Silicon is the basic material of modern microelectronics. However, this semiconductor is transparent to photons with wavelengths longer than 1 μm, which impedes the use of silicon photonics to fabricate efficient emitters and detectors of radiation. A significant breakthrough in this field could be brought about by making use of defect-free Ge/Si heterostructures with Ge quantum dots (QDs) capable of detecting near infrared radiation (wavelengths of 1–2 μm) owing to band-to-band optical transitions in QDs. Recent reviews on the fundamental and applied aspects of semiconductor QDs can be found in [1, 2]. The widespread use of Ge QDs is limited because the light absorption coefficient is small due to a low density of states associated with QDs [3–6], the spatial separation of electrons and holes at the Ge–Si heterointerface [7–10], and a large effective mass of charge carriers. It was recently found that embedding Ge/Si QD layers in a two-dimensional photonic crystal (PhC) leads to a significant (up to a factor of 5) enhancement of photocurrent in the near infrared range [11]. The PhC was a regular two-dimensional array of air holes in the Si/Ge/Si heterostructure with a period much shorter than the radiation wavelength. The results were explained by the combination of two effects: (i) the excitation of plane PhC modes propagating along the Ge/Si layers, which efficiently interact with band-to-band transitions in the QDs, by the incident light wave and (ii) a decrease in the reflection coefficient of the microstructured surface owing to a lower effective refractive index of the structure with air holes. Metasurfaces where normally incident light is redirected laterally because of optical diffraction were called photon-trapping structures [12–19]. The propagation of light in a PhC and its relation to the PhC structure were discussed in detail in [20, 21]. In general, achieving an increase in the light absorption coefficient in a photon-trapping structure does not necessarily requires the PhC with its characteristic band structure [22]. It is only necessary to have waveguide modes in the lateral direction, i.e., along the QD layers. However, recent results on the modeling of photonic nanostructures demonstrated that the character of light wave propagation in PhCs can significantly affect the photocurrent generation and bring about resonance absorption of electromagnetic radiation in solar cells [23].

A feature of the PhC is the spatially periodic modulation of the dielectric constant with a period smaller than the wavelength of optical radiation [24]. Micro- or nanoholes act, as a rule, as potential barriers for both photon and electron waves. From the quantum-mechanical point of view, electron Bloch waves are formed in a two-dimensional periodic lattice. By analogy with electrons, Bloch modes can also occur for photons as a result of interference of waves reflected from the holes; these modes form Brillouin zones. The dispersion relations ω(k||) (where ω and k|| are the frequency and in-plane wave vector, respectively) for these Bloch waves are characterized by the occurrence of nearly flat regions, where the photon group velocity \({{v}_{{\text{g}}}}\) = dω/dk|| is close to zero (“slow” light) [25, 26]. The long lifetime of slow photons leads to an increased efficiency of light–matter interaction and can result in a significantly increased optical absorption in weakly absorbing materials. Here, we measure for the first time the dispersion characteristics of optical excitations in Ge/Si QD layers embedded in the PhC. We find that the maximum increase in the photocurrent (up to a factor of 60) in a hybrid structure with the PhC is attained owing to the interaction of band-to-band transitions in QDs with slow Bloch modes.

The samples were grown by molecular beam epitaxy on silicon-on-insulator substrates and represented vertical p–i–n photodiodes (Fig. 1a). A detailed description of the electronic structure of these Ge/Si QDs and the growth conditions, such as temperature and growth rate, was presented in [10, 22]. The thickness of the buried SiO2 layer was 2 μm. The active area of the detectors consisted of ten layers of Ge QDs separated by 10-nm-thick silicon barriers. The barriers were grown in a two-stage process. The Germanium QD arrays were grown using the phenomenon of self-organization of semiconductor nanostructures in the course of heteroepitaxy of materials with a large lattice-constant mismatch (the Stranski–Krastanov growth mechanism). Germanium layers with a nominal coating thickness of 0.9 nm were deposited at a temperature of 250°C at a rate of 0.04 nm/s. According to the scanning tunneling microscopy data, Ge QDs had the shape of hut clusters [10]. The height and base length of the QDs were ~1 and ~10 nm, respectively, and their density was 5 × 1011 cm–2. At the final stage, a two-dimensional PhC was formed from the grown structures. An array of circular air holes with a diameter of 0.85 μm was fabricated using reactive ion etching of Ge/Si layers through a metal mask. The mask represented a 30‑nm-thick perforated Cr film formed at the heterostructure surface by electron beam lithography, metal deposition in vacuum, and subsequent lift-off process. The holes were arranged in a rectangular lattice with a period of p = 1.3 μm and their depth was 0.68 μm (Fig. 1b). To identify photocurrent features associated with the excitation of PhC modes, conventional Ge/Si p–i–n photodiodes with Ge QDs were also manufactured. Their difference from microstructured samples was only the absence of air holes.

Fig. 1.
figure 1

(Color online) (a) Cross-sectional layout of a vertical p–i–n photodiode with Ge quantum dots in the Si matrix on a silicon-on-insulator substrate embedded in a two-dimensional photonic crystal. (b) Scanning electron microscopy image of the regular array of microholes in Si/Ge/Si layers that forms a two-dimensional photonic crystal. The array period is 1.3 μm, and the hole diameter and depth are 0.85 and 0.68 μm, respectively. (c) Measurement configuration. Here, θ is the angle of incidence of light, k0 is the wave vector of the incident wave, and E is the electric field vector.

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The photocurrent spectra were measured at room temperature using a Bruker Vertex 70 infrared Fourier-transform spectrometer in the fast-scan mode with a resolution of 10 cm–1 in combination with an SR570 low-noise current preamplifier (Stanford Research Systems). A halogen lamp was used as the source of radiation. The photocurrent spectra obtained after the Fourier transform were normalized to the emission spectrum of the halogen lamp measured by a DLaTGS pyroelectric detector. Measurements of the dependences of the photocurrent on the reverse bias Ub showed that the photoresponse is voltage-independent for Ub ≥ 1 V, which indicates that all photoexcited charge carriers are efficiently collected [27]. The dispersion relations for the PhC modes were obtained by measuring the spectral characteristics of photocurrent for different angles of incidence of light θ [28]. Let the sample surface coincide with the (x, y) plane. Then, by rotating the samples around the y axis, we can capture the dispersion relation of Bloch waves in the direction kx = |k0|sinθ (Fig. 1c). In our experiments, the samples were mounted on a rotating platform with electronic control. The angle θ varied from 0° to 70° with an accuracy of 1°. The incident light was linearly polarized perpendicular to the axis of rotation.

Figure 2a shows the spectra of the current response of two Ge/Si photodiodes with Ge QDs for θ = 0. The first sample (reference) does not contain the air-hole array, while the only difference of the second is the presence of a two-dimensional PhC, in which Ge QDs are embedded (Fig. 2a, line 2). The oscillations of the photocurrent in the spectrum of the reference sample result from vertical resonances of the Fabry–Perot type caused by the interference of waves reflected from the air/Si and Si/SiO2 interfaces [22, 29]. The photodiode with the PhC is characterized by weaker vertical resonances, which indicates the successful transformation of the vertically incident plane wave into a set of laterally propagating modes; the absorption of them by the QDs leads to a significant increase in the photocurrent. This effect is most prominent in the long-wavelength part of the spectrum, in the range between 2 and 3 μm. Figure 2b shows the spectrum of the photodiode responsivity gain K by the PhC in this region. The value of K(λ) was determined as the ratio of the spectral characteristics of the photocurrent in the samples with and without the PhC. One can see a series of narrow resonances corresponding to the excitation of the Bloch PhC modes; the most intense of them are indicated in Fig. 2b by letters A–E. It is important that the photocurrent gain is as high as K ~ 50 and significantly exceeds the previously observed values of K ~ 5 [11].

Fig. 2.
figure 2

(Color online) (a) Photocurrent spectra of a reference Ge/Si photodiode without a photonic crystal and a hybrid sample where the quantum dot layers are embedded in a two-dimensional photonic crystal (lines 1 and 2, respectively) for θ = 0. The diode bias voltage is 2 V. (b) Spectrum of the photocurrent gain K in a structure with a photonic crystal in the wavelength range from 2 to 3 μm.

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Figure 3 shows the spectra of the current responsivity in the wavelength range from 2 to 3.1 μm for different angles of incidence of light on the surface of the photodiode with the PhC. It is seen that resonances B and E at wavelengths λ = 2.5 and 2.8 μm, respectively, exist only in a narrow range of angles near θ = 0 and disappear at θ > 10°. This behavior suggests that these resonances result from the interaction of optical transitions in the Ge/Si QDs with PhC modes near the Γ point of the Brillouin zone [23]. The behavior of mode C turns out to be the most interesting. The amplitude of this resonance is the smallest at small angles and increases greatly with the angle θ. For θ ≥ 30°, it becomes the dominant resonance in the structure, with the photocurrent gain as high as K ~ 60 (Fig. 3c).

Fig. 3.
figure 3

(Color online) Photocurrent spectra of a photodiode embedded in a photonic crystal for the angles of incidence of light (a) from 0° to 30° and (b) from 35° to 70○. The spectra are shifted vertically by 0.3 units for clarity. (c) Spectrum of the photocurrent gain K for θ = 55°.

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Figure 4 shows the dependences of the normalized angular frequency ωp/(2πc) of resonances A, C, and D on the in-plane wave vector kx. Here, ω = 2πc/λ and c is the speed of light in free space. The dispersion relation of mode D is oscillatory. This implies that this mode arises from the crossing of neighboring interacting branches of Bloch waves [24, 30]. The energy of mode C increases with kx, approaches mode A, and does not change at kx > 0.34(2π/p) (θ > 40°). In this range of wave vectors, modes A and C become degenerate, the photon group velocity \({{v}_{{\text{g}}}}\) tends to zero, and the photocurrent increases greatly (up to a factor of 60) in comparison to the photodiode without the PhC (Fig. 3c). Therefore, it can be concluded that the maximum photocurrent enhancement in Ge/Si heterostructures with Ge QDs embedded in a PhC is due to the interaction of QDs with slow Bloch modes of the PhC.

Fig. 4.
figure 4

(Color online) Normalized frequency of the optical resonances in a photodiode with Ge/Si quantum dots embedded in a two-dimensional photonic crystal versus the in-plane component of the photon wave vector.

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In conclusion, it should be noted that embedding QDs in nanostructured materials can lead to other unusual effects, for example, nonlinear ones [31].