The temperature dependence of the Seebeck coefficient of homoepitaxial metal organic vapor phase grown, silicon doped *β*-Ga_{2}O_{3} thin films was measured relative to aluminum. For room temperature, we found the relative Seebeck coefficient of $S\beta -Ga2O3-Al=(\u2212300\xb120)$ *µ*V/K. At high bath temperatures *T* > 240 K, the scattering is determined by electron-phonon-interaction. At lower bath temperatures between *T* = 100 K and *T* = 300 K, an increase in the magnitude of the Seebeck coefficient is explained in the frame of Stratton’s formula. The influence of different scattering mechanisms on the magnitude of the Seebeck coefficient is discussed and compared with Hall measurement results.

In the past years, *β*-Ga_{2}O_{3} crystals and thin films have proved to be promising materials for high power devices.^{1–5} Therefore heat transport mechanisms are of great interest. The energy dissipation sums up by Peltier, Fourier, and Joule heating terms, which are a function of the Seebeck coefficient *S*, the electrical conductivity *σ*, and the thermal conductivity *λ*, respectively. While the temperature dependent electrical^{6,7} and thermal properties^{8,9} are known and understood, the thermoelectric parameters remain to be investigated. For this purpose, values of the Seebeck coefficient of *β*-Ga_{2}O_{3} must be known.

*β*-Ga_{2}O_{3} is a transparent material, with a high bandgap *E*_{g} = 4.7 − 4.9 eV at room temperature.^{10–14} The majority charge carrier type is n-type, and the effective mass has been experimentally determined to be in the order of *m*^{*} = 0.25 − 0.28 free electron masses.^{14–16} *β*-Ga_{2}O_{3} has been intensively studied in terms of charge carrier transport with a maximum mobility so far being *μ* = 153 cm^{2}/Vs^{17} at room temperature.

In this work, we implement a microlab, based on metallic lines on the homoepitaxial metal organic vapor phase (MOVPE) grown (100) silicon doped *β*-Ga_{2}O_{3} thin films, and perform temperature-dependent Seebeck measurements between *T* = 100 K and *T* = 300 K. We compare the results with calculated room temperature Seebeck coefficients based on Hall charge carrier density by using Stratton’s formula.^{18}

The thin films have been grown on substrates prepared from Mg-doped, electrically insulating bulk *β*-Ga_{2}O_{3} single crystals, which were grown along the [010]-direction by the Czochralski method.^{17,19} The substrates with (100)-orientation have been cut with a 6° off-orientation to reduce island growth^{19,20} and increase the structural quality of the thin films. The MOVPE process used triethylgallium and pure oxygen as precursors. Silicon doping has been realized by tetraethyl orthosilicate. The substrate temperature was between 750 °C and 850 °C and the chamber pressure was between 5 and 100 mbar during growth.^{21}

The material parameters of the samples are listed in Table I. Both *β*-Ga_{2}O_{3} thin films have the same magnitude of charge carrier densities but a rather big difference in mobility. The thin films have been investigated with AFM measurements, which exhibit step-flow growth for the high mobility sample and two-dimensional island growth for the low mobility sample. This causes a low and rather high density of twin boundaries in the high and low mobility samples, respectively. The sample growth and structural analysis was previously described in detail in the literature,^{21,23} and the influence of twin boundaries was discussed by Fiedler *et al.*^{22} and Ahrling *et al.*^{6}

. | Sample 1 . | Sample 2 . |
---|---|---|

β-Ga_{2}O_{3} thin film
. | High mobility . | Low mobility . |

d (nm) | 185 | 212 |

n(T = 300 K) (cm^{−3}) | (5.5 ± 0.1) × 10^{17} | (6.2 ± 0.1) × 10^{17} |

μ(T = 300 K) (cm^{2}/V s) | 103 ± 1 | 29 ± 1 |

n(T = 100 K) (cm^{−3}) | (1.6 ± 0.1) × 10^{17} | (2.1 ± 0.1) × 10^{17} |

μ(T = 100 K) (cm^{2}/V s) | 233 ± 1 | 48 ± 1 |

. | Sample 1 . | Sample 2 . |
---|---|---|

β-Ga_{2}O_{3} thin film
. | High mobility . | Low mobility . |

d (nm) | 185 | 212 |

n(T = 300 K) (cm^{−3}) | (5.5 ± 0.1) × 10^{17} | (6.2 ± 0.1) × 10^{17} |

μ(T = 300 K) (cm^{2}/V s) | 103 ± 1 | 29 ± 1 |

n(T = 100 K) (cm^{−3}) | (1.6 ± 0.1) × 10^{17} | (2.1 ± 0.1) × 10^{17} |

μ(T = 100 K) (cm^{2}/V s) | 233 ± 1 | 48 ± 1 |

The microlabs have 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 metal lines of the microlab are isolated due to a Schottky contact relative to the *β*-Ga_{2}O_{3} thin film. Ohmic contacts with the *β*-Ga_{2}O_{3} thin film were achieved by direct wedge bonding with an Al/Si-wire (99%/1%) on the deposited metal structure,^{6} 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.

Figure 1 shows the microlab, which allows the measurement of the Seebeck coefficient, charge carrier density, and conductivity.

Figure 1(a) displays a scheme of the microlab. It consists of a two-point line heater at the top, where an electrical current is imprinted to create Joule heating, resulting in a temperature difference Δ*T* across the sample. Two four-point metal lines close to (hot) and far from (cold) the heater line serve as thermometers. The temperature dependent measurement of their resistances follows the Bloch-Grüneisen law^{24} and can be used to calculate the temperature difference in the area of the four-point resistances.

For the determination of the Seebeck coefficient, the exact temperature difference between the contacts must be known. Therefore, the Al-*β*-Ga_{2}O_{3} Ohmic contacts were localized directly on the thermometers at positions I and II (see the supplementary material).

The experimental procedure involves measurement of the thermovoltage for as long as it takes to stabilize the temperature difference across the sample. Afterwards, the four-point resistances of the thermometers are measured. This procedure is repeated for several heating currents before the bath temperature is changed in intervals of 10 K.

Figure 1(b) illustrates a cross-sectional view of the samples. The Mg-doped electrically insulating *β*-Ga_{2}O_{3} substrate has been used to grow the Si-doped *β*-Ga_{2}O_{3} thin film on top thereof. The Ti/Au metal lines of the microlab have then been deposited on the surface of the thin film, and electrical contacts have been prepared either by wedge-bonding with Al-wire (Ohmic contact) or attachment of Au-wire with indium (Schottky contact).

To ensure that the experimental setup has no influence on the Seebeck measurements, different measurement configurations have been developed. Details of the two contact configurations investigated are described in the supplementary material.

The thermovoltage *U*_{th} was measured as a function of temperature difference Δ*T* for several bath temperatures and *β*-Ga_{2}O_{3} thin films.

In the right graph of Fig. 2(a), the measurements of the thermovoltage as a function of time for different temperature differences can be seen. The measurements are performed for 180 s, and the last 10% of the measured data (marked by the vertical dashed line) are used to evaluate the thermovoltage as a function of temperature difference (left graph). Due to the weaker thermal conductivity and diffusivity of *β*-Ga_{2}O_{3} at higher temperatures,^{8,9} it takes longer to evolve a stable temperature difference and thermovoltage as compared to lower temperatures.

The left graph of Fig. 2(a) shows the thermovoltage as a function of the temperature difference. These data can be well fitted with a linear equation, where the slope equals the Seebeck coefficient

Figure 2(b) shows the normalized thermovoltage as a function of normalized temperature difference for bath temperatures of 290 K, 150 K, and 100 K with subtracted offsets for the same sample as Fig. 2(a). The change of the Seebeck coefficient as a function of bath temperature can be observed by the change in the slope of the linear fits.

Figure 3(a) displays the measured Seebeck coefficients of different *β*-Ga_{2}O_{3} thin films as a function of inverse bath temperature between 100 K and 300 K. The Seebeck coefficients are in the range of *S* = −280 *μ*V/K to *S* = −500 *μ*V/K above 100 K. At room temperature, the Seebeck coefficients are *S* = −(300 ± 20) *μ*V/K for all investigated *β*-Ga_{2}O_{3} samples. The precision of the Seebeck coefficient is mainly determined by the certainty of the measurement of the temperature difference. The correction of the Seebeck coefficient due to the thermovoltage of the aluminum wire is less than 1% and does not need to be considered since it is smaller than the uncertainty of the measurement result. The high mobility *β*-Ga_{2}O_{3} thin film shows a stronger increase in the Seebeck coefficient for lower temperatures than the low mobility *β*-Ga_{2}O_{3} thin films. We find that the investigated measurement configuration has no detectable influence on the Seebeck coefficient. The lower magnitude of the Seebeck coefficient for the 212 nm thick low mobility sample at low temperatures is expected to be due to different charge carrier scattering mechanisms.

These results are similar to those of other transparent conducting oxides like ZnO or In_{2}O_{3}, where Seebeck coefficients *S* in the range from *S* = −20 *μ*V/K to *S* = −500 *μ*V/K have been measured.^{25,26}

Figure 3(b) shows the mobility *μ* and Hall charge carrier density *n* as a function of temperature *T* for the investigated samples. The Hall charge carrier density is similar for all samples over the entire temperature range. The plot of the mobility shows electron-phonon (EP) interaction to be the dominant scattering mechanism for *T* > 250 K for all samples. For lower temperatures, a transition of the dominant scattering mechanism can be observed. For *T* < 120 K, electron scattering with ionized impurities becomes dominant. We explain the drop of mobility over the entire temperature range for the low mobility sample by increased scattering on twin boundaries, which has previously been reported.^{6,22}

The study of the temperature dependent Seebeck coefficient for *β*-Ga_{2}O_{3} thin films gives an insight into the scattering mechanisms within the material.^{27–29} Here we discuss the obtained Seebeck coefficients and their description by the Stratton formula,^{18}

with the elemental charge e, the scattering parameter *r*, the reduced electron chemical potential *η* = (*E*_{C} − *E*_{F})/(*k*_{B}*T*) and the conduction band energy *E*_{C}, the Fermi energy *E*_{F}, and the Boltzmann constant *k*_{B}. The scattering parameter is related to the scattering mechanisms within the sample. In the relaxation time approximation, the scattering time and mobility are related to energy by *τ* ∝ *E*^{r} and *μ* ∝ *E*^{r}, respectively. A first approximation of formula (2) has been plotted in Fig. 3. A linear fit with constant *r* = 0.3 ± 0.2 and *E*_{C} − *E*_{F} has been performed for *T* > 130 K and plotted in Fig. 3 (dashed lines). For the 185 nm thick high mobility sample, *E*_{C} − *E*_{F} ≈ 24 meV and for the 212 nm thick low mobility samples *E*_{C} − *E*_{F} ≈ 13 meV have been obtained. These different electron chemical potentials *E*_{C} − *E*_{F} explain the different temperature dependencies but only give a rough estimation since *r* = *r*(*T*) and *E*_{F} = *E*_{F}(*T*).

To calculate theoretical values for the Seebeck coefficient at room temperature, we use the analytical expression after Nilsson,^{30} which interpolates the range between non-degenerated and degenerated semiconductors. The reduced electron chemical potential *η* is then given by

with *N*_{C} being the effective density in the conduction band,

With an effective mass of *m*^{*} = (0.23 ± 0.02) · m_{e}, which we obtained by fitting the measured temperature dependent charge carrier density with the neutrality equation, and a Seebeck scattering parameter of *r* = −1/2,^{29,31,32} which applies for electron-phonon scattering, we calculated theoretical values using the Stratton formula (2). This calculation yields a value of *S*_{th}(*T* > 250 K) = (−300 ± 20) *μ*V/K.

The measurement of the Seebeck coefficient and charge carrier density allows a calculation of the scattering parameter *r* if the effective mass *m*^{*} is known. One has to consider that the measurement of the charge carrier density using Hall measurements is influenced by the scattering of the free electrical charges as well. This is usually described by the Hall scattering factor *r*_{H},^{32} if the relaxation time *τ* can be expressed by *τ* ∝ *E*^{r},

with Γ(*x*) being the gamma function, *μ*_{H} being the Hall mobility, and *μ* being the drift mobility. In order to correctly calculate the general scattering parameter *r*, one needs to know the Hall scattering parameter *r*_{H} or vice versa. Commonly for electron phonon (EP) scattering, *r*_{H} = 1.18 and for ionized impurity (II) scattering *r*_{H} = 1.93 are assumed.^{32} This allows a calculation of the scattering parameter *r*, if we assume that only II or EP scattering dominates the charge transport. Figure 4 shows the calculated *r* as a function of bath temperature. The calculations were performed with Eq. (2) considering the measured Seebeck coefficients and measured charge carrier densities, as well as various Hall scattering factors and effective masses for the 185 nm thick high mobility *β*-Ga_{2}O_{3} thin film.

For *r*_{H} = 1 and *m*^{*} = 0.23*m*_{e} (open diamonds), no correction of the charge carrier density has been done and the effective mass has been used as obtained from temperature dependent charge carrier densities fits. If we consider Hall scattering factors^{32} of *r*_{H} = 1.93 and *r*_{H} = 1.18 for ionized impurity scattering and electron-phonon interaction, respectively, and assume an effective mass of *m*^{*} = 0.25*m*_{e} as reported in the literature,^{14} we obtain the blue dots and red squares for II and EP scattering, respectively. The obtained results near room temperature are in agreement with the expected value of *r* = −0.5 for electron-phonon-scattering (Fig. 4, lower dashed line).^{29,31,32} For *β*-Ga_{2}O_{3} thin films, electron-phonon-scattering is expected to be the dominant scattering mechanism at these temperatures.^{6} Here, the scattering mechanism dominating the charge transfer caused by electrical potential gradients is the same as caused by temperature differences.

For low temperatures, we obtain values close to *r* = 3/2 as expected for ionized-impurity scattering (upper dashed line). This is also in agreement with values reported in the literature.^{29,31,32} Ionized impurity scattering has been reported^{6,33} to become the dominant scattering mechanism in *β*-Ga_{2}O_{3} below 100 K, depending on layer thickness and doping. This explains why we obtain values close to 1.5 if we assume that the charge carrier scattering mechanisms influencing the mobility and the Seebeck effect are the same.

The Seebeck coefficients in *β*-Ga_{2}O_{3} thin films at room temperature are mainly dependent on the doping level since electron phonon scattering is the dominant scattering mechanism, whereas at low temperatures the different dominant scattering mechanisms have a major impact on the magnitude of the Seebeck coefficient. These observations are consistent with previous studies on thermal^{8,9} and electrical^{6,7} conductivity where the dominant scattering mechanisms at room temperature are phonon-based. For applications at higher temperatures, the room temperature magnitude of the Seebeck coefficient gives an upper limit. Furthermore we can estimate that *S* approaches a value of 2*k*_{B}/*e* in the intrinsic regime.

In conclusion, the temperature dependent Seebeck coefficient of homoepitaxial *β*-Ga_{2}O_{3} thin films can be explained by the Stratton formula. *β*-Ga_{2}O_{3} thin films have Seebeck coefficients relative to aluminum of $S\beta -Ga2O3-Al=(\u2212300\xb120)$ *µ*V/K at room temperature with an increase in magnitude at lower temperatures. This leads to a room temperature Peltier coefficient of Π ≈ 0.1 V. The dependency of the Seebeck coefficient on the dominant scattering mechanism gives the possibility of Seebeck coefficient engineering by growing *β*-Ga_{2}O_{3} thin films with, for example, an increased concentration of neutral impurities or ionized impurities.

See supplementary material for details of the two contact configurations investigated.

This work was performed in the framework of GraFOx, partially funded by the Leibniz Association and by the German Science Foundation (Grant Nos. DFG-FI932/10-1 and DFG-FI932/11-1). The authors thank M. Kockert for fruitful scientific discussions.

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_{2}O

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_{2}O

_{3}for wide-bandgap electronics and optoelectronics

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_{3}thin films

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_{3}micro flakes

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_{3}single crystals by the Czochralski method

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