Open Access
Add to BibSonomy
Abstract
We propose the far-infrared and terahertz emitting diodes (FIR-EDs and THz-EDs) based on the graphene-layer/black phosphorus (GL/b-P) and graphene-layer/MoS2 (GL/MoS2) heterostructures with the lateral hole and vertical electron injection and develop their device models. In these EDs, the GL serves as an active region emitting the FIR and THz photons. Depending on the material of the electron injector, the carriers in the GL can be either cooled or heated dictated by the interplay of the vertical electron injection and optical phonon recombination. The proposed EDs based on GL/b-P heterostructures can be efficient sources of the FIP and THz radiation operating at room temperature.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The optically or electrically pumped graphene-layer (GL)-based heterostructures can be used in the far-infrared (FIR) and terahertz (THz) sources, detectors, and other devices exploiting the interband transitions [1–9]. One of the pumping techniques is the lateral injection from the n$^+$- and p$^+$- side contacts into the GLs [10–12]. The GL-heterostructure emitting diodes and lasers with the side injecting contacts can be used in both the edge and front-face configurations. Such heterostructures demonstrated a broadband amplified emission in the range 1 - 5 THz. Similar heterostructures with the edge configuration with the grating providing the distributed feedback generated a single-mode radiation with the frequency $5.2$ THz (at 100 K) [13,14]. However, the lateral injection of both electrons and holes becomes ineffective if the spacing between the injecting contacts is relatively large. A large separation limits the output power because of the potential sag in the middle of the GL resulting in relatively low electron and hole densities preventing the interband population inversion necessary, in particular, for the super-luminescence and lasing [15]. This drawback might be avoided by implementing the combined injection, for example, using the lateral injection of holes from both p$^+$-side contacts and the vertical injection of electrons from the n$^+$-contact on the heterostructure top [16]. However, each vertically injected carrier captured by the GL contributes the energy on the order of the conduction band offset, $\Delta _C$. The energy injected into the GL rises the effective temperature, $T$, of the two-dimensional electron-hole plasma (2DEHP) in the GL due to the frequent carrier-carrier collisions. As a result, the 2DEHP temperature can exceed the lattice temperature $T_0$. A rise of the 2DEHP temperature negatively affects the characteristics of the THz and FIR sources, in particular, thwarting the population inversion [16–18]. The carrier heating by the injection current is also harmful in the standard heterostructure light-emitting diodes (LEDs) and lasers, but in the devices with the gapless energy spectrum, such as incorporating GLs, the heating might be more detrimental. Using the layers surrounding the GLs made of the materials with decreased $\Delta _C$, for example, black-phosphorus (b-P) [17,18] (see also [19–33]) could diminish the carrier heating by the vertically injected electrons. The conduction band offset at the GL/b-P interface is about of $\Delta _C \simeq 200$ meV. This value is close to the GL optical phonon energy $\hbar \omega _0$, which is beneficial for limiting an excessive carrier heating. Moreover, in the EDs with such heterojunction, the interplay between the electron injection and optical phonon recombination can result in the 2DEHP cooling promoting the population inversion. In the EDs with $\Delta _C > \hbar \omega _0$, for example, based on the GL/MoS$_2$ heterostructures, in which the injection heating can essential, the carrier heating can be weakened using a sufficiently strong doping of the GL by acceptors [34] (see also [35]). This approach might be used for the FIR and THz emitting diodes (FIR-EDs and THz-EDs) and super-luminescent diodes (SLDs) as well as for the lasers with the GL-heterostructure active region.
In this paper, we propose, model and analyze the FIR-EDs and THz-EDs based on the GL/b-P and GL/MoS$_2$-heterostructures with the combined injection and the front-face configuration. In principle, in such a configuration, both the interband spontaneous and stimulated emission are possible. The latter requires the interband population inversion, a relatively weak intraband (Drude) absorption in the GL, and high quality top reflectors [1,36]. The implementation of the GL-based EDs is easier since it does not necessitate a sufficiently low Drude absorption and a resonant cavity. Figure 1(a) shows the sketch of an ED with the vertical electron and lateral hole injection and the front-face radiation output. Such EDs generate incoherent THz and FIR radiation. The LEDs with GLs (mainly using GLs as highly conducting transparent contacts) were reported in a number of publication [37–40]. Suspended electrically biased GLs heated by the electric current can also be used for emission of mid-infrared and visible light [41–45]. The optical emission from GLs induced by a strong THz field resulting in the carrier interband Landau-Zener transitions and their heating was reported recently [46]. In contrast to these works, we focus on the GL-based EDs mainly for the FIR and THz spectral ranges.
Fig. 1. The heterostructure device under consideration with the p$^+$–side hole injector and vertical top n$^+$–contact electron injector: (a) cross-section view, (b) band diagram at a relatively small bias voltage ($V < V_{bi}$) - the barrier limited injection, and (c) band diagram at a high ($V > V_{bi}$) bias voltage - the space-charge or scattering limited injection, where $V_{bi}$ is the built-in voltage between the p$^+$- and n$^+$- contact regions. The black and open circles correspond to electrons and holes, respectively. The wavy arrows show the propagation of the photons emitted in the GL.
The EDs under consideration are based on GL/b-P and GL/MoS$_2$ heterojunctions (with suitable band off-sets [19,21,32,33]). We focus on the carrier cooling and heating effects and analyze the role of the GL doping, in particular, demonstrating that the GL doping weakens the carrier cooling in the EDs with a relatively small band off-set (with the GL/b-P heterojunction) and heating in the EDs with a relatively large band off-set (with the GL/MoS$_2$ heterostructure). It affects the GL conductivity diminishing the lateral nonuniformity of the 2DEHP, arising due the current crowding effect [47,48], and promotes increasing ED efficiency.
2. Device and mathematical models
The ED device configuration includes the undoped, p-doped, or p$^+$-doped GL sandwiched between the top vertical n- and n$^+$- emitter layers (n-EL and n$^+$-EL) and the wide-gap substrate transparent for the THz and FIR radiation, for example, an hBN substrate. Two p$^+$ side contacts inject holes into the GL. The ED operation (similar to the operation of the standard LEDs) is associated with the capture into the GL of the electrons vertically injected from the emitter, lateral injection of holes from the side contacts, and the electron-hole radiative recombination in the GL. Depending on the bias voltage $V$, the electron injection can be limited by the potential barrier in the n-EL or by the space-charge or scattering on impurities and acoustic phonons [see Figs. 1(b) and 1(c), respectively].
Closing the famous THz gap by providing efficient, compact, and powerful enough sources of the terahertz radiation (see, for example [49,50]). Graphene and graphene based heterostructures have emerged as the most promising candidates to meet this challenge [51–55]. The key problems in implementing the THz sources based on graphene heterostructures are the efficient electron-hole injection and minimizing the contact resistance. These problems are addressed in the lateral-vertical geometry as discussed in [16]. Such structures allow for optimizing the position of the Fermi level in the device channel by adjusting doping and provide means of optimizing the vertical transport.
Our previous work [16] dealing the lateral-vertical graphene based heterostructures focused on the surface plasmon amplification. The results of the analysis presented in [16] showed that the lasing of the plasmons propagating in the lateral direction (along the GL) can be achieved in such structures. In this work, we focus on rather different devices, namely, the FIR and THz vertically emitting diodes and show that they could reach a high emission efficiency using the carrier cooling in the GL due to the interplay of the vertical electron the recombination processes involving optical phonons. Recent breakthroughs in the fabrication of the graphene-black phosphorus heterostructures make the fabrication of the proposed devices possible [53–55].
Our mathematical model (which generalizes the model developed in [11,16,17]) includes the equations governing the balance of the interband and intraband energy relaxation and the recombination-generation processes in the GLs. At room temperature, these processes are associated with the optical phonon emission and absorption in the GLs [56]. Hence, each act of the interband and intraband optical phonon emission/absorption in the GL decreases/increases the energy of the 2DEHP by the quantity $\hbar \omega _0 \simeq 198$ meV. The injection of an electron from the n-EL into the GL associated with a loss of the kinetic energy due the carrier-carrier collisions and the interaction with phonons [57] is similar to that in the standard quantum-well infrared photodetectors [58]. It leads to an increase of the 2DEHP energy by the quantity $\Delta _i$, which is determined by the conduction band offset $\Delta _C$ [see Figs. 1(b) and 1(c)]. The rates of the electron injection and the electron energy injection into the GL are given by $j/e$ and $j\Delta _i/e$, respectively, where $j$ is the density of the vertically injected electron current (in A/cm$^2$ units) and $e = |e|$ is the electron charge.
A relatively short time of the optical phonon decay into acoustic phonons in the GLs [59] and a high GL heat conductivity [60,61] allow us to disregard the heating of the optical phonon system. Hence, the optical phonon distribution is given by ${\cal N}_0 =[\exp (\hbar \omega _0/T_0) - 1]^{-1}$. The 2DEHP can be far from equilibrium with the optical phonon system. This is different from the situation in the suspended GLs [43] when both these systems are heated.
Since the ED operation assumes high carrier densities in the GLs, the inter-carrier collisions are fairly frequent and the quasi-Fermi electron and hole distribution functions have the common effective temperature $T$. However, the transition of the 2DEHP to full equilibrium requires a relatively long time. This implies that the electron and hole distributions have by generally different quasi-Fermi energies, $\mu _e$ and $\mu _h$ (as in super-luminescent and semiconductor laser diodes) with the quasi-Fermi distribution functions $f_e(p) = [\exp (pv_W - \mu _e)/T + 1]$ and $f_p(p) = [\exp (pv_W- \mu _h)/T) + 1]$, where $p$ is the carrier momentum and $v_W \simeq 10^8$ cm/s is the characteristic velocity of electrons and holes in GLs. Taking into account the above, rather common assumptions, the system of the equations governing the density and energy balance in the GL can be presented as
The thermal energy spread of the electrons in the n-EL, $\Delta _i$ can be presented as $\Delta _i = \Delta _C + 3T_E/2 - \Delta _i^*$ with $T_E$ being the electron temperature of the injected electrons (in the energy units). We set $T_E \simeq T_0$, where $T_0$ is the lattice temperature, neglecting deviations of $T_E$ from $T_0$, which might be caused by the electron Peltier cooling (at $V < V_{bi}$) or the Joule heating in the n-EL (at $V > V_{bi}$). The quantity $\Delta _i^*$ characterized the fraction of the injected electron energy which transferred to the optical phonon system. Following [11,17], we set $\Delta _i^* = K\hbar \omega _0/(1 + K \eta _{cc})$, hence
The radiative processes associated with the absorption and emission of photons (or emission of plasmons [62–64] followed by their transformation to output photons) also affect the density and energy balance. Since the latter processes are characterized by relatively long time $\tau _r$ (much longer that the characteristic times pertinent to the optical phonon processes $\tau _0^{inter}$ and $\tau _0^{intra}$ [65,66]), we have disregarded the contribution of the radiative processes to the density and energy balance. However, these processes determine the radiation emission from the 2DEHP with the parameters determined the optical phonon and injection processes.
3. Carrier cooling and heating by the injected current
Equations (1) - (3) yield
Figures 2 and 3 show the dependences of the normalized effective temperature $T/T_0$ and the average quasi-Fermi energy $\mu = (\mu _e + \mu _h)/2$ of the carriers in the GLs with different doping levels on the injection current density $j$ numerically calculated using Eqs. (9) and (10) for $T_0 = 26$ meV (300 K). It is set $\Delta _i = 140$ meV and $\Delta _i = 290$ meV for the EDs with the GL/b-P and the MoS$_2$ heterostructures, respectively, $\hbar \omega _0 = 198$ meV, $\eta _{cc} = 1$, and $j_G = 1.6\times 10^2$ A/cm$^2$.
Fig. 2. Normalized temperature $T/T_0$ versus injection current density $j$ in EDs with (a) $\Delta _i = 140$ meV (GL/b-P heterostructures) and (b) $\Delta _i = 290$ meV (MoS$_2$ heterostructures), different GL doping levels [different values of $\mu _a$ - the same for (a) and (b)], and $T_0 = 26$ meV (300 K).
Fig. 3. Average quasi-Fermi energy $\mu = (\mu _e + \mu _h)/2$ versus injection current density $j$ in EDs with (a) $\Delta _i = 140$ meV (GL/b-P heterostructures) and (b) $\Delta _i = 290$ meV (MoS$_2$ heterostructures), for the same GL doping levels as in Figs. 2(a) and 2(b): solid lines $T_0 = 26$ meV (300 K).
As follows from Fig. 2, the 2DEHP effective temperature $T$ markedly decreases (if $\Delta _i = 140$ meV, i.e., $\Delta _i < \hbar \omega _0$) and increases (if $\Delta _i = 290$ meV, i.e., $\Delta _i > \hbar \omega _0$) with increasing injection current. A decrease in $T$ below $T_0$ (the 2DEHP cooling down) with increasing injection current at $\Delta _i < \hbar \omega _0$ seen in Fig. 2(a) is due to the following. The injection of an electron into the GL adds the energy about $\Delta _i$ to the 2DEHP, while each recombination act with the emission of an optical phonon decreases the 2DEHP energy by the quantity $\hbar \omega _0$ (with $\Delta _i < \hbar \omega _0$ in the case of GL/b-P heterojunction). Thus, there is a predominant removal of the hottest carriers. However, the 2DEHP cooling leads to a dramatic decrease in the number of high energy carriers able to emit optical phonons. As a result, the optical phonon mechanism under consideration can step aside to other relaxation-recombination mechanisms due to the carrier interaction with surface optical phonons and Auger processes. The inclusion of such mechanisms can stop further cooling of the 2DEHP [16] below the temperatures corresponding to the dashed lines in Fig. 2(a) [and in Figs. 3(a) and 5(a) as well]. As seen from Fig. 2(a), at higher doping levels (larger $\mu _a$), the 2DEHP cooling down is weaker in the range of moderate injection currents. Moreover, the $T - j$ relations exhibit the tendency to the S-shape behavior corresponding to possible filamentation of the temperature distribution. This can be attributed to a non-monotonic dependence of the net carrier density in the doped GL on $\mu = (\mu _e + \mu _h)/2$ [see Eq. (8)]. In contrast, as seen from Fig. 2(b) corresponding to $\Delta _i > \hbar \omega _0$ , the electron injection increases the 2DEHP energy despite the optical phonon emission. This leads to a gradual rise of $T$ with increasing $j$. In the EDs with a higher doping, the effect of heating is less pronounced. This is because the injected power is distributed over a large number of the carriers at higher doping levels.
As shown in Fig. 3(a), the 2DEHP cooling down is accompanied by a fairly strong increase in $\mu$. The latter implies the reinforcement of the 2DEHP degeneration ($\mu > 0$). The ambiguity of the $T - j$ dependences is reflected also in the shape of the $\mu - j$ characteristics at elevated doping levels. It is interesting that in the case of the undoped GL, the injection leading to the 2DEHP heating can result in $\mu < 0$. This implies that the carrier Fermi energies in the GL ($\mu _e = \mu _h =0$ without the injection) become negative, i.e., the injection reliefs the degeneration.
4. Spectral characteristics and output power
Using the dependences obtained above, one can find the output intensity of the spontaneous radiation $P$. The probability of the interband radiative transition of an electron between a state in the GL conduction band with the momentum $p$ to a state in the valence band with the same momentum (direct transition) is given by [65,66]
We assume that the GL is in the thermostat surrounding the device and having the temperature $T_0$. The majority of the nonequilibrium photons emitted by the GL leaves the device through the outer surface [see Fig. 1(a)]. A fraction of the photons reflected from this surface crosses the GL. The GL absorption (due to both interband transitions and the Drude absorption) is weak. The interband absorption might lead to a photon recycling effect [67–69], although it is a rather small. As a larger fraction of the reflected photons passes the GL, it is absorbed in the top n$^+$-EL and the top electrode. Thus, we neglect the photon recycling and the resonant properties of the cavity between the outer surface and the top of the structure.
In such a situation, the photon emission from the 2DEHP in the GL is associated with the spontaneous interband radiative transitions and the interband transitions stimulated by the thermal (equilibrium) photons. The thermal radiation is characterized by the distribution function ${\cal N}_{ph}(\hbar \omega ) = [\exp (\hbar \omega /T_0) - 1]^{-1}$ with the temperature of the lattice and of the surrounding thermostat $T_0$. The indirect radiative interband and intraband (Drude) transitions associated with the photon emission accompanied by the carrier scattering on disorder and acoustic phonons, can also contribute to the net radiation output [70]. However, such a contribution is relatively small in the photon energy range under consideration, and, therefore, could be disregarded.
Taking into account that the rate of the interband radiative transitions is proportional to [64,65]
For the output power of the radiation emitted at the interband transitions (both spontaneous and stimulated by the thermal photons corresponding to the thermostat temperature $T_0$), we have
Here, $A < 1$ is the fraction of the emitted photons passed the outer surface (not reflected back to the GL).Taking into account that $\mu _e$ and $\mu _h$ (at sufficiently strong injection when $\mu _e, \mu _h > T$) are expressed via $\mu$ and $\mu _a$ as
from Eq. (12) we obtainFigure 4 shows the emission spectra of the GL/b-P and GL/MoS$_2$ EDs for different injected current densities $j$. As seen from Figs. 4(a) and 4(b), the spectral characteristics, $S_{\hbar \omega }$, of the EDs exhibit pronounced maxima at certain photon energies. The height of these maxima steeply rises with increasing the injection current density. The maximum values of $S_{\hbar \omega }$ are markedly larger in the GL/b-P EDs than in GL/MoS$_2$ EDs. The EDs with doped GLs have a much lower $S_{\hbar \omega }$ in comparison with the EDs having the undoped GLs. This implies that the most promising EDs are those based on the GL/b-P heterostructures with the undoped GLs.
Fig. 4. ED spectral characteristics: (a) for $\Delta _i = 140$ meV (GL/b-P heterostructures) and (b) $\Delta _i = 290$ meV (MoS$_2$ heterostructures) with different acceptor doping levels [different values of $\mu _a$ common for (a) and (b)]: solid lines - $j=250$ A/cm$^2$, dashed - $j=500$ A/cm$^2$, and dotted - $j=750$ A/cm$^2$.
The maxima positions somewhat depend on the GL doping: an increase in the doping level (an increase in $\mu _a$) leads to a moderate shift of the maxima toward higher photon energies.
The position of the maximum, $\hbar \omega _{max}$, of the spectral characteristic of the most interesting version of the GL/b-P ED (with the undoped GL) exhibits a marked “blue” shift with increasing injection, in particular, from $\hbar \omega _{max} \simeq 100$ meV at $j = 250$ A/cm$^2$ to $\hbar \omega _{max} \simeq 180$ meV at $j = 750$ A/cm$^2$. This can be attributed to a substantial increase of the quasi-Fermi energy $\mu$ when the injection becomes stronger [see Fig. 3(a)]. It is instructive that the latter is accompanied by a drop of the effective temperature $T$ [shown in Fig. 2(a)]. An increase in $\mu$ combined with a decrease in $T$ implies the reinforcement of the 2DEHP degeneration.
Figures 5(a) and 5(b) show the output THz power $P$ generated by the GL/b-P and GL/MoS$_2$ EDs as a function of $j$ (for $A = 1$). In these figures, we used the same parameters as in Figs. 2, 3, and 4.
Fig. 5. Normalized output power as a function of injected current densities for EDs based on (a) GL/b-P heterostructures and (b) GL/MoS$_2$ heterostructures with different acceptor doping levels.
As seen from Fig. 5(a), an increase in $j$ results in a significant rise in $P$ for the GL/b-P EDs (with $\Delta _i < \hbar \omega _0$). This increase could become fairly large, particularly, in the case of undoped GL. This is associated with a marked increase in $\mu$ and in a pronounced decrease in $T$ (and, hence, in a strong increase in $\mu /T$), i.e., in the reinforcement of the interband population inversion. This, in turn, leads to a substantial dominance of the photon emission compared to the absorption.
Comparing the ED emission power $P_0$ at $\mu _a = 0$ and $j=0$ ($\mu =0$ and $T=T_0$) with the power, $P_0^{GB} = \displaystyle \epsilon \biggl (\frac {\pi ^2}{60\hbar ^3c^2}\biggr )T_0^4$, of a gray-body emission at $T_0$ with the emissivity $\epsilon = 0.1$ (four times larger than that estimated in [41,42]), we find $P_0 \simeq P_0 \sim P_0^{GB}$. However, at elevated injection current densities, as follows from Fig. 5(a), $P \gg P_0 \sim P_0^{GB}$. Moreover, the output power density of the EDs with undoped GL might markedly exceed the black-body thermal radiation power density at room temperature $P_0^{BB} \simeq 45.9$ mW/cm$^2$.
In the case of GL/MoS$_2$ EDs, $P_0 \sim P_0^{GB}$ as well. At elevated injection current densities, $P$ might exceed $P_0^{GB}$, but the ration $P/P_0^{GB}$ is rather moderate [see Fig. 5(b)], the injection provides either $P/P_{GB}(T_0) < 1$ (if the GL is undoped) or $P/P_{GB}(T_0) \gtrsim 1$ (for doped GLs). This is explained by the negative values of $\mu$ in the case of the undoped GL (the 2DEHP is nondegenerate) and by relatively small values of $\mu /T$ when the GL is doped (weakly degenerate 2DEHP). In both these situations, the rates of the photon emission and absorption are close to their values under the thermal equilibrium. As a result, the output radiation power is comparable with the gray-body thermal radiation.
5. Comments and discussion
5.1 Emission efficiency and its droop
Since the power density receiving by the 2DEHP in the GL by the injection is about $P_{i} = j\Delta _i/e$, the emission efficiency for the GL/b-P ED is $E = P/P_i = eP/j\Delta _i$. Using the data of Fig. 3(a), for the range $j = 300 - 600$ A/cm$^2$, we arrive at the estimate $E \simeq (3-6)\times 10^{-4}$. These values are more than one order of magnitude larger than the emission efficiency of the electrically biased GL supported by hBN and SiO$_2$ [41,42], although it is smaller than that based on a suspended GL [43]. For the suspended GL, the electric power per unit of the GL area substantially exceeds the maximum values predicted our calculations.
The quantum efficiency, $Q = eP/\hbar \omega _{max} j$ (number of the emitted photons per injected electron), is approximately $Q \simeq (3 - 4)\times 10^{-3}$ photon/electron. The quantity $Q$ is relatively small because the characteristic recombination time associated with the optical phonon emission ($\propto \tau _{0}^{inter}$) is shorter than the characteristic time of the radiative recombination ($\propto \tau _r$).
As was pointed out above, the 2DEHP cooling down in the GL/b-P EDs, leads to a gradual diminishing of the GL optical phonon role. As a result, mechanisms, such as the recombination and energy relaxation or the interaction with the GL/substrate (in particular, GL/hBN) surface might become dominant. The inclusion of the interaction with the surface optical phonons with the energy $\hbar \omega _S , \hbar \omega _0$ and characterized by a weaker interaction with the carriers in the GL, prevents a further drop of the effective temperature and the quasi-Fermi energy rise [shown in Figs. 2(a) and 3(a) by the short dashed lines]. This implies that the output power as a function of the injected current should saturate. This might be revealed as the emission efficiency droop [71–73].
The inclusion of other relaxation mechanisms, for example, of the Auger processes might also somewhat modify the ED characteristics affecting the density and energy balance in the 2DEHP [16] (the specifics of these processes in GLs are considered in [74,75]).
5.2 THz emission
Although the position of the emitted power maxima corresponds to the FIR range, the emission in the THz range can also be sufficiently strong to enable many applications.
Let us estimate the power density emitted by the EDs in the THz range, $\omega < \omega _{THz} = 5 - 10$ THz. For the GL/b-P EDs with the undoped GL, this power density [see Eq. (16)] is given by
Since the 2DEHP generated in the GLs due to the electron injection can exhibit the interband population inversion $\mu > 0$ (particularly, in the GL/b-P EDs), the 2DEHP net dynamic conductivity can be negative in a certain photon frequency range (typically on the order of a few and several THz) and the EDs can generate the stimulated emission. Taking into account that the maximum contribution of the interband transitions to the dynamic conductivity is equal to $\sigma _{\hbar \omega }^{inter} = - e^2/4\hbar$, the net dynamic conductivity can be negative if the 2DEHP intraband (Drude) conductivity $\sigma _{\hbar \omega }^{intra} < e^2/4\hbar$. This requires sufficiently perfect GLs with moderate carrier momentum relaxation time (see, for example, [1,11,16]). In principle, both vertical and lateral lasing EDs can be implemented depending on the quality of the radiation reflectors and wave guiding structures. The appearance of the stimulated emission in sufficiently pumped EDs might limit the non-coherent emission considered above and presents an additional mechanism of the ED efficiency droop [68,69].
5.3 Multiple ED structures - current crowding
As shown in the Appendix, the potential drop between the side contacts can limit the injection efficiency. This can be avoided if the spacing, $2L$, between the side contacts is smaller than the characteristic length, $L_{cc}$. The estimates in the Appendix show that $L_{cc} \sim 10~\mu$m, If the radiation power density $P \sim 100$ mW/cm$^2$, the ED width in in the $y$-direction $H = 10^4~\mu$m and $2L = 10~\mu$m, the net emitted power of a single ED is about $2LHP \simeq 0.1~$ mW. To achieve output higher powers $\cal P$, one might use multiple-EDs with periodic structures having the lateral multiple interdigitated fingers as electrodes to the n$^+$ and p$^+$ contacts with the separation on the order on $2L_{cc}$ similar to that shown in Fig. 6. In particular, for ${\cal P} \sim 1$ mW, the pertinent structure should include a dozen of the periods.
5.4 Some assumptions of the model
Variations of the parameter $\eta _{cc}$ [introduced in Eq. (4)] in a wide range does not result in a qualitative change in the dependences shown in Figs. 2 and 3. The trends shown in Figs. 2 and 3 do not change if the sign of $\Delta _i/\hbar \omega _0 - 1$ does not change. In the case of GL/b-P heterojunction, this is true if $\eta _{cc} < 4$. In the opposite case, $\Delta _i - 1 >0$, and the $T$ and $\mu$ dependences on the injection current for the GL/b-P heterojunction [Figs. 2(a) and 3(a)] become similar to those shown in Figs. 2(b) and 3(b)).
The indirect interband radiative emission processes in GL (disregarded above) are generally weaker than the direct processes [70]. However, in the graphene bilayers (GBLs) such processes can be crucial [76]. The difference in the role of the indirect radiative transitions in GLs and GBLs is due to a substantial difference of the density of states near the Dirac point. Nonradiative recombination mechanisms (not included in our model due to their weakness in comparison with the optical phonon recombination) might decrease the efficiency of the EDs under consideration [73].
6. Conclusions
We proposed the FIR and THz EDs based on the GL/b-P and GL/MoS$_2$ heterostructures and evaluated their characteristics. Such EDs can work for efficient emitters of the FIR radiation and be useful room temperature compact sources of the THz radiation. The crucial mechanism providing high emission efficiency of the proposed EDs is cooling of the carriers in the GL due to the interplay of the vertical electron the recombination processes involving optical phonons. Considering the progress in fabrication the devices based on analogous and more complex heterostructures formed by the similar materials (including, inparticular, black-phosphorus) [53–55,80–86], we believe that this paper could actually stimulate the realization of such devices and serve as a guide for their designing.
Appendix. Current crowding
The continuity equation for the current along the GL reads as
Here $\sigma _{GL}$ is the GL conductivity and $j_0 = eN^{+}_{EL}v_T\exp [e(V-V_{bi})/T_0]$ is the injection electron density at $x = \pm L$, $2L$ is the spacing between the side contacts (the GL width).The GL conductivity depends on the carrier densities and, hence, on the quasi-Fermi levels $\mu _e$ and $\mu _h$, as well as on the carrier momentum relaxation mechanisms. At relatively high carrier densities, the short-range scattering on the strongly screened donors and mutual carrier-carrier scattering dominates [77–79]. In this case, the GL conductivity is given by [79]:
where $\xi = [\exp (-\mu _e/T) + 1]^{-1} + [\exp (-\mu _h/T) + 1]^{-1}$ and $\tau _i$ is the characteristic momentum relaxation time on the neutral and screened impurities and defects. Equation (12) demonstrates a specific dependence of the GL conductivity on $\mu _e$ and $\mu _h$ associated with the short-range scattering at high carrier densities, so that the factor $\xi$ in Eq. (19) becomes constant (equal to 2). This implies, a virtual independence of $\sigma _{GL}$ on $\mu _e$ and $\mu _h$ at large their values (i.e., on the doping level and the injection current). A decrease in $\mu _e$ and $\mu _h$, particularly the change of the sign, can lead to a marked drop of the conductivity.The electron injected current density is given by $j =j_S\exp [e(V - V_{bi} +\varphi )/T_0]$ and $j \simeq eb_{EL}N_{EL} (V - V_{bi} +\varphi )/l_{EL}$ at low ($V < V_{bi}$) and elevated ($V < V_{bi}$) bias voltages, respectively (with $j < j_S$). Here $\varphi = \varphi (x)$ yields the electric potential distribution along the GL, $j_S = eN_{EL}^{+} v_T$ is the saturation current density of the electron injector, $v_T$ is the thermal electron velocity, $N_{EL}^+$ and $N_{EL}$ are the donor densities in the n$^+$-EL and n-EL, and $b_{EL}$ is the electron mobility in the latter.
Limiting our consideration to the case of $V > V_{bi}$ when the injection current is close to its maximum, we have $eb_{EL}N_{EL}d\varphi /dx = d j/dx$. Using Eq. (18), we arrive at
At $\mu _e, \mu _h > T$,Setting $N_E= 10^{15}$ cm$^{-3}$, $l_{EL} = 10^{-4}$ cm, $b_{EL} = (250 - 500)$ cm$^2$/V$\cdot$ s, $\tau _i = 10^{-12}$ s, $T_0 = 26$ meV, we obtain $L_{cc } \simeq (13 - 18)~\mu$m. If $\tau _i = 10^{-13}$ s, $L_{cc} \simeq (4 - 6)~\mu$m.
Funding
Office of Naval Research; Army Research Laboratory (Cooperative Research Agreement); Research Institute of Electrical Communication, Tohoku University (H31/A01); Russian Foundation of Fundamental Research (19-07-00683A); Japan Society for the Promotion of Science (KAKENHI #16H06361).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamic conductivity of graphene with optical pumping,” J. Appl. Phys. 101(8), 083114 (2007). [CrossRef]
2. V. Ryzhii, M. Ryzhii, A. Satou, T. Otsuji, A. A. Dubinov, and V. Y. Aleshkin, “Feasibility of terahertz lasing in optically pumped epitaxial multiple graphene layer structures,” J. Appl. Phys. 106(8), 084507 (2009). [CrossRef]
3. T. Otsuji, S. B. Tombet, A. Satou, M. Ryzhii, and V. Ryzhii, “Terahertz wave generation using graphene: toward new types of terahertz lasers,” IEEE J. Sel. Top. Quantum Electron. 19(1), 8400209 (2013). [CrossRef]
4. F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4(9), 611–622 (2010). [CrossRef]
5. V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Terahertz and infrared photodetection using p-i-n multiple-graphene layer structures,” J. Appl. Phys. 107(5), 054512 (2010). [CrossRef]
6. V. Ryzhii, N. Ryabova, M. Ryzhii, N. V. Baryshnikov, V. E. Karasik, V. Mitin, and T. Otsuji, “Terahertz and infrared photodetectors based on multiple graphene layer and nanoribbon structures,” Opto-Electron. Rev. 20(1), 15–25 (2012). [CrossRef]
7. Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012). [CrossRef]
8. A. Tredicucci and M. S. Vitiello, “Device concepts for graphene-based terahertz photonics,” IEEE J. Sel. Top. Quantum Electron. 20(1), 130–138 (2014). [CrossRef]
9. F. H. L. Koppens, T. Mueller, P. Avouris, A. C. Ferrari, M. S. Vitiello, and M. Polini, “Photodetectors based on graphene, other two-dimensional materials and hybrid systems,” Nat. Nanotechnol. 9(10), 780–793 (2014). [CrossRef]
10. M. Ryzhii and V. Ryzhii, “Injection and population inversion in electrically induced p–n junction in graphene with split gates,” Jpn. J. Appl. Phys. 46(No. 8), L151–L153 (2007). [CrossRef]
11. V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110(9), 094503 (2011). [CrossRef]
12. A. A. Dubinov, A. Bylinkin, V. Ya. Aleshkin, V. Ryzhii, T. Otsuji, and D. Svintsov, “Ultra-compact injection terahertz laser using the resonant inter-layer radiative transitions in multi-graphene-later structures,” Opt. Express 24(26), 29603 (2016). [CrossRef]
13. D. Yadav, S. Boubanga Tombet, T. Watanabe, S. Arnold, V. Ryzhii, and T. Otsuji, “Terahertz wave generation and detection in double-graphene layered van der Waals heterostructures,” 2D Mater. 3(4), 045009 (2016). [CrossRef]
14. D. Yadav, G. Tamamushi, T. Watanabe, J. Mitsushio, Y. Tobah, K. Sugawara, A. A. Dubinov, A. Satou, M. Ryzhii, V Ryzhii, and T. Otsuji, “Terahertz light-emitting graphene-channel transistor toward single-mode lasing,” Nanophotonics 7(4), 741–752 (2018). [CrossRef]
15. V. Ryzhii, I. Semenikhin, M. Ryzhii, D. Svintsov, V. Vyurkov, A. Satou, and T. Otsuji, “Double injection in graphene p-i-n structures,” J. Appl. Phys. 113(24), 244505 (2013). [CrossRef]
16. V. Ryzhii, T. Otsuji, M. Ryzhii, V. E. Karasik, and M. S. Shur, “Negative terahertz conductivity and amplification of surface plasmons in graphene–black phosphorus injection laser heterostructures,” Phys. Rev. B 100(11), 115436 (2019). [CrossRef]
17. V. Ryzhii, M. Ryzhii, V. Mitin, A. Satou, and T. Otsuji, “Effect of heating and cooling of photogenerated electron-hole plasma in optically pumped graphene on population inversion,” Jpn. J. Appl. Phys. 50(9), 094001 (2011). [CrossRef]
18. M. Yu. Morozov, V. V. Popov, M. Ryzhii, V. G. Leiman, V. Mitin, M. S. Shur, T. Otsuji, and V. Ryzhii, “Optical pumping through a black-As absorbing-cooling layer in graphene-based heterostructure: thermo-diffusion model,” Opt. Mater. Express 9(10), 4061–4069 (2019). [CrossRef]
19. Y. Cai, G. Zhang, and Y.-W. Zhang, “Layer-dependent band alignment and work function of few-layer phosphorene,” Sci. Rep. 4(1), 6677 (2015). [CrossRef]
20. F. Xia, H. Wang, and Y. Jia, “Rediscovering black phosphorous as an anisotropic layered material for optoelectronics and electronics,” Nat. Commun. 5(1), 4458 (2014). [CrossRef]
21. Ling Xi, “The renaissance of black phosphorus,” Proc. Natl. Acad. Sci. U. S. A. 112(15), 4523–4530 (2015). [CrossRef]
22. R. W. Keyes, “The electrical properties of black phosphorous,” Phys. Rev. 92(3), 580–584 (1953). [CrossRef]
23. H. Asahina and A. Morita, “Band structure and optical properties of black phosphorus,” J. Phys. C: Solid State Phys. 17(11), 1839–1852 (1984). [CrossRef]
24. A. Morita, “Semiconducting black phosphorus,” Appl. Phys. A 39(4), 227–242 (1986). [CrossRef]
25. M. Batmunkh, M. Bat-Erdene, and J. G. Shapter, “Phosphorene and phosphorene-based materials - prospects for future applications,” Adv. Mater. 28(39), 8586–8617 (2016). [CrossRef]
26. M. Engel, M. Steiner, and P. Avouris, “A black phosphorus photo-detector for multispectral high-resolution imaging,” Nano Lett. 14(11), 6414–6417 (2014). [CrossRef]
27. Y. Deng, Z. Luo, N. J. Conrad, H. Liu, Y. Gong, S. Najmaei, P. M. Ajayan, J. Lou, X. Xu, and P. D. Ye, “Black phosphorus-monolayer MoS2 van der Waals heterojunction p-n diode,” ACS Nano 8(8), 8292–8299 (2014). [CrossRef]
28. A. Castellanos-Gomez, “Black phosphorus: narrow gap, wide applications,” J. Phys. Chem. Lett. 6(21), 4280–4291 (2015). [CrossRef]
29. F. Ahmed, Y. D. Kim, M. S. Choi, X. Liu, D. Qu, Z. Yang, J. Hu, I. P. Herman, J. Hone, and W. J. Yoo, “High electric field carrier transport and power dissipation in multilayer black phosphorus field effect transistor with dielectric engineering,” Adv. Funct. Mater. 27(4), 1604025 (2017). [CrossRef]
30. K. Kim, S. Larentis, B. Fallahazad, K. Lee, J. Xue, D. C. Dillen, C. M. Corbet, and E. Tutuc, “Band alignment in WSe2–Graphene heterostructures,” ACS Nano 9(4), 4527–4532 (2015). [CrossRef]
31. E. Leong, R. J. Suess, A. B. Sushkov, H. D. Drew, T. E. Murphy, and M. Mittendorff, “Terahertz photoresponse of black phosphorus,” Opt. Express 25(11), 12666–12674 (2017). [CrossRef]
32. Y. Deng, Z. Luo, N. J. Conrad, H. Liu, Y. Gong, S. Najmaei, P. M. Ajayan, J. Lou, X. Xu, and P. D. Ye, “Black phosphorus-monolayer MoS2 van der Waals heterojunction p-n diode,” ACS Nano 8(8), 8292–8299 (2014). [CrossRef]
33. Tian He, “Novel field-effect schottky barrier transistors based on Graphene-MoS2 heterojunctions,” Sci. Rep. 4(1), 5951 (2015). [CrossRef]
34. V. Ryzhii, M. Ryzhii, T. Otsuji, V. E. Karasil, V. Leiman, V. Mitin, and M. S. Shur, “Multiple graphene-layer-based heterostructures with van der Waals barrier layers for terahertz superluminescent and laser diodes with lateral/vertical injection,” Semicond. Sci. Technol. 35 (2018) (accepted May 15, 2020, doi: 10.1088/1361-6641/ab9398).
35. A. Kuruvila, P. R. Kidambi, J. Kling, J. B. Wagner, J. Robertson, S. Hofmann, and J. Meyer, “Organic light emitting diodes with environmentally and thermally stable doped graphene electrodes,” J. Mater. Chem. C 2(34), 6940–6945 (2014). [CrossRef]
36. A. A. Dubinov, V. Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-Perot resonator,” Appl. Phys. Express 2(9), 092301 (2009). [CrossRef]
37. F. Withers, O. Del Pozo-Zamudio, A. Mishchenko, A. P. Rooney, A. Gholinia, K. Watanabe, T. Taniguchi, S. J. Haigh, A. K. Geim, A. I. Tartakovskii, and K. S. Novoselov, “Light-emitting diodes by bandstructure engineering in van der Waals heterostructures,” Nat. Mater. 14(3), 301–306 (2015). [CrossRef]
38. G. Jiang, He Tian, X.-F. Wang, T. Hirtz, F. Wu, Y.-C. Quao, G.-Y. Gou, Yu-H. Wei, J.-M. Yang, S. Yang, Yi Yang, and T.-L. Ren, “An efficient flexible graphene-based light-emitting device,” Nanoscale Adv. 1(12), 4745–4754 (2019). [CrossRef]
39. J. Zheng, H. Y. Hoh, Y. Zhang, and Q. Bao, “Graphene-based light-emitting diodes,” in “Graphene Photonics, Optoelectronics, and Plasmonics”, Q. Bao, H. Hoh, and Y. Zhang eds., Chapter 9 (Jenny Stanford Publ., New York, 2017).
40. C. Palacios-Berraquero, M. Barbone, D. M. Kara, X. Chen, I. Goykhman, D. Yoon, A. K. Ott, J. Beitner, K. Watanabe, T. Taniguchi, A. C. Ferrari, and M. Atatüre, “Atomically thin quantum light-emitting diodes,” Nat. Commun. 7(1), 12978 (2016). [CrossRef]
41. S.-K. Son, M. Šiškins, C. Mullan, J. Yin, V. Kravets, A. Kozikov, S. Ozdemir, M. Alhazmi, M. Holwill, K. Watanabe, T. Tanigichi, D. Charazaryan, K. S. Novoselov, V. I. Falko, and A. Mishchenko, “Graphene hot-electron light bulb: Incandescence from hBN-encapsulated graphene in air,” 2D Mater. 5(1), 011006 (2017). [CrossRef]
42. M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos, and P. Avouris, “Thermal infrared emission from biased graphene,” Nat. Nanotechnol. 5(7), 497–501 (2010). [CrossRef]
43. Y. D. Kim, H. Kim, Y. Cho, J. H. Ryoo, C.-H. Park, P. Kim, Y. S. Kim, S. Lee, Y. Li, S.-N. Park, Y. S. Yoo, D. Yoon, V. E. Dorgan, E. Pop, T. F. Heinz, J. Hone, S.-H. Chun, H. Cheong, S. W. Lee, M.-H. Bae, and Y. D. Park, “Bright visible light emission from graphene,” Nat. Nanotechnol. 10(8), 676–681 (2015). [CrossRef]
44. H. M. Dong, W. Xu, and F. M. Peeters, “Electrical generation of terahertz blackbody radiation from graphene,” Opt. Express 26(19), 24621–24626 (2018). [CrossRef]
45. R.-J. Shiue, Y. Gao, C. Tan, C. Peng, J. Zheng, D. K. Efetov, Y. D. Kim, J. Hone, and D. Englund, “Thermal radiation control from hot graphene electrons coupled to a photonic crystal nanocavity,” Nat. Commun. 10(1), 109 (2019). [CrossRef]
46. I. V. Oladyshkin, S. B. Bodrov, Y. A. Sergeev, A. I. Korytin, M. D. Tokman, and A. N. Stepanov, “Optical emission of graphene and electron-hole pair production induced by a strong terahertz field,” Phys. Rev. B 96(15), 155401 (2017). [CrossRef]
47. A. Blicher, “Field-effect and Bipolar Power Transistor Physics,” (Academ Press, New York, 1986), p.104.
48. L. Wang, Z.-H. Zhang, and N. Wang, “Current crowding phenomenon: Theoretical and direct correlation with the efficiency droop of light emitting diodes by a modified ABC model,” IEEE J. Quantum Electron. 51(5), 1–9 (2015). [CrossRef]
49. A. Yazgan, L. Jofre, and J. Romeu, “The state of art of terahertz sources: A communication perspective at a glance,” in 2017 40th International Conf. on Telecommunications and Signal Processing (TSP), 2017: IEEE, pp. 81–816.
50. T. Otsuji and M. Shur, “Terahertz plasmonics: Good results and great expectations,” IEEE Microwave 15(7), 43–50 (2014). [CrossRef]
51. M. Y. Morozov, V. G. Leiman, V. V. Popov, V. Mitin, M. S. Shur, V. E. Karasik, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Optical pumping in graphene-based terahertz/far-infrared superluminescent and laser heterostructures with graded-gap black-PxAs1x absorbing-cooling layers,” Opt. Eng. 59(6), 061606 (2019).
52. V. Ryzhii, T. Otsuji, M. Ryzhii, D. S. Ponomarev, V. E. Karasik, V. G. Leiman, V. Mitin, and M. S. Shur, “Electrical modulation of terahertz radiation using graphene-phosphorene heterostructures,” Semicond. Sci. Technol. 33(12), 124010 (2018). [CrossRef]
53. J. Kang, D. Jariwala, C. R. Ryder, S. A. Wells, Y. Choi, E. Hwang, J. Ho Cho, T. J. Marks, and M. C. Hersam, “Probing out-of-plane charge transport in black phosphorus with graphene-contacted vertical field-effect transistors,” Nano Lett. 16(4), 2580–2585 (2016). [CrossRef]
54. Y. Liu, B. N. Shivananju, Y. Wang, Y. Zhang, W. Yu, S. Xiao, T. Sun, W. Ma, H. Mu, S. Lin, H. Zhang, Y. Lu, C.-W. Qiu, S. Li, and Q. Bao, “Highly efficient and air-stable infrared photodetector based on 2D layered graphene-black phosphorus heterostructure,” ACS Appl. Mater. Interfaces 9(41), 36137–36145 (2017). [CrossRef]
55. T. Wu, Yu Ma, Z. Qu, J. Fan, Q. Li, P. Shi, Q. Xu, and Y. Min, “Black phosphorus-graphene heterostructure-supported Pd nanoparticles with superior activity and stability for ethanol electro-oxidation,” ACS Appl. Mater. Interfaces 11(5), 5136–5145 (2019). [CrossRef]
56. F. Rana, P. A. George, J. H. Strait, S. Sharavaraman, M. Charasheyhar, and M. G. Spencer, “Carrier recombination and generation rates for intravalley and intervalley phonon scattering in graphene,” Phys. Rev. B 79(11), 115447 (2009). [CrossRef]
57. V. Ya. Aleshkin, A. A. Dubinov, M. Ryzhii, V. Ryzhii, and T. Otsuji, “Electron capture in van der Waals graphene-based heterostructures with WS2 barrier layers,” J. Phys. Soc. Jpn. 84(9), 094703 (2015). [CrossRef]
58. E. Rosencher, B. Vinter, F. Luc, L. Thibaudeau, and J. Nagle, “Emission and capture of electrons in multiquantum-well structures,” IEEE J. Quantum Electron. 30(12), 2875–2888 (1994). [CrossRef]
59. D. Golla, A. Brasington, B. J. LeRoy, and A. Sandhu, “Ultrafast relaxation of hot phonons in graphene-hBN heterostructures,” APL Mater. 5(5), 056101 (2017). [CrossRef]
60. A. A. Balandin, “Thermal properties of graphene and nanostructured carbon materials,” Nat. Mater. 10(8), 569–581 (2011). [CrossRef]
61. E. Pop, V. Varsney, and A. K. Roy, “Thermal properties of graphene: Fundamentals and applications,” MRS Bull. 37(12), 1273–1281 (2012). [CrossRef]
62. V. Ryzhii, A. Satou, and T. Otsuji, “Plasma waves in two-dimensional electron-hole system in gated graphene heterostructures,” J. Appl. Phys. 101(2), 024509 (2007). [CrossRef]
63. A. A. Dubinov, V. Ya. Aleshkin, V. Mitin, T. Otsuji, and V. Ryzhii, “Terahertz surface plasmons in optically pumped graphene structures,” J. Phys.: Condens. Matter 23(14), 145302 (2011). [CrossRef]
64. T. Otsuji, V. Popov, and V. Ryzhii, “Active graphene plasmonics for terahertz device applications,” J. Phys. D: Appl. Phys. 47(9), 094006 (2014). [CrossRef]
65. F. T. Vasko and V. Ryzhii, “Photoconductivity of intrinsic graphene,” Phys. Rev. B 77(19), 195433 (2008). [CrossRef]
66. A. Satou, F. T. Vasko, and V. Ryzhii, “Nonequilibrium carriers in intrinsic graphrene under interband photoexcitation,” Phys. Rev. B 78(11), 115431 (2008). [CrossRef]
67. F. Stern and J. M. Woodall, “Photon recycling in semiconductor lasers,” J. Appl. Phys. 45(9), 3904–3906 (1974). [CrossRef]
68. T. Baba, R. Watanabe, K. Asano, F. Koyama, and K. Iga, “Theoretical and experimental estimations of photon recycling effect in light emitting devices with a metal mirror,” Jpn. J. Appl. Phys. 35(Part 1, No. 1A), 97–100 (1996). [CrossRef]
69. N. Tsutsui, I. Khmyrova, V. Ryzhii, and T. Ikegami, “Analysis of photon recycling in light emitting diodes with nonuniform injection,” J. Appl. Phys. 88(6), 3613–3617 (2000). [CrossRef]
70. D. Svintsov, V. Ryzhii, and T. Otsuji, “Negative dynamic Drude conductivity in pumped graphene,” Appl. Phys. Express 7(11), 115101 (2014). [CrossRef]
71. J. Mickevičius, J. Jurkevičius, M. S. Shur, J. Yang, R. Gaska, and G. Tamulaitis, “Photoluminescence efficiency droop and stimulated recombination in GaN epilayers,” Opt. Express 20(23), 25195–25200 (2012). [CrossRef]
72. A. A. Efremov, N. I. Bochkareva, R. I. Gorbunov, D. A. Larinovich, Y. T. Rebane, D. V. Tarkhin, and Y. G. Shreter, “Effect of the joule heating on the quantum efficiency and choice of thermal conditions for high-power blue InGaN/GaN LEDs,” Semiconductors 40(5), 605–610 (2006). [CrossRef]
73. A. Žukauskas, M. S. Shur, and R. Gaska, “Introduction to Solid State Lighting,” (John Wiley and Sons, 2002).
74. M. S. Foster and I. L. Aleiner, “Slow imbalance relaxation and thermoelectric transport in graphene,” Phys. Rev. B 79(8), 085415 (2009). [CrossRef]
75. G. Alymov, V. Vyurkov, V. Ryzhii, A. Satou, and D. Svintsov, “Auger recombination in Dirac materials: A tangle of many-body effects,” Phys. Rev. B 97(20), 205411 (2018). [CrossRef]
76. D. Svintsov, T. Otsuji, V. Mitin, M. S. Shur, and V. Ryzhii, “Negative terahertz conductivity in disordered graphene bilayers with population inversion,” Appl. Phys. Lett. 106(11), 113501 (2015). [CrossRef]
77. V. Vyurkov and V. Ryzhii, “Effect of Coulomb scattering on graphene conductivity,” JETP Lett. 88(5), 322–325 (2008). [CrossRef]
78. D. Svintsov, V. Ryzhii, A. Satou, T. Otsuji, and V. Vyurkov, “Carrier-carrier scattering and negative dynamic conductivity in pumped graphene,” Opt. Express 22(17), 19873–19886 (2014). [CrossRef]
79. V. Ryzhii, D. S. Ponomarev, M. Ryzhii, V. Mitin, M. S. Shur, and T. Otsuji, “Negative and positive terahertz and infrared photoconductivity in uncooled graphene,” Opt. Mater. Express 9(2), 585–597 (2019). [CrossRef]
80. S.-J. He, D.-K. Wang, Z.-X. Yang, J.-X. Man, and Z.-H. Lu, “Integrated tandem device with photoactive layer for near-infrared to visible upconversion imaging,” Phys. Lett. 112(24), 243301 (2018). [CrossRef]
81. A. Gao, Z. Zhang, L. Li, B. Zheng, C. Wang, Y. Wang, T. Cao, Y. Wang, S.-J. Liang, F. Miao, Y. Shi, and X. Wang, “Robust impact-ionization field-effect transistor based on nanoscale vertical graphene/black phosphorus/indium selenide heterostructures,” ACS Nano 14(1), 434–441 (2020). [CrossRef]
82. T. Sakthivela, X. Huanga, Y. Wuc, and S. Rtimi, “Recent progress in blackphosphorus nano structures as environmental photocatalysts,” Chem. Eng. J. 379, 122297 (2020). [CrossRef]
83. F. Liu, C. Huang, C.-X. Liu, R. Shi, and Y. Chen, “Black phosphorus-based semiconductor heterojunctions for photocatalytic water splitting,” Chem. - Eur. J. 26(20), 4449–4460 (2020). [CrossRef]
84. L. Zhang, B. Wang, Y. Zhou, C. Wang, X. Chen, and H. Zhang, “Synthesis techniques, optoelectronic properties, and broadband photodetection of thin-film black phosphorus,” Adv. Optical Mater. 202000045 (2020).
85. Y. Wakafuji, R. Moriya, S. Masubuchi, K. Watanabe, T. Taniguchi, and T. Machida, “3D manipulation of 2D materials using microdome polymer,” Nano Lett. 20(4), 2486–2492 (2020). [CrossRef]
86. S. Masubuchi, M. Morimoto, S. Morikawa, M. Onodera, Y. Asakawa, K. Watanabe, T. Taniguchi, and T. Machida, “Autonomous robotic searching and assembly of two-dimensional crystals to build van der Waals superlattices,” Nat. Commun. 9(1), 1413 (2018). [CrossRef]
