Tunable cavity-enhanced terahertz frequency-domain optical Hall effect

The optical Hall effect (OHE) is a phenomenon in which the optical response of a conductive material is altered by the presence of an externally applied magnetic field.¹ This effect can be measured with generalized ellipsometry at oblique angles of incidence and at terahertz (THz) frequencies. Previously, the THz-OHE has been proven as a viable non-contact method to obtain the free charge carrier properties of semiconductor heterostructures using high-field superconducting magnets.^2–8 This approach allows the extraction of a sample’s carrier concentration, mobility, and effective mass parameters by using a THz-transparent substrate as a Fabry–Pérot cavity to resonantly enhance the THz-OHE signal. Recently, it has been shown that these properties can be conveniently obtained with permanent magnets (PMs).^9–12 Using low-field permanent magnets to provide the external field significantly decreases the magnitude of the THz-OHE signal. However, one can compensate for this by exploiting an externally coupled Fabry–Pérot cavity to further enhance the signal. In Ref. 9, different external cavity thickness values are achieved by simply stacking multiple layers of adhesive spacers between the sample and the magnet. This method is useful because it is straight-forward and low cost, but only large increments of cavity thickness can be produced. The cavity-tuning optical stage described in this work improves on this previous approach and is capable of finely tuning the cavity thickness, thus providing a new measurement dimension. As an example of the cavity-tuning stage described here, a sample-permanent magnet arrangement is placed inside a gas flow cell to improve sensitivity to small variations of free charge carrier parameters under varying gas flow conditions. This experiment highlights the advantage of the small footprint of this enhancement technique.

In this work, we discuss the concept of THz-OHE signal enhancement due to an externally coupled cavity. An optical model is used to choose desirable measurement parameters, such as angle of incidence, frequency, and external cavity thickness. Details of the instrument design and data acquisition are explained. Experimental and model-calculated data are presented and compared with data for the case of no cavity enhancement. It is demonstrated that the cavity-enhancement technique allows extraction of the free charge carrier properties of a two-dimensional electron gas (2DEG) at THz frequencies.

We refer to the OHE as a physical phenomenon that describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths, transverse and longitudinal to the incident electric field, and analogous to the static electrical Hall effect.^1,13 We have previously described data acquisition and analysis approaches for the method of the OHE in the mid-infrared (MIR), far-infrared, and THz spectral regions.^1,2,8,11

The OHE can be calculated by use of appropriate physical models. The models contain two portions, one describes a given material’s dielectric function under an external magnetic field, and the second portion describes the wave propagation within a given layer stack. We have provided a recent review on this topic in Ref. 13. Briefly, in the THz spectral range, the dielectric function can be approximated by a static contribution due to phonon excitations and higher energy electronic band-to-band transitions. Contributions due to free charge carriers can be well described by the Drude quasi-free electron model.^14,15 In the presence of a static external magnetic field, an extension of the Drude model predicts magneto-optic anisotropy,¹⁶ which is the cause of the classical OHE in conductive materials (for quantum effects, see, for example, Refs. 17–19).

The principle of the enhancement of the OHE in a layer stack is the constructive interference between the multiple outgoing beams created by the multiple reflections off of each layer interface in the stack, where each outgoing portion of the beam has undergone magneto-optic polarization conversion due to the externally applied magnetic field. Each time of passage, the electromagnetic field components undergo an additional polarization rotation caused by the magneto-optic anisotropy created by the response of the free charge carriers under the influence of the Lorentz force and within the conductive layer(s). The magnitude of the enhancement can be significant, which depends on the free charge carrier properties in a given sample configuration. Examples are discussed in this work.

The experimental configuration of the tunable cavity-enhanced OHE method is shown in Fig. 1. All configurations require the layer stack to be supported by a THz-transparent substrate. The substrate’s backside must be: flat, plane-parallel to the front surface, and backside roughness must be sufficiently small compared to the wavelength of the incident THz beam. The thickness of the substrate d_sub should be such that spectrally neighboring Fabry–Pérot interference maxima and minima within the substrate can be sufficiently resolved with a given spectroscopic setup. Plane wave electric field (E_i) components incident under an angle Φ_a then pass the sample layer structure multiple times due to multiple internal reflections within the substrate cavity. The frequencies of such maxima are controlled by the angle of incidence, the substrate thickness, and the substrate index of refraction. In principle, the substrate thickness is adjustable by depositing the sample layer stack onto different substrates. This, however, requires multiple fabrication steps. If a second cavity is created by the introduction of a mirror, placed at distance d_gap, the portion of electromagnetic waves lost at the backside of the substrate is fed back into the substrate and introduces additional fractions of plane wave components passing the layer stack. The two Fabry–Pérot cavities (substrate, gap) couple and produce coupled Fabry–Pérot resonances. The frequencies of the coupled interference maxima can then be tuned by d_gap, for any given but fixed d_sub. Then, the magneto-optic signal enhancement occurring at interference maxima, limited to certain frequencies for a given d_sub without an external cavity, can be tuned spectrally. Thereby, a new magneto-optic spectroscopy method is created where in addition to frequency, the external cavity is tuned by changing its thickness, d_gap.

Also shown in Fig. 1 are the limiting cases, when the external cavity is zero [d_gap → 0, Fig. 1(b)], infinite [d_gap → ∞, Fig. 1(d)], and when both the cavity and substrate are infinite [d_sub → ∞, Fig. 1(e)]. The case d_gap → 0 requires deposition of a metal layer onto the backside of the substrate, as demonstrated in Ref. 20, for example. The case d_gap → ∞ occurs when the substrate is THz-transparent and has parallel interfaces. The case d_sub → ∞ occurs when the layer stack is deposited onto a non-transparent substrate or when the backside of a transparent substrate is not parallel, for example, if the substrate consists of a wedge or a prism.

The generalized ellipsometry concept,²¹ its spectroscopic extension,²² and the Mueller matrix formalism²³ are employed in this work. The Mueller matrix connects Stokes vector²⁴ components of electromagnetic waves before and after interaction with the sample upon reflection or transmission. For the use of the Mueller matrix concept in spectroscopic generalized ellipsometry, we refer the reader to recent reviews (see, e.g., Refs. 25 and 26). For use of the Mueller matrix formalism in the OHE²⁷ and, in particular, for data format definition, we refer to Ref. 2.

Non-linear parameter regression analysis methods are used for data analysis. The experimental data are compared with the calculated OHE data. The calculated data are obtained with appropriate physical models and model parameters. Parameters are varied until a best-match is obtained minimizing an appropriately weighted error sum. The error sum takes into account the systematic uncertainties determined during the measurement for each experimental data value. Best-match model parameter uncertainties are obtained from the covariance matrix using the 90% confidence interval.² 

The tunable cavity and sample must be placed within a THz spectroscopic ellipsometer system and subjected to an external magnetic field. Ellipsometer, sample stage, and magnetic field designs are discussed in this section.

Two THz frequency-domain ellipsometer instruments are used in this work. The instruments described in Refs. 2 and 11 are used for the high electron mobility transistor (HEMT) structure measurements and epitaxial graphene measurements, respectively. Both instruments operate in the rotating-analyzer configuration that enables the acquisition of the upper left 3 × 3 block of the 4 × 4 Mueller matrix. The frequency-domain source is a backward wave oscillator (BWO) with GaAs Schottky diode frequency multipliers. Technical details are described in Refs. 2 and 11.

A principle design of the sample holder for the tunable cavity-enhanced THz-OHE is shown in Fig. 2. In this design, the permanent magnet serves both as the mirror as well as for providing the external magnetic field. The mirror properties of the magnet surface must be characterized by THz spectroscopic ellipsometry measurements at multiple angles of incidence prior to its use in the sample stage. The permanent magnet (PM) sits flush inside a hole in the mirror housing plate (MHP). The micrometer adjustment screw (AS) rests inside a brass bushing in the back plate. The rounded tip of the AS is made of ferromagnetic material that attracts the magnet providing synchronous PM–AS movement. The stepper motor (SM) is attached to the back of the setup and is connected to the AS by a flexible bellows shaft coupler. The flexible coupler allows a d_gap range of approximately 0 μm–600 μm. The stepper motor is operated by using a commercially available motor controller (Thor Labs, Inc.), which uses LabVIEW programming. The minimum d_gap increment is 1.6 μm, corresponding to the one step of the motor. The back plate in Fig. 2 is removable and allows the user to flip the magnet to the opposite pole face and redo experiments without disturbing the sample alignment.

For simplifying the tunable cavity-enhanced sample stage, non-magnetic adhesive spacers can be placed between the sample and the mirror surface dispensing with the need for the stepper motor in Fig. 2. This option is suitable for in situ measurements when limited space is available. However, no tuning of the cavity after sample mounting can be performed.

For use of the sample holder with an external electromagnet, the permanent magnet can be replaced by a non-magnetic insert with a THz mirror at the front toward d_gap and the sample backside. The normal reflectance properties of the mirror can be evaluated by performing THz spectroscopic ellipsometry measurements at multiple angles of incidence prior to its use. The external magnetic field can be provided by electromagnets, for example, by placing the stage within a Helmholtz coil arrangement.

The permanent magnet mounted on the sample holder is a high-grade neodymium (N42) magnet. With the use of a permanent magnet, the change in the magnetic field strength at the sample surface upon variation of d_gap can be substantial. Hence, it is necessary to implement the magnetic field as a function of distance in the optical model. Using a commercially available Hall probe (Lakeshore), the magnetic field is measured at multiple d_gap values. For our instrument, within ∼1 mm of the magnet surface, the field is approximately linear and can be approximated using

where the units of B are Tesla and the plus and minus sign refers to the two respective pole orientations of the magnet. The parameters d_gap and d_sub are in units of micrometers, and therefore, the linear slope (5.1 × 10⁻⁵) is in units of Tesla/micrometers.

Separate ellipsometry experiments are performed in the mid-infrared spectral range to determine the optical properties of the metallic permanent magnet surface as the mirror. Data analysis is performed using the classical Drude model parameters for a static resistivity of ρ = (9.53 ± 0.04) × 10⁻⁵ Ω cm and the average-collision time of τ = (1.43 ± 0.08) × 10⁻¹⁶ s. These parameters are used here to model-calculate the optical reflectance of the magnet surface for the model analysis in the THz spectral range for the cavity-enhanced measurements. No other contributions to the dielectric function of the metallic magnet surface are included in the optical model. The magnet surface behaves as an ideal metallic “Drude” mirror characterized by metal electron carrier scattering time and resistivity, and no magneto-optic polarization coupling occurs because the metal electron effective mass is too large and the mean scattering time is too short in order for the free charge carriers to respond to the external magnetic field producing measurable magneto-optic birefringence.

The mirror surface is aligned first and then the sample is mounted and aligned. The mirror surface is aligned with the ellipsometer’s coordinate system with the use of a laser diode mounted such that the laser diode beam is parallel to the plane of incidence, perpendicular to the sample surface, and coincides with the center of the THz beam at the sample surface. A gap value d_gap is selected in the middle of the range of values anticipated for experiments. To align the mirror, the alignment laser diode beam is reflected off the mirror surface, and the mirror is adjusted until the beam reflects back into the laser aperture. The adjustment is performed by moving the entire stage relative to the ellipsometer system.

Once the mirror is aligned, the sample is mounted on the sample tip–tilt plate (STP) (Fig. 2). The STP serves as an adjustable frame to mount the sample. The sample can be mounted via adhesive, for example, or mechanical clamps. The STP contains 3 μm screws secured against the MHP by using springs, creating a tip–tilt ability. This is necessary to ensure that the sample surface is also aligned with the ellipsometer’s coordinate system. The sample surface is aligned using the same alignment laser as that for the mirror.

After mounting the sample stage onto the ellipsometer system, data are acquired in a selected spectral range, for selected angles of incidence Φ_a, and gap distance d_gap. Figure 3 depicts a flow chart describing the data acquisition process. First, the sample–mirror air gap distance d_gap is set. Next, the frequency-domain source frequency is set. Then, the polarizer angle is set, and the intensity at the detector is recorded. This process is repeated for all polarizer settings. The stored intensity data are then analyzed using the process described in Ref. 2, and the elements of the upper 3 × 3 block of the Mueller matrix are obtained. The procedure is repeated for different settings of gap distance, frequency, or angle of incidence, for example. The acquisition process can be repeated with the magnetic field direction reversed, for example, by reverting the magnet direction, or by reverting the currents in external electromagnetic coils. The experiment can also be repeated with a mirror without a magnet for the acquisition of zero-field ellipsometry data. The minimum number of measurements needed as functions of d_gap, frequency, angle of incidence, or magnetic field direction depends on the sample’s free charge carrier properties and the overall sample structure. The measurement process must be reiterated over these different settings until reasonable model parameter error bars are obtained and a convincing fit to the experimental data is observed in the best-match model analysis.

Data measured by the tunable cavity-enhanced THz-OHE are analyzed using model calculations and numerical regression procedures. Multiple data acquisition modes are available, which will be discussed by examples further below.

Data obtained as functions of gap thickness are compared with the calculated data.

Data obtained at a fixed gap and/or substrate thickness but as functions of frequency are compared with the calculated data.

Data over a two-dimensional parameter set can be obtained tuning both gap thickness and frequency and are compared with the calculated data.

Field-reversal OHE data obtained at opposing magnetic field directions, ΔM_ij = M_ij(B) − M_ij(−B), are taken, and the difference data are compared with the calculated data.

Here, we discuss two sample systems as examples for the application of the tunable cavity-enhanced OHE. Both samples contain 2DEGs. The characterization of their free charge carrier properties is demonstrated. One sample is comprised of a transistor device structure for a high electron mobility transistor (HEMT) based on group-III nitride semiconductor layer structures. The second sample is an epitaxial graphene sample grown on a silicon carbide substrate.

The sample investigated is an AlInN/AlN/GaN HEMT structure grown using an AIXTRON 200/4 RF-S metal–organic vapor phase epitaxy system. The HEMT structure consists of a bottom 2 μm thick undoped GaN buffer layer, a 1 nm thick AlN spacer layer, followed by a 12.3 nm thick Al_0.82In_0.18N top layer.^28,29 The substrate is single-side polished c-plane sapphire with a nominal thickness of 350 μm.

All sample constituents are optically uniaxial, and the layer interfaces are plane-parallel. In a separate experiment, the HEMT structure was investigated using a commercial (J. A. Woollam Co., Inc.) mid-infrared (MIR) ellipsometer from 300 cm⁻¹ to 1200 cm⁻¹ at Φ_a = 60° and 70° at room temperature in order to determine phonon mode parameters of the AlInN top layer. No distinct phonon features are seen in the THz measurements. However, the MIR analysis is used to help determine the dielectric function of the HEMT structure constituents in the THz spectral range. Phonon parameters for the substrate, GaN buffer layer, and AlN spacer layer are taken from Refs. 6, 30, and 31, respectively. The thicknesses of the AlInN and AlN layers are found by growth rate calculations and not varied in the analysis. The best-match model layer thickness for the GaN layer is (2.11 ± 0.01) μm. For the AlInN top layer, the best-match model frequency and broadening parameters for the one-mode type E1- and A1-symmetry are ω_TO,⊥ = (625.4 ± 0.8) cm⁻¹, ω_LO,⊥ = (877.8) cm⁻¹, γ_⊥ = (40.8 ± 1.5) cm⁻¹, ω_TO,∥ = (610) cm⁻¹, ω_LO,∥ = (847.8 ± 0.4) cm⁻¹, and γ_∥ = (11.3 ± 0.4) cm⁻¹, which are in good agreement with previous workss.³² Note that certain phonon parameters are functionalized according to Ref. 32 and were not varied in the analysis. In order to obtain an excellent match between experimental and model-calculated THz-OHE data, a low-mobility electron channel was included in the AlInN top layer. This same low-mobility channel was also included in our previous model analysis for the same HEMT structure.^9,29 A mobility value of μ = 50 cm²/Vs for a similar HEMT structure is adopted for this sample, and an effective mass parameter of 0.3 m₀ is taken from density function calculations in Ref. 33. The volume density value for the low-mobility channel was previously determined to be N = 1.02 ×10²⁰ cm⁻³. This value is not varied in our analysis.

In Ref. 29, we reported field-reversal high-field OHE measurements on the same HEMT sample without an external cavity. The high-field measurements were performed in a cryogenic superconducting magnet setup. In Ref. 9, we reported field-reversal cavity-enhanced OHE measurements using a permanent magnet and various adhesive spacers (discrete settings for d_gap) on the same sample. The results reported in this work are in excellent agreement with those reported previously. All THz-OHE results further compare well with the Hall effect and C–V measurements done on similar samples.^34,35

Figure 4 shows experimental and best-match model data for field-reversal cavity-enhanced OHE data as functions of d_gap for the HEMT sample. The experiment was performed at two different, fixed frequencies of ν = 860 GHz and 880 GHz for a d_gap range of 120 μm–520 μm in increments of 3 μm. The experimental data for both frequencies are analyzed simultaneously. The layer stack optical model for the best-match model calculation is AlInN/AlN/GaN/sapphire substrate/external cavity/mirror (magnet surface). The external magnetic field is oriented normal to the sample surface. All off-block-diagonal Mueller matrix elements are zero for the HEMT sample structure without the external magnetic field. To begin with, the solid blue lines in Figs. 4(a) and 4(b) are model-calculated data for the same HEMT structure in the absence of the cavity enhancement [Fig. 1(e)], d_sub = ∞ and d_gap = ∞), where we assumed that the field at the layer stack is B = ±0.55 T. Specifically, ΔM₁₃ = ΔM₃₁ = 0.0004 and ΔM₂₃ = ΔM₃₂ = −0.004. Data are below our current instrumental uncertainty limit for the individual Mueller matrix elements of δM_ij ≈ ± 0.01. Hence, the 2DEG within the HEMT layer structure would not be detectable. In the cavity-enhanced mode, however, large off-diagonal Mueller matrix elements appear, far above the current instrumental uncertainty level, upon variation of the gap thickness d_gap. Features in Figs. 4(a) and 4(b) are due to Fabry–Pérot interference enhanced cross-polarized field components after reflection at the layer stack. Minima and maxima occur as functions of gap thickness. The cross polarization is produced only by the free charge carrier gas within the HEMT structure under the influence of the Lorentz force. The potential to use the variation of gap thickness as a new parameter variation measurement configuration is obvious. In particular, for this sample, and for the scans shown in Figs. 4(a) and 4(b), compared with the blue lines (no cavity enhancement), cavity thickness parameters can be adjusted where the OHE signal enhancement reaches 0.124 at ν = 860 GHz in Fig. 4(a) for ΔM_13,31 (an increase of ×220) and −0.088 at ν = 880 GHz in Fig. 4(b) for ΔM_23,32 (an increase of ×31). The best-match model 2DEG sheet density, mobility, and effective mass parameters obtained from the OHE data in Figs. 4(a) and 4(b) are N_s = (1.23 ± 0.13) × 10¹³ cm⁻², μ = (1250 ± 60) cm²/V s, and m^* = (0.272 ± 0.013)m₀, respectively. The results compare well with electrical measurements performed on similar samples^34,35 and with our previous THz-OHE experiments.⁹ 

Three-dimensional rendering of model-calculated data of cavity-enhanced field-reversal THz-OHE data vs frequency, gap thickness, and angle of incidence is shown in Fig. 5. The color type indicates positive or negative values, and the color intensity indicates the magnitude of the OHE data. The three-dimensional rendering is insightful as it indicates distinct regions within which data rapidly switch signs and regions within which data take very large values. All three parameters, frequency, gap thickness, and angle of incidence, influence the OHE data, and proper selection may result in strong OHE data, while poor choices may result in the disappearance of the OHE data. Since the OHE data are also dependent on the sample’s free charge carrier properties (which are unknown prior to the experiment and analysis), there is no direct procedure for determining the exact ranges for frequency, gap thickness, and angle of incidence needed for the experiment. However, it is useful to examine model-calculated OHE data that are generated using anticipated values for the free charge carrier properties, in order to estimate the optimum ranges. The horizontal plane indicated at 45° in Fig. 5 identifies the angle of incidence at which experiments are performed in this work. Two-dimensional rendering of experimental and model-calculated data at an angle of incidence of 45° is shown in Fig. 6(a) as a function of frequency and gap thickness. An excellent agreement between both the experiment and the model calculation is obtained. Figure 6(b) shows data at a fixed cavity thickness. Data are similar to those in Figs. 4(a) and 4(b); except now, the frequency is tuned. Figures 6(a) and 6(b) identify frequency and gap regions where the OHE data are very small. The blue solid lines are identical to those in Figs. 4(a) and 4(b) for the case of no cavity enhancement. All experimental data in Fig. 6 are analyzed simultaneously, and the resulting best-model sheet density, mobility, and effective mass parameters for the 2DEG are N_s = (1.22 ± 0.12) × 10¹³ cm⁻², μ = (1260 ± 60) cm²/V s, and m^* = (0.268 ± 0.012)m₀. The results are identical within the error bars to those obtained from the single-frequency gap thickness scans as well as to our previous OHE investigation reports.^2,9

The central results obtained from this section are (i) the demonstration of the strong enhancement obtained by use of multiple interferences through the substrate and external cavities, and (ii) that our OHE measurements can be performed both as functions of frequency and gap thickness, which is supported by our experiments and clearly apparent in Fig. 5. Note that the enhancement of THz-OHE signatures due to the substrate and external cavity is observed over a wide spectral range and is not limited to the spectral range shown here. For example, Ref. 9 depicts cavity-enhanced THz-OHE data for the same HEMT structure, where the signal magnitude continually increases toward lower frequencies, but begins to vanish near 1 THz. Also shown in Ref. 9 is that the OHE signal is largest near the reflection minima of the sample/cavity/magnet Fabry–Pérot mode. This is important to consider when performing cavity-enhanced OHE measurements and demonstrates that the reflection minima can be used as an indicator where strong OHE enhancement occurs.

In this section, we demonstrate the use of the cavity-enhanced OHE method for detecting changes in the properties of a 2DEG upon exposure to various external gas compositions. The purpose of this section is to demonstrate the use of this method when transient physical changes to a sample limit the time durations during which spectroscopic scanning measurements can be performed.

The sample studied here is graphene epitaxially grown on Si-face (0001) 4H–SiC by high-T sublimation in Ar atmosphere.³⁶ Reflectivity, low-energy electron microscopy mapping, and scan lines verify the primary one monolayer coverage across the 10 × 10 mm² sample surface.

The sample is optically modeled by considering the graphene monolayer as a 1 nm highly conductive thin film on top of the SiC substrate, as described in Ref. 36. All sample constituents have plane-parallel interfaces. No free charge carriers are detected in the SiC substrate. Due to the ultrathin layer thickness of the graphene, THz ellipsometry data cannot differentiate between the thickness and the dielectric function of the layer. Instead, a new parameter emerges, the sheet free charge carrier density. This parameter takes a constant ratio with the assumed layer thickness and, hence, can be determined independently and accurately (further details are discussed in Refs. 13 and 36).

The schematics of the in situ gas flow cell used in this work is shown in Fig. 8(a). The THz ellipsometer instrument is schematically indicated by source, polarizer, analyzer, and detector at an angle of incidence of Φ_a = 45°.¹¹ The cell is equipped with a humidity and temperature sensor, gas inlets, and gas outlets. The side walls of the flow cell are made from Delrin, and the cover and base portions are made from acrylic. THz-transparent windows are produced from homopolymer polypropylene. The thickness of the transparent sheets is 0.27 mm. Normal ambient gas is pushed through the cell using a vacuum pump (Linicon). Nitrogen and helium flows were provided by additional purge lines. The background pressure in the cell was 1 atm throughout the experiment. The flow rate was 0.5 l/min.

The sample consists of a 2DEG (graphene) at the surface of a THz-transparent substrate (d_sub = 355 μm). The substrate is placed with its backside using adhesive spacers onto the permanent magnet [Fig. 8(a)]. The sample is mounted with the neodymium (N42) magnet into the gas cell. The magnetic field near the permanent magnet surface is found to be 0.548 T by an additional Hall probe measurement. The gap thickness d_gap was fixed at 100 μm. OHE data acquisition is identical to the procedure in Fig. 3 with a fixed cavity thickness.

Figure 7 shows model-calculated THz-OHE data for M₂₃ as functions of ν, d_gap, and Φ_a for epitaxial graphene grown on SiC. Figure 7 can be used as a guide to find optimal values for ν, d_gap, and Φ_a to perform a THz-OHE measurement. The black sphere and three intersecting lines illustrate the point chosen to perform the in situ THz-OHE measurement. Only a single set of measurement parameters is chosen to minimize time between measurements in order to resolve sharp dynamic changes in the Mueller matrix data during the gas flow experiment. For practical reasons, d_gap = 100 μm and Φ_a = 45° were chosen. Therefore, ν = 428 GHz was selected for the in situ measurement. Unlike Fig. 5, Fig. 7 does not show THz-OHE difference data since the gas flow experiment is only performed using the north pole face of the permanent magnet.

Figures 8(c) and 8(d) depict experimental and best-match model data for single-field-orientation cavity-enhanced OHE data as functions of frequency for the graphene sample. These measurements were performed at two different points during the gas exposure experiment: the first spectral measurement was after three hours of exposure to helium and the subsequent spectral measurement after two hours of exposure to ambient air [labeled “T₁” and “T₂” in Fig. 8(b), respectively]. Indicated in Figs. 8(c) and 8(d) are also the THz-OHE data without cavity enhancement (blue lines: d_gap,sub = ∞). Comparing the cavity-enhanced data to the case of no cavity enhancement indicates that large changes in the Mueller matrix data are entirely due to interference enhancement in the substrate and external cavity. For the “T₁” spectral measurement, the M₂₃ element shows a maximum increase in the magnitude of ×16.7 at ν = 430 GHz (from 0.007 to −0.117). Note that the off-block-diagonal element M₂₃ is selected to show the OHE signature enhancement and would equal zero in the absence of an external magnetic field. The M₁₂ varies only slightly with the application of the external magnetic field and is nonzero with or without the magnetic field.

Figure 8(b) shows in situ cavity-enhanced THz-OHE data taken at a single frequency, ν = 428 GHz, for M₁₂ and M₂₃. Analyzing the data allows the extraction of the graphene’s free charge carrier properties N_s, μ, and charge carrier type as functions of time. The carrier type is determined to be n-type during each gas phase. It is found that N_s increases with helium and nitrogen exposure and decreases with air exposure. An inverse relationship is observed for μ and N_s throughout the gas flow experiment. The lowest N_s occurred at the end of the second air phase, where N_s = (8.40 ± 0.72) × 10¹¹ cm⁻² and μ = (2600 ± 220) cm²/V s. The highest N_s occurred at the end of the helium phase, where N_s = (2.31 ± 0.29) × 10¹² cm⁻² and μ = (2000 ± 250) cm²/V s. Further details on the in situ THz-OHE gas exposure experiment can be found in our previous publication.³⁶ Sensitivity to the free charge carrier properties as a function of gas flow is entirely dependent on the cavity-enhancement effect, as demonstrated in Figs. 8(c) and 8(d). The variations in free charge carrier properties upon exposure to He and Air, indicated by the blue lines in Figs. 8(c) and 8(d), would not have been detectable with the same instrument without the external cavity stage since the changes in Mueller matrix elements are below the detection limit.

We demonstrated a tunable cavity-enhanced THz frequency-domain OHE technique to extract the free charge carrier properties of 2DEG layers situated on top of THz-transparent substrates. A HEMT structure grown on sapphire and epitaxial graphene grown on SiC are studied as examples. For the HEMT structure sample, the OHE signatures are enhanced by tuning an externally coupled Fabry–Pérot cavity via a stepper motor. Data measured as functions of external cavity size and frequency are analyzed to obtain the carrier concentration, mobility, and effective mass parameters of the 2DEG located within the HEMT structure. For the epitaxial graphene on the SiC sample, an external cavity with a fixed size is used to enhance the OHE signal during a gas flow experiment. This enhancement effect allows the extraction of the graphene’s carrier concentration and mobility as a function of time throughout the experiment. In our experiments, the Fabry–Pérot cavity enhancement is made possible by the THz-transparent substrates as well as the external cavity (air gap) between the sample’s backside and the reflective metal surface. The magnetic field necessary for the OHE experiments is provided by a permanent magnet, for which the metallic coating also provides the reflective surface for the external cavity. Our enhancement technique can be expanded upon by using superconducting magnets to measure samples with much lower free charge carrier contributions. In general, our technique is a powerful method for materials characterization and can be used to study even more complex sample structures.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors thank Dr. Craig M. Herzinger for helpful discussions. Professor Nikolas Grandjean is gratefully acknowledged for providing the AlInN/GaN HEMT structure. The authors thank Dr. Chamseddine Bouhafs and Dr. Vallery Stanishev for growing the graphene sample and Professor Rositsa Yakimova for providing access to her sublimation facility for graphene growth. This work was supported in part by the National Science Foundation under Award No. DMR 1808715, the Air Force Office of Scientific Research under Award No. FA9550-18-1-0360, the Knut and Alice Wallenbergs Foundation supported grant “Wide-bandgap semiconductors for next generation quantum components,” the Swedish Agency for Innovation Systems under the Competence Center Program (Nos. 2016-05190 and 2014-04712), the Swedish Foundation for Strategic Research (Nos. RFI14-055 and EM16-0024), the Swedish Research Council (No. 2016-00889), the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO Mat LiU No. 2009 00971), and the J. A. Woollam Foundation.