1. Introduction

Heterostructures with graphene layers (GLs) are promising building blocks for infrared and terahertz photodetectors [1–14], optical modulators [15–18], plasmonic and frequency multiplication devices [19–28], and lasers and light-emitting diodes [29–44] (including those based on hybrid GL/black phosphorous devices [45,46]). The realization of the on-chip monolithic nanoscale relatively simple light sources for high-bandwidth inter- and intra-chip connections is still a challenging problem [47]. There are several proposals and realizations of the compact and simple GL thermal light sources using the electrical carrier heating in the GLs [48–55]. Such GL-base thermal light sources can be fairly effective and very fast [54]. The operation and, in particular, the operation speed of the GL-based thermal sources are determined by complex relaxation and recombination mechanisms, namely, of the carrier-carrier scattering, the interaction of the hot carriers in the GL with the GL optical phonons and the interface optical phonons and the optical phonons in the media surrounding the GL. The carrier heating in the GLs in the devices in question can be associated with the Joule heating by the electric current flowing in the GL between the side contacts to the GL [48–55]. Another option is to use the combined carrier injection (the injection of relatively cold carriers from the side contact(s) and the vertical injection of the not too hot carriers from the top contact). This concept was recently applied to the GL heterostructures [44–46] aiming to realize the interband population inversion in the GLs for its use for far-infrared and terahertz lasing or superluminescence (the combined carrier injection is widely used in the standard vertical-cavity surface emitting heterostructure lasers). In such heterostructure devices the vertical injection should not lead to a marked heating of the two-dimensional electron-hole plasma (2DEHP) in the GL. Due to this, the black phosphorus (BP) (or similar materials) having relatively small band off-sets at the heterointerface [56,57] were chosen for the emitter contact layer. In contrast, the thermal electrically driven light sources (IDLEs) require strong heating of the 2DEHP in the GLs. The thermal IDLEs, in particularly those based on the GLs encapsulated by the hBNLs, exhibit the following features. First of all, both the optical phonon and acoustic phonon systems can be relatively hot (their effective temperatures substantially exceed the temperature of the thermostat) when the carrires are hot. Second, the Auger recombination-generation processes [58] can be much more effective than those at moderate or low carrier temperatures.

This is why, the use of the emitter contact layer made of the black phosphorus (BP) (or similar materials) having relatively small band off-sets at the GL/black phosphorus [56,57] interface was proposed. However, in contrast, in the injection driven thermal light sources (IDLEs) a strong heating of the 2DEHP in the GLs is desirable. These thermal light sources, in particular those based on the GLs encapsulated by the IDLEs, exhibit the following features. First of all, at sufficiently hot carrier, both the optical phonon and acoustic phonon systems can also be relatively hot (their effective temperatures substantially exceed the temperature of the thermostat). Second, the Auger recombination-generation processes [58] can be much more effective than at moderate or low carrier temperatures.

In this paper, we study the thermal IDLEs based on the GL/hBN heterostructures with the combined lateral/vertical carrier injection providing an effective heating due to rather large the GL/hBN band off-sets. We focus on the dynamic properties of these sources, which can be used in different opto-electron systems. In particular, we compare the characteristics of the IDLE in question with the similar IDLE using the vertical double injection and the Joule heating and demonstrate that the IDLEs with the combined and double vertical injection can be faster due to a higher sensitivity to the variations of the controlling voltage.

The paper is organized as follows. In Section 2, we present the proposed IDLE device structure and formulate the pertinent mathematical model. In Section 3, we use this model for the calculations of the steady-state effective temperatures of the carrier, optical phonon, and lattice (acoustic phonon) systems as functions of the injected carrier current and the structural parameters. Apart from the numerical solution of the equations of the model, we analytically analyze their asymptotic behavior in the limiting cases. Section 4 deals with the calculations of the spectrum of the light emitted by the IDLEs and the output power at the steady-state carrier injection. For this, we use the data for the effective temperatures obtained in Section 3. In Section 5, we analyze the modulation of the output light by the ac injection current characteristics and evaluate the IDLE maximum modulation frequency. Section 6 is devoted to the derivation of the light modulation depth as a function of the modulation ac voltage. In Sections 7 and 8, we comment the obtained results and formulate the conclusions. Some mathematical details are singled out to the Appendix.

2. Device structures and mathematical model

Figure 1 shows the schematic views of the IDLEs based on the hBNL/GL/hBNL heterostructures with the combined (vertical-lateral) and double vertical injection. The GLs in the IDLE with the combined injection are supplied by the side contacts providing the electron injection to the GLs. The heterostructure top comprises an undoped or lightly doped hBN layer and a cap, which is heavily doped by acceptors. Such a p$^+$-p- hBNL region serves as an injector of holes into the GL. Here we focus on the IDLE with the vertical hole injection and the lateral electron injection [see Figs. 1(a) and 1(c)]. The obtained results can be easily extended for the IDLEs with the combined injection can be based on the heterostructures n$^+$-n-hBN vertical injector and the lateral hole injection and the IDLEs with the vertical double injection (see below). The IDLEs under consideration can comprise the multiple- and single GLs. The device structure can be placed on a SiO$_2$ substrate.

 

Fig. 1. Schematic views of the IDLE based on heterostructures (a) with the top p$^+$-p-hBN vertical hole injector and the GLs separated by the hBNLs and supplied with the n$^+$ side contacts and (b) with the top p$^+$-p-hBN hole and bottom n$^+$-n-hBN vertical injectors and the band diagrams of (c) the IDLE with vertical-lateral injection and (d) vertical double injection under applied voltage $U$. Opaque and open circles correspond to electrons and holes, respectively. Arrows show the carrier movement directions.

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Our mathematical model (which is the generalized version of those used previously [44–46], see also more early works [31,59]) corresponds to the following scenario:

(i) The injected hot holes transfer their energy partially to different optical phonon modes (intra-valley and inter-valley phonons in the GL and the interface phonons. (emitting the cascades of these phonons) and partially to the 2DEHP via the injected hole collisions with the 2DEHP electrons and holes;

(ii) The 2DEHP heated by the injected holes transfer the excessive energy to the optical phonon systems [60];

(iii) The optical phonon system transfers the energy to the acoustic phonons (lattice) due to the anharmonic optical and interface phonons decay [61–65];

(iv) The hot acoustic phonons bring their thermal energy primarily [55] to the top or bottom contact serving as the heat sink;

(v) The deviation of the electron and hole densities from the equilibrium ones is controlled by the interband recombination-generation processes involving the optical phonons [31,59,60] and the Auger processes [58,66].

It is assumed that the carrier-carrier interactions in the 2DEHP are sufficiently strong, so that both the electron and hole components have the common effective temperature $T$, which is generally different from the effective optical phonon temperature $\Theta$, the acoustic phonon temperature $T_L$ [the lattice temperature around the GL(s)], and the sink (vertical contact, thermostat) temperature $T_{C}$. Due to the latter, the electron and hole energy distribution functions are $f_e(\varepsilon _e) = \{1 + \exp [(\varepsilon _e - \mu _e)]/T\}^{-1}$ and $f_h(\varepsilon _h) = \{1 + \exp [(\varepsilon _h - \mu _h)]/T\}^{-1}$ with $\varepsilon _e$, $\varepsilon _h$, $\mu _e$, and $\mu _h$ being the pertinent kinetic carrier energies and quasi-Fermi energies counted from the GL Dirac point. This assumption is particularly reasonable for the IDLEs due to their operation at elevated carrier densities. For simplicity, we do not distinguish different optical modes assuming that the optical phonon system is characterized by a single effective phonon energy $\hbar \omega _O$, which accounts for the intra-valley (with the energy $\sim$ 200 meV), inter-valley ($\sim$ 165 meV), and interface ($\sim$ 100 meV) optical phonons.

Considering the above-mentioned processes, namely the intraband and interband emission and absorption of optical phonons [62], the decay of these phonons into the acoustic phonons, and the Auger recombination-generation processes as the main mechanisms determining the carrier density and energy balances in the 2DEHP, the main equations of the model can be presented in the following form:

Here $\Sigma$ and ${\cal E}$ are the 2DEHP density (the net density of the electrons and holes in the GL) and their energy, respectively, $j$ is the density of the vertically injected current into a GL (in the cases of the IDLEs with a single GL and multiple GLs, $j =j_i$ and $j = j_i/n$, respectively, where $j_i$ is the density of the net injected current and $n$ is the number of the GLs [44]), $\overline \Delta _i$ is the energy transferred by one vertically injected holes directly to the 2DEHP (taking into account that a fraction of the injected hole energy $\Delta _i$ goes to the optical phonons), and $e =|e|$ is the electron charge. The terms $R_{0}^{inter}$, $R_{0}^{intra}$, and $R_A$ are the rates of the interband and intraband transitions associated with the pertinent optical phonons and the Auger recombination-generation processes; their explicit forms are presented in the Appendix. The average energy brought to the 2DEHP by an injected hole is estimated as [58]

Here $\Delta _i =\Delta _V + 3T_{0}/2$, $T_0$ is the hole temperature in the p$^+$ injector, coinciding with the metal contact temperature (which serves as the heat sink). Equation (3) shows that each injected hole energy (with the average injected kinetic energy $\varepsilon _i = \Delta _V + 3T_{0}/2$) transferred directly to the 2DEHP is smaller than $\varepsilon _i$ because a portion, $k_0\hbar \omega _0$, of this energy goes immediately to the optical phonons. If we disregard the energy dependence of the time, $\tau _0$, of the optical phonon emission, the average numbers of the optical phonons of the given sort generated by the injected hole is equal to $k_{0} = \displaystyle \frac {1}{\tau _0/\tau _{cc}}\left[1 - \frac {1}{(1 +\tau _0/\tau _{cc} )^{K_{0}}}\right ]$, where $K_0$ is the maximum numbers of the optical phonons in the cascade emitted by the hole after its injection (see the Appendix). Since the density of states in GLs virtually linearly increases with the carrier energy, one might assume that the optical phonon intraband emission decreases with the energy. In this case, $k_{0} = K_{0}/(1 + \tau _0/\tau _{cc})$. Setting $\Delta _V = 1200$ meV, $T_{0} = 25$ meV, $K_0 = 8$, one obtains $\Delta _i \simeq 1240$ meV and $\overline {\Delta _i} \simeq 1090 - 1210$ meV (i.e., for the above two cases, $\Delta _i \lesssim \Delta _V$).

Considering the deviation optical phonons distribution function ${\cal N}_{0} = [\exp (\hbar \omega _{0}/\Theta ) - 1]^{-1}$ from the distribution function of optical phonons, ${\cal N}_{0}^{eq} = [\exp (\hbar \omega _{0}/T_L) - 1]^{-1}$, in equilibrium with the acoustic phonons in the vicinity of the GL (where $T_L$ is the lattice temperature or the effective temperature of acoustic phonons in the vicinity of the GL), for ${\cal N}_{0}$ we obtain

Here $\overline {\Sigma _{0}} = (1/2\pi )(\hbar \omega _0/\hbar \,v_W)^2$ is the characteristic carrier density determined by the energy dependence of the density of state in the GL (see the Appendix), $\hbar$ is the Planck constant, and $v_W \simeq 10^8$ cm/s is the electron and hole velocity in GLs. The quantity $\overline \Sigma _0/\tau _0$ is estimated using the data on the carrier interband generation-recombination rate [59] and $\tau _D$ is the time of the optical phonon decay due to the anharmonic lattice processes at $T_L = T_0$. Theoretical and experimental studies for $\tau _D$ in GLs at room lattice temperature [60–63] yield the values in the range from 1 to 5 ps. However, this time can be much shorter: $\tau _D \simeq 0.20-0.35$ ps in the GLs encapsulated by the hBNLs due to the role of the interface optical phonons. The factor $T_L/T_0$ reflects an increase in the optical phonon decay rate (a decrease in the decay time) with rising lattice temperature (the effective temperature of acoustic phonons) [67]. This temperature dependence arises from the dependence of the anharmonic decay process on the "daughter" acoustic phonon modes population [67,68]. The lattice temperature TL is determined by the power injected into the GL and by the thermal conductivity the per unit area, C, of the layers surrounding the GL, i.e., the hBNLs and the metal contact. Neglecting the lateral heat transfer to the side contacts [48–50], the lattice temperature, $T_L$, around the GL can be determined from the following equation:

Here $\tau _L$ is the characteristic time of the lattice heating and cooling down that is proportional to the lattice heat capacitance $c_L$.

The efficiency of the 2DEHP, optical phonon systems, and the lattice temperature crucially depends on the thermal conductivity $C$, i.e., on the the hBNLs thermal conductance of $c$ ($C \propto c$). If the metal contact really serves as the heat sink, i.e., if the contact temperature is close to room temperature, considering the hBNLs thermal conductance of $c \simeq 20$W/m$\cdot$ K [69], for the hBNLs of the thickness $L_{hBN} = (1 - 2)~\mu$m, we obtain $C =c/L_{hBN} \simeq (1 - 2)$ kW/cm$^2$K. However, when the cooling down of the top metal contact is limited by the heat transfer to the surrounding air, the quantity $C$ can be several orders of magnitude smaller.

3. Steady-state effective temperatures

3.1 Numerical analysis

Consider first the case of the dc injection current $j = j_0 = const$. At $j_0 = 0$, Eq. (4)–(6) naturally yield ${\cal N}_{0} = {\cal N}_{0}^{eq}$, i.e., $\Theta = T_L =T_{0}$, and $\mu _e + \mu _h = 0$. Using the steady-stater versions of Eq. (1) and (2) (see the Appendix), we obtain

Here $i_0 = j_0/\overline {j}$, where $\overline {j} = e\overline {\Sigma _0}/\tau _0 = eG_0\exp {(\hbar \omega _0/T)}$, $\eta _{0} \simeq (\hbar \omega _{0}/\pi \,T)^2$ [59]. The quantity $\eta _{0}$ characterizes the relative contributions to the 2DEHP energy relaxation due to the interband and intraband transitions mediated by the optical phonons. In the case of rather hot carriers, the intraband processes involve wide energy ranges, so that $\eta _{0}$ can be close to unity. The characteristic injection current density $\overline {j}$, determined by the parameters of carrier scattering on different optical phonon modes, can be treated as a main fitting parameter of the model under consideration. For definiteness, in the following assuming that the rate of the electron-hole pairs interband generation due to the absorption of the thermal optical phonons at $T_0 = 25$ meV and the characteristic ("average") photon energy are equal to $G_0 = 2\times 10^{21}$ cm$^{-2}$s$^{-1}$ and $\hbar \omega _0 = 150$ meV, respectively (see the Appendix). We also set $\overline {\Sigma _0}/\tau _0 = 2\times 10^{24}$ cm$^{-2}$s$^{-1}$, so that $\tau _0 \simeq 0.42$ ps and $\overline j \simeq 320$ kA/cm$^{2}$.

Due to a large ratio $\Delta _i/\hbar \omega _0$ and a smallness of $\tau _A/\tau _0$ in the situations under consideration, the right-hand side of Eq. (A6) can be simplified, so that from Eq. (6) and (7) we obtain

Accounting for the explicit expression for the terms in its right-hand side (see the Appendix) Eq. (4) yields

Here $T_L$ obeys the following equation [see Eq. (5)]:

Equations (8)–(11) allow to find the the effective carrier and optical phonon temperatures, $T$ and $\Theta$, as well as the lattice temperature $T_L$ near the GL as functions of the dc injected current density for the IDLEs with different structural parameters.

Figure 2 shows the dependences of the effective temperatures of the carriers $T$, optical phonons, and $\Theta$ on the normalized injected current density $i_0$ (i.e., actually on the injected power density $\Delta _i\,j_0/e$) for different values of $\tau _D/\tau _0$ calculated using Eqs. (8)–(12). It is assumed that $T_{0} = 25$ meV, $\hbar \omega _0 = 150$ meV, $\Delta _i = 1240$ meV, $\overline {\Delta _i} = 1210$ meV, $C = 1$ kW/cm$^2$K, and $\beta \simeq 2.76\times 10^{-2}$.

 

Fig. 2. Effective carrier temperature $T$ (solid lines) and optical phonon temperature $\Theta$ (dashed lines) versus the normalized injected current $i_0$ for $\Delta _i = 1240$ meV, $\overline {\Delta _i} = 1210$ meV, $C = 1$ kW/cm$^2$K, $T_0 = 25$ meV, and different values of the optical phonon decay time $\tau _D/\tau _0 = 0.25 -0.75$ ps.

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Figure 3 shows $T$, $\Theta$, and $T_L$ versus the normalized injected current density $i_0$ for $\Delta _i = 1240$ meV, $\overline {\Delta _i} = 1090$ and 1210 meV (corresponding to $\tau _0/\tau _{cc} = 1 - 5$), assuming $C = 1$ kW/cm$^2$K. One can see that $T$ at $\overline {\Delta _i} = 1210$ meV only slightly exceeds $T$ at $\overline {\Delta _i} = 1090$ meV, while the values of $\Theta$ for these cases are indistinguishable. As seen from Fig. 3, the lattice temperature $T_L$ is much lower than $T$ and $\Theta$ (for the chosen value of $C$). The lattice temperature around of the GL is independent of $\tau _D$. As follows from Eq. (11) (and fig. 3), it increases from $T_L = 25$ meV (300 K) at $i_0 = 0$ to $T_L \simeq 59$ meV (about 710 K) at $i_0 =1$, and to $T_L \simeq 93$ meV (about 1116 K) at $i_0 =2$. Hence, in the injected current densities range under consideration and the relatively high $C$ characteristic for the hBNLs, $T_L \simeq < \hbar \omega _0$. However, at small values of the hBNL thermal conductivity $C$ (in the devices with far from ideal heat removal), $T_L$ can be very close to $\Theta$ and only slightly lower than $T$. Figure 4 shows $T$, $\Theta$, and $T_L$ calculated as functions of the hBNL thermal conductivity (i.e., the hBNL thickness). One can see that in a wide range of $C$, the temperatures $T$, $\Theta$, and $T_L$ fairly weakly depend (at the given injection current densities). However, at small values of the hBNL thermal conductivity $C$(the devices with far from ideal thermal removal), the temperatures $T$, $\Theta$, and $T_L$ can become quite high with $T_L$ being very close to $\Theta$ and only slightly lower than $T$. The $T$, $\Theta$, and $T_L$ versus $C$ relations are shown only in the range $C$ ($C \geq 0.10 - 0.17$ kW/cm$^2$K) corresponding to $T_L$ smaller than the hBNL melting temperature $T_L \lesssim 250$ meV ($T_L < 2973$ K).

 

Fig. 3. Effective carrier temperature $T$ (solid lines), optical phonon temperature $\Theta$ (dashed line), and lattice temperature $T_L$ (dotted line) versus the injected current $j_0$ for $\tau _D/\tau _0 = 0.25$ at different $\overline {\Delta _i}$, setting $\Delta _i = 1240$ meV, $C = 1$ kW/cm$^2$K, and $T_0 = 25$ meV.

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Fig. 4. Effective carrier temperature $T$ (solid lines), optical phonon temperature $\Theta$ (dashed lines), and lattice temperature $T_L$ (dotted lines) as functions of the hBNL thermal conductivity $C$ at the injected current $i_0 = 1$ and $i_0 = 2$ for $\Delta = 1240$ meV, $\overline {\Delta _i}= 1210$ meV, $\tau _D/\tau _0 = 0.25$, and $T_0 = 25$ meV.

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3.2 Asymptotic analytic relations: weak heating

One can get analytic formulae for $T$ and $\Theta$ in the limits of small and relatively high injection currents. At small injected current densities $j_0$ ($i_0 \ll 1$), the temperatures $T, \Theta , T_L \ll \hbar \omega _0$ and Eq. (10) yields

As for the carrier effective temperature $T$ at relatively low injection current, from Eq. (8) we obtain

3.3 Asymptotic analytic relations: strong heating

At a strong injection (large $i_0$), the 2DEP, optical phonon system and the lattice (near the GL) can be fairly hot. At large values of $i_0$, from Eq. (8)–(12) we obtain the following relations:

(A) High thermal conductivity $C$ (small $\beta$). In this case,

(B) Low thermal conductivity $C$ (large $\beta$). In such a case,

In the case "B", $T \gtrsim \Theta \gtrsim T_L \gg T_0$ in agreement with the $T$, $\Theta$, and $T_L$ versus $C$ relations shown in Fig. 4. Since $T_L$ is limited by the hBNL melting temperature $T_L^{melt}$, Eq. (17) provides a limitation on $i_0 < i_0^{melt} = T_L^{melt}/\beta \Delta _i$. Setting $T_L^{melt} \simeq 256$ meV ($\sim 2973$ K), $\Delta _i = 1240$ meV, and $C = 0.1$ kW/cm$^2$K ($\beta = 2.76$), we find $i_0^{melt} \simeq 0.74$.

It is instructive that according to Eqs. (15)–(18), both $T$ and $\Theta$ are linear functions of $i_0$ when the latter is sufficiently large. In both cases "A" and "B", the difference $T-\Theta$ [the second terms in the right-hand sides in Eqs. (17) and (18)] is independent of $i_0$. As follows from the above equations that $dT/di_0 > d\Theta /di_0$ in the range of small values of $i_0$, whereas $dT/di_0 = d\Theta /di_0$ when $i_0$ is large. The asymptotic relations of $T$ and $\Theta$ on $i_0$ are in line with the above numerical data.

4. Steady-state spectral characteristic and output optical power

The probability, $\nu _R$, of the interband radiative transition in the GL is given by [71,72]

Here $\kappa _{HBN}$ is the dielectric constant of hBN, $c$ is the speed of light in vacuum, and $p =\hbar \omega /2c$ is the carrier momentum in the initial and final state, where $\hbar \omega$ is the energy of the emitted photon. In the range of the photon frequencies ($\omega > 1/\tau$) the processes associated with the indirect intraband transitions with the absorption (the Drude absorption) and emission of the photons are much weaker than the interband processes. Therefore, we account solely for the interband radiative transitions. Since the rate of the interband radiative transition [71,72]

Here $S_0 = \displaystyle \frac {2A}{3\pi }\left(\frac {e^2\sqrt {\kappa _{hBN}}\,T_{0}^3}{\hbar ^4c^3}\right )$, $\zeta = \exp (-\mu /T)$, and $A <1$ is the fraction of the emitted photons not reflected by the outer surface. According to Eq. (11), $\zeta = (1 + \beta \,i_0\tau _A/\tau _0)^{-1/2}$. For $\sqrt {\kappa _{hBN}} \simeq 2.5$, $T_{0} = 25$ meV, and $A \lesssim 1$ one obtains $S_0 \simeq 3\times 10^{17}$ cm$^{-2}$s$^{-1}$.

In deriving Eq. (22), we have neglected the photon recycling and the photon accumulation inside the device (i.e., the resonant properties of the heterostructure). Equations (20)–(22) explicitly account for the spectral dependence of the GL emissivity in the device under consideration. In the case of small $(\mu _e + \mu _h)/T$, Eqs. (21) and (22) coincide with the pertinent formulae in our recent paper [46] [in which the injection of relatively low energy carriers was considered, so that$(\mu _e + \mu _h)/T$ can be large].

Figure 5 shows the spectral dependence of the output radiation $S_{\hbar \omega }$ for different normalized injection current densities $i_0$. Figure 6 presents the output power $P$ versus the normalized injection current density $i_0$ for the same parameters as in Fig. 5. As follows from figures (5) and (6), the IDLEs can be effective sources of near-infrared and visible light.

 

Fig. 5. Emitted radiation spectral characteristic $S_{\hbar \omega }$ for different normalized injection current densities $i_0$ ($\Delta _i= 1210$ meV, $\overline {\Delta _i}= 1210$ meV, $\tau _D/\tau _0 = 0.25$, and $T_0 = 25$ meV).

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Fig. 6. Output radiation power (per unit of square) as a function of normalized injection current densities $i_0$ for different values of the thermal conductivity $C$. ($\Delta _i= 1240$ meV, $\overline {\Delta _i}= 1210$ meV, $\tau _D/\tau _0 = 0.25$, and $T_0 = 25$ meV).

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The output radiation power as a function of the 2DEHP effective temperature $T$ can be estimated by evaluating the integral in Eq. (22). As a result, we arrive at

The output power, $P^{BB} = (\pi ^2/ 60 \hbar ^3c^2)T^4$, of the thermal radiation emitted by a similar structure considered as the black-body with the same temperature $T$ is given by

Comparing $P$ and $P^{BB}$, given by Eqs. (23) and (24), respectively, we obtain the following estimate for the IDLE emissivity (the net emissivity, which accounts for the whole spectrum of the emitted photons): $\epsilon _{GL} = P/P^{BB} \simeq 0.114$. The latter is several times larger than that estimated in [48,49] and close to the value obtained in [46].

The power $P^{GB}_L$, emitted by the heated hBHL(s) considering it as a gray body with the emissivity $\epsilon _{hBN}$ and setting that the average temperature $\overline T_L =(T_L - T_0)/2$, where $T_L$ is given by Eq. (11), is estimated as

For $\Delta _i = 1240$ meV, $C = 1$ kW/cm$^2$K ($\beta = 0.0276$), and $\epsilon = 0.1$ (this estimate follows from Eq. (23) and (24 ) as stated above), Eq. (25) yields $P^{GB}_L \simeq 1\times i_0^4$ mW/cm$^2$. Taking into account that $T > \overline T_L, T_L$, we find $P^{GB}_L \ll P$. At low value of the thermal conductivity $C$, the lattice can be strongly heated, so that $\overline T_L \simeq T_L \simeq T$, and the net emitted power can be fairly high being only limited by the melting of the device.

The emitted optical power is much larger than that in the case of the "cold" injection [46], but the emissivity values are close to those for the source considered in [46].

5. Dynamic response: modulation characteristics

We consider the temporal variations (modulation) of the applied voltage, $U = U_0 + \delta U_m\exp (-i\omega _m t)$, so that $j = j_0 +\delta j_m\exp (-i\omega _m t)$, $\delta \,i_m = \delta \,j_m/\overline j$, where $V_0$ and $j_0$ are the dc bias voltage and current density, $\delta V_m$ and $\delta j_m$ are the pertinent amplitudes of the ac signal, and $\omega _m$ is the modulation frequency.

In the most interesting situations when the modulation frequency $\omega _m$ is sufficiently high [$\omega _m \gg 1/\tau _L$, see Eq. (5) )], the lattice temperature $T_L$ fails to follow the variations of the injected current. This implies that we can neglect the value $\delta \,T_L$. Taking this into account, from Eq. (4) we obtain

If

Here the terms in the brackets are taken at the steady-state values of $T$ and ${\cal N}_0$. Considering Eq. (8) and Eq. (2) with $\cal E$ given by (see the Appendix)

Using Eq. (27) for $\delta T_m$ and Eq. (22) for the output optical power, we obtain the following estimate for the radiation modulation depth $\delta P_m$:

The power modulation depth described by Eq. (30) is much larger that for the devices in which the electron and lattice temperatures are close. The latter one applies to the IDLEs with the weak remove of the lattice heat (small $C$, large $\beta$). This is because when the steady state temperatures $T \sim T_L$, the thermal emission of the lattice (as a gray body system) makes the denominator in Eq. (30) too large, while $|\delta T_m| \gg |\delta T_L|$. This emphasizes the necessity to provide a high thermal conductance of the hBNLs and of the top metal contact serving as effective heat sinks. A similar situation takes place in the standard incandescent lamps.

6. Injected current-voltage characteristics

The injected current density $j$ is determined by the voltage $U$ applied between the n$^+$-side contacts to the GL(s) and the p$^+$-region of the vertical contact (see Fig. 1). When $U < U_{bi}$, where $eU_{bi} \sim \Delta _G^{hBN}$ is the built-in voltage and $\Delta _G^{hBN}$ is the hBNL energy gap [as shown in Figs. 1(c) and 1(d)], the hole injection into the GL(s) is controlled by the energy barrier at the p-hBNL/GL interface. High carrier densities in the GL(s) result in high efficiency of the carrier-carrier scattering, and, therefore, the injected holes are more likely to be captured into the GL(s) than rejected back to the injector. This is in contrast to the situations in the standard quantum-well and GL-based structures [74,75] with the injected carriers mainly passing across such structures, in which the capture probability is relatively small. Hence, in the IDLEs under consideration, the density of the injection current to a single- or multiple-GL is given by

Here $j^{max} = e N^+v_T$ with $N^+$ and $v_T$ being the hole density in the p$^+$-contact region and the hole thermal velocity, respectively. For an effective 2DEHP heating ($i_0 \gtrsim 1 - 2$) the injected current density $j_i$ should exceed $n\overline j = e\overline \Sigma _0/\tau _0$, where $n$ is the number of the GLs in the device. Assuming the latter be equal to $\overline j = 320$ kA/cm$^2$ (see section 3), we arrive at the following condition: $N^+ \gtrsim 2n\times 10^{17}$ cm$^{-2}$. Taking into account the energy of the injected holes, we estimate the injected power density $P_i$ for $i_0 = 0.5 - 2.0$: $P_i \simeq 200 - 800$ kW/cm$^2$. These values of the injected power density are of the same order of magnitude as, for example, in [55].

Equations (30) and (31) yield

7. Comments

Depending on the contact metal, roughness of of its surface and the surrounding air pressure, $C$ and $\beta$ can be in ranges of (1 - 15) W/cm$^2$K and $2 - 30$, respectively. However, in the latter case, the IDLE energy balance essentially depends on the radiated power [50], which is disregarded in our model.

Similar IDLEs can exploit the lateral hole injection combined with the vertical electron injection. In this case, the electrons captured into the GL acquire the energy $\Delta _C$, so that in the above equations one needs to put $\Delta _i \simeq \Delta _C +3T_{0}/2$. Due to $\Delta _C > \Delta _V$ at the hBNL/GL interface, a relatively strong 2DEHP heating can occur at markedly smaller injection current densities. Analogously, in the IDLEs with the vertical double injection of both the electrons and holes [see Fig. 1(b) and 1(d)], the quantity $\Delta _i$ can be larger: $\Delta _i \simeq \Delta _C + \Delta _V+3T_{0} \gtrsim \Delta _G^{hBN}$. However, in this case, $\overline \Delta _i$ can be markedly smaller than $\Delta _i$ because of a larger factor $k_0$ in Eq. 3).

The operation of the IDLSs under consideration is associated with the injection of rather hot carriers. This is in contrast to the thermal light emitters using the double injection of relatively cold carriers, which are heated by the lateral current (via the Joule heating). The comparison of the IDLE under consideration and the emitters using the Joule heating (for example, [54,55]) demonstrates comparable or better characteristics (the consumed power and the emission efficiency) of the former. An advantage of the IDLEs can be associated with more uniform spatial distribution of the injected power in comparison with the devices with the Joule heating. In the latter, the Joule power cannot be distributed along the GL fairly uniformly due to the essential non-linearity of the lateral current-voltage characteristics [48,76]. Also, the in the emitter with the Joule heating having a short GL, the hot carrier can quickly pass through the GL and release a substantial power in the collecting contacts. The IDLEs can exhibit higher modulation efficiency due to higher values of $\delta j_m/\delta U_m$. This is because in the case of the lateral Joule heating, the injected current as a function of the applied voltage tends to saturation [55] (see also [76]) that results in smaller $\delta j_m/\delta U_m$.

The developed IDEL mathematical model can also be applied for more detailed analyzing of the thermal light sources using other method of the carrier heating, including those with the Joule heating.

8. Conclusions

The transparent mathematical model of the proposed IDLEs using the combined lateral-vertical carrier injection into the GL(s) encapsulated by the hBNLs allows to analyse, evaluate, and optimize the device characteristics. The efficiency of the IDLE sources increases with the input power and device temperature. The characteristic that allows a more direct comparison of the overall efficiency in comparison with similar devices using graphene is emissivity, which is 0.023 for the bare graphene (see [54], supplemental information) and 0.114 for IDLE (i.e. about a factor of five higher than that reported in [48,49]). The device model under consideration showed that the IDLEs can serve as simple efficient light emitters. Due to the possibility of fast variations of the hot carrier injection and temperature by varying voltage, these emitters can generate near-infrared and visible light modulated in the GHz range that is not possible for the standard incandescent lamps. This feature enables IDLEs be used in different communication and signal processing optoelectronic systems.