Potential thermoelectric performance of hole-doped Cu2O

High thermoelectric performance in oxides requires stable conductive materials that have suitable band structures. Here we show based on an analysis of the thermopower and related properties using first-principles calculations and Boltzmann transport theory that hole doped Cu2O may be such a material. We find that hole-doped Cu2O has a high thermopower of above 200 microV/K even with doping levels as high as 5.5x10 20 cm-3 at 500 K, mainly attributed to the heavy valence bands of Cu2O. This is reminiscent of the cobaltate family of high performance oxide thermoelectrics and implies that hole-doped Cu2O could be an excellent thermoelectric material if suitably doped.


I. INTRODUCTION
Thermoelectrics as a field has been attracting increasing attention in recent years.
Thermoelectrics are perhaps the simplest technology for direct thermal to electric energy conversion and can be used in both refrigeration and power generation devices. The current interest is stimulated in large measure by energy applications, where efficiency is a key parameter. The efficiency of thermoelectric devices is characterized by a material-dependent figure of merit ZT = S 2 σT/κ, where T is the temperature, S is the Seebeck coefficient or thermopower, σ is the electrical conductivity, and κ is the thermal conductivity, including both the electronic and lattice contributions. High ZT is a counter-indicated property of matter, i.e. a metric that requires a combination of properties that do not normally occur together. In particular, (1) one needs a combination of high conductivity and high thermopower and (2) one needs high conductivity and low lattice thermal conductivity. It has long been understood that optimization plays a critical role. In particular, as was emphasized long ago by Ioffe, 1 the carrier concentration needs to be optimized in order to obtain the best balance between thermopower and conductivity in a given semiconductor system. With such optimization, a variety of materials with ZT ~ 1 have been discovered over the years. It is recognized that one way forward is through unusual electronic structures. To obtain both high conductivity and high thermopower, one seeks materials in which both light band high mobility behavior and heavy band high thermopower behavior occur at the same time. Approaches that have been proposed include dimensional reduction, as in quantum wells, 2 multiple valley band structures and band structures with sharp structure superimposed, 3 Kondo physics, 4 modification of the density of states by resonant interaction with impurities, 5 and materials with complex band structures mixing heavy and light bands 6,7 or with highly non-parabolic bands that combine heavy and light features. The state-of-the-art materials, PbTe and PbSe, may be in this latter category. 5,[8][9][10][11][12] Currently, the most widely used thermoelectric materials are conventional "heavy-metalbased" semiconductor alloys, including Bi 2 Te 3 13-15

II. APPROACH AND RESULTS
The electronic structure calculations were performed using the full-potential linearized augmented plane-wave (LAPW) method, 28 as implemented in the WIEN2K code. 29 We employed the experimental lattice constant of 4.27 Å, and the radii of Cu and O LAPW spheres were chosen to be 1.84 and 1.63 bohr, respectively. We did convergence tests for the basis set and Brillouin zone sampling. Based on these we used RK max = 9 (R is the smallest LAPW sphere radius, i.e. 1.63 bohr and K max is the cut-off for the interstitial planewave sector of the basis) with a 12×12×12 k-point mesh for the self-consistent electronic structure calculations (a denser final mesh was used for the transport calculations, as discussed below). We used the Perdew-Burke-Ernzerhof (PBE) 30  However, Cu 2 O is a well-studied material, and a key point that emerges from past work is that the shapes of valence bands near the VBM obtained with different functionals are very similar although the band gaps of course differ. 35,37 For thermoelectric properties, the crucial part is within a few (~5-10) kT of the band edge, where T is the temperature. This can be simply rationalized in terms of the electronic structure (see below). Convergence of the Brillouin zone sampling is important for obtaining reliable thermoelectric properties. Therefore, for the transport calculations we employed the electronic structure derived from PBE-GGA, except that the band gap was corrected to the experimental value using a scissors operator to prevent bipolar transport at high temperature.
Calculated valence band masses along the Γ-X direction for the top three bands are 3.5, 3.5, and 0.2 m 0 , where m 0 is the free electron mass. As is well known, such heavy masses are highly favorable for the thermopower and thermoelectric performance, 38 and a mixture of heavy and light bands is also favorable. 6,39 As shown in Fig To better elucidate this, we have also performed DOS calculations within a hybrid functional approach, using the Heyd-Scuseria-Ernzerhof (HSE) functional 40 as implemented in the VASP code 41,42 and compare with the PBE results and experimental photoemission data. Figure 3 shows the calculated results for the valence band edge region that is important for transport (note that thermoelectric properties will depend on the electronic structure within ~5-10 kT of the VBM) along with the Cu 2 O photoemission data of Shen and co-workers, 43 which agrees with other photoemission studies of Cu 2 O. [44][45][46] Note that Shen and co-workers state an experimental resolution for their ARPES experiment of 0.25 eV and we have therefore incorporated a Gaussian broadening, using this 0.25 eV value, into both density-of-states plots. For this reason the calculated broadened DOS do not vanish at the energy of the VBM.
As shown in Fig. 3, both PBE and HSE DOS agree remarkably well with the ARPES intensity data in the first 0.5 eV below the VBM, which is the relevant region for p-type transport in the doping and temperature range considered here. Furthermore, angle resolved photoemission 46 show both the heavy bands and a light band at the band edge as predicted. Such good agreement is noteworthy considering that we have not made any adjustments of the experimental data, and strongly suggests that the calculated PBE GGA transport results will be reliable.
Returning to the calculated PBE band structure, following ligand field theory, the Cu d derived bands are formally anti-bonding combinations of Cu d and O p states, with the strongest anti-bonding character anticipated at the valence band edge. This is seen in the partial O 2p character of the DOS, which is larger between the -2 eV and 0 eV relative to the band edge than at higher binding energy. As mentioned, such mixing is generally favorable for conductivity. In any case, the transport properties of hole-doped Cu 2 O are closely related to the mixture of Cu 3d bands hybridized with O near the VBM. As is depicted in Fig. 1, the highest valence band shows a small band width of less than 0.75 eV. As mentioned, this is in accord with photoemission experiments, which show a peak with the corresponding width at the valence band edge. This leads to a high DOS near the band edge.
Such low dispersion bands are very favorable for thermoelectric performance provided that the material can be doped and is electrically conductive. This is qualitatively similar to the situation in the high ZT oxide thermoelectric Na x CoO 2 , 47 although in addition to narrow bands derived from hybridized 3d and 2p orbitals, that system is also near magnetism. 48 Returning to Cu 2 O, the DOS in the valence bands has a prominent peak at approximately 0.5 eV below the VBM. This can be attributed to the remarkably flat band along the ,  and  directions. In fact, such heavy mass bands and high DOS near the Fermi level strongly imply large thermopower of hole-doped Cu 2 O. Also, we note that such a sharp maximum in the DOS near the band edge has been discussed as an "ideal" electronic structure for a thermoelectric, 3 although considering the likely temperature range for application of Cu 2 O it would be better if the peak were closer to the band edge.
We calculated transport properties within Boltzmann transport theory. We did this based on the converged electronic structures, using the BoltzTraP 49 code. For this purpose the electronic states were calculated on a much more dense 34×34×34 k-point mesh in Brillouin zone. The use of dense grids is important for obtaining reliable transport properties. The constant relaxation time approximation was used in these calculations. With this approximation the thermopower can be determined as a function of doping and temperature without any adjustable parameters. This approximation, which consists of assuming that energy dependence of the scattering rate can be neglected at fixed doping and temperature, has been successfully used to provide a good description of transport properties in a variety of thermoelectric materials. 6,11,12,27,[50][51][52][53][54][55] We note that it does not involve any assumption about the dependence of the scattering rate on either temperature or doping level.
We present the calculated thermopower as a function of doping at various temperatures from 200 K to 900 K in Fig. 4. We note that the thermopower increases with temperature over the whole range even at low doping levels and high temperature, indicating that there is no bipolar conduction as expected from the substantial band gap. It is clear from Fig. 4  with heavy hole-doping has a good thermoelectric potential in a wide temperature range. Obviously, since lattice thermal conductivity decreases with T, while because of the large band gap, the thermopower and almost certainly the power factor S 2 increases with T, the best ZT will be at high temperature, i.e. 500 K, or higher if the material is stabilized at high T. This is because in the usual regime for heavily degenerate doped materials at moderate to high temperature (lattice thermal conductivity controlled by anharmonic Umklapp phonon scattering yielding  l ~ 1/T, conductivity controlled by electron phonon scattering,  ~ 1/T, and electronic thermal conductivity following Wiedemann-Franz,  e =L 0 T, = e + l ), with S increasing as calculated here, ZT will be a strongly increasing function of T. We note that since the decomposition of Cu 2 O in oxygen environments is by oxidation (related to Cu +  Cu 2+ ), it may well be that heavy p-type doping, which is essentially oxidation, will stabilize Cu 2 O to higher T.
While the thermopower can be obtained without adjustable parameters using the so we need to treat this as a parameter. However, this scattering time is not likely to depend on energy as strongly as v 2 which for a parabolic band is proportional to E 3/2 , where E is measured from the band edge) or df/dE, which is a sharply peaked function around the chemical potential.
Hence, as a first approximation for optimizing the actual power factor S 2 (T)(T), we here optimize the quantity S 2 (T)(T)/. Given the general lack of experimental information regarding the doping dependence of the hole mobility of Cu 2 O, further approximations concerning  are not likely to yield significant additional insight. We do make an effort to account for the change in  (or, equivalently carrier mobility) with doping in the next section.
In order to present more information on thermoelectric behavior, we calculate the electrical conductivity and the power factor divided by the inverse scattering rate, τ at 300 K and 500 K, as illustrated in Figs. 5 and 6, respectively. It can be clearly seen from Fig. 5 that the optimal doping level at 300 K corresponding to the peak power factor is 5.

III. DISCUSSION
The current results discussed above suggest that heavily hole-doped Cu 2 O possesses high thermopowers at a wide temperature range, suggesting good thermoelectric performance if appropriately doped. Compared to its neighboring compound ZnO, 22 the well-known oxide thermoelectric, Cu 2 O has much lower experimental lattice thermal conductivity and a much more favorable electronic structure. However, to achieve a high ZT value for practical applications, the mobility of Cu 2 O needs to be high enough to ensure a good electrical conductivity. Whether this is the case is not known. Quantitatively, if a mobility of 95 cm 2 /Vs were achieved at the heavy doping of 2.5×10 20 cm -3 , an electrical conductivity of 3.8×10 5 (m) -1 would result. High mobility near 100 cm 2 /Vs at room temperature have been reported on Cu 2 O nanowires (> 95 cm 2 /Vs), 58 polycrystalline thin films, 59, 60 and single crystals, 61 but these are at a doping level far from those discussed here, and so can only be taken as indicating a possibility of high mobility. In combination with lattice thermal conductivity of 4.5 W/mK 62 and our predicted room-temperature thermopower of 200 μV/K at this doping level, we can obtain a high ZT value of ~0.6. Considering the 1/T dependence of κ l and mobility, at 500 K κ l may be reduced to 2.7 W/mK, and the mobility should be 57 cm 2 /Vs according to the above assumed room-temperature mobility of 95 cm 2 /Vs. Then, a higher ZT value of 1.1 could be obtained at 500 K at a heavy doping of 5.5×10 20 cm -3 .
As mentioned, the actual value of the mobility at the heavy dopings envisioned here is a significant source of uncertainty regarding the ZT estimates, which we now discuss. In general, carrier mobility decreases with increasing carrier concentration for degenerately doped semiconductors (see, for example, Fig. 7 of Ref. 63) and then reaches a limiting value in the nondegenerate limit. The non-degenerate limit is approached as the temperature-dependent chemical potential moves into the band gap and is typically quantified by a parameter =µ/T, where  is the chemical potential measured from the band edge; we follow the usual convention 64 that  is negative when the chemical potential lies within the gap. Typically the degenerate limit is approached for

>2.
However, for the doping level (5.5×10 20 cm -3 ) and temperature (500 K) specified above, Cu 2 O is rather far from the fully degenerate limit - takes the intermediate value 0.3 (see Fig. 7), which is neither fully degenerate nor non-degenerate. Hence it is unclear exactly how much of a carrier mobility reduction, relative to the non-degenerate regime, would occur.
To put this quantitatively, let us assume a 75 percent reduction in the mobility relative to the nanowire case, and retain the assumed 1/T dependence of the mobility. Then the 1.1 500 K ZT  ZT values will result (the limit with S=200 μV/K is ZT~1.6). It is important to note that Cu 2 O is in a different regime than e.g. PbTe, i.e. an order of magnitude higher doping level. In this limit, the electrical conductivity may be higher, which would set a higher scale for the lattice thermal conductivity.

IV. CONCLUSION
In summary, a detailed study of the transport properties for hole-doped Cu 2 O is carried out using the full-potential LAPW method and Boltzmann transport theory.