We propose far-infrared photodetectors with the graphene nanoribbon (GNR) array as the photosensitive element and the black phosphorus (bP) base layer (BL). The operation of these GNR infrared photodetectors (GNR-IPs) is associated with the interband photogeneration of the electron–hole pairs in the GNR array followed by the tunneling injection of either electrons or holes into a wide gap bP BL. The GNR-IP operating principle is akin to that of the unitraveling-carrier photodiodes based on the standard semiconductors. Due to a narrow energy gap in the GNRs, the proposed GNR-IPs can operate in the far-, mid-, and near-infrared spectral ranges. The cut-off photon energy, which is specified by the GNR energy gap (i.e., is dictated by the GNR width), can be in the far-infrared range, being smaller that the energy gap of the bP BL of ΔG ≃ 300 meV. Using the developed device models of the GNR-IPs and the GNR-IP terahertz photomixers, we evaluate their characteristics and predict their potential performance. The speed of the GNR-IP response is determined by rather short times: the photocarrier try-to-escape time and the photocarrier transit time across the BL. Therefore, the GNR-IPs could operate as terahertz photomixers. The excitation of the plasma oscillations in the GNR array might result in a strong resonant photomixing.
The quantization of the energy spectrum in the graphene nanoribbon (GNR) arrays, obtained by partitioning of the graphene layer (GL), results in the transformation of the two-dimensional gapless spectrum of the electrons and holes to the spectrum consisting of the sets of one-dimensional subbands. Such a quantization also leads to the formation of the energy gap in the GNRs.1–8 The transformation of the energy spectrum brings about substantial modifications of the radiation absorption characteristics of the GNRs compared with the uniform GLs. These modifications include the appearance of the absorption resonances associated with the radiative transitions between the hole sub-bands tops and the electron sub-bands bottoms and the appearance of the absorption “red” edge.6,8 These features of the GNR radiation absorption can be used in different infrared optoelectronic devices. The incorporation of the black-phosphorus (bP) layers, exhibiting the interband absorption of the photons with the energies from to 2000 meV depending on the bP layer thickness, opens new prospects.9–19 The heterostructures, in which GLs or GNR arrays are integrated with the bP layers, can exhibit additional functionalities.20–26 This is due to the special GL and GNR and the bP layers’ band alignment (the Dirac point in the GLs and GNRs corresponds to the energy gap in the bP) and relatively narrow energy gap () in the bP layers with a sufficiently large number of atomic sheets.27 The GL- and GNR-heterostructures with the black-arsenic (bAs) layers having even smaller band gaps28–32 demonstrate similar properties.
The GNR infrared photodetectors have been (GNR-IPs) proposed and analyzed previously6,8,33 using the interband transitions. In those IPs, the carrier photogeneration leads to an increase in the GNR longitudinal conductivity. In contrast, here, we consider the GNR-based vertical IPs using the radiation interband absorption in the GNR arrays. We focus our analysis on the GNR-IPs with the undoped (weakly doped) GNR array sandwiched between the top cap layer (CL) doped by donors (with the donor surface density ) and the bP base layer (BL) doped by acceptors (with the surface density ). The BL is lying on the contact layer (substrate) of -type (see Fig. 1). The CL can be either the n-bP BL or made of other materials, in particular, hexagonal boron nitride (hBN). An example is the GNR-IP based on n-bP/GNR/p-bP/-bP heterostructure or on an n-hBN/GNR/p-bP/-bP heterostructure. The GNR array is supplied by the side n-type contact to each GNR, so that the GNR array with the contact forms a fork-like structure as shown in Fig. 1. The donors in the CL and acceptors in the BL can be spatially distributed uniformly across these layers or be concentrated in the delta-doped sheets. The GNR array plays the role of the photoemitter replacing the GL-photoemitter.26 The GNR-IP operation is associated with the emission of the photoexcited holes from the GNR due to tunneling into p-BL contributing to the terminal current.
For the detection of radiation with , one can use more or less standard photodiodes using the interband transitions in bulk bP layers.16 We limit our consideration to the photon energies , where is the energy gap in the bP BL. The band diagram of the GNR-IP with n-CL and p-BL is shown in Fig. 2. Such a choice of the device structure is associated with a relative smallness of the valence band off-set at the GNR/bP interface in comparison with the pertinent conduction band off-set . This implies that the IPs of the n-bP/GNR/p-bP/-bP type can have smaller cut-off photon energy (in the far-infrared or even in the terahertz ranges) compared to the IPs of the -bP type. The GNR-IPs based on heterostructures of the latter type can be considered analogously.
At , where and are the BL dielectric constant and thickness, respectively, and is the bias voltage, the Fermi level in the GNRs is in the conduction band, i.e., [see Fig. 2(a)].
The operation principle of the GNR-IPs resembles that of the unitraveling-carrier photodiodes34–36 with a bulk absorbing layer serving as the emitter with the diffusive supply of the carriers injected into the BL.
Since the electric field in the p-BL is high, all the holes injected from the GNR can reach the substrate. As a result, the current is determined solely by the injection from the GNR [as schematically shown in Fig. 1(a)], so that the specific of the hole transport in the p-BL (ballistic or drift) does not affect the steady-state characteristics. However, the hole transit time across the p-BL can be fairly different in the cases of ballistic and drift transport (see Sec. 4). In this regard, the situation is analogous to that in the unitraveling-carrier photodiodes.
Main Equations3 one obtains and ,7 where is the electron mass in vacuum.
Under the steady-state irradiation by the photon flux with the photon energy , the average photocurrent density and the GNR-IP responsivity are given by (see also the Appendix)26 see also references therein). For the normal radiation incidence, the GNR array interband absorption coefficient is given by6,8
The quantity in Eq. (2) is the tunneling escape probability of the photoexcited holes having the energy to escape from the GNR.
We estimate as37 we arrive at the following equation:
Setting and ,10 we obtain . For and , we obtain .
Spectral Characteristics of the GNR-IP Responsivity and Detectivity
According to Eq. (2), the GNR-IP responsivity —one of the main IP characteristics—can be presented as
Figure 3 shows the spectral characteristics of the GNR-IP responsivity calculated for parameters of the GNR-IP structure (, , , and ). The following parameters are assumed: , (), , , (), and . The temperature is assumed to be () except for Fig. 3(d). The parameter was assumed to be and 5 meV, which corresponds to the realistic momentum relaxation times and , respectively. At , the range of the Fermi energies to 50 meV corresponds to the . For the degenerate electron gas in the GNRs (sufficiently large ), the Fermi energy in the GNRs (see the Appendix).
At , the range of the Fermi energies to 50 meV corresponds to .
As seen from the comparison of the dependences for GNR-IPs with those for the GL-IP (without the partitioning of the GL into the GNRs, i.e., with , and presented by the dotted lines) shown in Figs. 3(a) and 3(b), the former can exhibit much higher when .
As follows from Figs. 3(c) and 3(d), an increase in the Fermi energy and in temperature leads to a smaller responsivity. This is attributed to stronger restrictions for the radiative interband transitions imposed by the Fermi principle [described by the factor given by Eq. (4)].
The GNR-IP detectivity is given byFig. 3(b). This indicated that the selection of the CL and BL parameters, where first of all their doping opens the possibilities of the GNR-IP detectivity enhancement. One needs to stress that the proper choosing of the bias voltage might also be important for the GNR-ID responsivity and detectivity.
Detection of Modulated Radiation
If the incident radiation comprises the steady-state and modulation components, and , respectively, where and are the amplitude and frequency of the modulation components, considering the Shockley–Ramo theorem,38,39 the ac terminal current density is given by [see Eqs. (29) and (30) in the Appendix]
As follows from Eqs. (8), (15), and (16), for the responsivity to the modulation signals, one can arrive at
Depending on the sign of , the one-dimensional electron or hole gases are formed in each GNR. The collective response of these electron or hole systems can lead to pronounced plasmonic phenomena.
The ac current of the photoexcited holes emitted from the GLs causes ac hole injection from the side contacts to the GNRs. The ac current along the GNRs (in the direction, see Fig. 1) could lead to the excitation of the plasmonic oscillations, i.e., the self-consistent spatial–temporal variation of the electron or hole density and the GNR array potential in the GNR array .33,40–42 This plasmonic oscillation is the standing wave formed by the plasmonic waves propagated along the GNR in the opposite directions (i.e., with the wave numbers ). As a result, accounting for the plasmonic oscillations in the GNR, the averaged current density can be presented as [see the Appendix, Eq. (37)]Appendix. Equation (21) yields the following equation for the responsivity:
If , , one obtains , and Eq. (22) turns to Eq. (17). At the combined resonance and , the responsivity exhibits a particularly high maximum.
Figure 4 shows examples of the modulation characteristics as functions of the modulation frequency . These characteristics account for the delay in the photoexcited holes escape from the GNRs, the hole transit-time delay, and the plasmonic effects.
The following parameters are assumed: and , , , and (corresponding to and , respectively). The frequency dependences shown in Fig. 4 reflect the interplay of the transit-time and plasmonic effects.
As seen from Fig. 4, and hence exhibit the maxima corresponding to the plasmonic resonance . The responsivity becomes particularly large at the combined resonances and . The resonance effects under consideration are pronounced when the carrier momentum relaxation time is sufficiently long.
The inclusion of the plasmonic effects enables the possibility of the GNR-IPs operation in the terahertz range of the modulation frequencies, in particular, as the terahertz photomixers.
Both the realization of the resonant interband intersubband radiative transitions and the possibility of the pronounced plasmonic oscillations require relatively perfect GNRs with high carrier mobilities along the GNRs or their ballistic transport, which are achievable in real GNR-based heterostructures. The electron and hole mobilities on the order of measured in GNRs43,44 are sufficiently large to support resonant interband transitions and plasmonic oscillations.
As seen from Figs. 3(a) and 3(b), the comparison of the vertical GNR-IP and GL-IP responsivities’ dotted lines in [Figs. 3(a) and 3(b)], and , at yields . The same is valid for . Thus, the GNR-IPs can markedly surpass the GL-IPs in the responsivity if the GNRs are sufficiently perfect with long electron and hole momentum relaxation times, in particular, [corresponding to ], assumed in Figs. 3(a) and 3(b).
Despite a similarity of the GNR-IP and the standard unitraveling-carrier photodiodes, there are the following distinctions enabling the advantages of the former due to the following: the operation in the far-infrared range because of a narrow energy gap in the GNRs and the BLs, an increase photoexcitation efficiency associated with the resonant intersubband transitions, and a shortened escape time of the photoexcited holes.
Comparing the responsivity of the vertical GNR-IP under consideration with the responsivity of the lateral GNR-IP with the p–i–n structure having the depleted GNR array (like those considered in Refs. 8 and 45–47) and exploiting the variation of the GNR conductivity under irradiation (with the same lateral sizes), at low- and high-modulation frequencies one can find, respectively, and . Here, is the ratio of the lateral (along the GNRs) and vertical (across the BL) transit times. The responsivity of the lateral GNR-IPs with the n-contacts and the energy barrier formed by the GNR between these contacts6,8,33 can include the photoconductive gain increasing their responsivity, but at the expense of a lower speed operation.
The structure of the GNR-IP can be inverted placing the BL on the device top. Such GNR-IPs can be made on the hBN substrate.
Due to the similarity of the band alignment of the GNR with the bP and bAs layers, the obtained results can be applied to the GNR-IPs with the b-As layers considering the pertinent quantitative difference in and .
The resonant response of the GNR-IPs is revealed when the smearing of the carrier spectra is relatively small. This requires a smallness of (i.e., sufficiently long momentum relaxation time ) and a small overlap of the wave functions in the neighboring GNRs. The latter implies that the spacing between these GNRs should not be too small. This condition looks like . For the GNR-IPs operating at the photon energies , , the above inequality is satisfied if . This means that the spacing between the GNRs can be fairly small, so that and the potential sag in the inter-GNR region is small. All these indicate that the lateral nonuniformity of the potential in the GNR array plane can be very small.
We proposed and evaluated interband far-infrared photodetectors with the GNR array and the bP BL—GNR-IPs. Due to the specifics of the energy spectra of graphene and bP, the GNR-IPs can operate at photon energies much smaller than 300 meV. Due to a strong interband resonant absorption in the GNRs, the GNR-IPs can surpass the interband GL-IPs and different intersub-band IPs in responsivity and detectivity. The GNR-IPs are analogous to the unitraveling-carrier photodiodes and can operate at fairly high-modulation frequencies of the incident radiation. The possibility of the excitation of plasmonic oscillations in the GNR array of the device might lead to a pronounced response when the radiation modulation frequency is close to the frequency of the plasmonic oscillations, which is typically in the terahertz range. This enables the use of the GNR-IPs as terahertz photomixers. The GNR-IP response can be particularly strong at the combined resonance—the radiation frequency corresponds to the resonant interband transitions between the sub-bands, whereas the radiation modulation frequency corresponds to the plasmonic resonance.
Considering that the incident radiation comprises the steady-state and modulation components, and , respectively, where and are the amplitude and frequency of the modulation component, the dc and ac components of surface density, and , of the photoexcited carriers in the GNR array averaged in the lateral directions is governed by the following equations:
The dc components of the average current density are given by
Photocurrent at Modulated Radiation
The ac component of the average terminal current density is equal to38,39 Equation (28) does not account for the possible excitation of plasma oscillations (spatiotemporal variations of the carrier density in the GNR-array and of the electric field in the BL), which could result in the temporal variation of and in the displacement terminal current. Such a simplification of Eq. (28) is justified when is far from the characteristic plasma frequencies.
Equations (26) and (28) yield
Plasmonic Oscillations in GNR Arrays
The amplitude of the plasmonic mode potential is governed by the equation, which is the consequence of the electron transport equation along the GNR and the Poisson equation (similar to that used previously for different plasmonic structures)33,40,417 is the hole density in the GNR array, is the effective mass in the GNRs, and is the length of the GNRs (spacing between the side contacts to the GNRs).
Since the ac current apart from the current due to the variation of the radiation intensity includes the component arisen due to the variations of the tunneling probability associated with the change of the electric field at the GNR array in the case of the plasmonic oscillations, Eq. (29) should be replaced by
Adding the density of the displacement current to , we find
Fermi Energy in the GNR Array
If , i.e., at a stronger doping of the CL than the BL and at not too high bias voltages, the GNRs are occupied primarily by the induced electrons. At sufficiently large , the electron gas in the GNRs can be degenerate. In this case, the linear electron density in GNRs is given by
At not too high electron densities when the electrons occupy only the lowest sub-band , considering Eq. (1), we have
The work at RIEC, UoA, IUHFSE, and UB was supported by the Japan Society for Promotion of Science (KAKENHI Grant No. 16H06361), and the RIEC Nation-Wide Collaborative Research (Projects Nos. R03/A12 and H31/A01), Japan. The work at MIPT was supported by the Russian Foundation for Basic Research (Grant Nos. 18-07-01379 and 18-29-02089), Russia. The work at RPI was supported by the US Army Research Laboratory Cooperative Research Agreement (project monitor Dr. Meredith Reed) and the US Office of Naval Research (project monitor Dr. Paul Maki).
Maxim Ryzhii is currently a senior associate professor at the University of Aizu, Aizu-Wakamatsu, Japan. His research activity includes physics and computer modeling of optoelectronic and terahertz nanostructure devices, and computational modeling of biophysical systems. He is the author or coauthor of more than 180 journal publications and numerous conference papers. He is a senior member of the IEEE and a member of the APS.
Victor Ryzhii is currently a principle researcher at the Institute of Ultra-High Frequency Semiconductor Electronics and a visiting professor at the Research Institute of Electrical Communication, Tohoku University, Japan. He is the author and coauthor of more than 400 research publications in the theory and computer modeling of micro- and nanostructures and electronic and optoelectronic devices and holds 11 patents. He is a corresponding member of the RAS and a fellow of the IEEE and the APS.
Petr P. Maltsev is currently a scientific director at the Institute of Ultra-High Frequency Semiconductor Electronics of RAS, Moscow, Russia. He graduated from Tashkent University of Information Technologies in 1971. Since 1998, he has been a professor at Moscow State Institute of Radio Engineering, Electronics, and Automation. He has more than 250 papers in refereed journals, 40 patents, and 7 monographs.
Dmitry S. Ponomarev works as a deputy director at the Institute of Ultra-High Frequency Semiconductor Electronics of RAS, Moscow, Russia. He is also a leading researcher at A. M. Prokhorov General Physics Institute of RAS and Moscow Institute of Physics and Technology, Moscow, Russia. His research interests are focused on photonic devices, terahertz spectroscopy, and plasmonics.
Vladimir G. Leiman is currently a professor in the Department of General Physics and a principle researcher at the 2-D Material’s Optoelectronics Laboratory, Moscow Institute of Physics and Technology, Russia. His research interests include the theory of intensive electron beams in vacuum and plasma devices and plasmonic phenomena in electronic and optoelectronic devices. He is the author and coauthor of more than 100 journal publications and numerous conference papers. He is a member of the OSA.
Valery E. Karasik is a professor in the Department of Radioelectronics and a director at the Center of Photonics and Infrared Engineering of Bauman Moscow State Technical University. His interests include the research and design of laser devices with the focus on the laser target detection and location. He is a member of the Russian Academy of EII.
Vladimir Mitin is currently SUNY distinguished professor in the Department of Electrical Engineering at the University at Buffalo, the State University of New York. His fields of specialization are microelectronic, nanoelectronic, and optoelectronic devices and materials with special emphasis on sensors and detectors based on nanostructures. He has more than 300 publications including 10 patents, 4 monographs, and 6 textbooks. He is a fellow of the IEEE, the APS, and other professional societies.
Michael S. Shur is the Roberts professor in the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA. He is a cofounder of Electronics of the Future, Inc. He is a fellow the IEEE, APS, Electrochemical Society, and a foreign member of the Lithuanian Academy of Sciences and several other professional societies.
Taiichi Otsuji is a professor at the Research Institute of Electrical Communication, Tohoku University, Sendai, Japan. His current research interests include terahertz electronic and photonic materials/devices and their applications. He has authored and coauthored more than 400 research publications. He is a senior member of the Optical Society of America, a fellow of the IEEE and the APS, and several other professional societies.