Weakly-correlated nature of ferromagnetism in non symmorphic CrO$_2$ revealed by bulk-sensitive soft X ray ARPES

Chromium dioxide CrO$_2$ belongs to a class of materials called ferromagnetic half-metals, whose peculiar aspect is to act as a metal in one spin orientation and as semiconductor or insulator in the opposite one. Despite numerous experimental and theoretical studies motivated by technologically important applications of this material in spintronics, its fundamental properties such as momentum resolved electron dispersions and Fermi surface have so far remained experimentally inaccessible due to metastability of its surface that instantly reduces to amorphous Cr$_2$O$_3$. In this work, we demonstrate that direct access to the native electronic structure of CrO$_2$ can be achieved with soft-X-ray angle-resolved photoemission spectroscopy whose large probing depth penetrates through the Cr$_2$O$_3$ layer. For the first time the electronic dispersions and Fermi surface of CrO$_2$ are measured, which are fundamental prerequisites to solve the long debate on the nature of electronic correlations in this material. Since density functional theory augmented by a relatively weak local Coulomb repulsion gives an exhaustive description of our spectroscopic data, we rule out strong-coupling theories of CrO$_2$. Crucial for the correct interpretation of our experimental data in terms of the valence band dispersions is the understanding of a non-trivial spectral response of CrO$_2$ caused by interference effects in the photoemission process originating from the non-symmorphic space group of the rutile crystal structure of CrO$_2$.


Introduction
Among the transition metal dioxides with rutile structure, chromium dioxide (CrO 2 ) is the only one possessing a ferromagnetic conducting phase. Its ground-state Fermi surface (FS) is composed of 100% spin polarized electrons, resulting from the so-called "half-metallic" nature of CrO 2 . For almost 30 years, the half-metallicity has been correctly predicted within density functional theory (DFT) using the local spin-density approximation (LSDA) to electron exchange-correlation [1,2]. A clear experimental demonstration was obtained by point contact Andreev reflection, showing a spin polarization of the conductive electrons higher than 90% [3] and later in subsequent studies higher than 98% [4]. The half-metallicity of CrO 2 finds important practical application in spintronics. Furthermore, it was demonstrated the exciting possibility to inject a spin-triplet supercurrent into CrO 2 [5] which sets up interesting connections between spintronics and superconductivity. However, one of the biggest limitations to fully exploit the device potential of the half-metallicity of CrO 2 is its dramatic spin depolarization with temperature, which is therefore considered as a property restricted to the ground state. Several depolarization mechanisms have been suggested [6,7], including those where electronic correlations might play an important role [7,8].
The electron correlation effects in CrO 2 beyond the mean-field approach within the local density approximation are still under debate, because the experimental data reported so far diverge concerning the degree of their contribution. In the DFT-LSDA framework, the calculated density of states (DOS) does not reproduce the angle-integrated photoemission spectra of the valence band [9,10,11]. However, a better agreement can be obtained by introduction of an on-site Coulomb interaction term within the LSDA+U approach on the Crd orbitals using theoretically derived values of U = 3 eV and J = 0.87 eV [12]. On the other hand, optical conductivity [13] and magnetic anisotropy [14] are better modeled by just static LSDA; furthermore magneto-optical Kerr spectroscopy data required only the gradient corrections within the generalized gradient approximation (GGA) to the exchangecorrelation [15]. CrO 2 has also been the subject of dynamical mean-field theory (DMFT) based calculations, including the many-body effects of the electron-electron interaction, in combination with DFT (LSDA+DMFT). In Ref. [16], the authors claimed an improved quantitative agreement in the interpretation of photoemission data with an LSDA+DMFT (U = 5 eV and J = 1 eV) treatment of the t 2g orbitals as compared to LSDA or LSDA+U as well as semi-quantitative agreement with thermodynamic and direct current transport measurements. A further LSDA+DMFT work [8] found the same Coulomb interaction parameters as Ref. [12] and demonstrated the appearance of non-quasiparticle states in the minority spin channel near the Fermi level (E F ). Their presence was claimed to be essential for the correct quantitative description of the spin polarization temperature dependence [8].
A recent comprehensive study employing static as well as dynamical treatments of correlations to investigate the mechanism leading to the peculiar half-metallic ferromagnetism in CrO 2 draws a complex picture [17]. According to this investigation static methods, like LSDA+U, were able to reproduce the magnetic ground state and some of its properties, whereas the mechanism underlying magnetic ordering could only be understood by considering dynamical correlations, inter-atomic exchange, as well as the polarization of the oxygen p states. In addition, these calculations would suggest a bit smaller "standard" Slater-type Hubbard parameters: U = 1.80 eV and J = 0.91 eV (see Appendix: Methods for the relation between the Hubbard parameters and the so-called Kanamori parameters used by the authors, U = 2.84 eV and J =0.7 eV, to describe the interactions of the t 2g Wannier orbitals) [17].
These results emphasize the fact that a correct treatment of electron correlations in CrO 2 is not just a pure theoretical issue but has important implications on its exotic transport properties. Different scenarios for the transport phenomena were suggested, depending on how the correlations were modeled [8,13,18]. However, a complete understanding of the nature of electron correlations in CrO 2 requires momentum-resolved experimental data of the electronic structure of this material. As we will show, these spectra are of paramount importance to judge the adequacy of the theoretical approach.
Angle-resolved photoemission spectroscopy (ARPES) represents a natural experimental technique to directly probe the momentum-resolved electronic structure.
However, the conventional ARPES in the photon energy range 20-200 eV has a very small probing depth as characterized by photoelectron mean free path much below 1 nm. This technique is therefore inapplicable to CrO 2 because its surface is metastable at normal conditions and, immediately after the synthesis, develops an amorphous layer with a composition close to the antiferromagnetic Cr 2 O 3 and a thickness of roughly 2 nm [19], well above the conventional ARPES probing depth. On the other hand, it has been demonstrated that the use of photons with higher energies towards 1200 eV [11] or lower energies towards 8 eV [20] deliver a probing depth around 2 nm, which gives the possibility to detect the photoelectrons emitted from CrO 2 through the Cr 2 O 3 overlayer.
In this work, we demonstrate that the momentum-resolved electron dispersions and FS of CrO 2 can be explored by bulk sensitive soft-X-ray ARPES (SX-ARPES) using photon energies in the range of 320-820 eV. Moreover, the increase of the photoelectron mean free path in this energy range reduces, by the Heisenberg relation, the intrinsic uncertainty of the momentum k z perpendicular to the surface [21], allowing an accurate mapping of the 3D electron dispersions. We find that the experimental FS appears as composed mainly of an electron pocket around the Γ point and a hole pocket around the Z point of the tetragonal Brillouin zone. Our first-principles DFT calculations, taking into account non-trivial matrix element effects in the ARPES response of CrO 2 originating from its non-symmorphic rutile space group, demonstrate that the spin-polarized GGA+U approximation (in what follows GGA implies an explicitly spin-polarized functional also in abbreviations such as GGA+U and GGA+DMFT unless stated otherwise) with the on-site parameter U eff (U eff = U -J) [22] equal to 0.4 eV (1 eV with the use of LSDA) delivers an accurate description of the CrO 2 band structure. Additional calculations within the DFT+DMFT framework allowed us to clarify this point, by showing almost no modification with respect to the DFT+U band structure. We conclude therefore that the occupied band structure of CrO 2 , below the magnetic ordering transition temperature, is compatible with a rather weakly correlated scenario, with electron correlation effects being essentially exhausted by static mean-field theory.

Fermi surface and photoemission interference effects
In Fig. 1 we gather our theoretical and experimental information on the FS of CrO 2 .
The FS obtained from our GGA+U calculations with U eff = 0.4 eV (for the determination of U eff see below) is reported in Fig. 1(a) inscribed into the first Brillouin zone (BZ). The theoretical FS is fully spin polarized and characterized by a quasi-isotropic electron pocket around the Γ point (violet surface), a hole pocket along the Γ-Z direction (yellow) that barely closes near the Z points, and another electron pocket around the A point (violet). Different colored planes p1-p4 in Fig. 1(a) show FS cuts explored in our SX-ARPES experiment, the surface-parallel cut p1 under sample rotation and the surface-perpendicular cuts p2-p4 under variation of photon energy. The top maps in the respective panels p1-p4 report the corresponding ARPES intensity rendered into the electron momentum coordinates k x , k y and k z corrected for the incident X-ray photon momentum. The sharpness of the FS contours in the k z direction confirms sharp definition of k z resulting from the photoelectron mean free path increase in the SX-ARPES energy range [21].
The bottom maps in the p1-p4 panels in Fig. 1 report the same experimental data overlaid with the FS contours obtained from GGA+U calculations. The contours shown on the k x,y >0 side of these maps (marked "2-Cr BZ") correspond to the full rutile unit cell of CrO 2 including two Cr atoms, and the ones on the k x,y <0 side (marked "1-Cr BZ") unfold these contours onto a reduced body-centered tetragonal unit cell including one Cr atom (see below). The most striking feature of our data, in comparison with the full unit cell calculations, is particularly evident in the p2 cut: the electron pocket around Γ (violet contours) is experimentally present only in every second BZ, centered at the surfaceperpendicular momenta k z equal to even integers n of 2π/a (in our case n = 8 and 10), and disappears in those centered at odd integers (n = 7 and 9 designated as Γ').
Complementarily, in the next BZ along the surface-parallel momentum k x represented in the cut p3, the electron pockets (around Γ, violet contours) are visible at k z equal to the even integers of 2π/a but disappear at odd ones. Furthermore, in the p2 cut we distinguish also the hole pockets (yellow contours) which show the same odd-even alternation. In this way, our data exhibit a periodicity in reciprocal space larger than expected, meaning that in real space this periodicity should be related to an effective unit cell smaller than the nominal one. Similar mismatch between the real space periodicity and that obtained in ARPES spectra has recently been observed in iron pnictides [23,24,25] as well as in a series of materials whose crystal structure possesses non-symmorphic space group such as graphite [26], BiTeCl [27], decagonal Al-Ni-Co quasicrystal [28] and Ruddlesden-Popper iridates [29].
In the case of CrO 2 the apparent twice as large periodicity of the ARPES response in k-space originates from the fact that its rutile-type space group (D 14 4h : P4 2 /mnm) is nonsymmorphic. The Cr atoms form a body-centered tetragonal lattice but are surrounded by distorted octahedra of oxygen atoms (see Fig. 1b) with the octahedron in the center being rotated by 90° around the c-axis with respect to the octahedra at the corners of the tetragonal cell. The presence of this screw axis reduces the accessible final states in the photoemission process belonging to different irreducible representations (even or odd with respect to the screw axis symmetry) along the high symmetry lines, which appears as alternating visibility of the spectral features through the successive BZs [26,30]. For reproducing this selection rule in the photoemission process, a possible way is to perform a formal unfolding of the band structure from the full to an effective unit cell, as has recently been demonstrated for iron pnictides [23] and for the Weyl semimetal WTe 2 [31]. Briefly, this procedure is based on projection of the Bloch wave functions onto a basis set that is just even or odd with respect to the screw axis symmetry. In our case, a simple basis set is the Having understood this crucial aspect of the ARPES response of CrO 2 we will discuss other FS cuts in Fig. 1. Due to the unit cell symmetry between the k x and k z directions, the (k y , k x ) cut in the panel p1 is identical to the (k y , k z ) one in the panel p2, although the latter shows the FS contours clearly affected by the intrinsic k z broadening [21]. Furthermore, the p4 panel clearly reveals the small electron pocket around the A point in perfect agreement with the GGA+U predictions, Fig. 1 (a).
Sharp contrast and excellent statistics of our SX-ARPES data confirm that this technique is indeed capable of digging out the electronic structure of CrO 2 through the Cr 2 O 3 overlayer. The experiment reveals the FS topology as composed of two electron pockets around the Γ and A points and one hole pocket between the Γ and Z points, on TiO 2 and interpreted as switching from Γ to the next Γ' point [30]. We note that the band calculations without unfolding ("2-Cr BZ") demonstrate the absence of hybridization between different bands in their intersections at the X and Z points (in particular, in the cone at the Z point near E F ). This effect appears because the two intersecting bands belong to different irreducible representations (even or odd) of the non-symmorphic space group

Determination of the static effective Coulomb interaction
For comparison with experiment, we have used the simplified approach by Dudarev et al. [22] (DFT+U eff ) since its one-parameter form facilitates fitting with the data. The effect of the on-site Coulomb interaction U eff on the majority spin band structure calculated within the GGA+U scheme is reported in Fig. 3(b). Higher U eff values push the d xy band to higher binding energy (most visible along the Γ-Z direction, as noted before, in the interval from -0.6 to -1.1 eV) and, at the same time, push the d yz+zx band closer to E F . The most satisfactory match between the GGA+U calculated and experimental data (grey points obtained by fitting of the spectral peaks) is achieved as the best compromise between these two trends reached with U eff = 0.4 eV. Furthermore, in Fig. 3(b) we also report the calculations using LSDA for the static exchange-correlation. The same good agreement with experiment is achieved, but in this case with larger U eff = 1 eV (the difference is attributed mainly to higher accuracy of the equilibrium lattice in GGA than in LSDA [33,34]). The effect of U eff on the FS contours is reported in Fig. 3(c). Their modifications with U eff are essentially restricted by the neighborhood of the Z and A points since the on-site Coulomb interaction do not much affect energies of the d yz+zx and d yz-zx orbitals.
Such small values of U were used before to describe the magnetocrystalline anisotropy of CrO 2 [14], where the authors concluded that correlation effects might be important although they are strongly screened out. A full description of screening mechanisms [35,36,37,38] in CrO 2 is particularly complicated because of the presence of many different screening channels (d-d, p-d and others). In addition to the screening channels, the choice of orbital basis and the corresponding polarization processes is another crucial point [39]. A complete discussion of the screening in CrO 2 is beyond the scope of this work.
For completeness, we complemented the simplified approach by Dudarev et al. [22] with calculations using the approach by Liechtenstein et al. [40], which includes the full matrix structure of the atomic Coulomb interaction. In this case, the same good agreement with experiment can be achieved within GGA+U using U = 1.0 eV and J = 0.87 eV.
These rather smaller interaction values with respect to the recently derived ones from first principles in Ref. [17] find a reasonable explanation by the presence of a large exchange splitting [14] already in the spin-polarized exchange-correlation functional used here. To illustrate this important point, we have performed an additional GGA+U calculation within Liechtenstein's scheme utilizing a spin-averaged GGA part (S AVG -GGA). Here, the calculation as a whole is spin-polarized, however, the GGA exchange correlation functional is spin-averaged and the spin polarization stems only from the Coulomb interaction. This is in the spirit of the usual DFT+DMFT scheme, where one starts from a spin-degenerate DFT calculation. Using this approach we obtain U = 2.0 eV and J = 0.87 eV, which are remarkably close to the derived ones [17].
Beside the specific values of the static effective Coulomb interaction, the comparison between theory and the measured k-resolved photoemission data for CrO 2 clearly indicates that a static mean-field method gives a very good description.

Dynamical mean-field theory investigation
The electronic band structure has also been investigated within the DFT+DMFT framework (see Appendix: Methods for details). We used a t 2g model derived once from Our spectrum in fact becomes similar to that of VCA with the same poor agreement with ARPES if we make the Coulomb interaction larger [41]. However, the data by Huang et al. [42] suggesting a rapid drop of the spin polarization above the Fermi level would be better explained in S AVG GGA+DMFT. It thus appears that the "dualistic electronic nature" of CrO 2 , a phrase coined in Ref. [42], is also present in our calculations in the sense that the occupied electronic structure measured in ARPES is very well reproduced within GGA+U and GGA+DMFT with small Coulomb interaction. However, the unoccupied electronic structure reported in Ref. [42] is better modeled in S AVG GGA+DMFT. A unified description capable of describing both occupied and unoccupied states is apparently beyond the t 2g model used here.
There are also other theoretical points that have to be considered, also touched upon in Ref. [17]. Another point that finally could be of importance is the momentum dependence of the self-energy, which is neglected hitherto. These additional non-local terms could lead to a reduced impact of the purely local correlation effects and thus to an improvement in the description of the d yz+xz spectrum also within an approach based on a non-spin-polarized one-particle starting guess such as GW.
The picture of weak electron correlations in the magnetically ordered phase of CrO 2 has important implications on the debated mechanism of its depolarization at finite temperature [6,7]. Indeed, this picture would support depolarization induced by singleelectron mechanisms such as formation of sublattices with different spin or phonon interactions rather than electron correlation effects such as non-quasiparticle or orbital Kondo interactions [6,7,8]. However, it is important to note that our data are only on the occupied part of the electronic structure, therefore we cannot exclude the alternative scenario of a "dualistic electronic nature" [42], where the occupied electronic structure contains weak correlation and the unoccupied electronic structure involves stronger ones.
Since the GGA+U calculations are well capable of reproducing the experimental data, they serve as a good starting point for discussing the half-metallic ferromagnetic behavior.
Obviously, CrO 2 eventually ends up in a half-metallic state because the exchange splitting of the t 2g bands (of about 3 eV) is considerably larger than the widths of the d yz+zx and d yz-zx bands (of about 2.5 and 2 eV for the spin majority and minority bands, respectively), which themselves are wider than the d xy band (of about 1.5 eV width). In this context, it is important to point to the small hybridization between the one-dimensional d xy bands and the three-dimensionally dispersing d yz+zx and d yz-zx bands as has been clearly revealed by orbital-weighted band structures for CrO 2 [14] and the neighboring rutile-type dioxides [44,45,46]. Consequently, these two types of bands (the 1D d xy and the 3D d yz+zx and d yz-zx ) disperse rather independently from each other and are coupled only via the common Fermi energy, i.e. by charge balance. The positions and widths of the d yz+zx and d yz-zx bands are rather determined by the π-type overlap of the t 2g orbitals with the O p states. In contrast, the d xy orbitals are particularly susceptible to both strong metal-metal overlap and strong (magnetic) correlations, which have strong impact on the band formation. In CrO 2 , the peculiar localized character of these bands causes a high density of states close to the Fermi level (however without participating in the Fermi surface itself) and thus is of fundamental importance for the ferromagnetic stabilization. In contrast, isoelectronic MoO 2 , rather than displaying a ferromagnetic phase, experiences a distinct lattice distortion away from the rutile structure to a monoclinic structure [46]. The different deviations of the two d 2 oxides away from a non-magnetic rutile-type ground state are thus reasonably related to the different band widths of the respective d states. Eventually, we are thus left to conclude that the half-metallic-behavior is mostly determined by the intra-atomic exchange coupling.

Conclusions and perspectives
Our study bears a number of new perspectives on the spectroscopic and theoretical include Dirac, Weyl and node-line semimetals (for entries see, for example, [48] and the references therein), nodal-chain metals [49] and organic metals [50]. Moreover, the sharp k z definition achieved in SX-ARPES due to enhanced probing depth allows accurate resolution of the topological band configurations in 3D k-space such as the Weyl cones [51,52,53].
The fundamental scientific and application perspective of the topological systems has been

Samples and SX-ARPES measurements
Epitaxial thin films of CrO 2 (100) were grown by chemical vapor deposition in oxygen atmosphere on top of a TiO 2 (100) substrate [58] at the MINT Center, University of Alabama, USA. The SX-ARPES experiments at different polarizations of incident X-rays were performed at the SX-ARPES endstation [59] of the ADRESS beamline [60] at the Swiss Light Source synchrotron facility, Villigen-PSI, Switzerland. The samples were transferred for SX-ARPES measurements ex-situ without any treatment. The sample temperature during the measurements was around 12 K to quench suppression of the coherent spectral weight due to thermal effects [61]. The combined (beamline and analyzer) energy resolution was set to vary between 40 and 100 meV through the incident photon energy range 300-900 eV. The sample surface was oriented normal to the analyzer axis, and the grazing incidence angle of photons was 20°. Details of the experimental geometry as well as photon momentum corrected transformation of the emission angles and energies into k values can be found in [59].

Data processing
The experimental FS maps in Fig. 1 were obtained by the integration of the spectral intensity within ±0.05 eV around E F . The data of each slice composing the FS maps were normalized to the integral intensity over 90% of their angular range, this allows to compensate the photoexcitation cross-section variation over the large photon energy range of panels p2, p3 and p4, and the slight variation of the probed region in the panel p1 due to the angular scan.
In the panels p3 and p4, the small blank region around k z = 12.5 Å -1 has not been probed due to the strong intensity signal coming from Cr 3d core levels excited by second-order radiation from the beamline monochromator. The non-dispersive spectral component in

Theoretical calculations
The DFT calculations on CrO 2 were performed using the Vienna Ab-initio Simulation Package (VASP) [62,63] with a plane-wave cutoff of 350 eV and 6 × 6 × 9 Monkhorst-Pack grid sampling for charge-density integration. The LDA and the GGA-PBE exchange-correlation functionals were used. In both cases, the on-site Coulomb interaction (U eff ) was tuned to obtain the best agreement with the experimental data, focusing on the most sensitive Γ-Z direction. We adopted the experimental lattice parameters a = b = 4.421 Å, c = 2.917 Å and u = 0.301 [64].
We performed DFT+DMFT calculations [65] using a t 2g -only model constructed from DFT via maximally localized Wannier functions [66,67]. The DFT in this case was performed using the VASP package with a plane-wave cutoff of 400 eV and a 16 × 16 × 32 Γ-centered Monkhorst-Pack grid to ensure a faithful representation of the hybridization function also at low temperature. For taking into account the local symmetry of the Cr sites we have used a different local coordinate system on each Cr site. A disentangling procedure with an energy window encompassing the 3d bands of Cr was used, resulting in three t 2g orbitals per Cr site, which agree reasonably well in their shapes and transfer integrals with previous calculations [17,68]. The DMFT part was performed using the code W2DYNAMICS [69] with a continuous-time quantum Monte Carlo impurity solver [70]. We used the density-density Hamiltonian for the t 2g orbitals as well as the SU (2) The results for the occupied bands are not influenced by this choice at all.

Relationship between Slater and Kanamori parametrizations of the Coulomb interaction
The Kanamori parameters for the t 2g orbitals U and J are connected to the Hubbard parameters for the full d shell U and J as used by, e.g., Liechtenstein et al. [40] via the Slater integrals F 0 , F 2 , and F 4 (for d states). Since this relationship is somewhat intricate and not well known in certain communities we outline it below in detail. More can be found in the classical works of Condon and Shortley [74], Slater [75] and Griffith [76]. These matrix elements are computed using multipole expansion etc., which is treated in detail in e.g. Ref. [75]. In the course of this calculation the Slater parameters F n arise as values of the radial parts of the integrals over the wave functions. For the d shell only F 0 , F 2 , and F 4 contribute by symmetry and furthermore we use the approximate relation F 4 =0.625F 2 [77,78]. These are the only atom dependent parameters remaining, the spherical integrals being universal for the d shell, since the spherical parts of the orbitals are given by spherical harmonics.
The tensor can be completely parameterized via the F n . As we do throughout our paper, primarily to facilitate comparison with other DFT+U work, a very common notation for the Slater integrals is via the parameters U and J as follows, see e.g. Ref. [72] = 0 , = ( 2 + 4 )/14 Once the tensor is computed it can enter the Hamiltonian for the two -article interaction full = where † and are the creation and annihilation operators, respectively.
In the DFT+U and often also in the DFT+DMFT methodologies an approximation of above Now we can write the interaction matrices for a d shell, with the real orbitals ordered as , , 3 2 − 2, , 2 − 2 in the following compact form, see e.g. Refs. [76,79,80,81] for similar expressions would violate the Pauli principle they are put to zero in the matrix. It is clear that the interorbital interactions are different between certain orbitals in the full atomic picture.
Thus, there is no single J . One also realizes, that the interactions within the t 2g orbitals ( , , ) relevant for CrO 2 can be parameterized completely by U and J 1 . In the manuscript we thus identified J 1 as our J for simplicity.

Unfolding procedure
The unfolding procedure is obtained by the projections on two basis sets constructed from the d xy , d yz-zx and d yz+zx atomic orbitals (for the two different Cr sites in their local coordinate frame) in a way to be even or odd with respect to the screw axis symmetry operator. The wave functions composing the even ("+") or odd ("-") basis set are: Being composed of only Cr d-states, these two basis sets of orbitals (with "+" and "-") do not form a complete one for the system, however, they are sufficient for describing the states near the FS (which is almost completely composed of the Cr d-states). For reproducing the matrix elements, the even basis set (composed of the three wave functions with the plus "+") is used for the p-polarization and the odd one ("-") for the s-polarization. As shown in Fig. 5, the projection on the even basis set is complementary to the one on the odd basis set. The partial spectral weight from the i-type d atomic orbital associated to the eigenstate , of the eigenfunction , is calculated as , ±, = |⟨ i ± | , ⟩| 2 , ("+" or "-"depending on even or odd basis set, and so corresponding to p-or s-polarization) and the total spectral weight is the sum of these three single contributions