1. Introduction

Recent years have seen the establishment of field-effect transistors (FETs) as sensitive detectors of electromagnetic radiation for the upper GHz and the lower Terahertz (THz) frequency range. This development was triggered to a substantial degree nearly three decades ago by the seminal theoretical work of M. Dyakonov and M. Shur who invesigated the rectification of high-frequency radiation in the channel of FETs [1,2]. With the hydrodynamic model presented by them, they went beyond the quasi-static resistive mixing which had been used before for rectification at microwaves and lower frequencies [3]. The new aspect of the process was that the electron motion of the two-dimensional charge density under the gate electrode is accompanied by charge density (plasma) waves, which have similar characteristics as waves in shallow water. They influence the rectification process both directly [2] and indirectly via a strong modification of the AC impedance of the channel with consequences for the impedance matching of the transistor to the radiation coupling environment [4]. The two influences tend to counteract each other, which has led to a debate about the relative importance of plasma waves for the rectification process [5,6]. That plasma waves exist in the channel at high radiation frequencies has been shown in the meantime beyond doubt for moderately damped waves at low temperature [7] and for overdamped ones at room temperature [8], in both cases using graphene FETs.

From a practical point of view, properties of prime importance are that the FET detectors can be fabricated entirely with standard semiconductor (foundry) technology and that the rectifying effect prevails at frequencies much above the transit-time or power amplification cut-off frequencies of the respective transistor. The demonstration of THz detection with Si CMOS FETs gave this detector approach a significant boost [3,9–13]. Since then, a vast number of papers have been published on detection with FETs across the terahertz range (${{100}{\textrm {GHz}}}$ to ${{10}{\textrm {THz}}}$) [4,11,14–23] into the infrared regime [24–26]. It has become apparent in the meantime – mainly by research on graphene and AlGaN/GaN FET detectors [27–30] – that the plasmonically modified distributed resistive mixing, often termed after their discoverers as Dyakonov-Shur detection mechanism, is not solely responsible for the ultrafast rectification process of FETs. Also the ultrafast hot-carrier thermoelectric effect [31], often termed photothermoelectric effect, must be considered. This voltage-generating effect is closely related to the Seebeck effect, both coming about by temperature differences between unequal materials and the ensuing diffusion of charge and thermal energy. While in the conventional Seebeck effect the charge carriers are in a local thermal equilibrium with the lattice and the lattice temperature is locally different, in the photothermoelectric effect the charge carriers are at a temperature higher than the local lattice temperature. In this sense, it is a purely electronic effect. For the hot-carrier assumption to be true, the energy exchange between charge carriers and lattice at any given position in the structure must be slow compared to the time scales of the relevant transport processes between the unequal material components. In nano-scale devices, this condition is rather easy to fulfill. With regard to state-of-the-art FETs, one finds experimentally, that the photothermoelectric effect is typically of significance if the transistor has ungated channel access regions between source and gate, or gate and drain, respectively, which is the case for HEMTs (in contrast to Si CMOS FETs, for which this effect seems to be negligible at least at room temperature [5,32]). While the classical Seebeck effect is comparatively slow as it requires lattice heating with at least nanosecond time constants, the photothermoelectric effect has been shown to exhibit ultrafast sub-ns response times [33].

In this work the FET detector response by the combined ultrafast non-quasistatic electronic effects (plasmonically modified distributed resistive mixing plus photothermoelectric effect) is found to decrease with increasing frequency. We show by experiments with picosecond-scale pulses at 3.9, 11.8 and ${{30}{\textrm {THz}}}$ that a third rectification effect plays a role and becomes dominant in the infrared. Because of its slow response time (as compared to the ultrafast picosecond-scale response of the electronic rectification effects) and its relative independence to the polarization of the radiation, we assign this mechanism to a combination of a classical Seebeck effect with a bolometric effect. The bolometric effect (i.e., temperature dependence of the resistance) of the channel of transistors is well studied and used for detector development [34,35]. Its observation requires a voltage or current bias to the channel. As we do not apply such a bias, a slow mechanism for self-biasing is needed. The best candidate for this is rectification by the Seebeck effect which has also been discussed before in research on graphene FET detectors for THz radiation [27,28,30,33,36–38]. This suggests that the slow response combines both mechanisms, with the Seebeck effect providing the required voltage, and the bolometric effect influencing the resultant current by the temperature-dependent channel resistance.

2. Experimental

We investigate Al$_{0.3}$Ga$_{0.7}$As/GaAs n-channel high electron mobility transistors (HEMTs), a special case of FETs which contain a remotely doped GaAs $n$-channel, grown by molecular beam epitaxy. The doping is situated within an Al$_{0.3}$Ga$_{0.7}$As barrier with a thickness between ${{15}{\textrm {nm}}}$ and ${{30}{\textrm {nm}}}$ similar to the ones presented in [24]. The electron mobility of all five investigated HEMT structures are listed in Table 1. The electron Hall mobility is between 4300 and 6000 cm$^2$/Vs.

Table 1. Parameters of the fabricated samples: $L_G$ denotes the gate length, $L_{eff}^{\infty }$ the approximate penetration depth of the THz wave into the channel in the high frequency limit calculated with the Dyakonov-Shur model [25], $L_{SD}$ distance between source and drain, including the gated areas and the ungated source and drain access regions, $N$ is the number of parallel connected FETs, $W$ the width of the FETs, $\mu$ the electron mobility, $n$ the charge carrier density for ${U_{GS}=0{\textrm {V}}}$, and $d_{}$ the channel depth corresponding to the thickness of the Al$_{0.3}$Ga$_{0.7}$As gate barrier. LAS-A, LAS-B and LAS-E are large-area arrays of HEMTs (LA-FETs) with an AC source-gate shunt capacitance for source-gate coupling, LA-E is a LA-FET device without shunt capacitance, but with a non-centric placement of the gate between source and drain; NG-A and NG-B are corresponding large area samples without gate contact.

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Here we study large-area field effect transistors (LA-FETs) consisting of, on the one hand, an array of $N$ FETs with widths of 300 $\mu$m and 1000 $\mu$m (see Table 1 and Fig. 1(a)) without antennas, as described in previous work [24], and, on the other hand, gateless control devices (NG-A and NG-B). Figure 1(a) depicts the large-area FET device LA-E schematically. Gate-less devices have been fabricated as reference samples to quantify contributions that are neither expected from a plasmonically modified distributed resistive mixing nor photothermoelectric effect.

 

Fig. 1. (a) Schematic of device LA-E for the 2D scan from the air side. The THz beam is raster-scanned across the LA-FET device. The black circle indicates the inner diameter of ${{1.5}{\textrm {mm}}}$ of an aperture placed above the 1$\times$ 1 mm$^2$ device that shields the wiring and the contact pads in order to prevent undesired incoupling. Note that all contacts are connected separately. The small overlap regions between gate (G) and source (S) at the left side, which naturally arise in the 2D layout, are separated with a 150 nm thick Al$_2$O$_3$ insulator. (b) The conical horn, with an opening of ${{5}{\textrm {mm}}}$ at the top, reflects beams impinging onto the walls of the conus into the center of the bottom aperture of ${{1.5}{\textrm {mm}}}$.

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In a symmetric detector design, the THz wave enters the gated channel equally from either side, resulting in rectification at both the source side and the drain side of the gate with equal amplitude but opposite sign and, consequently, no net rectified current exists [39]. The symmetry of the transistor itself and/or of the radiation coupling has to be broken by the design of the FET. We achieve this in two ways. Design LA-E features an asymmetric position of the gate, with the gate shifted towards the source contact. Designs LAS-A, LAS-B and LAS-E feature capacitively AC-shorted source-gate contacts by wiring the gate over the source contact, electrically (DC) separated by a thin insulating SiN layer. While the DC potential may differ, the source and gate feature the same AC potential with the consequence that the THz power couples in from the drain side only.

The large-area detectors (LA-FETs) are illuminated by focused THz/IR beams impinging from the top, i.e. the air side. In order to exclude impedance matching effects and the influence of the radiation resistance of any antenna, these devices do not use integrated antennas. Instead the THz field directly couples to the FET. We remark that antennas are typically employed in the research field for enhanced coupling of the THz radiation to the FET and thus increasing its responsivity [4,5,11,18–20]. We further opted for a very simple linear structure of very wide (300 ${\mathrm {\mu }}$m-1 mm) FETs where the geometry of source gate and drain are kept identical along the direction perpendicular to the source drain direction in order to ensure that only one polarization can interact with the transistor channel effectively according to the Dyakonov-Shur-like response. A conical horn aperture is mounted over the active area of the LA-FET to exclude or at least reduce parasitic coupling via bond wires or contact pads when raster scanning across the active device area: Fig. 1(b) shows that the horn shadows, to a large degree, the contact pads. They further simplify incoupling. Radiation which happens to hit the wall of the cone is reflected onto the active area of the detector as indicated in Fig. 1(c) by the dotted arrow. The LA-FETs are further integrated in a metal package in order to screen them from undesired radiation coming from outside the beam path.

The FEL generates pulses with a repetition rate of 13 MHz and a full width at half maximum (FWHM) from $\tau _{pls}=$31 ps down to $\tau _{pls}=$2.0 ps, inversely scaling with frequency for the corresponding frequencies of ${{1.3}{\textrm {THz}}}$ up to ${{30}{\textrm {THz}}}$. The FEL pulse length can be prolongated by a detuning of the optical cavity [40]. For recording the fast response, a Tektronix 80E08 head featuring a response time of about 7 ps on a DSA8200 oscilloscope is used. Including cables and parasitic effects, we estimate the system response time to 11-14 ps as calculated in [40] where a very similar post detection setup was used. For most measurements the FEL pulses are shorter than the post detection rise time leading to a suppression of the measured amplitude $\eta _a$ by the FETs as detailed in Ref. [40], where the impulse response of the measurement system, including the FETs, was calculated from the spectrometrically determined, Fourier-limited FEL pulse shape. The current response for measurements calculates as $\mathcal {R}^{sc}_I=U_{det}/[50\Omega \cdot P_{THz}\cdot \eta _a]$ with an oscilloscope input impedance of 50 $\Omega$. For the slow, time-integrated measurement the rectified current of the LA-FET was transformed into a voltage by a low-noise transimpedance amplifier (PDA-S from TEM Messtechnik) and then detected by a lock-in amplifier. The current response is calculated as $\mathcal {R}^{li}_I=I_{det}/\bar {P}_{THz}= [2.2 \cdot I_{Lock-in}]/\bar {P}_{THz}$ (unipolar square-wave modulation). A thermal power meter measures the average power of the FEL, $\bar {P}_{THz}$. The peak power calculates to $P_{THz}=\bar {P}_{THz}/(f_{rep} \tau _{pls})$, with the repetition rate of the FEL pulses, $f_{rep}$. The spatial intensity profile of the incident THz radiation in front of the conical horn aperture was characterized using a pyroelectric array (Ophir Spiricon, model PyroCam III). It followed approximately a Gaussian field distribution (see Supplement 1).

2.1 Fast non-quasistatic vs. slow Seebeck detector response

We aim at the discrimination between rectification by the device due to slow bulk heating on one hand, and the ultrafast electronic rectification in the channel. We discriminate the two rectification mechanisms by (i) the speed (IF bandwidth) of the response and (ii) by the different dependencies on the polarization of the radiation [34]. The ultrafast electronic effects depend on efficient coupling into the transistor’s channel and hence are polarization-sensitive because a strong AC voltage oscillation caused by the incident THz field is needed between gate and (in our case) drain. A uniformly distributed electric field oriented perpendicular to the source-drain direction would therefore not cause a Dyakonov-Shur-like response. In case of a non-uniform field distribution (e.g. a Gaussian field distribution), an ultrafast electronic response can in principle evolve due to the photon/plasmon drag effect [41]. However, since our beam spot size is large (> 500$\,\mu$m) compared to the SD distance ($\sim$ 10$\,\mu$m) (see chapter 3 of the Supplement 1 for details), this effect is considered negligible here . In contrast, the bolometric (Seebeck) effect only depends on heating by the absorbed radiation, which has no polarization sensitivity as long as the absorbance is the same for all polarizations. The latter may not completely be fulfilled as the metal contacts will act as a polarizer, i.e. the polarization along the source-drain direction will be rather transmitted, the one perpendicular to it rather reflected. Still, in either case, the incident fields will cause surface currents in the metal or channel currents in the FET.

In the following, we will mainly show results for device LA-E for a fixed gate bias of -0.4 V, approximately its threshold bias. We would like to emphasize at this point that the gate-bias-dependent measurements of the response patterns could not be performed within the limited beam time window of the FEL source. The gate-bias dependence of the LA-FET response was only investigated at fixed beam positions (e.g. in the active device region) for selective devices (see Supplement 1). The use of THz attenuators ensured that the LA-FET devices were characterized in the unsaturated THz response regime. For details on THz power saturation in LA-FETs, we refer the reader to [[24]]. The device features $N=48$ FETs with a period of $L_{FET} = 22\,\mu$m (as indicated as zoom-in in Fig. 1(a)). The length of the ungated regions are 3$\pm 1$ $\mu$m between source and gate, and 6$\pm 1$ $\mu$m between gate and drain, the gate length is $L_{G} =3\,\mu$m. The width of the LA-FET (i.e. array of FETs) is 1 mm in order to span a total area of approximately 1 $\times$ 1 mm$^2$. The gate is placed closer to the source contact. This way, the rectified current at the source side features a lower amplitude as that at the drain side. While for symmetrically placed gate, the two currents would cancel out, the asymmetric gate position gives rise to a net rectified current. In order to prevent any undesired absorption of THz power by the contact pads of the device, it is mounted below a conically tapered horn with an inner diameter of 1.5 mm that shadows the vast majority of gate, drain and source contact pads, as illustrated in Fig. 1(a) and Fig. 1(b). Its outer diameter is 5 mm.

2.1.1 Ultrafast electronic response

The ultrafast electronic response requires a polarization along the source-drain (SD) direction in order to drive an AC current. A polarization perpendicular to the transistor channel will – except at the edges of the active area – only drive charges along the metal electrodes and the semiconductor, but will not generate an AC current across the channel. This effect is very well visible in Fig. 2(a)) and b), measured at ${{3.9}{\textrm {THz}}}$. For parallel polarization (see Fig. 2(a)), there is a pronounced current response within the active LA-FET area (indicated by the black square). The response extends somewhat outside of the active area. This results from two contributions. First, from the convolution of the active device area with the beam waist of the THz spot, which has a diameter of $\rm {1/e^2} \approx 0.75\,$mm. Second, from radiation reflected from the conical wall of the aperture onto the active area. Well outside of the active area, there are four regions with a pronounced detector response. They line along the horizontal ($x$) and vertical ($z$) axes where the horn reflects the radiation onto the active area, as illustrated in Fig. 1(b), without altering the polarization state. The reflection of the incident THz radiation at the conical horn aperture at an angle of 45$^\circ$ rotates the polarization by 90$^\circ$. Despite THz radiation that hits the horn aperture along the diagonals is reflected onto the active device area, it does not generate a strong current response as the polarization arriving at the detector is orthogonal to the SD direction.

 

Fig. 2. Fast sub-ns-scale peak-peak current responsivity of LA-FET device LA-E. Panels (a), (c) and (e) are for radiation polarized along the transistor channel (Source-Drain (SD) direction) for 3.9, 11.8 and ${{30}{\textrm {THz}}}$ respectively. (b) and (d) represent measurements for THz polarization perpendicular to the transistor channel at 3.9 and ${{11.8}{\textrm {THz}}}$, respectively. The black squares indicate the position and size of the active area of the device with the dimension $1 \times 1\,\textrm {mm}^{2}$. Note the different logarithmic color code scalings of the five panels.

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For perpendicular polarization (see Fig. 2(b)), one observes the opposite picture: a vanishingly weak signal in the center of the active area, consistent with the expectation that only radiation polarized along the transistor channel can drive the required AC current. However, for radiation impinging along the diagonals of the horn flips the polarization from perpendicular to parallel and thus generates a pronounced response in the shape of a four-leaf clover. We remark that the responsivity in this case is a factor of about 3 weaker than for parallel polarization directly impinging on the active area, possibly because the beam is distorted by the conical shape of the horn (note the different scaling of the color code in Fig. 2(a)) and b)).

Figure 2(c) presents responsivity maps of a measurement with radiation at ${{11.8}{\textrm {THz}}}$ polarized along the transistor channel where the non-quasistatic electronic response suggests a strong detector response. The beam-waist estimated from far field beam patterns calculates to $\rm {1/e^2} \approx 0.25\,$mm, which results in a higher spatial resolution of the beam scans in comparison with the measurements at $3.9\,$THz. There is indeed a strong response within the active area of the LA-FET, however one notes that the overall signal is weaker by about a factor of 2.3 than at ${{3.9}{\textrm {THz}}}$, and that the signal is stronger at some parts of the edges than in the center of the active area. The reason for the inhomogeneous response – which is also present at ${{3.9}{\textrm {THz}}}$, but less pronounced there – is unknown. Possibly, an antenna effect of the gate contact pad, which is only partially covered by the concial beam stop, contributes at the right edge. The stronger signals at the upper and lower edges suggest that radiation reflected from the conical aperture onto the edges couples more effectively to the device than directly impinging radiation. Farther outside of the active area, one finds again – mainly along the $x$- and $z$-directions as was the case at ${{3.9}{\textrm {THz}}}$ – signals which can be attributed to reflections from the conus onto the active area.

Similar to the measurement at 3.9 THz, no signal is observed on the active device region at 11.8 THz for perpendicular polarization (see Fig. 2(d)), while at the diagonals the signal resurfaces with about a factor of 6 lower amplitude. The findings prove that the electronic response persists beyond the boundary of the THz to the infrared frequency range at ${{10}{\textrm {THz}}}$.

For radiation polarized along the channel, one even observes an electronic response in the mid-infrared at ${{30}{\textrm {THz}}}$, as shown in Fig. 2(e), though with significantly smaller responsivity. The response is confined to the active area. For the reflections that are usually present along the $x$ and $z$ axis, the signal amplitude is below the noise level except a small spot around $x=-1.2$ mm. As in the measurement at ${{11.8}{\textrm {THz}}}$, there is an enhanced signal at some parts of the active area’s edges, presumably for the same reason as given above. For a polarization perpendicular to the channel, no signal could be found. The responsivity for this polarization direction must therefore be lower by at least a factor of 2 as compared to the polarization along the channel.

2.1.2 Slow Seebeck/bolometric response

In order to capture the slow response, we measure the time-integrated source-drain current of the FEL-illuminated detectors with a transimpedance amplifier PDA-S (TEM Messtechnik, gain $3.3\times 10^5$ V/A) and a lock-in amplifier (Signal recovery 7265). The FEL pulse train is modulated by a mechanical chopper around 210 Hz in order to generate a frequency reference for lock-in detection. These measurements record both the time average of the fast electronic as well as the slow response at the same time. The total responsivity is calculated from the measured time-averaged current divided by the measured time-averaged power of the FEL radiation. The FEL beam is again raster-scanned across the active area of device LA-E.

Figure 3 presents maps of the time-averaged responsivity at ${{3.9}{\textrm {THz}}}$, ${{11.8}{\textrm {THz}}}$ and ${{30}{\textrm {THz}}}$ and for both directions of the radiation polarization. At ${{3.9}{\textrm {THz}}}$ and for polarization along the channel – see Fig. 3(a) –, one finds the strongest response in the active area, however slightly shifted along the $z$ direction. The origin may be the same effects that lead to pronounced spots in the fast response shown in and discussed in conjunction with Fig. 2(a). The map resembles the sub-ns response recorded by the oscilloscope in Fig. 2(a) including the cross-like pattern. We therefore conclude that the responsivity is dominated by the fast non-quasistatic electronic response. Two major differences, however, are prominent: first, the maximal responsivity value of the time-averaged response is a factor of 2 larger than that of the fast detector response. As the two types of measurements are very different, the normalization to the FEL power may have a noteworthy error bar on this order. Second, there are small negative signals, i.e. the rectified (DC) current reverses direction, at the diagonal positions with positive $z$ values outside the active area, where the polarization is rotated by $90^\circ$ by the conical aperture of the horn. We interprete these negative signal contributions to the slow Seebeck/bolometric effect, which dominates the detector response whenever the polarization of the incident light is perpendicular to the transistor channel, giving rise to an enhanced heating of charge carriers underneath the gold contact stripes and consequently a stronger Seebeck effect. The slow nature of this signal is further proven by the fact that is not observed at the diagonals in the oscilloscope maps for the same frequency and polarization (see Fig. 2(a)).

 

Fig. 3. Time-averaged, ms-scale current responsivity of LA-FET device LA-E: (a), (c) and (e) are for radiation at 3.9, 11.8 and ${{30}{\textrm {THz}}}$, respectively, with the polarization along the transistor’s channel. Panels (b), (d) and (f) show data for THz polarization perpendicular to the channel at the same frequencies. The black squares indicate the position of the active area. Each color scale is symmetrical for positive and negative values, with zero in neutral color. Therefore, parts of the scale may be unused.

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Figure 3(b) shows the results of the measurement at ${{3.9}{\textrm {THz}}}$ with polarization perpendicular to the transistor channel. Similar to the case of parallel polarization, the spatial pattern of the signal is strongly related to that of the fast response of Fig. 2(b), with one major exception: A strong signal with the opposite sign as those on the diagonals (that can be attributed to fast non-quasistatic electronic response) is observed at the position of the active device area as well as along the horizontal ($x$) and vertical ($z$) axes, where no signal is visible in the respective oscilloscope map. This suggests that the fast response features a positive sign in the time-averaged maps (according to our choice of the lock-in phase), while the slower Seebeck signal has opposite sign and dominates the detector response under perpendicular polarization conditions of the incident light. It is worth mentioning here, that we nulled the signal phase by an appropriate phase-shift in the reference channel when changing the FEL-frequency. This was done close to the response maximum for the polarisation parallel to the SD direction. We kept this configuration for all x-y beam scans at the respective FEL-frequency, so that we have a reliable basis for predicting the sign change of the observed signal within the responsivity patterns.

If we compare the ${{11.8}{\textrm {THz}}}$ measurement shown in Fig. 3(c) for parallel and Fig. 3(d) perpendicular polarization to the transistor channel together with the slow response at ${{3.9}{\textrm {THz}}}$ we observe that: (i) The signal patterns for both polarization directions shown in Fig. 3(c) and Fig. 3(d) are now very similar to each other, unlike the situation at ${{3.9}{\textrm {THz}}}$ and unlike the fast response at ${{11.8}{\textrm {THz}}}$. Still the polarization along the channel yields a higher signal. (ii) The four-leaf clover pattern is partially missing in Fig. 3(c) and Fig. 3(d) and can be observed for only two leaves in case of parallel polarization (Fig. 3(c), left and right from active device region) and one leaf for perpendicular polarization (Fig. 3(d), top right of the active device region). (iii) The maximum magnitude of the detected signal shifts from top to bottom of the active device region and the signal pattern within the active region is bipolar, which is in contrast to the ${{3.9}{\textrm {THz}}}$ measurement. From (i) and (ii) we can conclude that the polarization dependence at ${{11.8}{\textrm {THz}}}$ is partially lost, which suggest that the observed slow detector response pattern at this frequency is dominanted by the Seebeck/bolometric kind yet with a non-negligible fast electronic response. Reminding ourselves that the Seebeck effect is a diffusion effect, its driving term is a carrier density gradient. At ${{11.8}{\textrm {THz}}}$ the free space wavelength is 25.4 $\mu$m and 7.3 $\mu$m within the AlGaAs HEMT layer, which in contrast to ${{3.9}{\textrm {THz}}}$, is of the same order as the total transistor length ($L_{FET}$ = 22 $\mu$m, $L_{SD}$ = 12 $\mu$m). We speculate that the responsivity patterns observed for the two polarization directions are a consequence of the FET’s metallization, which acts essentially as an intensity mask for the radiation at that frequency, with the consequence that excitation and thus heating of the charge carriers of the semiconductor occurs only in the unmasked regions, and only weakly dependent on the polarization. Thereby the magnitude and sign of the Seebeck signal becomes strongly dependent to the details of light coupling to the FET metallization, contact pads and conus. The strong negative peak at the bottom and the positive peak at the top of the patterns for both polarizations could therefore originate from undesired coupling to the FET by scattering at the border of the inner conus. The contact pads, however, were left and right of the active structure. At ${{3.9}{\textrm {THz}}}$, grating aspects then would have a different impact on the polarization dependence of the absorption.

Based on the considerations outlined above, we conclude that we can only understand the data qualitatively if we assume that the Seebeck response dominates the response above the THz band, while the fast electronic detection mechanism plays only a minor role, though yet non-negligible.

Finally, Fig. 3(e) depicts the time-averaged response at ${{30}{\textrm {THz}}}$. Even more pronounced than at ${{11.8}{\textrm {THz}}}$, the spatial responsivity pattern has only a very weak to negligible dependence on the beam polarization, and it is quite similar to that observed at ${{11.8}{\textrm {THz}}}$. The strongest signals are measured at the top and bottom of the active area, where the absolute value is about a factor of 2 smaller than at ${{11.8}{\textrm {THz}}}$. For comparison, the fast response at ${{30}{\textrm {THz}}}$ was 4.5 times smaller than at ${{11.8}{\textrm {THz}}}$. The top and bottom responses feature again opposite signs. The similarities let us conclude also for ${{30}{\textrm {THz}}}$, that the Seebeck effect dominates over the fast electronic response. Figure 4(a) - Fig. 4(d) present the responsivity patterns of the AC shorted gate-source device LA-FET device LAS-E. We remark that the phase of Fig. 4(d) was nulled a second time and may differ from that of the other measurements. A similar polarization dependency is observed when a capacitive source-gate coupling is introduced to the FET. However, in case of the magnitude and signal direction we observe some discrepancies with regard to device LA-E. The fast (Fig. 4(a)) and the time-averaged detector sensitivity for parallel polarized light (see Fig. 4(c)) differ a factor of 6 not 2, while the observed maximum slow responsivity for LAS-E is 3 times higher than for the weakly coupled device LA-E. Similar results on further samples, also measured through backside illumination with a hyper-hemispherical silicon lens, are summarized in the Supplement 1.

 

Fig. 4. Fast sub-ns-scale current responsivity of the strongly coupled LA-FET device LAS-E at ${{3.9}{\textrm {THz}}}$ for a) parallel and b) perpendicular polarized light. c) and d) represent the respective time-averaged, ms-scale current responsivity for the given polarizations. e) Responsivity ratio of the THz signals for the two polarizations directions of the radiation along ($||$) and perpendicular ($\perp$) to the direction of the FET channel. The signals have been averaged over the active detector area prior to calculating the ratio. Diamonds display values for the time-integrated measurements, stars values for the fast signals. The different colors denote the different types of detectors, see legend. The horizontal dashed line represents a responsivity ratio equal to one.

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2.1.3 Quantitative evaluation of the responsivity as a function of the radiation polarization

Figure 4(e) evaluates the polarization dependence of both the slow and the fast detector response. The figure depicts the polarization-dependence of the different detectors for different frequencies via an averaged responsivity ratio $\eta _{pol} = |\mathcal {\bar {R}_{||,SD}}/\mathcal {\bar {R}_{\perp,SD}}|$, where the absolute detector responsivity for parallel ($\mathcal {\bar {R}_{||,SD}}$) and perpendicular ($\mathcal {\bar {R}_{\perp,SD}}$) polarization are averaged over the total active device area. The measured slow time-averaged signal at ${{11.8}{\textrm {THz}}}$ and ${{30}{\textrm {THz}}}$ for device LA-E (black diamonds) exhibits almost no polarization dependency ($\eta _{pol} \sim$ 1), which supports the notion that the slow response at high frequency is dominated by the Seebeck effect. In contrast to that, at ${{3.9}{\textrm {THz}}}$ slow signal response is characterized by strong polarization dependency ($\eta _{pol} =$ 6.6), which suggest that the dominant recitification mechanism at this frequency is the non-quasistatic electronic response.

We have performed related measurements with other devices from Table 1 (but due to the limited beam-time not systematically at all frequencies for all devices). Device LAS-E (red symbols) with strong capacitive source-gate coupling exhibits a slow response with an even larger value of $\eta _{pol}=13.9$ than device LA-E, indicating a stronger non-quasistatic electronic response relative to that of the Seebeck effect. LAS-A (blue symbols), also with strong source-gate coupling, shows a pronounced polarization dependence of the slow response, with $\eta _{pol} \approx 7$ at ${{11.8}{\textrm {THz}}}$, which shows that the non-quasistatic electronic response can also dominate the Seebeck effect (unlike device LA-E) at these high frequencies. Note that measurements of LAS-A have been performed for substrate-side illumination (backside illumination) with a hyper-hemispherical silicon lens (see Supplement 1) and not through a horn aperture from air-side as in case of LA-E and LAS-A. For strongly coupled devices such as LAS-A (and LAS-E) with an AC capacitive shunt between source and gate, only one plasma wave travels from drain to source contact and generates a sufficient rectified current signal. In contrast, for LA-E, two plasma waves with similar amplitudes and opposite signs compete with one another, which generates only a minor rectified net current between source and drain contact. Here the plasma wave amplitudes differ solely due to the unequal lengths of the ungated access regions between Drain-Gate and Source-Gate, which leads to different ohmic losses and therefore THz field amplitudes, that drive the plasma waves, at the respective gate entries. When calculating $\eta _{pol}$ for the fast detector response (star symbols) much larger values with regard to the slow response can be found for LAS-A (18.3) and LA-E (10.6) at ${{11.8}{\textrm {THz}}}$, which is related to the fact that the fast non-quasistatic electronic response is strongly dependent on the polarization. We conclude that the electronic, non-quasistatic detection mechanism based on plasmonically modified distributed resistive mixing and the photothermoelectric effect is active in the FETs at least into the low infrared frequency range. Note that no data point for the fast response can be presented at ${{30}{\textrm {THz}}}$, as the signal for perpendicular polarization is below the noise floor.

In order quantify the fast electronic signal contributions of the strong capacitive source-gate coupled devices (LAS) with regards to the slow Seebeck/bolometric effects, we also performed null measurements with the gate-less large-area devices NG-A (measured response pattern shown in Supplement 1) and NG-B which feature ohmic source and drain contacts, but no gate electrodes (cf. Table 1 for details). Despite there is no rectification effect that can be explained by non-quasistatic electronic response, these devices still showed a non-zero responsivity that is, however, at least an order of magnitude lower than those of the gated devices, with a larger difference at lower frequencies (see Supplement 1).

In summary, the polarization dependence of the detector responsivity in weakly-coupled LA-FETs (LA-E) is partially lost at ${{11.8}{\textrm {THz}}}$ and vanishing at ${{30}{\textrm {THz}}}$. The overall detector response above ${{11.8}{\textrm {THz}}}$ is dominated by the Seebeck/bolometric effect. For strongly SG-coupled devices (LAS-A, LAS-E) the polarization dependence persists at 11.8 THz and the plasmonically modified distributed resistive mixing plus photothermoelectric effect contribute mostly to the rectified detector signal. There is negligible undesired incoupling through bond wires or pads, as the $1 \times 1\,\textrm {mm}^{2}$ active area fits the beam waist for the three frequencies. Within the $1 \times 1\,\textrm {mm}^{2}$ active area of the LA-FET, the slow signal features a similar or up to factor 2 larger magnitude as the fast response. Therefore, it can be concluded that the integrated response is dominated by a slow detection mechanism, superposed by a fast detection mechanism. In the Al$_x$Ga$_{1-x}$As semiconductor, the charge carrier density, $n^{(2D)}$, and the mobility depend on the temperature. Therefore, the slow measurements being almost insensitive to a $90^\circ$ polarization rotation suggest a bolometric detection mechanism. The asymmetrical position of the gate with respect to source and drain causes an asymmetrical thermalization process, where the GaAs lattice reaches faster its equilibrium temperature after a pulse on the side of source and gate in the channel. The asymmetric layout of the electrodes with more metal on the source and gate position of the channel favors an enhanced transport of heat, as metal features a higher heat conductivity in comparison with Al$_x$Ga$_{1-x}$As.

3. Conclusions

We have characterized the sub-nanosecond and millisecond-scale response of large, antenna-less field effect transistor arrays by polarization-resolved measurements in order to discriminate Seebeck response and plasmonically modified distributed resistive mixing by charge carrier waves in the FET channel from ${{3.9}{\textrm {THz}}}$ up to ${{30}{\textrm {THz}}}$. As the Seebeck effect response requires thermalization it is not visible on sub-ns time scales. It further shows little polarization dependence. In contrast, plasma-wave rectification requires an electric THz field aligned to the source-drain direction. We find that in the Terahertz range (<10 THz), the plasma-wave rectification is dominant, in particular for devices, where the gate and source are capacitively shorted. Above 11.8 THz, all investigated structures show a dominant Seebeck effect, whereas the plasmonic rectification only gives a minor contribution. At 11.8 THz the results indicate a minor contribution of the plasmonic rectification.