Charge carrier density, mobility and Seebeck coefficient of melt-grown bulk ZnGa2O4 single crystals

The temperature dependence of the charge carrier density, mobility and Seebeck coefficient of melt-grown, bulk ZnGa2O4 single crystals was measured between 10 K and 310 K. The electrical conductivity at room temperature is about s = 286 S/cm due to a high electron concentration of n = 3.26*10^(19) cm^(-3), caused by unintenional doping. The mobility at room temperature is mu = 55 cm^2/Vs, whereas the scattering on ionized impurities limits the mobility to mu =62 cm^2/Vs for temperatures lower than 180 K. The Seebeck coefficient relative to aluminum at room temperature is S_(ZnGa2O4-Al) = (-125+-2) muV/K and shows a temperature dependence as expected for degenerate semiconductors. At low temperatures, around 60 K we observed a maximum of the Seebeck coefficient due to the phonon drag effect.

The temperature dependence of the charge carrier density, mobility and Seebeck coefficient of melt-grown, bulk ZnGa 2 O 4 single crystals was measured between 10 K and 310 K. The electrical conductivity at room temperature is about σ = 286 S/cm due to a high electron concentration of n = 3.26 · 10 19 cm −3 , caused by unintenional doping. The mobility at room temperature is µ = 55 cm 2 /Vs, whereas the scattering on ionized impurities limits the mobility to µ = 62 cm 2 /Vs for temperatures lower than 180 K. The Seebeck coefficient relative to aluminum at room temperature is S ZnGa 2 O 4 −Al = (−125 ± 2) µV/K and shows a temperature dependence as expected for degenerate semiconductors. At low temperatures, around 60 K we observed a maximum of the Seebeck coefficient due to the phonon drag effect. a) boy@physik.hu-berlin.de b) sfischer@physik.hu-berlin.de 1 arXiv:2001.10324v1 [cond-mat.mtrl-sci] 28 Jan 2020 Charge carrier density, mobility and Seebeck coefficient of melt-grown bulk ZnGa 2 O 4 single crystals Transparent conducting oxides (TCOs) have drawn attention due to their possible application in high power, optical or gas sensing devices [1][2][3][4][5][6][7][8] . Recently, β -Ga 2 O 3 and related semiconducting oxides with ultra-wide bandgaps of over 4 eV are in the focus, since they offer transparency in the visible spectrum, semiconducting behaviour and breakthrough electric fields of several MV/cm.
The fundamental research has been extended from binary to ternary and quaternary systems to find new substrate material for epitaxial thin film growth, as well as to make use of a higher degree of freedom in terms of doping 9 . ZnGa 2 O 4 is a novel ternary conducting oxide that crystallizes in the spinel crystal structure, which makes it interesting as a substrate for ferrite spinels 9 . Furthermore, the material might be promising for electric application, which gives rise to a study of the fundamental electric and thermoelectric transport properties. The isotropic thermal conductivity at room temperature is λ = 22 W/mK 9 , but many other material parameters remain to be clarified. Theoretical In this work, we investigate as-grown bulk ZnGa 2 O 4 of blueish coloration and perform temperaturedependent Seebeck-, van-der-Pauw-and Hall-measurements between T = 10 K and T = 310 K.
We discuss the results in terms of electron scattering processes and thermoelectric effects observed in a degenerate semiconductor.
The samples have been grown using the vertical gradient freeze method 9   been manufactured on the surface. Figure 1 shows a microscopic image and a schematic view of the microlab, which allows the measurement of the Seebeck coefficient, conductivity using the van-der-Pauw method, as well as the Hall resistance. The Ohmic contacts used for measuring the thermovoltage U th are located in the middle of the four-point metal lines (Ohmic contacts 1 and 3 in Fig. 1), which serve as thermometers and allow the measurement of the temperature difference ∆T .
The microlab has been manufactured by standard photolithography and magnetron sputtering of titanium (7 nm) and gold (35 nm), after cleaning with acetone and isopropanol and subsequent drying. The as-sputtered metal lines of the microlab are isolated due to a Schottky contact relative to the ZnGa 2 O 4 bulk crystal. Ohmic contacts with the ZnGa 2 O 4 bulk crystal were achieved by direct wedge bonding with an Al/Si-wire (99%/1%) on the deposited metal structure, creating point contacts. To keep some parts of the microlab isolated relative to the thin film, the electrical contacts were prepared by attaching gold wire with indium to the Ti/Au metal lines. This procedure can be compared to the one used with β -Ga 2 O 3 , see 23,24 . Figure 2 displays exemplary two-point I-V curves at room temperature.
The experimental procedure is carried out in a flow cryostat between T = 10 K and T = 320 K.
After the bath temperature is stabilized, the van-der-Pauw and Hall measurements are carried out.
Subsequently the Seebeck measurements are performed. The Seebeck measurements involve the creation of various temperature differences by imprinting different currents into the line heater.
The thermovoltage is measured simultaneously for approximately three minutes, which allows to create a stable temperature difference across the sample. Then, while keeping the heating current constant, the resistances of the thermometers are being measured. This procedure is repeated within bath temperature intervals of 10 K.
In the following, we present the measurement results of the electric and the thermoelectric trans- The temperature dependence of the electrical conductivity σ is shown in Fig. 3. The conductivity is between 285 and 315 S/cm for the entire temperature range. For higher temperatures T ≥ 260 K the conductivity σ is decreasing. A maximum can be identified around T = 150 K.
Hall measurements were performed to determine the charge carrier density, which is depicted as a function of temperature in Fig. 4. A linear fit has been added to the plot, showing the weak The dynamic resistance of the Schottky barriers in reverse bias is at least two orders of magnitude higher R dyn.,sch ≥ 1000 Ω. To understand the results of the mobility, we calculated the mean free path l e of the electrons as a function of temperature, shown in Fig. 6 b). The mean free path can be calculated with the following formula Here, E F is the Fermi energy and e is the elemental charge. The Fermi energy was computed from the reduced electron chemical potential η = E F /k B T with the Boltzmann constant k B . The reduced electron chemical potential η was calculated after Nilsson 25 . This method interpolates the range between non-degenerated and degenerated semiconductors and determines the reduced electron chemical potential η as follows with N C being the effective density of states in the conduction band, This can be seen in Fig. 6 a).
In order to determine the thermoelectric properties, the thermovoltage is measured as a function of temperature difference. Figure 7 shows the normalized thermovoltage as a function of nor- (U os < 50 µV) have been substracted from the data. The Seebeck coefficient is determined by The change of the Seebeck coefficient as a function of bath temperature can be observed by the change of slope of the linear fits. Furthermore the maximum achieved temperature difference ∆T max is depicted in Fig. 7.
The Seebeck coefficient S has been determined for temperatures between 30 K and 320 K. The results are shown in Fig. 8  In the following, we discuss all electrical and thermoelectrical properties in detail. The electrical conductivity shown in Fig. 3 has a weak temperature dependence when compared with β -Ga 2 O 3 23 . As can be seen, this originates partly from the very weak temperature dependence of the Hall charge carrier density shown in Fig. 4. The unintentionally doped ZnGa 2 O 4 is a degenerate semiconductor, which we conclude from the high magnitude and the weak temperature dependence of the charge carrier density and the calculated relative Fermi-level shown in Fig. 6 a), which is above the lower conduction band edge for all cases. Furthermore, this is in agreement with the Mott-criterium, which gives an approximation for a critical charge carrier density n c = m * e 2 16πε 0 ε sh The charge carrier mobility was calculated and shown in Fig. 5. For the high temperature regime, it is limited by optical phonon scattering (OP). OP scattering can deviate significantly from the T −3/2 dependence of acoustic deformation potential scattering 27 due to its inelastic nature.
In the chemically related TCOs β − Ga 2 O 3 28 and ZnO 29 it was shown that polar optical phonon scattering is the dominant scattering mechanism at high temperatures due to the partial ionic bonding. The following model after Askerov has proved to be useful for the interpretation of polar optical phonon scattering in degenerate semiconductors 29,30 E POP the average energy of the optical phonons, α the polaron coupling constant and ε ∞ is the high frequency dielectric constant. The high frequency dielectric constant of β − Ga 2 O 3 31,32 (ε ∞ = 3.57) has been used since the exact value for ZnGa 2 O 4 is unknown and it is expected in the same range as the values for ZnO 33,34 (ε ∞ ≈ 3.7).
The mean free path in Fig. 6 b) shows, that there is a temperature independent process that limits the mean free path at l e,max = 4 nm. This limit becomes clear for temperatures lower than T ≤ 180 K when electron-phonon interaction becomes weaker. Furthermore, the mean free distance between single ionized donors, assuming a simple cubic distribution, is shown. One can see, that there is an upper limit of l e,max = 4 nm and that the simple model for the donor distribution predicts a mean distance in the same range.
There are two approaches to explain the low temperature limit of the electron mobility. On one hand, it can be explained by the scattering of electrons with neutral impurities. The Hall charge carrier data in Fig. 4 suggests a constant ionization of the donors, acceptors and vacancies for the investigated temperature interval, so N II = const. This leads to the assumption, that also the neutral impurity density N NI = const. Electron scattering on neutral impurities can be described by 27 µ NI ∝ N −1 NI . Thus, if there is no change in concentration of the neutral impurities, there will be a temperature independent upper limit of the mobility, which can be observed in Fig. 5.
On the other hand, high electron concentrations in semiconductors mean, that there is either a high concentration of singly ionized impurities, or a lower density of ionized impurities with a higher degree of ionization. Furthermore, the scattering of electrons in degenerate semiconductors with ionized impurities can be described by the Brooks-Hering equation 29,35,36 with β (n) = 3 1/3 4π 8/3 ε 0 ε sh 2 n 1/3 e 2 m * .
Z is the degree of ionization and N II = N D + N A = 2N A + n is the density of ionized impurities, with the donator and acceptor densities N D and N A , respectively. The ionized impurity scattering is expected to be more dominant, since there is a high density of ionized donors due to the high Hall charge carrier density.
The fits shown in Fig. 5 have been calculated using the Matthiessen's rule and consider the scattering on polar optical phonons and ionized impurities after eq. (6) and eq. (8), respectively The measurement of the thermovoltage as a function of temperature difference, as shown in Fig. 7, reveals that there is a major change in the maximum reached temperature difference for different bath temperatures. This also correlates with the precision of the data points. The maximum reached temperature difference depends on the imprinted heating power, but is more strongly dependent on the thermal conductivity of the material. The higher the thermal conductivity of the material, the more difficult it is to create large temperature differences. This is the main reason for the increasing uncertainty of the Seebeck coefficient going to lower bath temperatures. From the change of maximum temperature difference, one can conclude the change of the thermal conductivity of the material. Having a look at the precision of the data in Fig. 8, which is correlated with the maximum temperature difference and therefore with the thermal conductivity, one can see, that the thermal conductivity seems to have a maximum around T bath = 60 K and decreases as the temperature decreases further. This could be due to a distortion of the lattice, which has been reported in 9 as particles revealing Moiré patterns in transmission electron microscopy bright field images.
The Seebeck coefficient (Fig. 8) is negative, which means that electrons are the majority charge carriers. This is in agreement with the Hall charge carrier results. The Seebeck coefficient is lower than the one reported 24 for β -Ga 2 O 3 in the same temperature regime. This can be understood, since the semiconducting oxide investigated here is degenerate.
The red area in Fig. 8 marks calculated Seebeck coefficients following the commonly used equation 27 for degenerate semiconductors assuming the effective mass to be between m * = 0.22 m e and m * = 0.66 m e and the scattering factor r to be between r = −0.5 and r = 1.5 S d (m * , n, r) = − k B e r + 3 2 The scattering parameter r is based on the assumption, that the electron relaxation time τ e follows a power law dependence τ e ∝ E r . In other investigations it was observed, that µ ∝ E r and r r + 1 holds in the investigated temperature interval. In general µ = µ(E, T ) and r is calculated by the assumption E = E(T ) = k B T and d ln(µ) d ln(T ) = r + ∂ r ∂ ln(T ) .
A calculated Seebeck coefficient with r = 1.0 and m * = 0.35 m e was added as a thin black dashed line, which fits the data for 100 K≤ T ≤ 200 K best. For T < 200 K the difference of the calculated Seebeck coefficient with r = 1.0, m * = 0.35 m e and the measured Seebeck coefficient is plotted as a green solid line. We account the observed deviation of the theoretical Seebeck coefficient to be due to the phonon drag effect.
In conclusion, we have shown, that as-grown, bulk ZnGa 2 O 4 single crystals show higher electrical conductivity at room temperature (σ ZnGa 2 O 4 ≈ 300 S/cm) than in earlier investigated ZnGa 2 O 4 ceramics (σ ZnGa 2 O 4,ceram. ≈ 30 S/cm 19 ), as-grown β −Ga 2 O 3 bulk (σ β −Ga 2 O 3 ≈ 3 S/cm 23 ) or as-grown ZnO bulk (σ ZnO ≈ 40 S/cm 37 ) due to the high charge carrier density. The donor mechanisms remain to be established. The wide band-gap of the material makes it suitable for application in high-power devices, which can become even more promising, if p-doped material becomes available. In terms of the power factor P f = σ S 2 for thermoelectric applications, ZnGa 2 O 4 has a room temperature value of P f ZnGa 2 O 4 ≈ 4.7 µW/K, being more than 5 times higher than that of β −Ga 2 O 3 with P f β −Ga 2 O 3 ≈ 0.8 µW/K. Therefore, ZnGa 2 O 4 is a more promising material for thermoelectric applications of transparent conducting oxides.

ACKNOWLEDGEMENT
This work was performed in the framework of GraFOx, a Leibniz-ScienceCampus partially funded by the Leibniz association and by the German Science Foundation (DFG-FI932/10-1 and DFG-FI932/11-1).