Enhancement of 2DEG effective mass in AlN

We report on the free charge carrier properties of a two-dimensional electron gas (2DEG) in an AlN/Al x Ga 1– x N high electron mobility transistor structure with a high aluminum content ( x ¼ 0.78). The 2DEG sheet density N s ¼ ð 7 : 3 6 0 : 7 Þ (cid:2) 10 12 cm (cid:3) 2 , sheet mobility l s ¼ ð 270 6 40 Þ cm 2 /(Vs), sheet resistance R s ¼ ð 3200 6 500 Þ X = (cid:2) , and effective mass m eff

Ultra-wide-bandgap semiconductors, such as Al x Ga 1Àx N with high Al-content, are very promising materials for next generation high-power electronic devices.A common way to compare the potential material performance are figures of merit, such as the unipolar and lateral figures of merit (UFOM and LFOM), indicating that materials minimize conduction losses in low-frequency unipolar and lateral device structures, respectively. 1,2The Johnson figure of merit (JFOM) indicates a material's suitability for high frequency power transistor applications.While the JFOM of AlGaN outperforms GaN for any Alcontent, 3 simulations of the UFOM and LOFM at room temperature for AlGaN show that this is only the case for an Al-composition of x > 0.7 and x > 0.9, respectively. 4,5However, the typical temperatures at which devices operate are much higher.For example, at 500 K, the UFOM and LFOM of AlGaN exceed the FOMs of GaN at x >0.3 and x >0.65, respectively. 4,6,7Furthermore, it is reasonable to assume that for very high Al-compositions (x >0.9), the conduction band offset at the AlN/AlGaN interface is not large enough anymore to provide sufficient confinement for the two-dimensional electron gas (2DEG) that can form at the interface in high electron mobility transistors (HEMTs). 8However, HEMTs with Al x Ga 1Àx N channels with slightly lower Al-content of 0:7 < x < 0:9 are of interest for future highpower and high-frequency devices.Though there are numerous publications reporting device structures with AlGaN 2DEG channels, [9][10][11][12][13][14][15][16][17][18][19] only few demonstrate structures with high Al-content x !][22] Despite progress in the growth of such devices, achievement of low Ohmic contact resistances, while maintaining the 2DEG in the channel, is still challenging. 23High Al-content AlGaN has a low electron affinity (in AlN the electron affinity is 0.6 eV) that naturally leads to large Schottky barriers. 24Methods to achieve Ohmic contacts include, among others, adding a heavily doped GaN cap layer, adding composition graded contact layers, 25 or using regrown Ohmic contacts. 2612]17 The optical Hall effect (OHE) is a contactless method for accurate determination of bulk free charge carrier and 2DEG properties and can be employed to study high Al-content AlGaN structures avoiding the issues with contacts. 27The OHE essentially combines generalized ellipsometry at long wavelengths with high static magnetic fields, allowing to characterize free charge carrier properties at frequencies relevant for device applications.Previously, the OHE was successfully employed for the determination of 2DEG properties in Al 0.26 Ga 0.74 N/GaN structures by Sch€ oche et al. 28 and for their temperature dependence in 0.25 Ga 0.75 N/GaN structures by Hofmann et al. 29 A cavity enhanced OHE was demonstrated on Al 0.82 In 0.18 N/AlN/GaN 30,31 and on Al x Ga 1Àx N/AlN/GaN (x ¼ 0.19 and x ¼ 0.25) HEMT structures. 32,33HE data analysis was discussed by Schubert et al. 27 In this work, we present contactless characterization of 2DEG properties in an AlN/Al x Ga 1Àx N HEMT with a high Al-content (x ¼ 0.78) channel.The free charge carrier parameters of the 2DEG are extracted from THz OHE measurements and discussed using results obtained in this Letter from self-consistent Poisson-Schr€ odinger calculations.
The HEMT structure was grown in a hot-wall metal-organic chemical vapor deposition (MOCVD) reactor on a semi-insulating 4H-SiC (0001) substrate with a nominal thickness of 500 lm.First, a 250 nm-thick AlN nucleation layer was grown at 1200 C, followed by a 100 nm-thick Al x Ga 1Àx N graded layer grown as the temperature was reduced from 1200 to 1040 C. 34 Then, a 400-nm-thick Al x Ga 1Àx N layer with x ¼ 0.78 was grown at 1040 C. Finally, an AlN barrier with the thickness of 10 nm was grown at 1040 C. A 2DEG forms at the AlN/Al x Ga 1Àx N interface for which the roughness is expected to be comparable to the surface roughness of an identical sample structure without the AlN barrier with the root mean square (RMS) of 0.67 nm determined over a 3 Â 3 lm 2 area.
The THz OHE measurements were employed to determine the free charge carrier properties of the 2DEG channel, using an in-house built THz ellipsometer. 33The sample was mounted in a cavity enhanced (CE) configuration 30 on an aluminum plate, using adhesive tape spacers providing a nominal cavity thickness of 100 lm.The sample was loaded into a superconducting 8 T magnet with optical ports and cooled to a temperature of T ¼ 5 K. Mueller matrix spectra 35 M 0 and M þ at magnetic fields of B ¼ 0 T and B ¼ 8 T, respectively, were recorded in the reflection configuration at an angle of incidence of U ¼ 45 , with the magnetic field parallel to the incoming THz beam.For data analysis, OHE difference spectra Figure 1 shows a high-resolution x-ray diffraction (HRXRD) reciprocal space map (RSM) around the 10 15 reciprocal lattice point.Analysis confirms the Al-content of x ¼ 0.78 in the channel layer.Also, in-and out-of-plain strain of e c ¼ 0:0003 and e a ¼ 0:0004, respectively, are determined, indicating that the Al x Ga 1Àx N layer is highly relaxed.The sample is crack-free thanks to the strain management via the graded AlGaN buffer layer.
OHE data analysis in the THz spectral range was performed employing stratified layer model calculations, using the 4 Â 4 matrix formalism, 35 in combination with the Levenberg-Marquardt algorithm and parametrized model dielectric functions. 27The model consists of a semi-infinite layer for the metal backside, a void gap with variable thickness for the backside cavity, a uniaxial layer with variable thickness for the 4H-SiC substrate, and a layer with a fixed thickness of d ¼ 1 nm for the 2DEG channel.Due to the large ratio of the wavelength to the product of index of refraction and layer thickness (k ) nd), all other layers present in the sample structure only add minor contributions to the ellipsometry spectrum and can, therefore, be neglected.The optical properties of the metal backside are modeled with a classic Drude dielectric function with free charge carrier density N ¼ 10 23 cm À3 , mobility l ¼ 5 cm 2 /(V s), and effective mass m eff ¼ m 0 , where m 0 is the free electron mass.The index of refraction of the backside cavity is set to n ¼ 1.The 4H-SiC substrate has a hexagonal crystal structure and is, therefore, optically anisotropic (uniaxial).Its THz optical properties are determined by measuring a bare substrate to be n ?¼ 3:13 and n k ¼ 3:20, for the in-and out-of-plane indices of refraction, respectively.The optical response of the 2DEG channel is modeled by a dielectric tensor based on the Drude model augmented by a term accounting for the influence of the Lorentz force.Instead of the volume carrier density, we determine the sheet carrier density N s ¼ dN, where d is a virtual optical thickness of the 2DEG and N is a virtual three dimensional charge carrier density.In our OHE experiment, we only detect information related to N s .This can be best understood when inspecting the mathematical formalism, which is used to analyze our OHE data.A partial 4 Â 4 transfer matrix is composed representing every physical layer in the layer stack, 35 T p ¼ exp i x c Dd À Á .For ultra thin layers, k ) d, e.g., for a 2DEG at THz frequencies.It can be shown that all resulting OHE spectra regardless of whether measured in reflection or transmission, for ultra thin layers, are only proportional to and dependent on the product dN, hence, N s , while no sensitivity is contained in the optical data to differentiate d and N. Further model parameters include a small offset parameter for the angle of incidence and a calibration offset for the polarizer.Best-model calculations are performed by varying the parameters named above together with the sheet carrier density N s , mobility l, and electron effective mass m eff of the 2DEG channel.More details on the principles of OHE data analysis together with relevant optical models are given in Ref. 27.
Experimental data together with best-model calculations are presented in Fig. 2. We find a good agreement between experimental (blue squares) and best-model calculated (red lines) data.The pronounced feature at % 0:8 THz is induced by a Fabry-P erot interference in the 4H-SiC substrate and backside cavity.The spectral location of the feature is determined by the substrate and backside cavity thicknesses, while the line shape and amplitude are strongly influenced by the 2DEG free charge carrier properties.The block-off diagonal elements dM þ 13;31 and dM þ 23;32 deviation from zero are indicative of crosspolarization emerging from the sample under linear polarized light illumination at the given frequency.Furthermore, these matrix elements vanish for B ¼ 0 T. Therefore, the appearance of cross polarization for B 6 ¼ 0, in a stack of optically isotropic layers, is caused by the magnetic field (Lorentz force).This confirms the existence of a channel with free charge carriers.Additionally, the sign of the features in the OHE data (e.g., the maximum in dM þ 32 ) indicates that electrons are in the sample.The best-model parameters for the angle of incident offset and the calibration offset for the polarizer are dU ¼ ð1:8 6 0:1Þ and H p ¼ ðÀ0:09 6 0:01Þ , respectively.The thicknesses of the backside cavity and the 4H-SiC substrate are d cavity ¼ ð114:3 6 0:4Þ lm and d SiC ¼ ð494:0 6 0:1Þ lm, respectively, which is in good agreement with the nominal values.The best model has a mean squared error (MSE) of MSE ¼ 1.44, it yields a sheet carrier density of the 2DEG of N s ¼ ð7:3 6 0:7Þ Â 10 12 cm À2 , a sheet mobility of l s ¼ ð270 6 40Þ cm 2 /(V Á s), a sheet resistance of R s ¼ ð3200 6 500Þ X=ٗ, and an effective mass of m eff ¼ ð0:63 6 0:04Þm 0 .
We conducted simulations of the HEMT structure under investigation employing self-consistent Poisson-Schr€ odinger calculations. 36,37The conduction band offset is assumed to be 65% of the difference between AlN and Al 0.78 Ga 0.22 N bandgaps. 38For Al 0.78 Ga 0.22 N, the bandgap energy is obtained by interpolation between values of the binary compounds at T ¼ 5 K with a bowing parameter of 0.8 eV.The bandgap of AlN and Al 0.78 Ga 0.22 N is corrected for the strain measured in the layers.Figure 3(a) displays the charge carrier distribution in the vicinity of the AlN/Al 0.78 Ga 0.22 N interface and (b) the conduction band profile together with the quantized energy levels in the triangular quantum well.In the case of a fully relaxed Al 0.78 Ga 0.22 N layer, the sheet carrier density is calculated as N s ¼ 2 Â 10 13 cm À1 , which is about twice as high as the value extracted from our OHE data.Furthermore, capacitance-voltage (C-V) measurements were performed using a Hg-probe setup with a 4284A LCR meter from Agilent.A sheet carrier density of N s ¼ 5:2 Â 10 12 cm À2 was determined at room temperature.This value is very similar to the low-temperature N s obtained from the OHE, and it further validates our results, since for polarization doping, only a small temperature dependency of 2DEG density is expected.
Comparing with the conventional Al 0.19 Ga 0.81 N/GaN HEMT structures, 32 the sheet mobility measured in AlN/Al 0.78 Ga 0.22 N HEMT is significantly lower.Note that here the 2DEG is located in the Al 0.78 Ga 0.22 N, where the mobility is mainly limited by alloy scattering. 4,23,39The mobility value that we measure is at T ¼ 5 K, while literature data for similar device structures are typically determined at or above room temperature.The alloy scattering is the limiting scattering mechanism up to room temperature. 4,5Its contribution to the total mobility has been predicted to follow a T À1=2 dependence. 40Using FIG. 2. Best-model (red lines) and experimental (blue squares) cavity enhanced OHE difference spectra dM þ ¼ M þ À M 0 (M þ : Mueller matrix data at positive field; M 0 : Mueller matrix data at zero field) for AlN/Al 0.78 Ga 0.22 N HEMT structure at U ¼ 45 , T ¼ 5 K and B ¼ 8 T, with the magnetic field parallel to the incident THz beam.The inset on top indicates the sample configuration during OHE measurements.
this relation, we extrapolate a room temperature mobility of l s;300 K ¼ 35 cm 2 /(V Á s), which is in good agreement with calculations for Al 0.78 Ga 0.22 N from Coltrin et al., 5 but lower than the value obtained by Bajaj et al. 4 In our case, the room temperature free charge carrier properties could not be determined due to the lower mobility value resulting in OHE signatures in Mueller matrix spectra below the signal-to-noise ratio of our system.
The effective mass determined here is significantly higher than the value measured recently by Sch€ oche et al. using mid-infrared OHE for epitaxial Si-doped Al 0.79 Ga 0.21 N layers (m eff ¼ 0:334 m 0 ). 41The discrepancy may be due to (i) the enhancement of the effective mass due to hybridization, conduction band non-parabolicity, the electronphonon coupling (polaronic effect) in the case of two-dimensional (2D) structures, 42,43 and/or (ii) the possible existence of physical mechanisms at THz frequencies in 2DEG channels, which are not accounted for by the Drude model for quasi-free electrons involved here in our analysis.
In order to quantify possible mass enhancement phenomena, self-consistent Poisson-Schr€ odinger calculations are performed.Our calculations provide the space distribution of the 2DEG at AlN/ Al 0.78 Ga 0.22 N interface as well as the probability p AlN for the electron wave function to penetrate into the AlN barrier.For a relaxed Al 0.78 Ga 0.22 N layer, we found that p AlN ¼ 13%.The hybridized effective mass can be calculated using 1=m 44 which leads to an increase by 2%.The effect of the conduction band non-parabolicity can be roughly estimated by m [44][45][46] where E 0 is the sum of the kinetic energy of the electrons in the triangular well and the Fermi energy and E g is the bandgap energy.The non-parabolicity is found to increase the effective mass by 7%.The polaronic contribution to the electron effective mass is estimated in the case of 2D electrons with a weak phonon coupling 44,47,48 , where a is the Fr€ ohlich polaron coupling parameter.To evaluate the magnitude of the polaronic effect, we use the Fr€ ohlich coupling parameter for AlN, a ¼ 0:75, 49 and obtain 36% increase in the effective mass.Combining all effects, the electron effective mass of m Ã eff ¼ 0:50 is estimated.However, this value serves as an upper limit, since the polaronic effect should be reduced due to screening at carrier concentration determined. 44,50Hence, our OHE result of m eff ¼ ð0:63 6 0:04Þm 0 is still substantially larger than the possible effective mass enhanced by polaronic, non-parabolicity, and hybridization effects.At this point, we have no explanation for this observation, and we believe that potentially, physical processes not presently included into our OHE model analysis may be responsible for the observed mass enhancement at THz frequencies at very large Al-content AlGaN/GaN HEMT 2DEG structures.Future work is proposed to study the temperature dependence of charge carrier properties and high frequency electrical properties.
We studied the free charge carrier properties of 2DEG in an AlN/ Al 0.78 Ga 0.22 N HEMT structure at low temperature ðT ¼ 5 KÞ employing the THz optical Hall effect and compared the results with selfconsistent Poisson-Schr€ odinger calculations.While the measured 2DEG sheet density of N s ¼ ð7:3 6 0:7Þ Â 10 12 cm À2 is in good agreement with our self-consistent Poisson-Schr€ odinger calculations, and the low temperature sheet mobility of l s ¼ ð270 6 40Þ cm 2 /(V Á s) falls within the expected range, the extracted electron effective mass value of m eff ¼ ð0:63 6 0:04Þm 0 is significantly higher than what is reported in the literature for bulk Al 0.78 Ga 0.22 N layers. 41The observed enhancement of the electron effective mass is attributed to the combined effects of the hybridization, conduction band non-parabolicity, and electron-phonon coupling.However, physical mechanisms not accounted for in our OHE model analysis cannot be ruled out.

FIG. 1 .
FIG. 1. HRXRD reciprocal space map around the asymmetric 10 15 reciprocal lattice point for the AlN/Al 0.78 Ga 0.22 N/SiC HEMT structure.The lateral (perpendicular to the c-axis) and vertical (along the c-axis) scattering vectors Q x and Q z are given in reciprocal lattice units (rlu) with dimension of 2p=k, where k ¼ 1:540 597 4 A ˚.The sample structure is indicated on the right.

FIG. 3 .
FIG. 3. (a) Charge carrier distribution in the vicinity of the AlN/AlGaN interface, obtained by the self-consistent solution of the one-dimensional Poisson-Schr€ odinger equation.Calculations for the charge carrier distribution are shown for strained and fully relaxed Al 0.78 Ga 0.22 N. (b) Conduction band profile and quantized energy levels for the same structure, where the Al 0.78 Ga 0.22 N layer is assumed to be relaxed.