1 Introduction

It is hardly necessary to emphasize the importance of thin metal films in modern science and technology. Metal films with thickness ranging from a few Angstroms to tens of nanometers have received great attention in recent years. They are used in solar cells, detectors, electronic semiconductor devices, magnetic memory devices, optical coatings (such as anti-reflective coatings), chemical and biological sensors, plasmonic and terahertz devices, etc. Optical properties of thin metal films with thickness substantially smaller than the light wavelength λ are usually characterized by light reflectance and transmittance measured at distances much larger than λ, in the so-called far field (for review, see e.g. [1]).

These properties are rather well researched. Recently, an understanding of electromagnetic field characteristics at distances from metal films substantially smaller than the radiation wavelength has become increasingly significant. It is primarily caused by the development of near-field optics (see e.g. [2,3,4,5,6,7,8,9]). Furthermore, metal films are often used in the fabrication of metallic gates, metamaterials, plasmonic and near-field opto-electronic devices based on 2D materials (e.g., graphene, the most prominent example), topological insulators, and low-dimensional heterostructures. In these cases, the electric field distribution in the subwavelength vicinity of metal films plays a crucial role. It is known that the rough surface of a well-conducting metal film enhances the interaction of electromagnetic radiation with objects located near the film. The most famous example of this phenomenon in the visible band is the surface-enhanced Raman scattering (SERS) of light by organic molecules on the rough silver surface, when an increase takes place in the scattering cross section up to 105 − 106 times presumably due to the plasma excitation in surface roughness features [10, 11]. In the terahertz band, a roughness of metal gate made of gold or aluminum film results in a nonlinear photoconductivity of a tunnel Schottky diode that corresponds to an increase in the local energy density of the electromagnetic field in some small regions of the gate relative to the energy density in the incident radiation up to 103 − 105 times due to the field enhancement in the near zone during radiation diffraction by surface irregularities [12,13,14,15].

For homogeneous highly conductive films, however, it is expected that the radiation is totally reflected and the electric field behind the film should decrease by orders of magnitudes.

Here, we report on the observation of anomalously strong penetration of terahertz electric field through tens of nanometers thick homogeneous perforation-free gold films deposited on top of GaAs/AlGaAs heterostructures. Using the two-dimensional electron gas (2DEG) in GaAs quantum well as detector, we show that the electric field of THz radiation acting on the 2DEG located beneath the highly conductive gold film is almost the same as it would be without the gold film, given that two conditions are fulfilled simultaneously: (i) the gold film has a thickness d less than skin depth δs and (ii) the field is detected at a distance much less than the radiation wavelength λ. Taking into account the high conductivity σ of gold at terahertz frequencies [16,17,18,19], this is a very surprising result.

Indeed, it follows from classical electrodynamics that, under condition ω/σ ≪ 1, where ω is the angular frequency and σ is the conductivity, a metal film has a reflection coefficient |r| ≤ 1 and a transmission coefficient |t|≪ 1, even at thicknesses much thinner than the skin depth \(\delta _{s} = c /\sqrt {2 \pi \omega \sigma }\), where c is the speed of light. The dependence of metal film penetrability on the thickness d for the film placed on a dielectric substrate with the refractive index nb is given under conditions of the classical skin effect by:

$$ \frac{E_{t}}{E_{i}} = \frac{ 2 \gamma t_{0} \mathrm{e}^{- \beta d}}{ (1 - \mathrm{e}^{- 2 \beta d} ) + \gamma (1 + \mathrm{e}^{- 2 \beta d} ) } . $$

Here, Et and Ei are complex amplitudes of the electric field in the transmitted and incident waves, t0 = 2/(1 + nb) is the amplitude transmission coefficient with no metal film:

$$ \beta = \frac{ 1 - i}{\delta_{s}} , \gamma = (1 - \mathrm{i}) \frac{1 + n_{b}}{2} \sqrt{\frac{f}{\sigma}} = (1 - \mathrm{i}) (1 + n_{b}) \pi \frac{\delta_{s}}{\lambda} , $$

f = ω/2π is the frequency, and λ is the wavelength.

Considering this equation for thin films characterized by 2|β|d ≪ 1, we obtain in accordance with the impedance expression provided by Tinkham [20, 21]:

$$ t =\frac{E_{t}}{E_{i}} \cong \frac{t_{0}}{1 + d\frac{\beta}{\gamma}} = \frac{t_{0}}{ 1 + \frac{\lambda}{(1+n_{b}) \pi \delta_{s}}\frac{d}{\delta_{s}} } = \frac{t_{0}}{ 1 + \frac{4 \pi d \sigma}{c(1 + n_{b})} } . $$

From here follows that in order to achieve a radiation transmission through the film close to unity, the film thickness d needs to be several orders of magnitude smaller than the skin depth δs, so that d/δs ≪ (1 + nb)πδs/λ. Accordingly, for the wavelength of about λ ≈ 100 μ m (frequency f ≈ 3 THz), an almost unchanged electric field amplitude is only expected for the gold film thickness d < 1 Å. Consequently, the electric field at the second boundary as well as the amplitude of the transmitted electromagnetic wave at any real film thickness should be much less than the field amplitude in the incident wave. Noteworthy, that this abrupt decrease in the amplitude of the electric field inside the metal occurs immediately at the vacuum-metal interface, where the electric fields of the incident and reflected waves are almost mutually canceled due to the high electrical conductivity of the metal. A further decrease in the field amplitude deep into the metal is characterized by the parameter δs and is of little significance under our conditions d < δs. A. E. Kaplan has made recently analogue conclusions by using also the impedance language [22]. Figure 1 shows the transmitted electric field amplitude, which has been calculated after Fresnel formulae as a function of the gold film thickness for three different frequencies used in our experiments (for calculation details see Appendix 1).

Fig. 1
figure 1

Ratio of transmitted and incident radiation electric fields as a function of the gold film thickness. Data are presented for three wavelengths and two temperatures

It demonstrates that for films with a thickness of 10 nm or more, the field magnitude should decrease by at least two orders of magnitude. This has been observed in a number of experiments on far-field transmission of THz radiation (see e.g. Ref. [16]). It was also detected in our reference experiments performed under conditions diλ and/or d > δs, where di is the distance from the film to the QW position at which the field is probed. The unchanged electric field amplitude, observed in our experiments for diλ and d < δs, discloses an obvious contradiction to fundamental textbook arguments given above. While the origin of this phenomenon is unexplained in its microscopic details yet, we believe that our experiments presented below may serve as basis for an understanding of a fundamental physical problem providing insight into a structure of electromagnetic fields at the metallic film-semiconductor interface.

The paper is organized as follows. In Section 2, we briefly discuss the essential features of our research methodology and the properties of the gold-film/semiconductor structures under study. The results on THz field penetrability are presented in Section 3. The last section summarizes the work. We also supply our paper by Appendixes 13 presenting description of the equations used for calculations of curves in Fig. 1, experimental methods, and additional details of growth and electric characteristics of the gold films.

2 Experimental Technique and Samples

Gold films are deposited on top of GaAs/AlGaAs heterostructures leaving either half of the sample uncovered or depositing gold only on a predefined rectangle (see Figs. 2a and 3). The anomalous penetrability of Au films has been observed by detecting the photoconductivity [14] or the linear photogalvanic effect (LPGE) [14, 23] excited by THz radiation in a GaAs/AlGaAs quantum well (QW) placed underneath the gold film at a distance di much smaller than the radiation wavelength λ in the GaAs (diλ). These effects yield photosignals with an amplitude proportional to the second power of the radiation electric field acting on the 2D electron gas in the QW (for details, see Appendix 2). We compared the photosignals obtained for light illuminating either gold-covered or gold-uncovered areas of the sample. Taking the ratio of these signals \(\tilde {R}\), we obtain the reduction of the radiation intensity caused by the gold film. The experimental geometry is shown in Fig. 2a. In particular, the beam having the spot size (diameter about 1.5 mm, see Appendix 2) smaller than the gold film area (about 3.5 × 3.5 mm2) was scanned across the sample. Linearly polarized laser radiation was applied at normal incidence. Illuminating the sample changes the conductivity of the QW, which for biased sample resulted in a voltage drop across the load resistance RL. In some samples, it also yielded a photovoltage caused by the linear photogalvanic effect. The signal in response to the modulated radiation was registered by a standard lock-in technique. Experiments were carried out applying continuous wave cw THz molecular laser operating at wavelengths of 118, 184, or 432 μ m [24, 25]. Corresponding frequencies (photon energies) are f = 2.54 THz (\(\hbar \omega =10.51\) meV), f = 1.62 THz (\(\hbar \omega =6.7\) meV), and f = 0.69 THz (\(\hbar \omega =2.87\) meV). Experiments were performed at room and liquid helium temperatures. More details are given in Appendix 2.

Fig. 2
figure 2

a Experimental setup. Vertical downward arrows and red spots illustrate the polarized THz radiation beam scanned across the sample. The beam was scanned following the middle line along the long side of the sample. Steps of 100, 200, or 250 μ m were used. The geometrical sizes, being of the same order for all samples, and typical gold film profile are exemplary shown for sample #B-20E. b Thickness profile d(x) of 20-nm-thick evaporated gold film measured by DektakXT Stylus Profiler (Bruker GmbH). Zero thickness d corresponds to the surface of GaAs top cap layer

Fig. 3
figure 3

Structure cross-section of wafers type #A, panel (a), and #B, panel (b), used for fabrication of samples. Single GaAs QW is grown at a distance di from the top cap layer surface. Structures are grown on 350-μ m-thick GaAs substrate

Two groups of molecular beam epitaxially (MBE) grown GaAs/GaAlAs heterostructures with single GaAs quantum well were used. Figure 3 shows a cross-section of the structure. Four wafers of the same heterostructure design, grown on (001)-oriented GaAs substrate and hosting a 25-nm-thick QW, were used for fabrication of type #A samples. Wafer #B, grown on a (110)-oriented GaAs substrate and hosting a 15-nm-thick QW, was used for fabrication of samples of type #B. The latter structure, with a reduced point group symmetry compared with (001)-grown structures, allows us to measure besides the photoconductivity also the photogalvanic effect excited by normal incident THz radiation [26] in unbiased samples (for details, see Appendix 2). Substrates are made of 350-μ m-thick semi-insulating GaAs and are transparent for THz radiation. The density and mobility of electrons in 2DEG, obtained by magnetotransport experiments performed at T = 4.2 K, are n ≈ 1011 cm− 2 and mobility μ ≈ 106 cm2/V ⋅s, respectively.

Gold films with thicknesses 10, 20, and 85 nm were fabricated either by MBE directly connected to the GaAs-MBE or thermal vacuum evaporation. For the latter, the samples were exposed to air before the Au deposition. The 3.5 × 3.5 mm2 gold films covered about half of the sample surface with a rectangular shape with about 5 mm width and 10 mm length. The shape and position of the gold layer are shown in Fig. 2a. For optoelectronic measurements, after the gold film deposition, a pair of ohmic contacts was prepared in the middle of short edges (see Fig. 2a).

MBE-grown gold films were used for preparation of samples made from wafers #A1, #A2, and #A4. For that subsequently to the semiconductor growth, the wafers were transferred in situ from the III-V to the metal/oxide MBE where the Au layers were grown on parts of the 2-in. wafer by a movable shadow mask between the sample and the Knudsen cell. Reflection high-energy electron diffraction (RHEED) was used to check the crystalline quality of the layer, and to crosscheck the growth rate by RHEED oscillations. In the following, these samples will be labeled as e.g. sample #A1-20M, which shows that the sample is fabricated from the wafer #A1, has a 20-nm-thick film deposited on top of the heterostructure cap layer (the number after hyphen), and it is grown by MBE (last letter). More details are given in Appendix 3A.

Thermal vacuum evaporation was used for preparation of samples made from wafers #A1, #A3, and #B. Gold films were deposited on an unheated wafer using a Leybold Univex system. The thickness of the film was monitored by a quartz-microbalance. The samples were not annealed after deposition. A metallic shadow mask was used to cover only half of the sample. The gold structure size was chosen such that all sample edges are not covered to avoid short-cutting the QWs (see Fig. 2a). These samples are labeled in the same way as MBE samples, e.g., sample #B-20E, but with last letter E in the notation, indicating that the film is made by vacuum evaporation. For some measurements aimed to study the gold penetrability for the distances of gold film to QW larger than the radiation wavelength (diλ), we prepared sample with 20-nm gold film deposited on the backside of the sample’s substrate instead of the deposition onto the top layer. This fact is reflected in the sample label by an additional letter “S,” i.e., sample #A3-20ES.

Figure 4 shows typical atomic force microscopy (AFM) images of the fabricated gold films. These data show that the thickness fluctuations in MBE-grown films do not exceed 20% and in evaporated films is less than 25%. Consequently, the AFM images ensure that there are no perforations of film at least up to the resolution limit of the AFM. Figure 4g–i show AFM image of sample #A2-20M. While being grown by MBE, due to unknown reason, this sample is characterized by large thickness fluctuations (at some points up to 80%). We included this sample in the study in order to check how important the surface roughness is for the gold film penetrability.

Fig. 4
figure 4

AFM profile of 20-nm gold layers measured for different samples. Images obtained on samples: ac sample #A1-20M (MBE grown gold); df sample #B-20E (evaporated gold); and gi sample #A2-20M (MBE grown gold)

Besides the AFM measurements, yielding gold film profiles on a micrometer scale, we checked the film thickness over the whole sample length. The results are shown in Fig. 2b. Here, the layer thickness was measured by DektakXT Stylus Profiler (Bruker GmbH). This is a contacting method using low-force low-inertia scanning needle and has an accuracy of 4 Ångström. The data show that the thickness variation is near ± 5 nm and the border of the gold film is rather sharp (\({\varDelta } x \simeq 10 \div 15 \mu \)m).

3 Results

3.1 Room Temperature Data

Applying normally incident linearly polarized cw radiation we detect an increasing resistivity of the QW’s in all samples upon illumination (negative photoconductivity). The change of the QW’s conductivity is caused by the electron gas heating due to Drude-like free carrier absorption and consequent change of the electron mobility (so-called hot electron bolometer or μ-photoconductivity effect). The signal strength is proportional to the square of the amplitude of the electric field acting on the 2DEG. The negative sign of the photoconductivity is due to the fact that at room temperature the mobility is governed by phonon scattering, which becomes more efficient upon heating. This phenomenon is well known and deeply studied for various bulk and low dimensional materials including GaAs QWs with characteristics similar to that used in our work, more details are given in Appendix 2. Thus, detailed discussion of the physical mechanisms of the photo-response is out of scope of the present paper aimed to anomalous penetrability observed in gold films.

Figure 5a and b show the main result of the work obtained for 20 and 10 nm thick gold films deposited on the top surface. Moving the beam across the surface from the area without gold film to that covered by gold, surprisingly the photoconductive signal does not change much. It remains almost the same even if the spot is located fully within the gold film area! This anomalous penetrability was observed for films fabricated by both methods and having different surface roughness. We emphasize that the data are obtained for gold thicknesses at which according to Fresnel formulae the amplitude of the electric field penetrating through the film should be decreased by orders of magnitude (see Fig. 1 and Section 1). A possible reason for this result could be film perforations or percolation effects. The latter was studied in detail by the transmission measurements of gold films of various thicknesses [16]. It demonstrated that while radiation transmission is substantially enhanced for film with thickness up to 7 nm, the percolation effects for 20-nm-thick film become absent. AFM images discussed above demonstrate that there are no perforations of film at least up to the resolution limit of the AFM. This makes near-field diffraction effect as a possible reason of the electric field enhancement very unlikely. We also note that the results for samples #A1-20M with the most homogeneous gold film, (see Fig. 4a–c) and #A2-20M with the most rough film surface (see Fig. 4g–i) are almost the same. Would the effect be caused by the film-surface roughness or near-field diffraction, we would expect to detect differences in the electric field penetration in these samples with essentially different gold surface profiles.

Fig. 5
figure 5

Coordinate dependence of the normalized photoconductivity signal in response to cw radiation with λ = 118 μ m obtained in samples with gold film of thickness; a 20 nm, b 10 nm, c 85 nm, and d 20 nm. For three first panels, (a)–(c) gold films were fabricated on top of the heterostructure. In these cases, normally incident THz laser radiation was applied from the top side. In the last panel, (d) gold film was deposited on the substrate side and normally incident radiation was applied from the substrate side. Note that semi-insulating GaAs substrates are transparent for THz radiation. The data are normalized on the signal for the area without gold film. The laser beam was scanned along the axis x. The areas covered by gold are marked by a yellow background. Solid lines in panels (c) and (d) show calculated integral intensity of a Gaussian beam scanned across the opaque film. Parameters of the Gaussian beam have been obtained applying pyroelectric camera; see Appendix 2

To justify that our weak laser radiation does not modify the gold film surface, e.g., drills holes in the films, we first took an AFM image at a marked position, then exposed the sample to the radiation for a long time (35 min), and finally again obtained the AFM image at a previously marked position. For that, we applied radiation with power levels even higher (115 mW) than those used for photoconductive measurements (40 mW). The results shown in Fig. 6 demonstrate that the exposure with cw laser radiation does not change the gold thickness and the surface roughness, excluding laser radiation-induced gold ablation in our experiments.

Fig. 6
figure 6

AFM profile of 20 nm MBE grown gold film (Sample #A4-20M). Panels ac, before illumination; and df, after 35-min exposition to cw-laser radiation powerP ≈ 115 mW (λ = 118μ m)

In the measurements discussed above, two conditions have been simultaneously fulfilled: (i) film thickness d was less than the skin depth δs (for used frequency about 65 nm at room temperature; see Appendix 3B), and (ii) the distance to QW-based detector, di, was much smaller than the wavelength λ in GaAs.

Furthermore, violation of any of these conditions results in a drastic reduction of the photoconductive signal demonstrating that the film becomes opaque. Figure 5c shows the results obtained for a 85-nm gold film which is thicker than the skin depth. Scanning the beam from uncovered parts to gold-covered parts results in the substantial signal reduction. The data can be well fitted by a curve describing the intensity reduction expected for a Gaussian beam moving across the boundary of an opaque material (see curve in Fig. 5c). The Gaussian beam parameters used for the calculation were determined applying a pyroelectric camera (corresponding image and discussion are presented in Appendix 2).

Placing a detector at a distance \(d \geqq \lambda \) and, consequently, measuring the far-field transmission through films we confirm that, as expected from Fresnel formulae, our 20-nm films are opaque for THz radiation. This has been verified by two methods. In the first one, we deposited gold film on the bottom side of the heterostructure and illuminated the film from the substrate side (see inset in Fig. 5d). In this case, the distance between the gold film and QW detector (of the order of 350 μ m) becomes larger than the radiation wavelength 118 μ m. Note that the latter is additionally reduced in GaAs by a refractive index of about 3.5. Figure 5d shows the result for 20-nm-thick gold film deposited on the backside of the substrate, sample #A3-20ES. It demonstrates that the photoconductive signal drastically reduces by shifting the beam to the gold film and the scan can be well fitted by intensity reduction at movement of the Gaussian beam across an opaque film. In the second method, we measured the the radiation transmission of a film for samples with 20-nm gold film placed on top of the heterostructure. Now, however, instead of measuring photoconductive signal, we measured a signal of a large area with a pyroelectric detector placed behind the sample at a distance os several millimeters, i.e., much larger than λ (see Fig. 7). The scans are shown in Fig. 7 and demonstrate that the signal and, correspondently, radiation transmission become zero for a laser spot located within the gold-film area of samples.

Fig. 7
figure 7

Coordinate dependence of the radiation transmission measured by pyroelectric photodetector placed directly after the sample #A4-20M with 20 nm (blue circles) gold film grown by MBE. The data are normalized on the signal for the area without gold film. Note that semi-insulating GaAs substrates are transparent for THz radiation. The data are obtained applying for cw laser radiation with λ118μ m. Solid lines show calculated integral intensity of a Gaussian beam scanned across the opaque film. Parameters of the Gaussian beam have been obtained applying pyroelectric camera; see Appendix 2

In all measurements described above, the gold film penetrability was studied by comparing photoconductive signals obtained for radiation illuminating the gold-film-free and gold-film-coated areas of samples. It is noteworthy that the same result was achieved by using a completely different effect—photogalvanic current excited in QWs by terahertz radiation. To obtain a significant signal, in these experiments, we used radiation of pulsed THz laser yielding single 100-ns pulses (see Appendix 2). The polarization dependencies of the photocurrent generated by pulsed THz radiation in unbiased sample #B for radiation spot positions at gold-film-free sample area and within the 20-nm gold film–coated area are shown in Fig. 8a and b, respectively. The data reveal that the magnitude of the signal and its polarization dependence are almost the same for both spot positions. This result is in accordance with the photoconductivity data obtained on the same sample (see Fig. 5a). Essentially, new information obtained by LPGE is that not only the radiation electric field magnitude acting on 2D electrons but also its polarization remains the same. Polarization dependence of the photocurrent is described by \(j/P = \chi _{-} + \chi _{+} {\cos \limits } 2 \alpha \), being in agreement with the phenomenological theory of the LPGE (see Eq. 10 in Appendix 2). The polarization dependence for the conditions relevant to our experiment, i.e., normal incidence excitation of (110)-oriented GaAs/AlGaAs QWs and the current measured along [110]-direction is discussed in Appendix 2 (see Eq. 10). The mechanisms of the LPGE in such QWs involve asymmetric scattering of photoexcited electrons; they are well known and will not be discussed here (for somewhat more details see Appendix 2).

Fig. 8
figure 8

a, b Photocurrent as a function of the azimuthal angle α measured in sample #B-20E. Data are taken for irradiation of gold-film-free part (a) of the sample (blue points) and gold-film-covered part (b) (red points, film thickness 20 nm). Data are obtained with radiation of pulsed laser operating at wavelength λ = 90 μ m. The total pulse duration about 100 ns and peak pulse power P ≈ 800 W. Solid lines are fits by phenomenological equation (10) of Appendix 2 describing LPGE and given by \(j/P = \chi _{-} + \chi _{+} {\cos \limits } 2 \alpha \). Panel ce show pulse traces that show photosignal for irradiation of gold-film-free-part (c), gold-film-covered part (d) of the sample and response of the reference photon drag detector (e)

To measure rather weak LPGE currents, we applied pulsed laser radiation with possibly low intensity allowing us to avoid radiation-induced gold film destruction. Using short radiation pulses additionally allowed us to analyze the time constant of the photoresponse. Characteristic pulse traces, shown in Fig. 8c–e, demonstrate that the photoresponse for both spot positions reproduces short spikes caused by the spontaneous mode-locking (time constant less than 10 ns). Note that the same time constants were obtained for photoconductive signals measured for illumination of areas without and with gold film (not shown). These short response times are in agreement with characteristics times of both of LPGE and μ-photoconductivity (see Appendix 2). The time scale as well as the coincidence of the time responses for the illumination of gold-film-free and and gold-film-coated sample parts additionally confirms that in both geometries the origin of the photosignals is identical.

3.2 Low-Temperature Measurements

Anomalous penetrability of terahertz electric fields through 20-nm gold films deposited on a GaAs quantum well structure was also observed at liquid helium temperatures. For these measurements, the sample was placed in an optical cryostat with z-cut crystal quartz windows. The photoconductive signal excited by normally incident radiation was measured as a function of an external magnetic field applied perpendicularly to the film. The data were obtained for three radiation wavelengths 118, 184, and 432 μ m corresponding to frequencies 2.54, 1.68, and 0.69 THz, respectively. We measured photoconductive signal for two sample positions at which radiation was focused on gold-film-covered or gold-film-free parts of the sample (see Fig. 9). The values of the ratio \(\tilde {R}\) were calculated as a ratio of these signals. The inset in Fig. 9 shows the frequency dependence of the intensity reduction obtained for zero magnetic field. In contrast to the room temperature data yielding for 20-nm-thick film \(\tilde {R} \thickapprox 1\) (see the inset in Fig. 9), at low temperature \(\tilde {R}\) varies from ≈ 0.6 to 0.8 by frequency decrease from 2.54 to 0.69 THz. Note that signals for both spot positions behave equally upon magnetic field variation showing that the signals are caused by the same mechanism. The observed increase of the photoconductive signal with increase of magnetic field is attributed to magnetic field-induced modification of scattering mechanisms and its discussion is beyond the scope of the present paper. We also note that, besides the drastically larger electric field compared with that expected from Fresnel formulas, the field reduction is characterized by frequency dependence opposite to that of the far-field gold film transmission (see Fig. 1).

Fig. 9
figure 9

Magnetic field dependence of the photoconductive signal obtained for two spot positions with radiation focused on sample areas without and with 20 nm gold film. The data are obtained at liquid helium temperature. Inset shows frequency dependence of the ratio \(\tilde {R}\) of the signals obtained at zero magnetic field for illumination of gold-film-covered and gold-film-free area of the sample

The observed temperature and frequency behavior of the ratio \(\tilde {R}\) qualitatively correspond to that of the skin depth. Being aware that it can not be directly applied to our case since the electron mean-free path is larger than the gold film thickness d we, nevertheless, like to pay attention to these similarities. The skin depth δs can be re-written as:

$$ \delta_{\mathrm{s}} = \sqrt{\frac{\rho}{\pi f \mu_{0}}}. $$

Due to the reduction of the film resistivity upon temperature decrease (see Appendix 3B), the skin depth at low temperatures is reduced compared with that at room temperature. Furthermore, the skin depth depends on frequency as \(\sqrt {1/f}\), i.e., becomes smaller for higher frequencies. In 20-nm-thick gold films at room temperature, the skin depth, being of the order of 65 nm for f = 2.54 THz (see Appendix 3B), is about three times larger than the film thickness. At 4.2 K, however, these values become comparable. Consequently, cooling the sample results in partial violation of the inequality δs > d, which was shown above to be crucial for the anomalous penetrability. Both tendencies, the decrease of the ratio \(\tilde {R}\) with the temperature decrease at constant frequency or with frequency increase at constant temperature, are detected in our experiments (see inset in Fig. 9).

4 Summary

To summarize, we observe an anomalously strong penetration of THz radiation electric fields through almost homogeneous, perforation-free gold films deposited on semiconductor heterostructures with a conductive quantum well placed beneath the film at a distance substantially smaller than the radiation wavelength. We detected no field amplitude reduction as long as two conditions are simultaneously fulfilled: (i) the thickness of a highly conductive film is less than the skin depth (about 65 nm at room temperature and for the frequencies used in the study) and (ii) the distance to the QW-based detector is much smaller than the wavelength. Our measurements demonstrate that a violation of any of these conditions results in a drastic reduction of the photosignal, showing that the film becomes opaque. In this case, the result is described by the Fresnel formula. So far, we have no microscopic picture for this striking result. We note that, while we did our best to reduce the surface roughness, we cannot exclude completely inhomogeneities of the metal film, particularly on its internal surface, which would result in the effective conductivity to be less than expected by general frequency scaling law of DC bulk conductivity of the material. While focusing only on THz/gold/AlGaAs, we bear in mind that the effect should be also present in other systems as far as as the two abovementioned conditions are fulfilled simultaneously. At terahertz frequencies, these conditions can be satisfied for various metals with high enough values of high-frequency conductivity, such as gold, silver, and aluminum. For higher frequencies, i.e., in near infrared and visible range, the conductivity decreases, and metals become less reflective, so that the field can penetrate through metal films of feasible thicknesses. As concerning the AlGaAs QWs used in our experiments, the only aim of it is a technical realization of local radiation electric field detection at nanometer scale distances from the metal film surface. Electrical photoresponse to terahertz radiation has been observed in a great number of low-dimensional systems, different QWs and heterostructures, graphene, 2D materials etc., which offer a number of ways to realize such experiments. The observed effect seems attractive for the development of THz devices based on 2D materials, which usually require robust top gates made of highly conductive metal films usually placed at less than 1 nm distance from the electron gas location.