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Abstract
We develop the model for the terahertz (THz) and infrared (IR) photoconductivity of graphene layers (GLs) at room temperature. The model accounts for the linear GL energy spectrum and the features of the energy relaxation and generation-recombination mechanisms inherent at room temperature, namely, the optical phonon absorption and emission and the Auger interband processes. Using the developed model, we calculate the spectral dependences of the THz and IR photoconductivity of the GLs. We show that the GL photoconductivity can change sign depending on the photon frequency, the GL doping and the dominant mechanism of the carrier momentum relaxation. We also evaluate the responsivity of the THz and IR photodetectors using the GL photoconductivity. The obtained results along with the relevant experimental data might reveal the microscopic processes in GLs, and the developed model could be used for the optimization of the GL-based photodetectors.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The gapless energy carrier spectrum and high carrier mobility in graphene layers (GLs) [1] enable unique optical and transport properties. The GL conductivity dependence on the carrier effective temperature has been analyzed and reported in a number of publications (for example, [2–9]). Due to the specifics of the carrier scattering in the GLs, the variation of the GL DC conductivity under the terahertz (THz) and infrared (IR) irradiation (THz and IR photoconductivity) can exhibit interesting behavior [6, 10]. The response of the carrier system in the GLs caused by the THz and IR radiation enables their utilization in the bolometric photodetectors [11–15]. Extensive studies of the carrier dynamics in the GLs in the response to ultrashort optical pulses (for example, [16–24]) stimulate the creation of novel ultrafast optoelectronic devices. The GL ac (dynamic) conductivity under the optical pumping can become negative in the THz range due to the population inversion [25–28] (see also [16,19]).
The THz and IR photoconductivity in GLs is associated with the interplay of the variations of the carrier effective temperature and density (the quasi-Fermi energy). Depending on the dominating scattering mechanisms and the photon energies, the effective temperature can either exceed or be smaller than the lattice temperature. This can be accompanied by a splitting of the quasi-Fermi levels [25,29]. The interplay of the intra- and interband absorption of the incident THz and IR radiation [30] adds complexity to the processes governing the GL conductivity.
The GL photoconductivity was experimentally studied in a number of works demonstrating the effect of negative photoconductivity [10,31,32]. However, it was also shown that the sign of the THz photoconductivity of the GLs changes with the radiation frequency depending on the environment [32].
Despite a variety of the works aimed to describe and explain the features of the THz and IR photoconductivity, there is still no transparent model of the phenomena. The previous studies mainly were focused on the deeply cooled GLs and related bolometric photodetectors. Due to this, the main energy relaxation and generation-recombination mechanisms were associated with the carrier interaction with acoustic phonons and ambient thermal radiation. However, in the uncooled GLs, the carrier intraband and interband transitions due to the recombination and emission of the optical phonons [33] (see also [10]) and the carrier-carrier interaction (Auger processes) [34, 35] can dominate. The inclusion of these mechanisms is necessary for the explanation of the nontrivial spectral characteristics of the GL photoconductivity. Namely, the observed inversion of the photoconductivity sign requires an adequate but transparent model. In this paper, we develop such a model, which accounts not only for the variations of the carrier temperature but also for the possible splitting of the quasi-Fermi levels associated with the THz and IR radiation absorption. This approach allows us to consider the effect of the THz and IR photoconductivity at room temperatures in more details. We also evaluate the application of this effect for uncooled THz and IR photodetectors. The obtained results can be used for the optimization of these photodetectors and promote a deeper insight into the GL microscopic properties.
We focus on both the intrinsic (or compensated) and p-doped GLs at relatively elevated temperatures (such as room temperature). We assume that the carrier momentum relaxation is associated with impurity scattering, scattering on defects, and and on the acoustic phonons. The carrier scattering on the clustered impurities inside the GLs and the substrate inclusions in the decorated GLs is accounted for as well. The intraband and interband transitions of the carriers due to the emission and absorption of the optical phonons are considered as the crucial mechanism of the energy relaxation and the generation-recombination. Despite the specifics of the Auger generation-recombination in the GLs associated with the carrier dispersion law [34] and the carrier collinear scattering (see [35] and references therein), these generation-recombination mechanisms are also included into the model under consideration
2. GL conductivity dependence on the carrier temperature
Due to relatively high carrier densities in GLs at the room temperature, their pair collisions lead to the establishment of the quasi-Fermi energy distributions, fe(ε) and fh(ε), with the same effective temperature T. Therefore, we use the following formula for the real part of the intraband conductivity of the GL- channel σΩ at the frequency Ω (see [2–4]):
Here it is assumed that the momentum relaxation time as a function of the momentum p is equal to τp = τ0(pvW/T0)l = τ0(T/T0)lξl, where τ0 is its characteristric value, ξ = pvW /T0 is the normalized carrier energy in the GL, vW ≃ 108 cm/s is the characteristic velocity in GLs, and ħ is the Planck constant.If the scattering on the weakly screened charged objects (weakly screened Coulomb, long-scale scattering) prevails, τp ∝ p. This case corresponds to the superscript l in Eq. (1) equal to unity (l = 1). In the case of scattering on the neutral impurities and defects, the strongly screened charged impurities, and the acoustic phonons, τ ∝ p−1 (i.e., l = −1). If several mechanisms contribute to the carrier scattering, τp can be described by a combination the linear and the inverse dependences. For a simplified description, in some papers τp = const is used.
The scattering processes can depend on the presence of large-scale charged defects, clustered impurities, [36,37] and various particles in the GL or near the GL interface with the substrate. The latter can be important in the so-called decorated GLs [38,39].
When the carrier momentum relaxation is determined by the short–range scattering, Eq. (1) yields for the pertinent conductivity σΩ is given by
Here σ00 = (e2T0τ0/πħ2) is the characteristic intraband conductivity (it is equal to the low electric-field conductivity in the case of short-range scatterers). In the intrinsic (or compensated) GLs, the electron-hole symmetry leads to the equality of the pertinent quasi-Fermi energies μe = μh = μ (this does not generally imply that the positions of the quasi-Fermi levels and coincide; such a coincidence only occurs if μ = 0). In this case, Eq. (2) transforms to the following: In the case of the short–range scattering, one can find for the characteristic time τ from Eq. (1) . If the long-scale momentum relaxation is the dominant (l = 1), we obtain from Eq. (1) where is the Fermi-Dirac integral. Equations (3) and (4) show that the variations of both the quasi-Fermi energy μ and the effective temperature T caused by the radiation absorption determine the variation of the conductivity, i.e., the effect of photoconductivity. As demonstrated in the following, the quasi-Fermi energy changes not only due to the carrier density change but due to the heating.3. Generation-recombination and energy balance equations
The equation governing the interband balance of the carriers can be generally presented as
The terms in Eq. (5) correspond to the interband Auger generation-recombination processes and the processes associated with optical-phonon, acoustic–phonon, and radiative transitions (in particular, indirect transitions, for which some selection restrictions are lifted). In the situations under consideration (in particular in the temperature range in question), the characteristic times of the processes associated with acoustic phonons and radiative transitions (as was mentioned in the Introduction) are much longer than those related to the optical-phonon and Auger mechanisms [6,9,33,35]. Due to this, we disregard the terms GAc and GRad in Eq. (5).Due to the fast processes of the optical phonon decay into acoustic phonons and their effective removal, confirmed by high values of the G-layer thermal conductivity [40–42], the optical phonon system in the GLs is assumed to be in equilibrium with the lattice having the temperature T0.
For the rate, GOpt, of the interband and intraband transitions assisted by the optical phonons emission and absorption one can use rather general equations accounting for the transitions matrix elements, the features of carrier energy spectrum in the GLs, and the conservation of the carrier momentum and energy [33]. As shown [29] (see also [11, 43]), at ħω0 ≫ μ, T, T0, where ħω0 ≃ 200 meV is the energy of optical phonons in the GL, the general equations for GOpt [33] can be simplified by separating of the exponential factors and relatively slow varying factors depending on μ and T0. Similar procedure leads to a simplified formula for GAuger. As the result, in the situation under consideration, we present Eq. (5) describing the carrier balance in the conduction and valence bands of the GL in the following form:
The equation governing the energy electron and hole balance is given by:
Here τOpt and are the characteristic recombination and intraband relaxation times associated with the carrier interaction with the optical phonons, τAuger is the Auger recombination time, [2] (for a small μ) and are the GL interband and intraband absorption coefficients (determined by the interband and intraband ac conductivities), respectively, Σ0 is the characteristic carrier density, α = (e2/cħ) ≃ 1/137 is the fine structure constant, c is the speed of light in vacuum, and κ is the background dielectric constant. The factors Σ0/τOpt, , and Σ0/τAuger are weak functions of μ and T (mentioned above, see, for example, [29]). These factor can be considered as phenomenological parameters of the model.The exponential terms in the left-hand sides of Eqs. (6) and (7) are proportional to the rate of the generation-recombination and the rate of the energy transfer into the lattice, respectively. The terms in the right-hand sides of these equations correspond to the carrier photogeneration and to the power obtained by the electron-hole system due to the radiation absorption. The latter is proportional to (βintra + βinter)ħΩ IΩ.
Limiting our consideration by low radiation intensities IΩ, Eqs. (6) and (7) are presented as
where, due to a smallness of IΩ, we put and and . Here D0 = (4T0τ0/πħ) is the Drude factor. At T0 = 25 meV and τ0 = (10−12 – 10−13) s, one obtains D0 ≃ 5.3 – 52.7.4. GL photoconductivity
Using Eqs.(8) and (9), we express the normalized carrier quasi-Fermi energy μ/T and temperature variation (T − T0)/T0 via the radiation intensity IΩ, its frequency Ω, and the characteristic times τOpt, , and τAuger as
Here is the characteristic relaxation-recombination time, which accounts for the interband transitions associated with the optical phonons and the Auger generation recombination processes, as well as the intraband energy relaxation on the optical phonons, and b = τOpt/τAuger. The parameters a and b can be estimated using the data found previously [29,33] and [33,35], respectively. Equations (10) and (11) can be rewritten as where The first term in the square brackets of Eq. (12) is negative when Ω < ω0/(1 + b). In this case, the interband transitions promote a decrease in the carrier temperature. The point is that every act of the photon absorption brings the energy ħΩ to the carrier system, while the interband emission of optical phonon decreases the carrier system energy by the value ħω0, which is a smaller value. If b ≪ 1, these processes lead to the carrier heating. The Auger recombination (in the gapless GLs) does not directly change the carrier energy, but decreases the cooling role of the optical photon emission.In the limit of very strong Auger processes (τAuger ≪ τOpt, i.e., b ≫ 1), Eq. (12) yields
In the latter case, the radiative interband processes supply the energy, while the interband optical phonon emission does not work (only the intraband optical phonon processes contribute to the carrier energy relaxation). In this limit, we obtain from Eq. (13) μ = 0. The latter implies that the electron and hole quasi-Fermi levels coincide ().Figure 1 shows the carrier temperature variation (T − T0)/T0 and the quasi-Fermi energy μ/T normalized by IΩ/ĪΩ (i.e., the quantities [(T − T0)/T0](ĪΩ/IΩ) and (μ/T)(ĪΩ/IΩ)), versus the photon energy ħΩ in the GLs with dominating short-range scattering calculated using Eqs. (12) and (13) for different b (i.e., different τAuger) and different τ0. We set ħω0 = 200 meV, T0 = 25 meV, a ≃ π2(T0/ħω0)2(1 + 2.19T0/ħω0) ≃ 0.2 [24].
Fig. 1 Upper panel: the normalized carrier temperature variation (T − T0)/T0 (dashed lines) for different values of the parameter b (upper panel) and τ0 = 1 ps and the quasi-Fermi energy μ/T (solid line) for b = 0.1 and τ0 = 1 ps. Lower panel: the normalized carrier temperature variation (T − T0)/T0 (dashed line) for b = 0.1 and τ0 = 1 ps and the quasi-Fermi energy μ/T (solid lines) for b = 1 and different τ0.
As seen from Fig. 1 that the photon energies where μ and (T − T0) change their signs do not coincide.
Using Eqs. (3) and (13) when |μ| ≪ 2T, for the difference ΔσΩ = σΩ→0 − σ00 between the dc conductivity under irradiation and the dc dark conductivity we obtain
so thatFigure 2 shows the GL photoconductivity (ΔσΩ/σ00) normalized by a factor IΩ/ĪΩ) calculated using Eq. (16) for different parameters. The GL photoconductivity as a function of the photon energy changes its sign twice at ħΩ = ħΩ0 and ħΩ = ħΩ1 which correspond to the points, where μ/T = εe = εh = 0. One can see that the absolute value of the GL photoconductivity steeply decreases with increasing b, i.e., with the decreasing Auger recombination time τAuger (and, hence, decreasing τR). If τAuger tends to zero, the quantities μ/T and Δσ0 also tend to zero, despite (T −T0) ≠ 0, as follows from Eqs. (13) and (16). All this is because at very intensive Auger processes the splitting of the quasi-Fermi levels vanishes, and the factor in the square brackets in the right-side of Eq. (2) is equal to unity. Hence, the conductivity becomes independent of the carrier temperature.
Fig. 2 The normalized GL photoconductivity ΔσΩ/σ00 as functions of the photon energy ħΩ (dominant short-range scattering) for different b = τOpt/τAuger and τ0 = 1 ps (left panel) and different τ0 and b = 0.1 (right panel).
As seen from Fig. 2, the GL photoconductivity is negative ΔσΩ < 0 in the photon energy ranges 0 < ħΩ < ħΩ0 and ħΩ1 > ħΩ. In the intermediate range ħΩ0 < ħΩ < ħΩ1, the GL photoconductivity is positive.
Taking into account that in reality D0 > 4T0(1 + a)/ħω0, i.e., ω0τ0 > π(1 + a), from Eq. (16) we obtain
andFor τ0 = (10−12 – 10−13) s−1 and a ≃ 0.2, from Eqs. (17) and (18) one obtains Ω0/2π ≃ (1.35 – 3.16) THz and Ω1/2π ≃ (59.29 – 60.92) THz. Thus, the frequencies, at which the GL photoconductivity changes sign, correspond to the THz range (Ω0/2π) and to the Mid-IR range (Ω1/2π).
It should be noted that if the momentum relaxation time τp is independent of the momentum p (as was assumed in some GL conductivity models [10, 44, 45]), the conductivty variation Δσ0/σ00 ≃ μ/4T [46] (compare with Eq. (15). Therefore, the results for the case of τp = const are qualitatively close to those obtained in this section.
5. Clustered impurities and decorated GLs - long-range scattering
In this case, using Eq. (4), at relatively low intensities of the incident radiation IΩ when the deviation of μ from zero is small, we obtain
Hence, The latter is quite different from Eq. (15), which is valid in the case of short-range momentum relaxation. Equation (19) with an equation similar to Eq. (11) results in (compare with Eq. (14))Figure 3 shows the normalized responsivity of the photodetectors based on the GLs with the dominant long-range scattering calculated using Eq. (21).
Fig. 3 The same as in Fig. 2 but in the case of dominant long-range scattering: for b = τOpt/τAuger and τ0 = 1 ps (left panel) and for τ0 and b = 0.1 (right panel).
As follows from the comparison of Eqs. (16) and (21) (see also Figs. 2 and 3), the photoconductivities involving the short- and long-range scattering mechanisms exhibit fairly different spectral dependences: the photoconductivity at long-range scattering is positive at low and high photon energy and negative in the range of the intermediate photon energies. This is because in the case of the dominant short-range scattering, the variation of the dc conductivity stimulated by the interband and intraband radiative transitions is associated primarily with the variation of μ (more exactly with the variation of μ/T), i.e., with the shifts of the electron and hole quasi-Fermi levels. The latter is in contrast with the case of the dominant long-scale scattering in which the variation of the dc conductivity is due to the variation of the carrier effective temperature. This can be seen from the comparison of Eqs. (3) and (19) describing a decrease in the conductivity with increasing −μ/T and an increase with increasing T, respectively.
6. Responsivity of the GL-based photodetectors
Using the obtained values of the photoconductivity, ΔσΩ, of the GLs one can find the responsivity RΩ = |ΔσΩ|E/ħΩ IΩ of the photodetectors based on undoped GL. Using Eqs. (16) and (21), for the cases of the dominant short- and long-scale scattering we arrive at
respectively, where Here E is the electric field along the GL plane and L is the spacing between the contacts. For example, setting τ0 = 10−12 s, τOpt = 10−12 s, κ = 4, L = 10−4 cm, and E = 100 V/cm, from Eq. (24) we obtain R ≃ 2.47 × 10−4 A/W. One needs to note that the expression for RΩ includes a large parameter D0. In particular, for the responsivity of the photodetector using high quality GLs (τ0 = 10−11 s) at ħΩ = 10 meV (i.e., Ω/2π ≃ 2.5 THz) Eq. (21) yields RΩ ≃ 1.8 A/W.Figures 4 and 5 show the RΩ/R versus ħΩ dependences calculated using Eqs. (12) and (22) for the GL-photodetectors with different τ0 and b at a = 0.2. As the GL photoconductivity, the detector responsivity RΩ also turns to zero at certain frequencies.
Fig. 4 The spectral characteristics of the responsivity, RΩ/R, of the GL-based photodetectors with dominant short-range scattering at different b and τ0 = 1 ps (left panel) and different τ0 and b = 0.1 (right panel).
The responsivity of the GL-based photodetectors is proportional to the fundamental constant πα ≃ 0.023, which determines the coupling of the carriers and the incident radiation. The inclusion of the plasmonic effects associated with the excitation of the plasmons in the optical couplers (metal gratings and so on) and in the GLs can lead to a substantial reinforcement of the carrier-radiation coupling (see, for example [47] and references therein). As a result, the GL-detector responsivity can be markedly enhanced. The responsivity spectral characteristic can exhibit pronounced plasmonic resonances superimposed on the spectral dependences obtained above.
7. Heavily doped GLs
Since the pristine GLs are usually p-type, we will consider the GLs with sufficiently high acceptor and hole densities ΣA. In the limit of short Auger recombination time, μe = −μh and the variations of the carrier temperature do not lead to the variations of the GL conductivity and the effect of the photoconductivity vanishes. In a more realistic situation one can to assume the long recombination time τAuger, therefore, the term with τAuger can be omitted. In this case, we can use Eqs. (6) and (7) but with 2μ replaced by (μe + μh). Apart from this, the quantities and should be replaced by
and respectively, and instead of Σ0 one needs to put Σ = ΣA ≫ Σ0. As a result, we arrive at the following equations: From Eqs. (25) and (26), we obtainThe hole Fermi energies μe and μh are related to the effective temperature T as
Here U ∝ ΣA = const.Taking into account that at μ ≫ T, ℱ1(x) ≃ (x2/2 + π2/6) + x ln(1 + exp(−|x|), at T ∼ T0 from Eq. (28) for μh = μ we obtain
The screening length of the charged scatterers in the doped GLs with μ ≫ T, is much shorter than that in the intrinsic GLs. Therefore, it is natural to assume that in the former GLs the short-range scattering dominates. Considering Eq. (2) with the carrier temperature and the quasi-Fermi energies given by Eqs. (27), (28), and (30), we get
If μ0 > ħΩ1/2 ≃ (1 + a)ħω0/2, Eq. (15) yields ΔσΩ < 0 at all the photon energies. This is in line with the experimental results for the p-GLs with ΣA = (2 – 11) × 1012 cm−2, i.e., with μ0 = 192 – 367 meV [10].Equation (31) resembles Eq. (16). However, the latter includes a small factor [exp(−μ0/T0)/ΣA], different frequency-dependent factor of the interband absorption coefficient, and much shorter . Due to the former, the responsivity of the photodetectors based on doped GLs is smaller than that of the intrinsic GLs. The factor [exp(−μ0/T0)/ΣA] reflects the facts that the GL conductivity is a rather weak function of the carrier temperature at μ0 ≫ T0 and that heat received by the carrier system from the absorbed radiation is distributed among a larger number of the carriers (ΣA ≫ Σ0).
8. Conclusions
We developed an analytical model describing the THz and IR photoconductivity in the GLs at room temperature. Using this model we showed that the energy relaxation and the generation-recombination associated with the optical phonons and the Auger processes are crucial for determining the behavior of GL photoconductivity. We found that in the GLs with dominating short-range scattering mechanism of the carrier momentum relaxation the photoconductivity of the intrinsic GLs changes sign at a certain photon frequency Ω0 in the THz range. Depending on the GL parameters (the transition from the negative to positive photoconductivity) occurs at a certain photon frequency Ω1 in the mid-IR range (the photoconductivity is negative at Ω < Ω0 and Ω > Ω1, while it is positive in the range Ω0 < Ω < Ω1. In contrast, in the case of the long-range scattering dominances, the photoconductivity is positive at Ω < Ω0 and Ω > Ω1 and negative at Ω0 < Ω1. The change in the GL environment might result in the change from the short-range to long-range scattering. The evaluated responsivity of the GL-based THz and IR photodetectors indicates their prospects for different applications.
Funding
Japan Society for Promotion of Science (16H06361); Russian Science Foundation (14-29-00277); Russian Foundation for Basic Research (16-37-60110, 18-07-01379); RIEC Nation-Wide Collaborative Research Project, Japan; Office of Naval Research (Project Monitor Dr. Paul Maki).
Acknowledgments
The authors are grateful to A. Arsenin, V. E. Karasik, V. G. Leiman, and P. P. Maltsev for useful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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