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Abstract
We study the optical near- and mid-infrared pumping of the heterostructure based on graphene with a black-As layer. This layer serves for the optical generation and cooling of the electron-hole pairs to be injected into the graphene layer. Due to the cooling of the electron-hole pairs, their energy in the case of the absorbing-cooling layer with the optimized thickness can be close to the energy gap of the black-As layer. Owing to a relatively narrow energy gap of the black-As layer $\Delta _G$, the energy of the injected electron-hole pairs can be smaller than the energy of optical phonons in in graphene ($\hbar \omega _0 \simeq 0.2$ eV. This can provide the formation of the cold electron-hole plasma in the graphene-layer that is beneficial for achieving of the interband population inversion and the interband terahertz lasing. The obtained results can be used for the optimization of the terahertz lasers with the optical pumping.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Graphene-based heterostructures exhibiting the zeroth energy gap [1] are very attractive for different optoelectonic applications [2–6]. As predicted previously [7–11], the optical pumping of graphene layer (GL), can result in the interband population inversion in this layer leading to its negative dynamic conductivity. The latter can be used for the implementation of the terahertz (THz) lasers. Despite the experimentally demonstration of the negative dynamic THz conductivity feasibility [12–16], the realization of such lasers with a sufficient output power and operating at the room temperature remains a challenge. The main effect, which hampers the lasing is associated with a relatively high effective temperature of the electrons and holes generated by the pumping by near- or mid-infrared (NIR or MIR) photons [17]. This adds the complexity to the formation of the interband population inversion. Indeed, the electrons and holes in the GL are created with a rather high energy $\varepsilon _0 = \hbar \Omega /2$, where $\hbar \Omega$ is the photon energy. Apart from this, the direct optical pumping exhibits a fairly low quantum efficiency determined by the interband absorption coefficient of the GLs $\beta =\pi \alpha \simeq 0.023$ ($\alpha \simeq 1/137$ is the fine structure constant). In principle, the pumping quantum efficiency can be enhanced by using multiple-GL heterostructures (consisting of a number of the non-Bernal stacked GLs), using different waveguide structures, such as plasmonic structures enhancing the photon-carrier coupling. The the electron-hole pairs in GLs can also be generated using a special absorbing layer, in which the electron-hole pairs are optically generated, followed by their injection into the GL [18]).
If the absorption layer has the energy gap characteristic for the standard semiconductor materials, such as GaAs, InGaAs, or InAs, this pumping method still provides high effective temperature in the electron-hole GL plasma. This obstacle can be avoided in the case of the black-As absorbing layer (and the black-As$_x$P$_{1-x}$ with $x$ close to unity) because of their relatively narrow energy gaps [19]. Both black-As and black-P, as well as their compounds are considered very promising for optoelectronic applications and their fabrication technology makes rapid strides [20–30]. Depending on the number of the atomic layers, the black-P stacks energy gap varies in a wide range: $\Delta _G \sim (0.3 - 1.2)$ eV. The energy gap in the bulk compounds black-As$_{x}$P$_{1-x}$ varies from $\Delta _G \sim 0.3$ eV in the black-P ($x =0$) to $\Delta _G \sim 0.15$ eV in black-As ($x \simeq 1$).
In this paper, we develop a model for the optical pumping of the GL through the bulk black-As absorbing-cooling layer (ACL). In this pumping scheme, the electron-hole pairs photogenerated in the black-As ACL diffuse across the layer and are injected into the GL. It is important that the kinetic energy of the carriers can drop from $\varepsilon _0^{*} = (\hbar \Omega - \Delta _G)/2$ to virtually the thermal energy determined by the lattice temperature $T_0$. In this case, each electron-hole pair brings to the GL the energy about of $\varepsilon _{GL} \simeq \Delta _G + 3k_BT_0$ if the carrier cooling is sufficiently effective. This implies that if $\Delta _G\,<\,0.2$ eV, $\varepsilon _{Gl}$ can be smaller than the energy lost by the electron-hole plasma due to the interband emission of optical phonons. When $\varepsilon _{GL}\,<\,\hbar \omega _0 \simeq 0.2$ eV, an effective cooling of the electron-hole plasma can occur, i.e., the carrier effective temperature $T$ can become smaller than $T_0$. The realization of this situation is possible in the black-As (and black-As$_{1 -x}$P$_x$ ACLs with $x\sim 1$). In the previous consideration of the pumping method in question a rather simplified diffusion model was developed [19]. It ignored that the diffusion of the photogenerated carriers across the ACL is affected by their fairly high initial energy (effective temperature), which decreases due to cooling, associated with a gradual transfer of the carrier energy to the ACL lattice. As a result, the spatial distribution of the carrier effective temperatures is strongly nonuniform. This affects the diffusion and might markedly influence the carrier effective temperature in the ACL at the GL interface. In this paper, the generalized model (thermo-diffusion model) is used for the evaluation of the pumping characteristics.
2. Device structure and equations of the thermo-diffusion model
We consider the heterostructure based on a GL placed on a wide-gap substrate and covered by a black- As ACL. The latter constitutes a stack of sufficiently large number of the As atomic layers (to provide a minimal energy gap [20]). This structure is illuminated by NIR or MIR radiation emitted by MIR/NIR light emitting diodes or lasers, such as the quantum-cascade lasers, and polarized in the heterostructure plane In particular, these optical pumping sources can be integrated with the heterostructure under consideration.
Figure 1 shows the band diagram of the device structure under consideration. This band diagram reflects the band alignment of the black-As ACL and GL.
Fig. 1. The band diagram of a heterostructure comprising the black-As- ACL and GL. A bulk arrow indicate the direction of incident pumping NIR/MIR with the photon energy $\hbar \Omega$, and a wavy vertical arrow shows the interband photogeneration of electrons and holes ((black and open circles) propagating (diffusing and cooling) from the illuminated surface of the ACL to the GL in which they are accumulated with the formation of the interband population inversion.
The carrier density in the absorbing-cooling layer, $n=n(x)$ and the carrier effective temperature $T=T(x)$ (in the energy units) are governed by the following equations (see, for example, Ref. [31]):
The boundary conditions are as follows:
3. Spatial distributions of the carrier density and effective temperature
Nonlinear system of Eqs. (1) - (3) with boundary conditions (4) and (5) was solved numerically. It is assumed that: the energy gap $\Delta _G = 0.15$ eV, the effective absorption coefficient $\alpha _0 = 10^{4}$ cm$^{-1}$, the lattice temperature $T_0 = 200$ K, the characteristic times $\tau _R = 550$ ps and $\tau _{\varepsilon } = 1$ ps, the capture rate $s = 10^{5}$ cm/s, the diffusion coefficient $D$ and the carrier mobility $b$ are related to each other according to the Einstein relation $D = b T/e$, with $b = 10^{4}$ cm$^{2}$/V$\cdot$s, and $\kappa = 1.6\times 10^{5}/T^{1.6}$. These parameters can be attributed to the black-As ACLs (or black- $_x$P$_{1-x}$ ACLs with $x \sim 1$. For the definiteness we set the pumping power $P_{\Omega }$ to be equal to $P_{\Omega } = 10$ W/cm$^{2}$.
Figure 2 shows the spatial distributions of the carrier density and effective temperature across the black-As ACL of the thickness $d = 2.0~\mu$m. calculated for different pumping photon wave lengths $\lambda _{pump} = 2\pi c/\Omega$.
Fig. 2. Spatial distributions of (a) the carrier density $n$ and (b) the carrier effective temperature $T$ at different pumping photon wavelength $\lambda _{pump}$ for the black-As ACL ($\Delta _G = 0.15$ eV).
As seen, the carrier density $n$ decreases across the ACL rather slow, so that the carrier density spatial distribution is virtually uniform. Higher values of the carrier density $n$ at the pumping by a longer wavelength radiation $\lambda _{pump}$ are attributed to an increase in the carrier photogeneration rate $\alpha _{\Omega }I_{\Omega } \propto P_{\Omega }\sqrt {(\hbar \Omega -\Delta _G)}/\hbar \Omega$ with increasing $\lambda _{pump}$ at the fixed pumping power $P_{\Omega }$.
In contrast to the spatial distributions of the carrier density, the carrier effective temperature $T$ drops steeply tending to the lattice temperature $T_0$. That is attributed to a relatively long diffusion length $l_D = \sqrt {D\tau _R}$ and to a rather short cooling length $l_T =\sqrt {D\tau _{\varepsilon }}$. Indeed, setting $T \gtrsim T_0 = 200$ K in the main part of the ACL and using the parameters assumed above we find $l_D \gtrsim 2 \mu$m, i.e., comparable with the ACL thickness. Simultaneously, we obtain $l_T \simeq 0.08 \mu$m. The fact that the effective temperature drops markedly at $x \gg l_T$ is associated with the relatively deep penetration of the pumping radiation into the ACL, so that the hot carriers are generated not only on the illuminated surface of the ACL but in the ACL bulk as well.
4. Injection quantum efficiency and injected power
As was mentioned in Sec. I, the quantum efficiency of the direct optical pumping is limited by the fundamental parameter $\beta \simeq 0.023$. The quantum efficiency of the optical pumping, i.e., the injection quantum efficiency into the GL in the devices with the ACL can be substantially higher.. This quantum injection efficiency is defined as a ratio of the carrier pair flux into the GL and the incident pumping photons flux $I_{\Omega }$: $\eta = j/eI_{\Omega }|_{x = d} = -[d (Dn)/dx]/I_{\Omega }|_{x = d}$.
The carrier energy injected from the ACL into the GL $Q = j\varepsilon _{GL}/e|_{x = d}$ is shown in Fig. 3(a) (all parameters are the same as in Fig. 2). This quantity determines the effective temperature, $T_{GL}$, and the quasi-Fermi energy, $\mu$, of the carriers in the GL. Larger values of the injected power $Q$ corresponding to longer pumping radiation wavelengths $\lambda _{pump}$, seen in Fig. 3(a), are associated with the following. As seen in Fig. 2(b), at not too thin ACLs, the injected carriers have the temperature $T|_{x = d}$ close to the lattice temperature $T_0$ for all considered $\lambda _{pump}$. This implies that for all values of $\lambda _{pump}$ under consideration, the pair of injected carriers brings to the GL virtually the same energy $\varepsilon _{GL} = \Delta _G + 3T|_{x = d} \simeq \Delta _G + 3T_0$. However, the injected carriers flux ($\propto sn|_{x=d}$) increases with increasing $\lambda _{pump}$.
Fig. 3. Dependences of (a) the injected power $Q$ and (b) the injection quantum efficiency $\eta$ on the black-As ACL thickness $d$ for different pumping wavelength $\lambda _{pump}$.
Figure 3(b) shows the injection quantum efficiency $\eta$ as a function of the black-As ACL thickness $d$ found for the same photon wavelengths $\lambda$ as in Fig. 2. As seen from Fig. 3(b), the injection efficiency $\eta$ could be about of $\eta \sim 0.5 - 0.75 \gg \beta$.
5. Parameters of the electron-hole plasma in the pumped GL
The injection of the carriers photogenerated in the ACL into the GL forms the nonequilibrium electron-hole plasma in the GL. Due to a high carrier density in the GL (expected in the heterostructures intended for the interband emission of the THz photons), an intensive carrier-carrier scattering enables the “Fermization” of the electron and hole sub-systems, characterized by the quasi-Fermi energies $\pm \mu$ counted from the GL Dirac point and by the common effective temperature $T_{GL}$. The latter generally differs from the lattice temperature $T_0$. As shown in the following, the carrier effective temperature can be both larger and lower than $T_0$ depending on the pumping conditions.
Considering that the intersubband and intraband relaxation of the carrier injected into the GL is primarily determined by the interaction with the GL optical phonons having the energy $\hbar \omega _0$ [17,19] (see also, Refs. [33–37]), taking into account that in each act of the interband and intraband emission/absorption of the optical phonons the net carrier kinetic energy changes by the quantity $\pm \hbar \omega _0$, and using the equation governing the balance of the carrier number and their energy, one can arrive at the following expressions:
Equations (6) and (7) account for the difference between the lattice temperature $T_0$ and the carrier effective temperature at the interface between the ACL and the GL $T_{x=d}$ as well as the carrier effective temperature in the GL $T_{GL}$. Apart from this, for simplicity, we have restricted our analysis to the case of not too strong pumping when the GL is not fully occupied by the carriers, i.e., $\mu\,<\,\textrm {min} \{\Delta _C, \Delta _V$}, where $\Delta _C$ and $\Delta _V$ are the barriers for the electrons and holes, respectively, determined by the band alignment of the black-As and the GL ($\Delta _C + \Delta _V = \Delta _G$).
As seen from Eq. (6), due to relatively small energy gap ($\Delta _G\,<\,\hbar \omega _0$) at sufficiently effective carrier cooling in the ACL (when $T|_{x=d}$ tends to $T_0$) and not too high lattice temperature $T_0$, the carrier effective temperature in the GL $T_{GL}$ can be smaller than $T_0$. This is in line with the previous prediction based on a simplified model [19]. This implies that the optical pumping in the devices under consideration can result in cooling down of the carrier in the GL despite relatively high energy of the pumping photons. The pertinent condition sounds as
For $\hbar \omega _0 = 0.2$ eV and $\Delta _G = 0.15$ eV, the latter inequality yields $T_0\,<\,0.017$ eV or $T_0^{T} \simeq 192$ K. If $T\,<\,T_0^{T}$, the carrier effective temperature $T$ monotonically decreases with increasing pumping intensity $I_{\Omega }$. However, when $T$ becomes much lower than $T_0$, the energy relaxation processes associated with the GL optical phonons weaken, so the subsequent carrier cooling can be limited by the inclusion of other relaxation mechanisms, in particular, by the relaxation on the phonons in the ACL and the substrate.The condition $\mu\,> \,0$ corresponds to the interband population inversion in the GL [7,19]. Equation (7) shows that in the case of total carrier cooling when $T|_{x = d} \simeq T_0$, the condition $\mu\,> \,0$ can be achieved if
Setting $\hbar \omega _0 = 0.2$ eV, $\Delta _G = 0.15$ eV, and $a = 0.2$, we obtain from Eq. (9) $T_0 \leq 0.03$ eV ($T_0^{\mu } \lesssim 300$ K). Comparing these two inequalities, one can see that $T_0^{T}\,<\,T_0^{\mu }$, i.e., the carrier cooling in the GL can occurs at somewhat lower lattice temperature than that required for the interband population inversion.At a weak optical pumping ($\eta (I_{\Omega }/I_0 \ll 1$) and a strong ($\eta (I_{\Omega }/I_0 \gg 1$) optical pumping, Eq. (7) yields the following linear and saturated $\mu$ versus $I_{\omega }$ dependences, respectively:
Figure 4 shows the normalized quasi-Fermi energy, $\mu /T_{GL}$, as a function of the ACL thickness $d$ calculated for different values of the normalized pumping intensity $(I_{\Omega }/I_0)$ and the pumping radiation wavelength $\lambda _{pump}$ using Eq. (7) and invoking the above results for $\eta$ and $T|_{x=d}$. In addition for the parameters used in the previous figures, we set $I_0 = 1.8\times 10^{19}$ cm$^{-2}$s$^{-1}$ and $a = 0.2$. The quantity $I_0$ was estimated interpolating the data presented in Ref. [34] using the following relation: $I_0 \propto \exp (- \hbar \omega _0/T_0)$. As follows from Fig. 4, the condition $\mu\,> \,0$, i.e., the condition of the interband population inversion, can be realized in a wide range of the pumping radiation intensities, but requires using sufficiently thick ACLs (about $1 \mu$m or slightly thinner). As seen, the thickness $d$ can be optimized to maximize $\mu /T_{GL}$. The obtained curves correspond to $T_{GL} \simeq T_0 = 200$ K. The values of $\mu /T_{GL}$ for $T_0 = 200$ K shown in Fig. 4 are in line with the previous results based on a simplified model [19]. This implies that the method of optical pumping in the heterostructures under consideration might provide the generation of the degenerate unheated electron-hole plasma. An increase in $\mu /T_{GL}$ is beneficial for the reinforcement of the population inversion and the realization of an effective interband THz lasing. It can be achieved when the electron-hole plasma in the GL is cooled down $T_{GL}\,<\,T_0$. As follows from Eq. (8), this is feasible at lower temperatures $T_0\,<\,T_0^{T}$. The calculations performed for $T_0 = 150$ K, $d = 1~\mu$m, $\lambda _{pump} = 0.808~\mu$m, and $I_{\Omega }/I_{0} = 10$ yield $T_{GL} = 120$ K (i.e., $T_{GL}\,<\,T_0$) and $\mu /T_{GL} \simeq 2$.
Fig. 4. The normalized carrier quasi-Fermi energy $\mu /T_{GL}$ in the GL as a function of the ACL thickness $d$ for (a) different normalized pumping radiation intensity $I_{\Omega }/I_0$ and $\lambda _{pump} = 0.808\mu$m and (b) different pumping radiation wavelengths at $I_{\Omega }/I_0 = 10$.
Conclusions
We propose the method of optical pumping of the GLs via the ACLs to achieve the elevated quantum efficiency of relatively cool carrier injection. The latter can be realized employing the black-As ACLs because of their relatively narrow energy gap. Using the developed device model, we demonstrated the feasibility of the interband poulation inversion and the THz lasing in the GL heterostructures with the black-As ACLs pumped by the NIR or MIR radiation sources. The latter can include different NIR and MIR light emitting diodes and the quantum cascade lasers (see, for example, Refs. [39–43]). The obtained results enable the design of the graphene based THz lasing and plasmon generating devices.
Funding
Russian Foundation for Basic Research (18-07-01379, 18-29-02089, 18-37-20004); Japan Society for the Promotion of Science (16H06361); RIEC Nation-Wide Collaborative Research Project Japan; Office of Naval Research; Army Research Laboratory.
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