The optical Hall effect (OHE) in semiconductor layer structures is the occurrence of magneto-optic birefringence detected in response to incident electromagnetic radiation, caused by movement of free charge carriers under the magnetic field-induced influence of the Lorentz force [1]. In general, this birefringence leads to polarization mode coupling that is conveniently detected by generalized ellipsometry at oblique angle of incidence and at terahertz (THz) frequencies, for example. THz-OHE has recently been demonstrated as noncontact and therefore a valuable tool for the investigation of free charge carrier properties in semiconductor heterostructures [2 –9]. The OHE discussed in this work is not to be confused with the “polarization-dependent Hall effect of light,” described for example in Refs. [10,11]. Previous instrumental approaches, discussed more detailed in Refs. [2,9], rely on high magnetic fields provided either by conventional, water-cooled or superconducting, liquid He-cooled electromagnets resulting in comparably large and costly experimental setups. In general, OHE configurations capable of detecting signals at low and conveniently obtainable magnetic fields are desirable. The use of small magnetic fields for THz magneto-optic measurements was demonstrated recently by Ino et al. for bulk-like InAs [12]. Due to low effective mass and high electron concentration, the low field still yielded large enough signals for detection. The signal-to-noise separations of the OHE signatures depend on many factors, the most important being low effective mass, high mobility, and high carrier density, but also crucial is the thickness of the physical layer that contains the charge carriers. The OHE signals are defined by and presented here as the differences between the off-diagonal Mueller matrix elements determined for opposing magnetic field directions [2]. The amplitude of the OHE signals is proportional to the percentage of cross-polarization at the given frequency [8,13] and is caused only by the Drude-like magneto-optic contribution of the two-dimensional electron gas (2DEG). Note, the off-diagonal Mueller matrix difference spectra are expected to be zero if there is no external magnetic field. These signals, in first approximation, scale linearly with the magnetic field amplitude. Hence, the first approach to detect OHE signatures in samples with very thin layers, or low-mobile, heavy-mass, and low-density charge carriers where the OHE signatures are weak is to increase the magnetic field amplitude.

In this communication, we demonstrate and exploit the enhancement of the OHE signal obtained from samples with plane parallel interfaces deposited on THz transparent substrates using an external and tunable optical cavity. We show that an OHE signal enhancement of up to one order of magnitude can be achieved by optimizing the cavity geometry, which is very useful for small magnetic field strengths. This signal enhancement allows the determination of free charge carrier effective mass, mobility, and density parameters using OHE measurements at low magnetic fields. An AlInN/GaN-based high-electron mobility transistor structure (HEMT) grown on a sapphire substrate is investigated as an example, while the cavity enhancement phenomenon discussed here is generally applicable to situations when a layered sample is deposited onto a transparent substrate. This cavity enhancement may be exploited in particular for layered samples grown on technologically relevant low-doped or semi-insulating substrate materials such as SiC, Si, or GaAs, etc.

For a thin-film layer stack deposited on a transparent substrate, where the substrate thickness is much larger than the combined layer thickness of all sublayers in the stack, and where the substrate thickness may be multiple orders of the wavelength at which the OHE signatures are detected, the fraction of the incident beam transmitted through the entire sample is coupled back into the substrate using an external cavity. For example, a highly reflective surface placed at a distance $d_{\mathrm{gap}}$ behind and parallel to the backside of the substrate, as shown in Fig. 1, permits THz radiation to be coupled back into the sample and thereby produce an enhancement of the OHE signal. The enhancement is due to the positive interference of wave components traveling back and forth within the coupled cavity-substrate while undergoing polarization conversion upon passing the magneto-optic birefringent 2DEG multiple times. Thereby, the amount of polarization-converted light increases, which gives rise in the measured off-diagonal Mueller matrix elements. In our example discussed below, we have achieved up to one order of magnitude enhancement by varying $d_{\mathrm{gap}}$.

 

Fig. 1. Schematic drawing of the beam path through the sample and the external optical cavity, shown for example for an AlInN/GaN/sapphire HEMT structure with 2DEG. The sapphire substrate and metallic cavity surface are parallel and separated by the distance $d_{\mathrm{gap}}$. Here, the magnetic field $\mathbf{B}$ is perpendicular to the sample surface with the positive magnetic field direction oriented into the sample.

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Figure 2 shows model-calculated contour plots of the THz-OHE signal (difference of the Mueller matrix elements calculated for $B=0.55\mathrm{T}$ and $B=-0.55\mathrm{T}$) illustrating the enhancement phenomenon. The structure used for the calculation is a AlInN/GaN HEMT structure deposited on a 350-μm-thick $c$-plane ${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate, similar to the HEMT structure discussed in Ref. [7]. The nontrivial off-block Mueller matrix components $\mathrm{\Delta }{M}_{13,31}$ and $\mathrm{\Delta }{M}_{23,32}$ show periodic resonances that depend on the frequency of the THz probe beam $\nu$ and $d_{\mathrm{gap}}$. The frequency is varied over the range from 600 to 1000 GHz and the $d_{\mathrm{gap}}$ ranges from 0 to 1000 μm to obtain overview over an experimentally feasible parameter range and to gain insight into the multiplicity of the occurrences of coupled-substrate-cavity mode enhancements of the OHE signal. In order to show that these occurrences are related to the minima in reflectance for the substrate and cavity modes, the maxima (minima) of the $s$-polarized reflectivity of the sample and the cavity are plotted as solid (dashed), vertical, and horizontal lines, respectively. The signatures follow a commonly observed anti-band crossing behavior, where the bands of the substrate reflectance minima couple with the OHE bands and induce strongest changes with frequency and cavity thickness. The resonance frequency of the sample Fabry–Perot mode is determined by the sample’s substrate thickness, which is much larger than the total HEMT thickness (see further below). The $p$-polarized modes that occur indistinguishably close to the $s$-polarized modes are omitted for clarity in Fig. 2. It is interesting to note that there are regions in frequency and $d_{\mathrm{gap}}$ where the OHE signal is very small or vanishes. Configurations where both frequency and $d_{\mathrm{gap}}$ can be varied over sufficiently large regions, that is, to cover at least one period of coupled substrate-cavity modes will be valuable for practical applications.

 

Fig. 2. Model-calculated contour plots of typical THz-OHE data, here for example $\mathrm{\Delta }{M}_{13,31}=M_{13,31}(+B)-M_{13,31}(-B)$ and $\mathrm{\Delta }{M}_{23,32}=M_{23,32}(+B)-M_{23,32}(-B)$ for the AlInN/GaN HEMT sample at ${\mathrm{\Phi }}_a=45°$ and $|B|=0.55\mathrm{T}$ are shown as a function of frequency and $d_{\text{gap}}$. The vertical solid and dashed black lines indicate the sample’s $s$-polarized reflection maxima and minima, respectively. The $s$-polarized reflectivity maxima and minima of the external cavity and that depend on $d_{\mathrm{gap}}$ are shown as horizontal solid and dashed black lines, respectively. Note that the $p$-polarized modes occur indistinguishably close to the $s$-polarized modes and are omitted for clarity.

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For the experimental verification of this enhancement effect, an AlInN/GaN-based HEMT structure was grown using metal-organic vapor phase epitaxy on a single-side polished $c$-plane sapphire substrate with a nominal thickness of 350 μm. Subsequent to the growth of a 2-μm-thick undoped GaN buffer layer, a 1-nm-thick AlN spacer layer was deposited, followed by a 12.3-nm-thick ${\mathrm{Al}}_{0.82}{\mathrm{In}}_{0.18}\mathrm{N}$ top layer [14]. The THz-OHE data presented here were obtained using a custom-built THz ellipsometer [2,15]. THz-OHE data were measured in the spectral range from 830 to 930 GHz with a resolution of 2 GHz at an angle of incidence ${\mathrm{\Phi }}_a=45°$ and for three different gap distances $d_{\mathrm{gap}}$ of 104.7, 194.5, and 280.7 μm.

 

Fig. 3. Panels (a) and (b) show the corresponding experimental (green lines) and best-model calculated (red solid lines) data $\mathrm{\Delta }{M}_{13,31}$ and $\mathrm{\Delta }{M}_{23,32}$ at three different $d_{\mathrm{gap}}$ values. The $\mathrm{\Delta }{M}_{13}$ ($\mathrm{\Delta }{M}_{23}$) and $\mathrm{\Delta }{M}_{31}$ ($\mathrm{\Delta }{M}_{32}$) spectra are shown as open and closed data points, respectively. The panels (a) and (b) also include best-model calculated data for $d_{\mathrm{gap}}\to \infty$ as blue solid lines for comparison.

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The measurements were facilitated by mounting the HEMT structure onto a Ni-coated, high-grade N42 neodymium permanent magnet using adhesive spacers to create a homogeneous air gap between the Ni-coated surface of the magnet that serves as the metallic cavity backside and the HEMT structure. The THz Mueller matrix measurements were carried out with the sample mounted on the north and on the south pole-face of the permanent magnet to obtain THz-OHE data (differences of the Mueller matrix elements $M_{13}$, $M_{23}$, $M_{31}$, and $M_{32}$ measured at opposing magnetic fields). Across the sample area illuminated by the THz probe beam, the magnetic field strength provided by the permanent magnet was $B=(0.55\pm 0.005)\mathrm{T}$. For values of $d_{\mathrm{gap}}$ used here, the change in the magnetic field magnitude at different gap values is negligible at the sample position.

In addition to the THz-OHE measurements, the sample as well as the metal magnet surfaces were investigated using a commercial (J.A. Woollam Co. Inc.) mid-infrared (MIR) ellipsometer in the spectral range from 300 to $1200{\mathrm{cm}}^{-1}$ at ${\mathrm{\Phi }}_a=60°$ and 70° in order to determine the HEMT layer thickness parameters and phonon mode parameters, and the optical constants of the magnet surface metal layer (Ni). All measurements were carried out at room temperature and analyzed simultaneously. The experimental and model calculated data are reported using the Mueller matrix formalism [16].

The experimental MIR-SE and THz-OHE data sets were analyzed simultaneously using an optical model composed of eight phases including a AlInN top layer/2DEG/AlN spacer/GaN buffer/${\mathrm{Al}}_2{\mathrm{O}}_3$ substrate/air gap/Ni cavity surface [7]. Nonlinear regression methods were used to match the lineshape of experimental and optical model calculated data as close as possible by varying relevant model parameters using parameterized model dielectric functions [16]. The THz and MIR dielectric function tensors of the optically uniaxial sample constituents GaN, AlInN, AlN and ${\mathrm{Al}}_2{\mathrm{O}}_3$ are composed of contributions from optically active phonon modes ${\epsilon }^L(\omega )$ and free-charge carrier excitations ${\epsilon }^{\mathrm{FC}}(\omega )$. Details on the parametrization approach are omitted here for brevity and we refer to previous publications [1,8,13,17,18]. The optical response of the magnet’s Ni mirror surface that forms the external cavity is described using the classical Drude formalism using the static resistivity parameter of $\rho =(1.72\pm 0.49)\times {10}^{-5}\mathrm{\Omega }\mathrm{cm}$ and the average-collision time parameter of $\tau =6.15\times {10}^{-16}\mathrm{s}$. $\rho$ is obtained as best-match model parameter from MIR-SE data analysis, and is within typical values for Ni [19,20]. The average-collision time parameter is taken from Ref. [19] and not varied in the model analysis.

Figure 3 shows experimental (green lines) and best-model calculated (red solid lines) THz-OHE spectra (differences of the Mueller matrix elements $M_{13}$, $M_{23}$, $M_{31}$, and $M_{32}$) measured at $B=0.55\mathrm{T}$ and $-0.55\mathrm{T}$. For this sample structure and the perpendicular magnetic field orientation, the magnetic-field-induced changes in the Mueller matrix elements $M_{13}$ ($M_{23}$) equal those in $M_{31}$ ($M_{32}$). For comparison, best-model calculated data for $d_{\mathrm{gap}}\to \infty$ is shown as blue solid lines. Based on the best-model analysis the low background free charge carrier densities of the AlInN, GaN, and AlN layers were found to have a negligible contribution to the THz-OHE signal. We find a good agreement between experimental and best-model calculated data for the different $d_{\mathrm{gap}}$ values.

The maxima and minima depicted in Fig. 3 are due to the coupling of the Fabry–Perot oscillations in the sample structure with those of the external cavity. The experimentally accessed range in frequency and $d_{\mathrm{gap}}$ was selected to sufficiently cover the response of the HEMT sample-substrate-cavity mode under the influence of a small magnetic field to detect the enhanced OHE signal. Depending on the distance $d_{\mathrm{gap}}$ between sample backside and cavity surface, the frequency-dependent response of the OHE signal changes where extrema occur in the vicinity of the intersection of the sample and cavity reflection extrema. Comparing $\mathrm{\Delta }{M}_{13,31}$ and $\mathrm{\Delta }{M}_{23,32}$ reveals distinct differences. $\mathrm{\Delta }{M}_{23,32}$ shows a derivative-like shape with a single pair of maximum and minimum in the range from 830 to 930 GHz, whereas $\mathrm{\Delta }{M}_{13,31}$ exhibits a single maximum or minimum and a strong amplitude variation, depending on $d_{\mathrm{gap}}$. The largest amplitude is observed for $d_{\mathrm{gap}}=194.5\mathrm{\mu m}$ where $\mathrm{\Delta }{M}_{13,31}$ is approximately 0.15. The smallest change is observed for $d_{\mathrm{gap}}=104.7\mathrm{\mu m}$ where $\mathrm{\Delta }{M}_{13,31}\approx 0.07$. The best-model calculated data excluding the cavity enhancement shown as solid blue line in Fig. 3(a) is almost vanishing and the cavity enhances the OHE signal by one order of magnitude. The largest amplitudes of $\mathrm{\Delta }{M}_{23,32}\approx 0.1$ can be observed for $d_{\mathrm{gap}}=104.7\mathrm{\mu m}$ and 280.7 μm [Fig. 3(b)]. The largest amplitudes in $\mathrm{\Delta }{M}_{23,32}$ without the cavity effect is approximately 0.05, which is a factor of two smaller than the OHE signal amplitude observed for $d_{\mathrm{gap}}=104.7\mathrm{\mu m}$ and 280.7 μm.

The best-model sheet density, mobility, and effective mass obtained for the 2DEG are $N=(1.02\pm 0.15)\times {10}^{13}{\mathrm{cm}}^{-2}$, $\mu =(1417\pm 97){\mathrm{cm}}^2/\mathrm{Vs}$, $m^{*}=(0.244\pm 0.020)m_0$, respectively. These results are in good agreement with the results of high-field ($B=7\mathrm{T}$) THz-OHE measurements on the same sample $N=(1.40\pm 0.07)\times {10}^{13}{\mathrm{cm}}^{-2}$, $\mu =(1230\pm 36){\mathrm{cm}}^2/\mathrm{Vs}$, $m^{*}=(0.258\pm 0.005)m_0$. Excellent agreement between THz-OHE and electrical measurement results were reported previously on similar samples [7]. The OHE response measured at multiple frequencies and multiple $d_{\mathrm{gap}}$ values provides sufficient information to determine the Drude model parameters independently. We find in our numerical data analysis that $N$, $\mu$, and $m^{*}$ are uncorrelated parameters. Ideally, the polaronic effects on the effective mass need to be considered. However, as discussed in Ref. [21], these corrections have been found negligible for GaN and are not considered in our present analysis. Note that increasing the number of data sets obtained at different cavity lengths into the numerical data analysis reduces the error bars on the 2DEG parameter set. Our findings demonstrate that the cavity enhancement of the THz-OHE signal allows the investigation of free charge carrier properties of two-dimensional free charge carrier gases at low magnetic fields that may be conveniently provided by permanent magnets.

A variation in $d_{\mathrm{gap}}$ provides a new degree of freedom to tune the experimental conditions so as to reach a maximum response for a given frequency range. According to Fig. 2, other gap lengths, e.g., 150 μm will provide even larger signals in this situation. For a given sample system, the experimental configuration can be optimized by calculating the coupled substrate-cavity modes and then selecting frequency range and $d_{\mathrm{gap}}$ accordingly. Furthermore, varying the $d_{\mathrm{gap}}$ at a fixed frequency may be used to maximize the OHE signal.