Electro-optical modulation of terahertz beam by drifting space-charge domains in n-GaN epilayers

The interaction of electromagnetic waves with free charge carriers (FC) enables terahertz (THz) radiation application for the diagnostic of semiconductors in, e.g., THz time-domain spectroscopy,^1,2 ellipsometry,^3,4 electro-modulation spectroscopy,⁵ pump-probe spectroscopy,⁶ and THz pulse excitation spectroscopy.^7,8 Modulation of THz waves is a common goal for various applications including wireless communications, quantum electronics, spectroscopic applications, and imaging.^9,10 Several mechanisms were proposed, including THz modulation by FC density in semiconductor,¹¹ implementation of microelectromechanical systems,^12,13 2D plasmons,¹⁴ intraband absorption in 2D materials,^15,16 and 2D optoelectronics.¹⁷ Ongoing research and developments have aim to improve the efficiency of THz modulators.

Acoustoelectric (AE) domains have been widely studied during last decades, with observations reported in many piezoelectric semiconductors such as, e.g., CdS, ZnO, and GaAs.¹⁸ The AE domains were used to shift the absorption edge, to rotate the plane of light polarization, to cause light emission and Brillouin-scattering in the visible spectrum range, and to attenuate the transmission of microwave radiation. Investigation of spectral dependence and polarization anisotropy of visible light transmission modulation by AE domains at the intrinsic absorption edge of the semiconductor revealed that modulation signals were due to a Franz–Keldysh effect originating from the high field over one wavelength, which resulted from fluctuating AE potentials inside the domain rather than from the domain field itself^19–21 The modulation was monitored using a standard polarizer–analyzer setup measuring the change of the refractive index of the material. In competition to photoelastic effects, the electro-optic (EO) effect due to the domain field was expected also to cause a change in the refractive index and give rise to light modulation with the magnitude proportional to linear or quadratic refractive index modulation change with the internal electric field.¹⁸ Nevertheless, the effect was considered to have an acoustical origin because the observation of reflected domains was not accompanied by large domain electric fields.

Positive growth rates of space-charge (SC) waves in the presence of negative differential resistance and of acoustic waves via piezo-electric interaction can be realized in the same semiconductor material as demonstrated by simulations in a number of papers.^22,23 The description of the propagating SC wave with two oppositely propagating acoustic waves in the semiconductor layer was given by Ridley. It is usually assumed that the AE domain as a high-field dipole layer of high resistance limits a total charge current to the value that corresponds to a drift velocity near to and just above the velocity of sound.^18,24 However, a consideration of impact of traveling high field domain of the SC wave on semiconductor performance was limited by lack of experiments until recently.²⁵ All samples with the distance between electric contacts larger than 0.12 mm were found to be suitable for the formation of drifting SC domains in GaN epilayers. In this work, the EO modulation of THz waves on the GaN epilayer operating in a regime of drifting SC domains was revealed, demonstrating good polarization selectivity and modulation depth values up to 50%. The interactions of SC and AE domains with THz radiation were investigated at 77 K temperature identifying experimentally the contribution of EO modulation due to the high internal electric field of the SC domain.

The EO terahertz beam (EOT) modulator was composed of a 10 μm thick lightly n-type doped GaN epilayer grown on a c-plane native semi-insulating substrate, with two ohmic contacts formed on opposite sides of the mesa.^25,26 Hall measurements demonstrated the low-field electron mobility and density values in the epilayer at 77 K temperature to be of about 2652 cm²/V⋅and 0.21 × 10¹⁶ cm⁻³, respectively. Two types of n-GaN epilayer mesas were developed for the research. First, the rectangular shape mesas were designed and the lateral electrical contacts were fabricated of one width but with different spacing, L, making a set of resistors needed for the estimation of contact resistance by standard transfer length model measurements.²⁷ Another group of mesas was developed by implementing a geometry of the dumbbell-shaped resistor (DR) having a large area in between electrical contacts used also for THz beam modulation experiments. The width of the conductive channel here was designed up to 31% narrower than that of ohmic contacts. In total, three sizes of DR samples were fabricated. Design parameters are summarized in Table I.

The conductivity of all samples in the short-pulse regime was measured using a high-voltage Hg-relay switch that generated the voltage pulses of 10 ns-duration with a rise time and repetition rate of approximately 0.2 ns and 100 Hz, respectively. The value of an external electric field was defined as E_DC = U_bias/L, where U_bias is a bias voltage applied to electrical contacts. The signals were recorded on a four-channel oscilloscope with a bandwidth of 2 GHz (Wavepro 7200 from LeCroy). Sample characterization was also carried out in a long-pulse regime using a custom design transistor-based switch generating high-voltage pulses with a duration of 4 μs. The amplitude of the bias voltage was also recorded employing a high-voltage probe connected to one of the oscilloscope channels. More details about setups used for sample current voltage characterization in the pulse regime can be found elsewhere.^25–27 

The schematic of the measurement setup is shown in Fig. 1. Linearly polarized radiation emitted from a frequency multiplier chain (manufactured by Virginia Diodes, Inc.) operating in a continuous wave regime was collimated by a 2″ high density poly ethylene (HDPE) lens with a focal distance of 12 cm to the Gaussian beam with diameter of approximately 15 mm at a full width of half maximum. The collimated beam was then focused with a 2″ off-axis parabolic (OAP) mirror on the sample in the area between electrical contacts. Transmission of the beam through the sample was additionally controlled by means of a thin circular metallic aperture (not shown in Fig. 1). The transmitted radiation was collimated with an OAP mirror and directed onto the THz detector based on a Schottky barrier diode coupled with a spherical silicon lens made for quasi-optical THz detection with a bandwidth of about 1 MHz (manufactured by ACST GmbH).

The DR3 sample was selected for the investigation of the EOT modulator operating in a high-field domain regime. It was mounted on an aperture with a diameter of 3 mm on a cold finger of the liquid nitrogen-cooled cryostat equipped with 2″ HDPE windows and with 50 Ω impedance coaxial lines for proper guiding of electric signals. The probe was normally incident on the sample positioned to have the electrical contacts at an angle of 45° with respect to polarization of the THz beam. The radiation polarized either parallelly (⁠ $e → ω Par$⁠) or orthogonally (⁠ $e → ω Ort$⁠) to the vector of the external electric field (⁠ $E →$⁠) was selected by a 2″ holographic wire-grid polarizer positioned in front of the sample. Thus, the THz detector was used to monitor the change in transmission ΔT of the n-GaN layer under bias voltage, i.e., ΔT = T(E_DC)/(T(0) √2), where T(0) is an average value of the THz detector signal recorded by means of a lock-in amplifier at E_DC = 0; and T(E_DC) is the peak signal value recorded on the oscilloscope screen after trace averaging up to 1000 times. Note that the factor of √2 is needed to consider a difference between average and peak values of the signals. Transmitted THz radiation was recorded synchronously to the external electric field pulses by an oscilloscope. Typical values of THz detector signals without and with DR3 sample on a 3 mm diameter aperture were approximately 200 and 32 mV, respectively. The detection scheme of THz signals in the setup had limited accuracy owing to the operation at the highest gain of the oscilloscope input (2 mV/div). Thus, detected signal traces were considered only in terms of pulse shape, kinetics, and relative amplitude values. Because of parasitic coupling of electrical signals to the THz signals, high-pass electronic filters were needed to limit the signal bandwidth to approximately 3 MHz.

Modulation experiments were performed at several frequencies of the THz source delivering highest output power, namely, at 587.56, 602.15, and 614.24 GHz. However, this study describes only the data obtained at the frequency of 614.24 GHz, where the largest values of ΔT were measured because of smaller attenuation of THz signals by Fabry–Perot interference effects in the sample with total thickness including the substrate of approximately 375 μm.²⁶ 

The investigation of the electrical conductivity of samples with different geometric configuration enabled identification of the operation regime for the EOT modulator by gradually turning on domain formation and drift processes in the n-GaN epilayer. The SC domain formation was not observed in the kinetic of charge current density when either the distance between electrical contacts or the duration of strong electric field pulse were too short. Meanwhile, all samples with L = 1 mm and larger demonstrated the current instabilities when the pulse of the electric field was long enough to support the incubation period and drift of the domains along the entire n-GaN layer.^18,25 An example of charge current density trace of the DR3 sample operating in the regime of drifting domains is shown in the inset of Fig. 2.

The conductivity of different samples obtained in the short-pulse regime is shown in Fig. 2 by solid symbols. The measured characteristic exhibits well-pronounced field regions:²⁶ almost constant conductivity (<40 V/cm); steady increase due to ionization of shallow impurities (from 38 to 450 V/cm) and the non-monotonic conductivity decrease resulting mainly from a mobility decrease due to electron and phonon heating effects (>1200 V/cm),^28–30 followed by a material electrical breakdown at fields well above 100 kV/cm. It is worth to note that the use of 10 ns-long voltage pulses allowed us to avoid thermal breakdown and to reach electric field values more than four times larger than those obtained when using pulses with a duration of 120 ns.²⁶ The electrical breakdown of samples in the short-pulsed regime most likely resulted from interfaces between the substrate and n-GaN epilayer used without any passivation. The electric field pulses of 120 ns and longer resulted in the thermal breakdown of the GaN epilayer already at 53 kV/cm due to inefficient heat dissipation.²⁷ Notably, the observed breakdown fields in the lateral direction were still an order of magnitude lower than the predicted value of critical electric field for GaN of 3.7 MV/cm.³¹ 

The incubation time, t_inc = L/ν_ac, for the DR3 sample with L = 3 mm was approximately 750 ns, which can be obtained using the velocity of a transverse acoustic wave of ν_ac ≈ 3.9 × 10⁵ cm/s. Calculated value was in agreement with experimental observations at the highest value of applied electric fields.²⁵ The DR3 sample demonstrated the maximum current density modulation when the duration of the applied bias was longer than incubation time and varied from 390 to 740 ns depending on bias voltage required to form high-field domains in the n-GaN epilayer. According to Fig. 2, the conductivity of samples before domain formation behaves similarly to the samples in which SC domain formation is prohibited either by short sample length or by short pulse duration.

The electrical conductivity deviated significantly in a long-pulse regime in the presence of SC domain.²⁵ The results for two selected samples, DR1 and DR3, are shown in Fig. 2 by open red and green symbols, respectively. Here, the conductivity values were obtained at the moment in time after the charge current density dropped to the saturation value, j_sat (see inset to Fig. 2) due to the SC domain drift at a constant velocity of acoustic wave, υ_ac. The saturation current density of the n-GaN epilayer demonstrated little dependence on E_DC exhibiting the value of approximately j_sat = 768 A/cm², which can also be estimated based on the parameters of the used material and equation j_sat = neυ_ac. As one can see, the conductivity values decreased threefold to the initial equilibrium value in the range of used fields. The conductivity decrease of the DR3 sample started earlier with a slope much steeper than that of the DR1 sample. For example, the deviation from short-pulse-conductivity characteristic for DR3 (green symbols) and DR1 (red symbols) started with the bias voltage corresponding to the electric field of 1 and 3 kV/cm, respectively. In addition, the conductivity always demonstrated less steep change with the field independently on the presence or absence of drifting domains in the n-GaN epilayer in the field regions just before electrical breakdown. The samples with all geometric configurations in different operation regimes approached similar value of the conductivity at the level of approximately 0.27 S/cm, indicated by the dashed horizontal line in Fig. 2. These results demonstrate that internal electric field accumulated in the vicinity of the SC domain reaches values above 200 kV/cm, sufficient for the electrical breakdown of n-GaN epilayers.²⁶ 

Experimental fact is that n-GaN epilayers in a long-pulse regime require application of significantly smaller external electric field to turn on SC domains with a high-field internal electric field required to achieve a nonlinear electron transport and hot phonon phenomena in the semiconductor.^28,30 It is widely accepted^18,24 that the AE domain in the n-GaN epilayer acts as a high resistivity ohmic region; thus, the peak internal electric fields could be extrapolated from the same conductivity value that was obtained in a short-pulse regime. Important fact from the standpoint of application is that only a fraction of the electric field is needed to drive the n-GaN epilayer into the regime of drifting SC domains whose high internal electric field can be employed for various applications, as for example, EO modulation of THz beam transmission.

Transmission modulation of THz beam polarized either parallelly (ΔT_Par) or orthogonally (ΔT_Ort) to the electric field vector was investigated in a long-pulse regime. Transmission signal traces for each polarization and the corresponding charge current density kinetics are shown in Fig. 3. The traces of transmission modulation in both polarizations and current density repeated the shape of applied voltage pulses at fields below 0.4 kV/cm; however, the signals at higher electric fields demonstrated the oscillating behavior resulting from the domain drift with the velocity limited by acoustic wave speed in the n-GaN epilayer. The signal modulation frequency for the DR3 sample was measured to be approximately 1.4 MHz. It is worth reminding that electric fields higher than 0.4 kV/cm tend to decrease the electrical conductivity of the DR3 sample when operating in the long-pulse regime in the presence of domains (see also Fig. 2).

Magnitude and sign of the transmission change in the region of low electric fields were invariant for two polarizations of the THz beam. This results in a zero difference in transmission modulation for different polarizations, ΔT_EO = ΔT_Par− ΔT_Ort (see left panel, bottom part of Fig. 3). Meanwhile, the onset of difference between transmission modulation for two polarizations was observed under electric fields exceeding 0.5 kV/cm. The amplitude of ΔT_EO signal oscillations increased rapidly with the applied electric field and oscillated synchronously with the current density. To be more specific, transmission modulation traces demonstrated only oscillations of one sign corresponding to the absorption increase for the case of the THz beam being polarized orthogonally to the applied electric field. Considerably different case was found by using the THz beam polarized parallelly: the transmission modulation demonstrated richer spectrum of oscillations corresponding to both darkening and bleaching of the sample. At 1.4 kV/cm, the difference between two polarizations was found to be as high as 45% of the equilibrium value, indicating a promising potential of the EOT modulator for many practical applications. Note that the highest observable modulation value of 50% was limited by the external electric field of approximately 1.65 kV/cm, corresponding to the maximum field that was possible to use before an electrical breakdown of the sample.

The THz transmission modulation during an applied dc-voltage pulse was composed of static and dynamic contributions. The static, time-independent component during the pulse was identified at low electric fields. The shape of transmission modulation traces was similar for both polarizations exhibiting zero difference in amplitude, i.e., ΔT_EO = 0. Therefore, it was attributed to the absorption by FC, in which the density increased with the field due to ionization of shallow impurities in the n-GaN epilayer. A controlled doping with silicon was used during a layer growth in order to reduce the amount of residual oxygen resulting in a low net free-electron density and high mobility values.^26,32 The degree of impurity ionization increased with the electric field until approximately 0.45 kV/cm, at which value the complete ionization of shallow impurities was assumed (see also Fig. 2). The change of charge carrier density with the application of bias voltage also manifested in detected ΔT_Par and ΔT_Ort signals, with the average value of static components increasing with the electric field until it reached the saturation level of approximately 11% at 0.45 kV/cm. Importantly, the FC contribution existed as a constant offset (time-independent component) in every ΔT_Par and ΔT_Ort curve recorded under higher values of applied electric fields. The field dependence of THz absorption was compared with the field-induced change of free-carrier density derived from the field dependence of electrical conductivity normalized to the conductivity value obtained under 10 V/cm (equilibrium conditions). The results are shown in Fig. 4. The curves in a tenfold-different scale are parallel, thus confirming a direct correlation between the attenuation of the transmitted THz signal and free-charge carrier density increase with the external electric field.

Another contribution to transmission modulation has demonstrated a dynamic character and was attributed to the formation and drift of AE and SC domains in the n-GaN epilayer. The EO modulation of THz transmission usually depends on polarization due to a scalar nature of interaction between the SC wave and incident electromagnetic waves. Largest EO modulation can be achieved for a case of TE polarization of the THz beam. In contrast to FC absorption, the EO interaction demonstrated an opposite modulation sign that corresponded to bleaching of the sample. While FC contribution was seen as the attenuation of THz transmission independently on THz beam polarization, the EO contribution was sensitive to beam polarization; therefore, it was visualized by calculating the difference between ΔT_Par and ΔT_Ort signals. The resulting traces of ΔT_EO signals are shown in the bottom part of Fig. 3 (left and right panels). Notably, the oscillations of the ΔT_EO signal stop immediately at the end of the bias voltage pulse. Conversely, the AE domains persisted in the epilayer longer and continued modulation of both ΔT_Par and ΔT_Ort signals after switching off an external electric field. The result is a direct confirmation of the SC domain presence in the n-GaN epilayer, demonstrating no EO interaction after the electric field was switched off.

The AE domains can be treated as a high-field dipole layer of high resistance limiting the total charge current density in the sample.^18,24 Therefore, the AE domains modulated THz beam transmission independently on polarization of the THz beam contributing to FC absorption in the sample as discussed elsewhere.¹⁸ The redistribution of free-charge carriers occurs during domain formation, which, in turn, forms a high-amplitude internal electric field in the vicinity of the SC domain that changes THz beam transmission via EO interaction. The term of EO modulation in the experiment was measured as sample bleaching oscillations observed in ΔT_Par and ΔT_EO signal traces during the domain drift through the GaN epilayer.

The peak of ΔT_EO signals was found at time moments when the charge current density dropped to the saturation value, i.e., after the formation of drifting SC domains in the n-GaN epilayer. To elaborate on the mechanism involved, the time-dependent electric resistivity of the DR3 sample was analyzed in the range of strong fields. The results are shown in Fig. 5.

A three-step increase in E_DC/j signal traces synchronously with the domain formation was observed in the experiment. After the second step, the E_DC/j kinetic obviously demonstrated very little amplitude variation due to the formation of a high resistance AE domain in the n-GaN epilayer. However, the amplitude of ΔT_EO signals continued to grow until the third small amplitude step on the E_DC/j kinetic was reached that indicated the formation of the drifting SC domain in the n-GaN epilayer. Later a gradual decrease in ΔT_EO signals was observed, which contrasted with almost steady behavior of E_DC/j. This time period we attributed to the decomposition of the SC domain. Afterward, a rapid decrease in E_DC/j signals occurred indicating the decomposition of the AE domain and extraction to an external circuit. Note that the modulation of ΔT_EO signals was not disturbed by the contra propagating AE waves that experienced multiple reflections on sample boundaries, the effect of which has been discussed elsewhere.^18,33,34

The field dependences of the peak difference in transmission change for two polarizations obtained at the first and second maxima were investigated separately. The results are shown in Fig. 6 by black and red symbols, respectively. The EOT modulator demonstrated a non-linear signal dependence on electric field with possibility to increase THz modulation even further with higher electric fields providing that the sample will withstand larger applied voltages. The threshold field for EO modulation was found to be of about 0.45 kV/cm.

In the region of E < 1 kV/cm, the peak ΔT_EO signals demonstrated strong dependence on applied electric field with a characteristic slope of 3.0, whereas in the range of higher fields, the slope decreased to 1.4. This result indicates a few possible nonlinear effects in the GaN epilayer, which can be related to the change in the carrier scattering mechanism to spatially anisotropic electron scattering on optical phonon,³⁰ generation of hot phonons,²⁸ electron streaming,^35,36 and scattering to higher energy valleys.^37,38 It is difficult to identify the dominating effect because the internal electric fields exceed manifold the field applied externally, value of which was estimated from the bias voltage and sample geometric parameters. Hydrodynamic models for dynamics of AE and SC waves in n-GaN epilayers under a strong electric field are required to explain all details observed in the dynamic of the electrical conductivity, ΔT_EO, and ΔT_FC signals. Hydrodynamic models have been developed to explain photo-refractive phenomena in dynamic Gunn-domain gratings in GaAs under heating electric fields.^39,40

Notably, the peak ΔT_EO signals demonstrate a nonlinear behavior in a similar manner to field dependences of the electrical conductivity. The field dependences of conductivity in the moment of time before domain formation and when drifting SC domain is formed, i.e., when the current density dropped to saturation value, are shown in Fig. 2.

Drude model is often used to describe the electrical conductivity of semiconductors.^36,41 Electron relaxation time from the material conductivity characteristics at 77 K temperature was estimated to be approximately τ_p ≈ 0.3 ps. Its product with the cyclic frequency of the THz beam was approximately unity, i.e., ωτ_p ≈ 1. This means that there is no simple analytical description but instead both drift (ohmic conductivity) and displacement (polarization) current terms have to be considered when modeling the modulation of total charge current under the strong electric field. The power density in the THz beam focused on the sample was below 1 mW/cm², resulting in the value of the ac electric field in the THz wave of approximately 0.6 V/cm. From differential electrical conductivity measurements, we assumed that the GaN epilayers for application as EOT modulator can be used for THz beam modulation with power density above 1 W/cm², which can be achieved by free-electron lasers and similar sources. Note that the dielectric relaxation frequency is approximately ω_c ≈ 10¹⁰ rad/s (ω_c = σ₀/ɛ₀ɛ); therefore, free-carriers are not able to follow AE and SC waves originating at the frequency of 0.6 THz.

Main result of our work was the identification of contributions of AE and SC domains into transmission of a linearly polarized THz beam through n-GaN epilayers under strong electric field. The discrimination between two contributions was possible because of FC generation via shallow impurity ionization occurred in a wide-bandgap GaN epilayer at electric fields smaller than those required to trigger domain formation. This fact and consideration of a resistive nature of AE domains allowed us to observe EO contribution of SC domains for the first time in the experiment. An existence of both SC and AE waves in the n-GaN epilayer was feasible in theory, but their discrimination by other experimental techniques was not possible so far.²⁴ 

We have previously found that drifting SC domains can form at T = 77 K in the used epilayer samples with lengths as small as 0.12 mm,²⁵ which indicates that the operational frequency of such an EOT modulator is limited to approximately 33 MHz. The maximum operational voltage required for efficient EOT modulation of the THz beam will depend on exact geometric parameters of the GaN epilayer sample, and its value can be estimated from the results shown in Fig. 2. The fields above 1.6 kV/cm lead to sample destruction due to the electrical breakdown of the material. Other calculations indicate that the electric field required for domain formation can be reduced by improving the carrier mobility in the sample.¹⁸ Moreover, the SC domains were also observed experimentally in GaN epilayers at T = 300 K, even though the electric fields required for the onset of domain formation and critical sample length were higher than those at T = 77 K. These findings suggest that the EOT modulator can perform at elevated temperatures higher than those reported in this work.

In conclusion, efficient electro-optic modulation of THz beam transmission was demonstrated employing space-charge domains in the GaN epilayer excited under a pulsed electric field. The maximum value of absorption attenuation by change of free charge carrier density owing to impurity ionization amounted to approximately 11%. This contribution could be increased in the entire range of electric fields by another 10% in presence of drifting space-charge domains in the sample. The electro-optic contribution, on the other hand, increased the THz transmission with electric field up to 50% at the maximum field of 1.6 kV/cm. Maximum modulation frequency limit was estimated at approximately 33 MHz. The possible directions of the performance improvement and some practical applications were outlined.

The authors thank Emilis Šermukšnis and Rimantas Miškinis for fruitful discussions of the various aspects of this work. The work was supported by the Research Council of Lithuania (Lietuvos mokslo taryba) through the “Hybrid plasmonic components for THz range (T-HP)” Project (Grant No. DOTSUT-184) funded by the European Regional Development Fund according to the supported activity “Research Projects Implemented by World-class Researcher Groups” under the Measure No. 01.2.2-LMT-K-718-03-0096. The work at Warsaw, including the growth and processing of test structures, was supported by the National Center for Research and Development (Grant No. TECHMATSTRATEG-III/0003/2019/EnerGaN).

The authors have no conflicts to disclose.

Roman M. Balagula: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Liudvikas Subačius: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Paweł Prystawko: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (equal). Irmantas Kašalynas: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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