Al Doped ZnO Thermoelectrics Determined from Electronic Structure

For the aluminium doped wurtzite ZnO, comparing the Boltzmann transport theory calculated results and existing experiments, we acquire a few properties that are inaccessible otherwise. We nd that the doping makes the samples metallic as the shifted Fermi levels are above the conduction band edge. We further nd that the contradictory conclusions from two experiments with similar formula can be attributed to the quite disparate carrier concentrations and carrier mobility and the carrier mobility strong relates to the sample preparation.


Introduction
As the climate change becomes severer, converting the wasted heat into electricity is very promising, except with an obstacle of low conversion e ciency. Thermoelectrics relies on the Seebeck effect, while a gure of merit ZT is used to describe the conversion e ciency. The ZT relates to other parameters through the equation ZT = σS 2 T/(κ e + κ l ), where σ, S, T, κ e, and κ l are respectively the electrical conductivity, Seebeck coe cient, temperature, electronic thermal conductivity, and lattice thermal conductivity. [1][2][3] High electrical conductivity and high Seebeck coe cient cannot coexist in a natural material. Furthermore, the electronic thermal conductivity is linear to the electrical conductivity, indicating the high electrical conductivity might cause high thermal conductivity. [4][5][6] There have been extensive studies on how to optimize thermoelectric performance.
The wurtzite ZnO is a large bandgap semiconductor, which should have large Seebeck coe cients and low electrical conductivities. Due to its potential high-temperature thermoelectric applications, there have been extensive studies on the aluminum-doped ZnO to increase the electrical conductivity, started by Ohtaki et al. 7 Cai et al obtain Al-doped ZnO samples by hot pressing ZnO-Al 2 O 3 powder to improve σ by two orders. 8 Jood et al nd that the Al-induced grain re nement ZnAl 2 O 4 nanoprecipitates presages lead to к ∼ 2 Wm − 1 K − 1 ,α∼ -300 µV K − 1 and σ ∼ 1-10 4 Ω −1 m − 1 at 1000 K. 9 However, with the same formula Al 0.01 Zn 0.99 O, the RF plasma processing 10 and codepositing 11 yield samples are quite discrepant in terms of Seebeck coe cient. This is very confusing. Another di culty is that Ohtaki et al and Cheng et al both claim the samples are metallic because of the high electrical conductivity. 7,10 The large bandgap ZnO becoming metallic seems unrealistic, while the semiconductingmetallic transition needs are proved rigorously from the Fermi level perspective. When a Fermi level is above the conduction band minimum (CBM), the material is metallic. Nevertheless, theoretically calculating a Fermi level for a doped model is very challenging, as the calculation may involve 100atom supercell of many con gurations.
Here, we adopt a different approach. We use BoltzTraP2 to solve the BTE when we adopt the rigidband approximation (RBA). 12 In the approximation, the band structure is not changed, even with doping or heating. This would lead to tremendous savings because we only use the band structure of the pure unit cell, rather than a supercell approach adopted in a rst principle study considering the doping levels. The Fermi energy and carrier concentrations are a function of doping level and temperature, i.e. the doping effect is incorporated within the varying Fermi level. The bandgap will not change, but for an n-type semiconductor, the Fermi level shifts toward the conduction band. When the carrier concentration is not too high and the doping does not introduce impurity states in the bandgap region, the RBA is valid, proven by the success of BoltzTraP2 application in thermoelectric materials.
By comparing with the experimental results, we retrieve the related properties. We nd that all the Fermi levels are indeed above the CBM, showing the semiconductingmetallic conversion. We also nd that the unexpected high Seebeck coe cients come from the low carrier concentrations. We analyze why different processing of giving different carrier concentrations.

Computational Details
We study a unit cell of 4 atoms. ZnO has a typical wurtzite structure and the lattice constants are a = b = 3.29 Å and c = 5.30 Å with α = β = 90 o and γ = 120 o , respectively. We perform the geometry optimization of the unit cell with VASP, choosing the recommended settings. [13][14][15][16] We do not consider the electron spin, because its effect on the thermoelectrics is negligible. We use Perdew-Burke-Ernzerhof (PBE) to exchangecorrelation functions for geometric optimization and energy calculation. In the energy calculation, 17 we set the following parameters: cut-off energy 520 eV (1.3 times larger), total frequency band 22, and kmesh 18 × 18 × 9. Using the resulting charge le, we calculate the density of states (DOS) with a denser 42 × 42 × 21 kmesh, while other settings are the same as in the energy calculation. We use a modi ed BeckeJohnson (mBJ) function to get the DOS le. The DOS le is the input for subsequent BoltzTraP2 calculations to solve the electronic Boltzmann transmission equation. 18 The BoltzTraP2 interpolation magni es the kmesh by 30 times. Using the generated BT2 le, we perform the integration throughout the con guration space. The data is then extracted from the trace le.

Results
The optimized geometry of P63mc wurtzite ZnO has lattice constants of a = b = 3.29 Å and c = 5.30 Å with α = β = 90 o and γ = 120 o , respectively, in a precise agreement with the experimental values. The experimental bandgap of 3.4 eV is di cult to reproduce by conventional rst principle studies, which is normally about 1.1 eV. This difference would cause severe estimation error at high temperatures, as bipolar effects occur for a narrow bandgap semiconductor of 1.1 eV bandgap, whereas the much large bandgap would suppress such effect. We use different CMBJ values of 0.225 and 1.008 to obtain 3.37 eV of the bandgap. The bandgap from other functions can be adjusted to this value, but the mBJ might yield good bands shape that is important in determining thermoelectrics. It is a direct bandgap semiconductor, with a sharp nongenerate parabolic conduction band. Though the initial Fermi energy locates near the top of valence bands for a pure model at absolute zero degrees Kelvin, the doping and heating change the Fermi levels.
According to the rst experimental data, 10 the maximum Fermi level and minimum ones are respectively 0.68 eV and 0.41 eV above the CBM shown in Fig. 1, which have a window of 0.27 eV. They are all above the CBM, meaning that the doping changes the larger bandgap ZnO into metallic due to very heavy doping. The lowest Fermi level and the highest one are respectively at 325K for x = 0.04 and at 1025 K for x = 0.02. This is against normal thermoelectric materials because Fermi energy is a function of temperature and doping level. As the doping level increase, the Fermi level increase, whereas the opposite trend occurs with the temperature change. Another abnormality is that the big span of Fermi energies should cause a large variety of carrier concentration, which will be shown later. The higher than the CBM unambiguously demonstrate the metallic behavior from the electronic structure perspective.
For the second experimental data, 11 the maximum and minimum Fermi levels are respectively 0.14 eV for x = 0.01 at 773K and 0.38 eV for x = 0.03 at 1073 K above the CBM. These two doping cases become metallic substantially, as for an electron, the room-temperature thermal energy is about 0.025 eV. If we adopt the FermiDirac distribution, the carrier concentration should be enormously large. We also notice with the same molecular formula Al x Zn 1−x O, for x = 0.01, the Fermi energies have at least 0.26 eV difference, which is huge.
With the existing experimental temperature-dependent Seebeck coe cient S (T), at each temperature, we search the calculated Seebeck coe cients around the experimental one. In our experience, there are similar values close to each other, but the carrier concentrations Ns are sometimes quite disparate. Therefore, putting N (T) as a checking tool by ensuring its smoothness. We acquire the corresponding S (T). In the process, the temperature dependence of the Fermi level, electronic DOS, electrical conductivity One can measure a carrier concentration experimentally with the Hall method. We check the applicability using the formula r = N*q*Rh, where r, q, and Rh are the Hall factor, carrier charge, and Hall coe cient, respectively. We nd that the r factor is essentially 1.0, the widely used constant in the literature. The reason is that we have one single parabolic conduction band at the Γ. For the n-type ZnO, all other bands are far away, even we integrate over the bands around the Fermi level. The single parabolic band is the foundation for the success of the Hall Effect.
The temperature-dependent electrical conductivity per relaxation time is shown in Fig. 3a. The trend of curves is remarkably similar to that of the N (T) curves. Given the constant relaxation time and electronic mobility, the electrical conductivity should be linear with the carrier concentration.
Even though the carrier concentrations of the 2nd experiment are one order lower than those of the 1st one, the electrical conductivities are a few times lower. This comes from different scattering mechanisms.
Using the calculated electrical conductivity per relaxation time σ/τ, the experimental electrical conductivity σ, and the formula τ = σ exp /(σ/τ) cal at each experimental temperature, we arrive at τ(T), which describes the relaxation time as a function of temperature. We show the τ (T) patterns in Fig. 3b. For the 1st experiment, the three lower τ(T) curves look similar to tellurium doped CoSb3 in terms of the size and trend, which indicates an acoustic phonon scattering dominance with inverse temperature dependence. The relaxation time re ects the obstacles encountered by a moving electron, while this cannot be visualized directly. The electronic mobility offers a more intuitive picture. Using the experimental electrical conductivity σ, the carrier concentration N, and the equation σ = Neµ, where µ is the mobility, we have a µ. The temperature dependence µ (T) curves are shown in Fig. 3c. The pattern looks similar to that of τ (T) curves as these two quantities are linear if the electron mass is unchanged. For the 2nd experiment, the mobility can reach 40 cm 2 /Vs, while the undoped ZnO has an electron mobility of 200 cm 2 /Vs. The high mobility of the 2nd experiment explains the lower but moderate electrical conductivity.
The calculated and experimental temperature-dependent Seebeck coe cient is shown in Fig. 4. For the 1st and 2nd experiments, the size difference is striking, given the similar doping levels. Because the calculated S (T) curves are in good agreement with the experimental ones and the calculated ones have much lower carrier concentrations for 2nd experiment, the low carrier concentration leads to a high Seebeck coe cient. Those much lower carrier concentrations are the reason why in the 2nd experiment, the doped S (T) curves are proximate to the one of pure ZnO. When we have high a Seebeck coe cient, a moderate electrical conductivity, and the corresponding temperature, we have a high power factor, de ned as PF = σS 2 T, which is an important indicator of thermoelectric performance. Based on the above observations, we conclude that the high Seebeck coe cient and high electrical conductivity are not mutually exclusive. When we have a low carrier concentration, we have a high Seebeck coe cient. If a material has a high relaxation time or high carrier mobility, the material would have a high electrical conductivity. This way, we may obtain a good thermoelectric material. Synthesizing this kind of material becomes the key, while the co-deposited Al x Zn 1−x O (x = 0.01 and 0.03) seem to achieve this goal until 673 K, after which the Seebeck coe cients plummet sharply, reducing the high-temperature performance. This trend is against the typical large bandgap semiconductor behavior that has to learn S (T) curves. The sharp decrease of S (T) curves for the 2nd experiment is due to the sharp increase in the carrier concentration after 673 K. The S (T) curves of the 1st experiment are consistent with large bandgap semiconductor whereas the ones of the 2nd are not.
Achieving a high power factor only ful lls part of success, while we also need a low thermal conductivity as low as 3 W/m•K. We calculated the electronic contribution to the thermal conductivity, while the calculation of the lattice contribution involves solving phonon BTE, which needs a different package. We focus on the electrical part and estimate the lattice contribution to be the difference between the experimental total one and the calculated electronic one. The temperature dependence is shown in Fig. 5a. While the experimental к can reach 5 W/m•K for the 2nd experiment, this does not yield a nal good thermoelectric performance as the Seebeck are substantially lower. The electronic contribution to the thermal conductivity is negligible in each doping, which indicating the tuning of the lattice contribution is decisive.

Discussion
A Fermi energy can determine intuitively whether the aluminum-doped wurtzite ZnO becomes metallic, but theoretically estimating the Fermi energy is di cult. For the undoped model, the Fermi level locates in the bandgap region. In any ab initio calculation, we need to build a supercell of a much large number of atoms. For x = 0.01 doping, a supercell has 100 atoms and many con gurations, from which we need to nd the one with the maximum formation energy. For each con guration, we search for the most stable structure, which consumes a huge amount of time. Using the RBA in BoltzTraP2, we tremendously reduce such burden, because we only consider the pure unit cell of ZnO. This concerns 4atom electronic structure, which can be calculated quickly and easily. On this calculated electronic structure, the calculation of varying Fermi levels is straightforward. As a Fermi level is a function of doping level and temperature, we can obtain them conveniently by comparing it with an experiment. For each of the doped case, the Fermi level is above the CBM, displaying that all Al x Zn 1−x O become metallic. We can obtain a high electrical conductivity and a high Seebeck coe cient simultaneously if we have a low carrier concentration and high carrier mobility. Codepositing components could achieve this goal shown in the 2nd experiment, while the instability at high temperature destroys the potential, as the mobility decreases sharply, and the carrier concentration increase sharply, leading to the steeping plummet of the Seebeck coe cient.
In the experiments, powders are used, while we assume a single crystal. The calculated results show that in the experimental temperature range, the thermoelectric coe cients are isotropic. Furthermore, we use the averaged values instead of vectors or tensors. The experimental powder should help to reduce the lattice thermal conductivity, but not the electronic properties, when the grains are not tiny enough to affect electrical structure.

Conclusions
The changed Fermi levels of aluminum-doped wurtzite ZnO indicate that the doping makes the sample metallic, which is unexpected because of the large bandgap. The RF plasma processing and codepositing give similar molecular formula but quite disparate microstructures that constrain conduction electrons quite differently, yielding striking lower carrier concentration for the latter one. The co-deposited ones have a similar Seebeck coe cient to the pure ZnO and much higher mobility and subsequent moderate electrical conductivity. This gives the co-deposited samples good thermoelectric performance until 673 K when the performance plummets. To have good thermoelectric material, the Seebeck coe cient and electrical conductivity need to be both high by having a low carrier concentration and high mobility.

Con icts of interest
There are no con icts to declare.  Temperature dependence of (a) electrical conductivity per relaxation time (σ/τ), (b) the relaxation time (τ) and (c) the mobility (μ). (a) Calculated and experimental temperature-dependent Seebeck coe cient as a function of temperature. The solid gure represents the calculation data, and the hollow gure represents the experimental data.
(b) DOS effective masses as a function of temperature.