a symmetric superposition over the neighbouring lattice sites. The additional bands, which must be orthogonal to the original state will destructively interfere in evaluation of the ARPES matrix element, preventing observation of many of the folded states. Ultimately, an SBP can mix these states; when the SBP is an essential feature of the potential landscape, as in graphene and the Febased superconductors,^{7,35} the folded bands can be observed with strong intensity over a range of momenta. When the SBP is weak or disordered, intensity from these folded bands vanishes away from the avoided crossings.
To demonstrate these effects, we consider the artificial example of a square lattice of Li 2s orbitals, into which we substitute some number of Na 3s orbitals. Allowing for nearest neighbour hopping alone, and imposing a \(\epsilon _{Na}\) = −0.35 eV impurity potential for the Na sites, we simulate the effect of local defects in this lattice and the resulting ARPES spectra. Kinetic terms and onsite potentials have been adapted from the phenomenological rules set out in ref. ^{36}. In an attempt to consider the impurity problem realistically, we populate a 30 × 30 supercell of Li with various concentrations of randomly distributed Na impurities. For each particular distribution, the density of states is integrated to fix the Fermilevel at halffilling, consistent with electron counting. We then compute the photoemission intensity at hν = 21.2 eV over the ΓM direction of the extended Brillouin zone. For clarity, we assume a constant intrinsic linewidth of 10 meV along the entire dispersion. Energy and momentum resolution are set to 10 meV and 0.005 Å^{−1}. The results, plotted in Fig. 5c have been averaged over 80 such configurations, corresponding to a nominal doping of Li_{0.9}Na_{0.1}. This can be compared against the pure Lisupercell in Fig. 5b. The full spectrum of the latter is displayed in Fig. 5a.
In each case, we compute the photoemission intensity from all states. However intensity from all folded bands is vanishing in the absense of the SBP; the full bandstructure is plotted in white over the spectra to demonstrate the large suppression of photoemission intensity. At the bottom of the band, the dipole selection rules suppress photoemission intensity from even the main band. In the disorderaveraged supercell, a substantial broadening of the spectral lineshape is observed.^{37,38} As indicated by the overlain bandstructures of Fig. 5a–c, the impurity potential introduces a high density of avoided crossings, where the eigenvector supports finite photoemission intensity. In this sense, the broadening can be associated with the relative phases within the tightbinding eigenvector to which the ARPES matrix element is sensitive.
One can demonstrate that the linewidth broadening is dependent on both concentration and strength of impurities. In Fig. 5e, f we plot energy distribution curves (EDCs) at k_{F} = 0.44 Å^{−1} for fixed concentration (Li_{0.9}Na_{0.1}) with variable attractive (negative) \(\epsilon _{Na}\), and fixed \(\epsilon _{Na} =  0.35\) eV with variable concentration. Each spectrum has been averaged over 80 similar configurations, and normalized to its peak intensity. The linewidth is observed to increase monotonically with both concentration and impurity potential, indicating the similar role these degrees of freedom play in modifying the spectra of disordered systems. Despite this lineshape broadening, the lowenergy dispersion is resilient against a high level of disorder, as illustrated by Fig. 5c. By applying an outofplane polarization sensitive to states near the bottom of the band, we also confirm an increase of the bandwidth for this attractive impurity potential, which grows quadratically with impurity potential for the modest \(\frac{{\epsilon _{\mathrm{Na}}}}{W} \;<\; 0.15\) considered here: at 10% Na and \(\epsilon _{\mathrm{Na}} =  0.35\) eV, the band bottom is extended 30 meV. Such detailed study of the impuritysubstituted ARPES spectra is not possible without consideration of the matrix elements, which allow for a straightforward disentanglement of the supercell bandstructure and an opportunity to achieve meaningful insights from disordered materials.
Surface vs bulk, and emergent k _{z} dispersion
To this point we have considered the bulkelectronic structure, but it is important to appreciate the surfacesensitivity of the ARPES experiment: the high scattering crosssection in the ultraviolet regime results in penetration depths of the order of 5–10 Å.^{26} This corresponds to the top few unit cells of the lattice, depending on experimental details. In many cases, the surface introduces modest corrections to the local electronic structure, facilitating a direct connection between the measured photoemission intensity and the bulk electronic structure.^{39} In others, details of the surface preparation result in reconstructions of the ARPES spectra which deviate profoundly from the bulk electronic structure.^{39,40,41,42}
This surface sensitivity becomes rather important in the context of three or even quasitwo dimensional materials, where the photoelectron escape depth and k_{z} information are intimately connected. For intermediate energies in the ultraviolet regime, where the penetration depth is of the order of 5 Å, the Δk_{z} required by the uncertainty principle becomes comparable to the size of the Brillouin zone. In the presence of finite k_{z} dispersion, this can result in anomalously broad linewidths, as the spectrum effectively integrates over the third dimension of momentum space. This is visualized well in Fig. 6, where we have projected our FeSe model onto a 20layer slab model along the (001) direction. While the slab bandstructure is by construction independent of k_{z}, signatures closely related to the bulk k_{z} dispersion are observed in photon energydependent matrix element calculations, as seen in Fig. 6b. We estimate the attenuation factor \(e^{  \xi /2z_i}\) of the escaping photoelectrons using the universal escape depth curve from ref. ^{26}. The k_{z} value probed is calculated using an inner potential of V_{0} = 12.2 eV.^{10} While at both low and high photon energies the penetration depth is sufficiently large that that Δk_{z} should be less than 0.05 Å^{−1}, at hν = 71 eV Δk_{z} = 0.11 Å^{−1}, and linewidth broadening is observed as a result (note here π/c = 0.57 Å^{−1}). Conversely, for larger ξ values, Δk_{z} becomes negligible, and something akin to the bulk electronic structure is recovered. Note that Δk_{z} is not explicitly included in these calculations, but emerges naturally from the combination of slab geometry with variable penetration depth. From these results, it becomes evident that the surface sensitivity can complicate successful estimation of the bulk electronic structure. This emphasizes the need for proper characterization of the k_{z} dispersion, accessible via photonenergydependent measurements, as in Fig. 6b.
Photoelectron interference, and spinARPES
Despite these challenges associated with surface sensitivity, the surface can also precipitate new states localized to the interface region which are not possible in bulk systems. Such is the case for example in the Shockley surface states observed along the (111)termination of noble metals,^{23,43} Fermi arcs on Weylsemimetals,^{44} and conductive surface states observed in topological insulators such as Bi_{2}Se_{3}.^{45}
To model the ARPES spectra from these surface states, an extended lattice basis is required, with the unit cell projected onto a slabgeometry. Our implementation of the slab generation is inspired by the algorithm in ref. ^{46}, allowing for nearly total automation of the slab Hamiltonian initialization. Given a surface Miller index, new lattice vectors can be defined which projects the new unit cell along the desired surface direction to the desired thickness. The bulk Hamiltonian can be propagated over this slab supercell. While we formally maintain periodic boundary conditions, rather than preserve the full translational symmetry of the bulk crystal, a vacuum buffer is defined, with a thickness sufficiently large to suppress hopping elements between neighbouring slab unit cells. The precise location of the crystalvacuum interface is tuned by the user to achieve the desired surface termination of the crystal. In the case of Bi_{2}Se_{3}, this termination must occur between the van der Waalsbonded layers of two adjacent quintuple layers (QL) to preserve the topological surface states. The procedure is illustrated in Fig. 7a–c.
Expansion of the basis set to a suitably large slab carries the caveat of a significant memory overhead, which can be to an extent mitigated in the calculation of ARPES intensity: as the finite penetration depth of the probe and photoemitted electrons limit the volume of the unit cell to which we are actually sensitive, the eigenvectors are truncated beyond a modest multiple of the meanfree path, allowing for both efficient and highfidelity surfaceprojected ARPES maps to be computed, as done for the 400orbital basis used for the simulation in Fig. 8a. As a result, it is the meanfree path more than the size of the basis which limits the ability to treat very large slab unit cells. As an example of this functionality, simulated and experimental ARPES intensity from Bi_{2}Se_{3} are plotted in Fig. 8. Many of the central tenets of a model strong TI have been confirmed in this material, such as the anticipated chiral spin texture, observed directly via spinresolved ARPES.^{45,47,48,49} Such spinresolved experiments^{50,51,52} can also be simulated within the chinook software, as shown in Fig. 8b, where we present the simulated spin polarization:
This result is in agreement with experiment,^{45} and can be compared favourably with the surfaceprojected expectation value of the spin \({\hat{\mathbf S}}\) operator, \(e^{  \frac{{{\hat{\mathbf z}}}}{\xi }}{\hat{\mathbf S}}\), plotted in Fig. 7e. The bulk states, which lack any discernible spinpolarization (Fig. 7d), vanish from the calculation of P_{y} and so do not appear in Fig. 8b.
While the topological surface states Ψ_{TSS} are primarily composed of p_{z} orbitals at the surface, a pronounced modulation of the photoemission intensity around the Dirac cone is observed as a consequence of the finite extension of Ψ_{TSS} into the crystal bulk. Hybridization with bulk states, in addition to interlayer photoelectron interference can be understood as the progenitor of this modulation, as explored in depth in ref. ^{8} The interpretation of this angular intensity pattern in ARPES measurements presents an essential experimental verification and explanation of the limitations of applying a simple \(\vec{k}\cdot\vec{p}\) model to the description of real topological insulators such as Bi_{2}Se_{3}. While localized within a finite region near the vacuum interface, the full threedimensionality of the surface state becomes apparent through consideration of this spectroscopic evidence. Convenient extension to a slabgeometry is then a critical functionality offered by the chinook package.
Variable experimental geometry
In practice, ARPES experiments rotate either the sample normal or spectrometer in order to access a broad set of emission angles. While some modern techniques such as photoemission electron microscopy (PEEM),^{53} angleresolved timeofflight (ARTOF),^{47,54} and deflectorbased ARPES^{55,56,57} apparatus avoid this complication, the assumption of a constant experimental geometry is not always possible. Furthermore, it is often advantageous to rotate the sample orientation in order to for example explore large regions of momentum space, or to achieve better momentum resolution available at higher emission angles.^{3} While the most direct complication is associated with variable photon polarization, in the case of SARPES, the relative orientation of the detector with respect to the sample is essential to interpret data correctly.
To exemplify the practical considerations associated with such experiments wherein the geometry is variable, one may consider an exploration of Rashbasplit spinpolarized surface states, as in for example PtCoO_{2}.^{58} Using the model presented in ref., ^{58} in Fig. 9, the Fermi surface is surveyed over several neighbouring Brillouin zones, accessed by rotation of the sample about the horizontal (i.e. k_{x}) axis. The ARPES intensity calculated with and without consideration of the rotated polarization vector can be compared in Fig. 9a, b. While the intensity in the first Brillouin zone is fairly homogeneous in either case, the higher order zones reflect more substantial variation. These rotations complicate the extraction of orbital character from the photoemission intensity, requiring explicit consideration of the polarization rotation.
In the context of spinresolved measurements, the spinprojection is measured within the laboratory frameofreference, which remains fixed for all sample orientations. As the sample is rotated, contamination of orthogonal spinchannels is inevitable. While such effects are minimal near Γ_{1}, a significant outofplane spinpolarization arises near Γ_{2}, with P_{z} over 36% of P_{x}, as demonstrated in Fig. 9d–f. It is important to note that the intrinsic spinpolarization is confined entirely to the plane; this apparent outofplane polarization exists only in the coordinate frame of the laboratory apparatus. Accounting for the rotation of the measurement coordinate frame for the given experiment, one can then redistribute this information into the channels associated with the sample’s intrinsic spin texture:
where here \(\vec P_{\mathrm{exp}}\) is the measured spin polarization from Eq. (18) and \(\vec S(\theta )\) the spinprojection axis measured at each emission angle. Although Fig. 9b indicates higher photoemission intensity is available in the second Brillouin zone, Fig. 9e, f illustrate the practical challenges associated with resolving the spintexture near Γ_{2}. By affording the user with an ability to encode a realistic experimental configuration in the simulation, such effects can be accounted for in detail, circumventing a significant experimental limitation which may otherwise restrict more general application of the techniques detailed here.
We have presented here a simple and powerful numerical framework implemented in Python for the simulation and interpretation of ARPES spectra for a broad variety of materials of interest. Designed with this specific purpose in mind, the opensource structure of the chinook software package is engineered to accommodate further extension beyond this application, as we have done recently for the study of resonant optical excitations in pumpprobe spectroscopy experiments.^{59} Through the development of these tools, we hope to motivate and facilitate the consideration of the great depth of information encoded in the matrix element of ARPES towards a better understanding of these experiments and the electronic structure of the materials under consideration.
Methods
All work presented here was performed in Python using the chinook package.
Data availability
Data presented here is available from R.P.D. on reasonable request.
Code availability
The chinook source code is available at https://www.github.com/rpday/chinook.
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Acknowledgements
We happily acknowledge many helpful conversations with George A. Sawatzky and Elia Razzoli. This research was undertaken thanks in part to funding from the Max PlanckUBCUTokyo Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program, in addition to the Killam, Alfred P. Sloan, and Natural Sciences and Engineering Research Council of Canada’s (NSERC’s) Steacie Memorial Fellowships (A.D.), the Alexander von Humboldt Fellowship (A.D.), the Canada Research Chairs Program (A.D.), NSERC, Canada Foundation for Innovation (CFI), British Columbia Knowledge Development Fund (BCKDF), and CIFAR Quantum Materials Program.
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R.P.D. designed and wrote the code with support from B.Z. and I.S.E. The manuscript was written, and calculations performed by R.P.D. All authors contributed to the interpretation of calculations and writing of the manuscript. I.S.E. and A.D. supervised the project.
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Day, R.P., Zwartsenberg, B., Elfimov, I.S. et al. Computational framework chinook for angleresolved photoemission spectroscopy. npj Quantum Mater. 4, 54 (2019). https://doi.org/10.1038/s4153501901948
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