Thermally and Optically Activated Migration of Charge Carriers in Alkali Metal Sesquioxides

Abstract The alkali metal sesquioxides A4O6 (A=K, Rb, Cs) are mixed‐valent with respect to oxygen and display several degrees of electronic and structural freedom, which give rise to diverse transport and ordering processes. We report on analyses of the respective underlying excitations by diffuse reflectance spectroscopy and thermally activated electron transport. Backed by DFT based band structure calculations we identify three possible mechanisms, inter valence charge transfer from peroxide to superoxide, excitation across the Jahn‐Teller gap of tilted superoxide anions, and polaron migration. The activation energies as found by the three different approaches are in a rather narrow range of 0.62–0.89 eV for Rb4O6 and 0.49–0.65 eV for Cs4O6, confirming opacity in the full range of visible light. The effect of the phase transition from cubic to tetragonal as demonstrated for the caesium representative corresponds to a marginal shift to higher activation energy.


Introduction
The alkali metal sesquioxides A 4 O 6 (A = K, Rb, Cs) constitute a small family of solids representing rare examples of mixed valency within an oxide anion substructure. [1][2][3] These binary oxides display all the same electronic and structural degrees of freedom known from open shell transition metal compounds, giving rise to diverse order/disorder transitions, as there are electron localisation, charge ordering, structural relaxation, orbital ordering and spin ordering (long range magnetic order). Thus, even representing kind of a singularity, the p electron open shell alkali metal sesquioxides are of general relevance in condensed matter sciences since they offer opportunities for exploring potential correspondences with the observations made on open shell d and f electron systems.
The charge disordered state of the sesquioxides is realised in a cubic crystal system, and derives from the thorium phosphide type of structure, which can be expressed formally as Th 3 P 4 !(O 2 ) 3 A 4 . Here, all oxygen atoms occupy the same Wyckoff position, thus establishing complete peroxide/superoxide disorder, i. e. charge disorder (Figure1a). [1,2] On cooling, a charge ordering transition takes place, which is accompanied by a diffusionless structural relaxation to a tetragonally distorted variant of the cubic structure ( Figure 1b). [4,5] Since this process is associated with a drop in electronic conductivity by more than two orders of magnitude, all phenomena taken together parallel those characterising the classical Verwey charge order transition, [6] first reported for prototypic Fe 3 O 4 . [7] As another result of that structural reorganisation, the open shell superoxide dumbbells, (O 2 ) À , are tilted with respect to each other, indicating orbital ordering, consistent with an intermolecular Jahn-Teller effect, lifting the degeneracy of the unequally populated π* molecular orbitals. [4] Recently, on cooling Rb 4 O 6 further, another displacive phase transition was encountered, effecting a change in the orbital ordering scheme. [8] Finally, only for the Rb representative, coupling of the unpaired spins at adjacent superoxide anions to spin dimers  has resulted in a fully diamagnetic ground state existing below 15 K. [8] Still, two intriguing phenomena have remained unresolved. Depending on the strength of electron correlation, the partly populated bands might drive valence electrons itinerant, causing metallic conductivity. However, this has not yet been observed up to pressures, applied in a diamond anvil cell, of 80 GPa and 180 GPa for Rb 4 O 6 and Cs 4 O 6 , respectively. The valence electrons stay localised at the (O 2 ) nÀ dumbbells, which on their turn are disordered in the cubic phase. Further, the deep black colour of all three representatives known is hard to rationalize since they consist of colourless or pale-yellow ionic building blocks. Charge ordering in alkali metal sesquioxides is associated with a diffusionless structural phase transition from a cubic to a tetragonal crystal system implying reorientation of the diatomic oxide anions and translational movements of electrons, exclusively. Thus, mobility of the electrons, its activation and underlying mechanism, are of crucial relevance for understanding chemistry and physics of these materials. First clues into this issue are the bulk electronic conductivity measured [6,8] and the deep black inherent colour, indicating absorption of the full range of visible light. Here we report on the results of UV/VIS/NIR spectroscopy, and a DFT based theoretical analysis.

UV/VIS/NIR Reflectance Spectroscopy
For both, Rb 4 O 6 (green spectrum in Figure 2) as well as Cs 4 O 6 (blue spectrum in Figure 2) the reflectance spectra at room temperature consist of a pronounced absorption edge, onset at around 6200 cm À 1 . This is corresponding to 0.77 eV for Rb 4 O 6 (intersection of dashed line and solid line in Figure 2) whereas the absorption drop is slightly shifted to lower energies for Cs 4 O 6 (0.68 eV, intersection of dotted line and solid line in Figure 2). These steep slopes of absorption were observed down to about 13000 cm À 1 and 15000 cm À 1 , while the absorption stays on a virtually constant level between 15000 cm À 1 and 30000 cm À 1 for both compounds. Additional absorption edge starting at about 30000 cm À 1 was observed for all samples. The latter has been assigned undoubtedly to transitions caused by BaSO 4 (added to the specimens for the purpose of optical dilution), which was evidenced by recording the reflectance spectra of the pure compounds BaSO 4 and Cs 4 O 6 ( Figure S1). Thus, the low energy absorption features are caused by the alkali metal sesquioxides and can be assigned to band-bandtransitions. On cooling the Cs 4 O 6 specimen to 250 K, the low energy absorption edge shifts to slightly higher energy (red spectrum in Figure 2) reflecting the phase transition from the cubic to the tetragonal polymorph, and the orbital ordering associated. In addition, a relatively narrow absorption peaking at 6900 cm À 1 (0.86 eV) is observed, which we assign to a charge transfer between local energy levels within the band gap of O 2 nÀ units in the tetragonal phase. At warming to RT, the absorption edge shifts back to lower energy (black spectrum in Figure 2), confirming that the phase transition is reversible. Further, the additional absorption band observed at 6900 cm À 1 disappears. For an accurate experimental determination of the band gaps for Rb 4 O 6 and cubic as well as tetragonal Cs 4 O 6 , Tauc-Plots were set up assuming indirect band gap transitions ( Figure 3) using equation (1), which relates the absorption coefficient α and the energy of the band gap E g . [9] (1) h represents the Planck constant, ν the photon frequency and B a proportionality constant. The exponent n indicates the nature of the electronic transition, n = 1/2 for a direct allowed, n = 2 for an indirect allowed transition. The absorption coefficient can be expressed by the Kubelka-Munk function F(R ∞ ), equation (2), (2) while transforming the reflectance spectra into absorption spectra, [10] equation (3), where K and S are absorption and scattering coefficients, respectively.
(3)  The linear fits of the respective plots ( Figure 4) resulting with n = 2 are indicative for an indirect allowed band transition. The band gaps amount to 0.84 eV for cubic Rb 4 O 6 and to 0.72 eV for cubic Cs 4 O 6 , respectively, while for tetragonal Cs 4 O 6 a value of 0.80 eV was obtained. After warming up and keeping at room temperature for 30 min the band gap relaxed to 0.76 eV. The absorption coefficient in the low energy range of the spectra obeys the exponential Urbach law [11] (Figure 5) and can be calculated using the relation (4) where A is a constant and E u is the Urbach-energy, which can be explained as the width of the tails of states in the band gap above and below the valence band and the conduction band, respectively. These states can be associated with structural defects (e. g. vacancies, F-centres) and/or disorder within the crystal. Here, the width of the Urbach-tail E u was determined by the reciprocal slope of the linear fit to E u = 0.09 eV for Rb 4 O 6 and E u = 0.13 eV for Cs 4 O 6 , respectively. ( Figure 5).

Activation Energy of Bulk Electronic Conductivity
The transport properties of bulk Rb 4 O 6 and Cs 4 O 6 have been reported earlier in the context of studies on charge and magnetic ordering phenomena. [6,8] Since the inter (O 2 ) nÀ hopping integrals are but small and the on-site electron correlations rather strong, both sesquioxides adopt an insulating ground state. [12] The Arrhenius plots for ac conductivity, as measured by impedance spectroscopy, evidence a thermally activated behaviour ( Figure 6).
In either case, the charge ordering phase transition from the cubic to tetragonal modification is associated with a drop in conductivity by two orders of magnitude. The activation energies for the cubic phases were determined as E A = 0.62 eV and 0.49 eV for Rb 4 O 6 and Cs 4 O 6 , respectively. The activation energy for the tetragonal phase is slightly increased to E A = 0.51 eV for Cs 4 O 6 , while staying virtually unchanged for Rb 4 O 6 . The pronounced hysteresis as function of temperature observed in the impedance data for both compounds reflect the martensitic nature of the cubic to tetragonal phase transition, and the concomitant variance in long range electron transport, without affecting the activation energies (band gaps) significantly. Thus, the hysteresis is not noticeable by diffuse reflectance.

Electronic Structure Calculations
For the DFT based band structure calculations the Hubbard correction (U) was applied to the p-orbitals of the oxygen atoms, which are the chemically active orbitals on the superand peroxide anions. In earlier studies a correction between 4 and 8 eV have proven to give a reasonable description of alkali super-and peroxides. [13,14]    (open circles). [6,8] Arrows indicate heating and cooling modes, respectively, the colors of the symbols stand for the cubic phase (red) and for the tetragonal phase (blue). The small jumps seen in the cooling mode for Rb4O6 are artefacts due to changes of electrical contact at varying the temperature.
However, the choice of the U value is not unique, and it depends on the investigated system (in this case, on the type of the alkali ion) as well as on the implementation of the U correction. A comparison with the measured band structure gaps yields U = 3.7 eV and U = 4.1 eV as the optimum values in the case of Rb 4 O 6 and Cs 4 O 6 , respectively; for the applied Ucorrection scheme see ref. [15]. Figures 7 and 8 [13] However, the occupied peroxide band is significantly broader for Rb 4 O 6 than for Cs 4 O 6 and extends below the respective occupied p x /p y bands of O 2 À . Splitting between the occupied peroxide and unoccupied superoxide states in Cs 4 O 6 is smaller, therefore the corresponding electron transfer is lower in energy. According to the calculated density of states and band structure, such transfer is relevant for Rb 4 O 6 , too, however, it is superimposed by an excitation across the Jahn-Teller gap for the superoxide anions. A more detailed insight into the electronic levels is given in terms of the band structures, Figures 9 and 10. The topologies of the lowest unoccupied bands in both systems are very similar, except that they are slightly narrower in the case of Rb 4 O 6 .
The difference is more pronounced for the bands below the Fermi level. Cs 4 O 6 clearly exhibits a direct gap, whereas a slightly indirect gap in Rb 4 O 6 might result in a weaker absorption. The intriguing difference between the band structures of the two systems might be to some extent ascribed to the difference in the lattice parameters and consequently to the influence of the crystal field acting on the oxygen dumbbells. Due to the smaller lattice parameter, the effect of the crystal field is more pronounced in Rb 4 O 6 than in Cs 4 O 6 , making the O 2 2À anions in the former more stable, which is reflected in lower energies of the corresponding occupied states, as it is evident from the calculated density of states. For the same reason the empirically determined Hubbard U value is lower for

Discussion
The alkali metal sesquoxides can be regarded as basically ionic compounds, consisting of alkali metal cations and diatomic anionic species, i. e., all valence electrons are located in the 2p derived π and π* type molecular orbitals of oxygen. Thus, the mixed-valency displayed, the unique physical properties and ordering phenomena like the Verwey type charge ordering on the anionic sublattice [6] and spin dimers constituting the magnetically ordered ground state [8] are driven by electronic processes between and within the diatomic oxygen species.
Obviously, for any kind of microscopic understanding of the phenomena mentioned, knowledge of electron mobilities and activation thresholds constitute a crucial condition. On the timescale of vibrational spectroscopy, the electrons appear localized, [16] corresponding to minimal lifetimes in the range of 100 fs to 1 ns, even if the charges of the diatomic oxygen species are disordered, e. g. in the cubic polymorphs. As recorded by impedance spectroscopy, the charge transfer within the cubic modifications takes place at a mobility of u(298 K) � 10 À 9 cm 2 /(Vs) [6,8] determining the border line for the thermally activated dynamics of electron migration. Computational analyses of the electronic structures of alkali metal sesquioxides have proven to be challenging, mainly due to uncertainties and lacking experience in proper treatment of electron correlations. In an early study based on density functional calculations within the local spin density approximation, half-metallic ferromagnetism and a magnetic ordering temperature of 302 K was predicted, [17] which later has been shown to be at odds with experiment. [18] Subsequent calculations correctly reproduced the semiconducting nature, however failed to predict the true magnetic exchange pattern. [12] Based on knowledge of the charge ordered crystal structures [4] and an appropriate choice of the Hubbard U, optical and thermal excitations in terms of DFT band structure calculations, UV/VIS/NIR and impedance spectroscopy have produced quite consistent results, in the study presented here.
The band structures of the tetragonal phases feature rather flat bands throughout and are underlining the pronounced localization of the valence electrons in the π and π* type MOs as well as the ionic character of the compounds. Accordingly, the relevant bands around the Fermi level are prevailingly of oxygen nature, while the alkali metal s electron related states are virtually irrelevant. Three optically and thermally activated processes can be assigned, inter-valence charge transfer (IVCT) (O 2 ) 2À !(O 2 ) À , excitation across the Jahn-Teller gap opened by lifting the degeneracy of the π* type MO of (O 2 ) À in the tetragonal low temperature polymorph, and polaron migration. As to be expected, the topologies of the band structures of the rubidium and cesium representatives are similar. Yet the oxygen p type bands are shifted against each other, which is basically a consequence of the stabilisation of the peroxide anion in Rb 4 O 6 due to the stronger crystal field in comparison to the Cs analogue. Among the possible transitions, the IVCT process contributes to long-range electron transport, but the contribution by a hole (electron) polaron migration should also be considered, as suggested by the fact that thermal activation energies found from conductivity measurements are slightly different from those obtained from UV/VIS/NIR, see Table 1. Therefore, we cannot rule out that the microscopic mechanisms for thermal and optical excitations are different. Especially as the Urbach tails seen by reflection spectroscopy indicate presence of inter band gap defect states above and below the valence band and conduction band. A thermally activated charge transport could proceed preferentially via defect states, while an optical activation would tend to a transfer directly into the conduction band at higher energies.
Basically, all findings are in accordance with the classical model for semiconductors, reflecting a gapped band structure, where thermal as well as optical excitations are enabling electron transport. However, this picture is not suited to provide an explicit understanding of the microscopic mechanism underlying the long-range electron transport or, to put it more correctly, polaron migration. At a first glance, the long intermolecular distances of about 3.30 Å would exclude to assume some direct intermolecular (O 2 ) nÀ interactions, since this distance coincides with the so-called van der Waals peak [19] were a maximum in the number of non-interacting OÀ O separations is observed in experimental data. However, the magnetic dimer coupling to happen across an intermolecular OÀ O separation as wide as 3.27 Å [8] encourages to assume existence of a small, however, finite hopping integral. Accepting this assumption would admit assigning the system as mixed valence of type II according to the classification by Robin and Day. [20]

Conclusions
In summary, for Cs 4 O 6 the relevant transitions dominating optical absorption and activation of polaron migration correspond to a plain IVCT from (O 2 ) 2À to (O 2 ) À . For Rb 4 O 6 , however, the IVCT appears to be superimposed by a vertical excitation across the Jahn-Teller gap, π*(x)!π*(y). Table 1. Activation energy of the electronic conductivity (E ae ), energy of the optical band gap (E g ) and the theoretical band gap (E gt ) in eV for the C-Phase (C) and T-phase (T).

Source of Materials
Rb 4 O 6 and Cs 4 O 6 were synthesized as micro-crystalline powders starting from the respective superoxides by controlled thermal degradation. A detailed prescription is published elsewhere. [21] Phase purity and identity of the samples investigated were confirmed by le Bail fits to the X-ray powder diffractograms, see Figure 11.
Reflectance spectra were measured with an UV/VIS/NIR photospectrometer Cary 5000 from Varian (Agilent Technology) equipped with an integration sphere for diffuse reflectance measurements. The deep black pristine sesquioxides were diluted with BaSO 4 in a ratio of 1 : 20 to 1 : 30 to avoid excessive absorption. Furthermore, samples of Cs 4 O 6 were cooled for one day at 250 K to investigate the tetragonal phase additionally to the cubic one. All samples were prepared in an Ar-glove box and inserted in a home-built, airtight cell for measurements.

Impedance Spectroscopy
All conductivity data included in this report are taken from earlier work on Rb 4 O 6 [8] and Cs 4 O 6 , [6] all experimental details are given in the respective supplementary informations.

Computational Details
The band structure calculations were performed in the framework of the density-functional theory (DFT) by applying the Quantum Espresso code [22] for the experimentally-determined [6,8] structures of the tetragonal phases (Tables S1 and S2). The exchange-correlation effects were calculated in the generalized-gradient approximation [23] with explicitly added Coulomb-repulsion U term for the oxygen p states. The electron-ion interactions were described with the norm-conserving pseudopotentials. The planewaves and charge-density cut-off parameters were set to 639 eV and 2543 eV, respectively, and a 3 × 3 × 3 k-points meshes were used for the Brillouin-zone integration. The self-consistent criterion was the total-energy difference between the two subsequent iterations being less than 10 À 6 Ry. The anti-ferromagnetic ordering of the O 2 À dumbbells was considered. The U parameters were determined empirically in order to achieve an agreement with the experimental results, which yielded U = 3.7 eV for the Rb, and U = 4.8 eV for the Cs system. Since for the considered magnetic ordering the difference in densities of states and the band structures for the two spin channels are very subtle, we do not make the distinction in the corresponding graphs.