1.

Introduction

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 $\hslash \mathrm{\Omega }\simeq 300$ 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 (${\mathrm{\Delta }}_G\simeq 300\mathrm{meV}$) 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 gaps^28–32 demonstrate similar properties.

The GNR infrared photodetectors have been (GNR-IPs) proposed and analyzed previously^6,8,33 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 ${\mathrm{\Sigma }}_d$) and the bP base layer (BL) doped by acceptors (with the surface density ${\mathrm{\Sigma }}_a$). The BL is lying on the contact layer (substrate) of ${\mathrm{p}}^{+}$-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/${\mathrm{p}}^{+}$-bP heterostructure or on an n-hBN/GNR/p-bP/${\mathrm{p}}^{+}$-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.

Fig. 1

Schematic view of the GNR-IP device structure.

For the detection of radiation with $\hslash \mathrm{\Omega }>300\mathrm{meV}$, 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 $\hslash \mathrm{\Omega }<{\mathrm{\Delta }}_G$, where ${\mathrm{\Delta }}_G\simeq 300\mathrm{meV}$ 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 ${\mathrm{\Delta }}_V$ at the GNR/bP interface in comparison with the pertinent conduction band off-set ${\mathrm{\Delta }}_C$. This implies that the IPs of the n-bP/GNR/p-bP/${\mathrm{p}}^{+}$-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 $\mathrm{p}\text{-}\mathrm{bP}/\mathrm{GNR}/\mathrm{n}\text{-}{\mathrm{bP}/\mathrm{n}}^{+}$-bP type. The GNR-IPs based on heterostructures of the latter type can be considered analogously.

Fig. 2

(a) The GNR-IP band diagram with tunneling of the photoexcited holes under bias voltage and (b) interband electron transitions between the sub-bands ($n=1\to n=1$ and $\mathrm{\Delta }<\hslash \mathrm{\Omega }<2\mathrm{\Delta }$) in the GNR valence and conduction bands. The open and opaque circles correspond to the holes and electrons, respectively.

At $\mathrm{\Sigma }={\mathrm{\Sigma }}_d-{\mathrm{\Sigma }}_a+{\kappa }_{\mathrm{BL}}V/4\pi eW>0$, where ${\kappa }_{\mathrm{BL}}$ and $W$ are the BL dielectric constant and thickness, respectively, and $V$ is the bias voltage, the Fermi level in the GNRs is in the conduction band, i.e., ${\epsilon }_{\mathrm{GNR}}^F>0$ [see Fig. 2(a)].

The operation principle of the GNR-IPs resembles that of the unitraveling-carrier photodiodes^34–36 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 ${\mathrm{p}}^{+}$ 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.

2.

Main Equations

The electron and hole, ${\epsilon }^{-}(p)$ and ${\epsilon }^{+}(p)$, energy spectra in GNRs can be presented as the sets of the one-dimensional sub-bands^6–8 [see Fig. 2(b)]

Under the steady-state irradiation by the photon flux $I_{\mathrm{\Omega }}$ with the photon energy $\hslash \mathrm{\Omega }$, the average photocurrent density $⟨j_{\mathrm{\Omega }}⟩$ and the GNR-IP responsivity $R_{\mathrm{\Omega }}=⟨j_{\mathrm{\Omega }}⟩/\hslash \mathrm{\Omega }{I}_{\mathrm{\Omega }}$ are given by (see also the Appendix)

The quantity ${\mathrm{\Theta }}_{\mathrm{\Omega }}$ in Eq. (2) is the tunneling escape probability of the photoexcited holes having the energy ${\epsilon }_{\mathrm{\Omega }}=(\hslash \mathrm{\Omega }-\mathrm{\Delta })/2$ to escape from the GNR.

We estimate ${\mathrm{\Theta }}_{\mathrm{\Omega }}$ as

Setting ${\mathrm{\Delta }}_V=100\mathrm{meV}$ and $m_z\simeq 0.28m_0$,¹⁰ we obtain $E_{\mathrm{tunn}}\simeq 1.19\times {10}^6\mathrm{V}/\mathrm{cm}$. For ${\mathrm{\Sigma }}_a=2.2\times {10}^{12}{\mathrm{cm}}^{-2}$ and ${\kappa }_{\mathrm{BL}}=4$, we obtain $E_a\simeq 1\times {10}^6\mathrm{V}/\mathrm{cm}$.

3.

Spectral Characteristics of the GNR-IP Responsivity and Detectivity

According to Eq. (2), the GNR-IP responsivity $R_{\mathrm{\Omega }}$—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 ($\mathrm{\Delta }$, $\mathrm{\Gamma }$, ${\epsilon }_{\mathrm{GNR}}^F$, and $T$). The following parameters are assumed: $\beta =1.15\times {10}^{-2}$, ($\kappa =4$), ${\mathrm{\Delta }}_V=100\mathrm{meV}$, ${\tau }_{\mathrm{esc}}/{\tau }_{\text{relax}}\simeq 0.1$, $E_{\mathrm{tunn}}\simeq 1.19\times {10}^6\mathrm{V}/\mathrm{cm}$ ($m_z\simeq 2.5\times {10}^{-28}\mathrm{g}$), and $E_{\mathrm{GNR}}=1\simeq {10}^2\mathrm{V}/\mu \mathrm{m}$. The temperature is assumed to be $T=26\mathrm{meV}$ ($T\simeq 300\mathrm{K}$) except for Fig. 3(d). The parameter $\mathrm{\Gamma }$ was assumed to be $\mathrm{\Gamma }=1$ and 5 meV, which corresponds to the realistic momentum relaxation times $\tau \simeq 6.25\times {10}^{-13}\mathrm{s}$ and $1.25\times {10}^{-13}\mathrm{s}$, respectively. At $\mathrm{\Delta }=100\mathrm{meV}$, the range of the Fermi energies ${\epsilon }_{\mathrm{GNR}}^F=25$ to 50 meV corresponds to the ${\mathrm{\Sigma }}_d-{\mathrm{\Sigma }}_a\simeq (1.8\text{to}4.4)\times {10}^{12}{\mathrm{cm}}^{-2}$. For the degenerate electron gas in the GNRs (sufficiently large $\mathrm{\Sigma }$), the Fermi energy in the GNRs ${\epsilon }_{\mathrm{GNR}}^F\simeq \mathrm{\Delta }{(\mathrm{\Sigma }{d}^2/2)}^2\propto \mathrm{\Sigma }/\mathrm{\Delta }$ (see the Appendix).

Fig. 3

The responsivity $R_{\mathrm{\Omega }}$ versus photon energy $\hslash \mathrm{\Omega }$ of GNR-IPs with (a) $\mathrm{\Gamma }=5\mathrm{meV}$ and (b) $\mathrm{\Gamma }=1\mathrm{meV}$ at different energy gaps $\mathrm{\Delta }$ (different GNR widths), (c) $\mathrm{\Delta }=75\mathrm{meV}$, $\mathrm{\Gamma }=5\mathrm{meV}$, and ${\epsilon }_{\mathrm{GNR}}^F=25\mathrm{meV}$ at different Fermi energies in the GNR ${\epsilon }_{\mathrm{GNR}}^F$, and (d) $\mathrm{\Delta }=75\mathrm{meV}$, $\mathrm{\Gamma }=5\mathrm{meV}$, and ${\epsilon }_{\mathrm{GNR}}^F=25\mathrm{meV}$ at different temperatures $T$. Dotted lines correspond to the GL-IP responsivity.

At $\mathrm{\Delta }=75\mathrm{meV}$, the range of the Fermi energies ${\epsilon }_{\mathrm{GNR}}^F=25$ to 50 meV corresponds to ${\mathrm{\Sigma }}_d-{\mathrm{\Sigma }}_a\simeq (2.4\text{to}5.9)\times {10}^{12}{\mathrm{cm}}^{-2}$.

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 $\mathrm{\Delta }=0$, and presented by the dotted lines) shown in Figs. 3(a) and 3(b), the former can exhibit much higher $R_{\mathrm{\Omega }}$ when $\hslash \mathrm{\Omega }\sim \mathrm{\Delta }$.

As follows from Figs. 3(c) and 3(d), an increase in the Fermi energy ${\epsilon }_{\mathrm{GNR}}^F$ and in temperature $T$ 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 $S_{\mathrm{\Omega }}$ given by Eq. (4)].

The GNR-IP detectivity is given by

4.

Detection of Modulated Radiation

If the incident radiation comprises the steady-state and modulation components, $I_{\mathrm{\Omega }}$ and $\delta I_{\mathrm{\Omega }}^{\omega }\exp (-i\omega t)$, respectively, where $\delta I_{\mathrm{\Omega }}^{\omega }$ and $\omega$ 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, $R_{\mathrm{\Omega }}^{\omega }=⟨\delta j_{\mathrm{\Omega }}^{\omega }⟩/\hslash \mathrm{\Omega }\delta I_{\mathrm{\Omega }}^{\omega }$ one can arrive at

5.

Plasmonic Response

Depending on the sign of $\mathrm{\Sigma }={\mathrm{\Sigma }}_d-{\mathrm{\Sigma }}_a-{\kappa }_{\mathrm{BL}}V/4\pi eW$, 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 $y$ 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 $\delta {\phi }_{\mathrm{\Omega }}^{\omega }(y)\exp (-i\omega t)$.^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 $\pm k_y$). As a result, accounting for the plasmonic oscillations in the GNR, the averaged current density can be presented as [see the Appendix, Eq. (37)]

If $\omega \ll {\tau }^{-1}$, ${\tau }_d^{-1}\ll {\omega }_p$, one obtains $P^{\omega }\simeq 1$, and Eq. (22) turns to Eq. (17). At the combined resonance $\hslash \mathrm{\Omega }/\mathrm{\Delta }\simeq 1$ and $\omega /{\omega }_p\simeq 1$, the responsivity $|R_{\mathrm{\Omega }}^{\omega }|$ exhibits a particularly high maximum.

Figure 4 shows examples of the modulation characteristics $|R_{\mathrm{\Omega }}^{\omega }/R_{\mathrm{\Omega }}|$ as functions of the modulation frequency $\omega /2\pi$. These characteristics account for the delay in the photoexcited holes escape from the GNRs, the hole transit-time delay, and the plasmonic effects.

Fig. 4

The responsivity $|R_{\mathrm{\Omega }}^{\omega }|$ normalized by $|R_{\mathrm{\Omega }}|$ versus the modulation frequency $\omega /2\pi$ for (a) ${\omega }_p/2\pi =1.5\mathrm{THz}$ and $\tau =0.625\mathrm{ps}$, (b) ${\omega }_p/2\pi =1.5\mathrm{THz}$ and $\tau =1.875\mathrm{ps}$, and (c) ${\omega }_p/2\pi =2.5\mathrm{THz}$ and $\tau =1.875\mathrm{ps}$. Dashed lines correspond to the modulation without the excitation of plasmonic oscillation.

The following parameters are assumed: ${\omega }_p/2\pi =1.5$ and ${\omega }_p/2\pi =2.5\mathrm{THz}$, ${\tau }_{tr}=0.5\mathrm{ps}$, ${\tau }_d=10\mathrm{ps}$, ${\tau }_{\mathrm{esc}}=\tau =0.625\mathrm{ps}$ and ${\tau }_{\mathrm{esc}}=\tau =1.875\mathrm{ps}$ (corresponding to $\mathrm{\Gamma }=\hslash /\tau =1\mathrm{meV}$ and $\mathrm{\Gamma }=1/3\mathrm{meV}$, respectively). The frequency dependences shown in Fig. 4 reflect the interplay of the transit-time and plasmonic effects.

As seen from Fig. 4, $|R_{\mathrm{\Omega }}^{\omega }/R_{\mathrm{\Omega }}|$ and hence $|R_{\mathrm{\Omega }}^{\omega }|$ exhibit the maxima corresponding to the plasmonic resonance $\omega /{\omega }_p$. The responsivity $|R_{\mathrm{\Omega }}^{\omega }|$ becomes particularly large at the combined resonances $\mathrm{\Omega }\simeq n\mathrm{\Delta }/\hslash$ and $\omega /{\omega }_p\simeq 1$. The resonance effects under consideration are pronounced when the carrier momentum relaxation time $\tau$ 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.

6.

Discussion

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 $(2000\text{to}2500){\mathrm{cm}}^2/\mathrm{Vs}$ measured in GNRs^43,44 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)], $R_{\mathrm{\Omega }}$ and $R_{\mathrm{\Omega }}$, at $\hslash \mathrm{\Omega }=\mathrm{\Delta }$ yields $R_{\mathrm{\Omega }}/R_{\mathrm{\Omega }}\gg 1$. The same is valid for $R_{\mathrm{\Omega }}^{\omega }/R_{\mathrm{\Omega }}^{\omega }$. 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, $\tau =(1.25\text{to}6.25)\times {10}^{-13}\mathrm{s}$ [corresponding to $\mathrm{\Gamma }=(1\text{to}5)\mathrm{meV}$], 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 $R_{\mathrm{\Omega }}^{\omega }$ of the vertical GNR-IP under consideration with the responsivity ${\overline{R}}_{\mathrm{\Omega }}^{\omega }$ 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 $\omega$ one can find, respectively, $R_{\mathrm{\Omega }}/{\overline{R}}_{\mathrm{\Omega }}\simeq 2$ and $R_{\mathrm{\Omega }}^{\omega }{\overline{R}}_{\mathrm{\Omega }}^{\omega }\simeq 2{\overline{\tau }}_{\mathrm{tr}}/{\tau }_{\mathrm{tr}}$. Here, ${\overline{\tau }}_{\mathrm{tr}}/{\tau }_{\mathrm{tr}}\simeq (L/W)\gg 1$ 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 contacts^6,8,33 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 ${\mathrm{\Delta }}_C$ and ${\mathrm{\Delta }}_V$.

The resonant response of the GNR-IPs is revealed when the smearing of the carrier spectra is relatively small. This requires a smallness of $\gamma$ (i.e., sufficiently long momentum relaxation time $\tau$) and a small overlap of the wave functions in the neighboring GNRs. The latter implies that the spacing between these GNRs $l=(D-d)$ should not be too small. This condition looks like $\exp [-2\sqrt{2m_z({\mathrm{\Delta }}_V-\mathrm{\Delta }/2)}(D-d)/\hslash ]\ll 1$. For the GNR-IPs operating at the photon energies $\hslash \mathrm{\Omega }\sim 100\mathrm{meV}$, $d\sim 4\mathrm{nm}$, the above inequality is satisfied if $D-d>0.4\mathrm{nm}$. This means that the spacing between the GNRs can be fairly small, so that $d/D\le 1$ 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.

7.

Conclusions

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.

8.

Appendix