aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Modulation characteristics of uncooled graphene photodetectors
V. Ryzhii , , M. Ryzhii , T. Otsuji , V. Leiman , V. Mitin , and M. S. Shur Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan Institute of Ultra High Frequency Semiconductor Electronics of RAS, Moscow 117105, Russia Department of Computer Science and Engineering,University of Aizu, Aizu-Wakamatsu 965-8580, Japan Center for Photonics and Two-Dimensional Materials,Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia Department of Electrical Engineering, University at Buffalo, SUNY, Buffalo, New York, 1460-192 USA Department of Electrical, Computer, and Systems Engineering,Rensselaer Polytechnic Institute, Troy, New York 12180, USA
We analyze the modulation characteristics of the uncooled terahertz (THz) and infrared (IR) de-tectors using the variation of the density and effective temperature of the two-dimensional electron-hole plasma in uniform graphene layers (GLs) and perforated graphene layers (PGLs) due tothe absorption of THz and IR radiation. The performance of the photodetectors (both the GL-photoresistor and the PGL-based barrier photodiodes) are compared. Their characteristics are alsocompared with the GL reverse-biased photodiodes. The obtained results allow to evaluate theultimate modulation frequencies of these photodetectors and can be used for their optimization.
I. INTRODUCTION
The non-equilibrium two-dimensional electron-holeplasma (2DEHP) in graphene layers (GLs) from equi-librium [1–9] enables the operation of different GL-basedterahertz (THz) and infrared (IR) electro-optical modu-lators, ultrafast thermal light sources with high carriereffective temperature, the superluminescent and lasingdiodes, and, particularly, the hot/cool carrier bolomet-ric detectors of electromagnetic radiation [10–20]. Suchdetectors use the effect of the conductivity variation dueto the carrier density and effective temperature varia-tion and the change associated with the radiation ab-sorption. The speed (in particular, the maximum modu-lation frequency) of the GL-based photodetectors is oneof their important characteristics. The maximum modu-lation frequency depends on not only the carrier energyrelaxation and recombination, but also on the 2DEHPheat capacities. Despite a relatively small value of thelatter compared to the heat capacity of the substrate andof the GL encapsulation layers in which GLs are encap-sulated, it can limit the modulation speed.In this paper, we study the modulation characteristicsof the uncooled THz and IR GL hotodetectors using theeffects of the interband carrier generation/recombinationand the carrier heating/cooling (the bolometric mecha-nism). We account for the essential interdependence ofthese processes. We clarify of the relative roles of the car-rier energy accumulation and the relaxation and recom-bination processes in the dynamic (modulation) prop-erties of the uncooled (operating at room temperature)photodetectors. Our analysis accounts for the correctvalue of the 2DEHP capacity, which substantially devi-ates from its classical value.The developed device model describes the high-speedoperation of the photodetectors with uniform and perfo-rated GLs.
II. DEVICE MODEL AND GENERALEQUATIONS
We consider the photodetectors based on the undopedGLs supplied by the side Ohmic (p- and n-) contacts.The Fermi levels in the contacts align with the GL Diracpoints shifted, by the dc bias voltage V . The GL regionscan be either uniform (in the GL photoresistors) or com-prise a perforated array in the perforated GL (PGL) bar-rier photodiodes. Figure 1 show the structures and theband diagrams of the devices under study. The normallyincident THz or IR radiation absorbed in the GL resultsin the extra electron-hole pair generation and affects theenergy of the 2DEHP. The variation of the carrier densityand their effective temperature leads to the variation ofthe GL conductivity and the current through the PGL re-gion and modulates the terminal current. In the perfora-tion array, the carrier current flows only in the nanocon-strictions [21–24] (GL quantum wires or nanoribbons)between the perforations. If the nanoconstrictions aresufficiently narrow, the quantization of the carrier energyspectrum results in the barriers for the carriers comingfrom the uniform parts of the GL, which limit the termi-nal current. The steady-state bolometric characteristicsof the uniform GL and the GLs with the region patternedinto the GL nanowires (nanoribbons) were evaluated pre-viously [9, 12]. As it was predicted a long time ago [21],the inclusion of the vertical and lateral barrier regionsbetween the photosensitive standard quantum wells [25–27] and between the GLs in the GL-based hot-carrierbolometers with the vertical structure [28, 29] generallyleads to the responsivity enhancement. The barriers be-tween the carrier puddles and the lateral quantum dotsin GLs [13, 16] can play a similar role.The 2DEHP density and effective temperature are de-termined by radiation absorption and the interband (re-combination) and intraband relaxation processes.Due to relatively frequent carrier-carrier collisions in N a n o c o n s t r i c t i o n s D P G L ( c )
G L ( a )
S u b s t r a t e ( b )
E l e c t r o n sH o l e sS u b s t r a t e ( d )
E l e c t r o n sH o l e s h W h W FIG. 1: Schematic view of the device structures and their band diagrams: (a) and (b) - for GL-photoresistor with uniform GLchannel, and (c) and (d) - for PGL-photodiode with the barrier-limited carrier transport via the perforated GL region (withthe barrier heights in the nanoconstrictions ≃ ∆ / ~ Ω. the GLs at room temperatures, the 2DEHP electron andhole subsystems can be described by quasi-Fermi en-ergy distribution functions f e ( ε ) and f h ( ε ) with com-mon effective temperature T (in energy units) : f e ( ε ) = (cid:20) (cid:18) ε − µ e T (cid:19)(cid:21) − , f h ( ε ) = (cid:20) (cid:18) ε − µ h T (cid:19)(cid:21) − .Here ε ≥ µ e and µ h arethe electron and hole quasi-Fermi energies, respectively.Due to a symmetry of the electron and hole subsystemsin the situations under consideration, µ e = µ h = µ . Inthe equilibrium (without an external irradiation), µ = 0.Generally, the absorption of the incident radiation canlead to µ = 0, although, except for the cases of a strongirradiation, | µ | ≪ T .Limiting our consideration to the room temperatures,we assume that the main mechanisms of the densityand energy relaxation are associated with the GL opticalphonons [30–32].The pertinent rate equations, which govern the densityand energy balances in the 2DEHP, are given by d Σ dt = G − R inter − R A , (1) d E dt = ~ Ω G − ~ ω ( R inter + R intra ) . (2)Here Σ and E are the net carrier density and the car-rier energy per unit square, respectively, in the entireGL or its parts separated by the constriction area, G , R inter , R intra , and R A are the pertinent rates of theelectron-hole pair generation by the incoming radiationand the interband and intraband transitions with thegeneration and absorption of optical phonons, and theAuger recombination-generation processes [9, 33–35], re-spectively, ~ Ω and ~ ω are the photon and optical phonon energies ( ~ ω ≃
200 meV), and ~ is the Planck con-stant. The right-hand side of Eq. (2) reflects the factthat each absorbed photon increases the 2DEHP energyby a quantity ~ Ω, while the absorption and emission ofoptical phonons change this energy by ± ~ ω . The Augerprocesses affect the carrier density, however, due to thezero energy gap, they do not vary the 2DEHP energy.The explicit expressions for the relaxation terms R inter , R intra , and R A are given (exemplified) in the AppendixA.The dispersion relations for electrons (upper sign) andholes (lower sign) in the GLs are presented as ε ± = ± v W p, (3)where v W ≃ cm/s is the characteristic carrier velocityin GL , p = | p | is the carrier momentum. Hence, for therate of the electron-hole pairs photogeneration we arriveat the following formulas: G = β tanh (cid:18) ~ Ω − µ T (cid:19) I Ω ( t ) . (4)Here β = π e /c ~ √ κ = ( π/ √ κ ), e and c are the elec-tron charge and light speed, respectively, κ is the dielec-tric constant of the layers surrounding the GL, T is the2DEHP temperature. The normally incident radiationflux, I Ω ( t ), comprises the steady-state component I Ω andthe modulation signal component I ω Ω exp( − iωt ): I Ω ( t ) = I Ω + I ω Ω exp( − iωt ) , where I ω Ω and ω are the amplitude and the frequency ofthe modulated signal.The absorption the THz and IR photons in GLs is dueto the interband (mainly direct optical transitions) andto the intraband (indirect transitions). At the photonfrequencies Ω ≫ τ − , where τ is the carrier momentumrelaxation time, the latter mechanism, which correspondsto the carrier Drude absorption, is relatively weak and,for simplicity, it is disregarded in Eq. (3) (although thismechanism was included into our model previously [9]).The carrier densities and their energy densities as func-tions of the effective temperature and the quasi-Fermienergy are given in the Appendix B, Eq. (B1).The GL conductivity is determined by the carrier den-sities, i.e., by the quasi-Fermi energy µ , and by theireffective temperature T . Assuming that the GL dcand low signal frequency conductivities, σ GL are deter-mined by the short-range scattering (on the defects andacoustic phonons), so that τ = τ ( T /pv W ) ∝ /p (forGLs) [3, 9], where τ is the characteristic short-rangescattering time [36], one can obtain the following for-mula: σ GL = 2 σ dark e − µ/T . (5)Here σ dark = ( e T τ /π ~ ) = σ is the intrinsic con-ductivity in the dark condition: T = T and µ = 0.At relatively weak irradiation, | µ | ≪ T , T , and Eq. (5)yields σ GL ≃ σ dark (cid:18) µ T (cid:19) . (6)According to Eq. (6), the terminal photocurrent acrossthe uniform GL, ∆ J GL = J GL − J darkGL , between the sidecontacts in the GL photoresistors can be presented as∆ J GL ≃ σ dark V H L µT , (7)so that its value normalized by the dark currents is givenby ∆ J GL J darkGL ≃ µ T . (8)Here 2 L is the spacing between the contacts and H is theGL width in the direction perpendicular to the current.The nanoconstriction [21, 22] constitutes a 1D chan-nel for electrons and holes (quantum wires or nanorib-bons [37–39]). Considering that the energy gap in thenanoconstriction as a function of the coordinate z alongthe carrier propagation direction is determined by thenanoconstriction length and width, 2 l and d , respectively,and the voltage drop between the nanoconstriction edges V pn . V : Φ( z ) = Φ max (1 − z /l ), can be presented asΦ max ≃ (∆ ∓ eV pn ) / eV np ≪ ∆) with ∆ = 2 π ~ v W /d .For an effective control of the net current the heightof the barrier (i.e., the energy gap ∆) should be suf-ficiently large to provide relatively small voltage drop across the uniform regions in the GL and GBL. In thiscase, V pn ≃ V , where V is the bias voltage between theside contacts. Accounting for this, the ratio of the netthermionic photocurrents currents, ∆ J P GL and ∆ J P GB ,in the photodiodes based on the perforated GLs (PGLs)over the barriers in the nanoconstriction [22], can be pre-sented as∆ J P GL J darkP GL ≃ (cid:18) ∆2 T (cid:19)(cid:20) µT + ( T − T )2 T (cid:21) . (9)One can see from the comparison of Eqs. (8) and (9)that the relative variations of the net currents under theirradiation in the case of the perforated GLs comprise thefactor ∆ / T , which can be large for sufficiently narrownanoconstrictions. III. STEADY-STATE CARRIER QUASI-FERMIENERGY, EFFECTIVE TEMPERATURE, ANDDC PHOTOCURRENT At | µ | ≪ T ≪ ~ ω , Eqs. (1) - (3) with Eqs. (A1)and (B1) result in the following values of the steady-state components of the quasi-Fermi energies µ and theeffective temperature T : µT = β (cid:18) a − Ω ω (cid:19) a + b + ab ) tanh (cid:18) ~ Ω4 T (cid:19) I Ω I , (10) T − T T ≃ β (cid:18) Ω ω − (cid:19)(cid:18) T ~ ω (cid:19) ( a + b + ab ) tanh (cid:18) ~ Ω4 T (cid:19) I Ω I . (11)Here a = (cid:18) π T ~ ω (cid:19) (cid:18) . T ~ ω (cid:19) , b = t /t A , t = Σ /I ,Σ = ( π/ T / ~ v W ) is the 2DEHP density at T = T , t A is the characteristic time of the Auger carrier genera-tion, I is the rate of the electron-hole pair generation inGLs due to the absorption of equilibrium optical phonons I ≃ (1 − × cm − s − [21]. At T = 25 meV, a ≃ .
196 and t ≃ −
168 ps.Equations (10) and (11) show that depending on theratio Ω /ω , the quasi-Fermi energies and the deviation ofthe effective temperatures from the lattice temperaturecan be both positive and negative. The radiation fre-quencies, at which the quasi-Fermi energies change theirsigns are the same (Ω = ω ), while the change of thesigns of ( T GL − T ) and ( T P GL − T ) occurs at somewhatdifferent Ω.Omitting the term b due to its smallness (see AppendixA), the dc photocurrents in the GL photoresistor andPGL photodiode can be presented as Photon energy, h _ Ω (meV) | R G L | ( µ m / µ W ) | R P G L | ( µ m / µ W ) × FIG. 2: Normalized dc responsivity of a GL photoresistor | R GL | (dashed line) and of a PGL photodiode | R PGL | (solidline) versus photon energy ~ Ω. ∆ J GL J darkGL ≃ β a (cid:18) a − Ω ω (cid:19) tanh (cid:18) ~ Ω4 T (cid:19) I Ω I , (12)∆ J P GL J darkP GL = β a (cid:18) ∆2 T (cid:19) (cid:20) a − Ω ω + (cid:18) T ~ ω (cid:19) (cid:18) Ω ω − (cid:19)(cid:21) tanh (cid:18) ~ Ω4 T (cid:19) I Ω I . (13)Equation (12) coincides with the pertinent formula ob-tained previously [9] for the case of the short-range carrierscattering. The expression for the GL photoconductivitygiven by Eq. (12) does not explicitly comprises the termassociated with the variation of the effective tempera-ture, i.e., the bolometric term. However, the variationof the quasi-Fermi energy and, particularly, the possiblechange of this quantity sign at a certain ratio Ω /ω , isrelated to the 2DEHP heating (or cooling). This impliesthat the bolometric effect in the GL photoresistors playsa role, although implicitly. In contrast to the GL photo-conductivity, the expression for the PGL photoconduc-tivity explicitly comprises the term associated with thebolometric effect. The bolometric contribution is pro-portional to a small parameter T / ~ ω = 0 . / T .As seen from Eqs. (12) and (13), the dc photoconduc-tivity changes the sign at a certain ratio Ω /ω . This isbecause at a large Ω, the radiation absorption leads to the2DEHP heating that results in a lowering of the electronquasi-Fermi level (it goes below the Dirac point to thevalence band) and in a elevating of the hole quasi- Fermilevel (which appears in the conduction band), while atΩ /ω <
1, the radiation absorption can give rise to the2DEHP cooling and, hence, to µ <
0. Figure 2 shows the spectral characteristics of the GLphotoresistor and PGL photodiode normalized dc re-sponsivities, | R GL | = | ∆ J GL /J darkGL ~ Ω I | and | R P GL | = | ∆ J P GL /J darkP GL ~ Ω I | , calculated using Eqs. (12) and (13).We set κ = 4, a = 0 . d ≃
10 nm).The most remarkable feature of the spectral character-istics of | R GL | and | R P GL | seen in Fig. 2 is that thesevalues turn to zero at a certain ratio Ω /ω . This isassociated with the equality of the energy received bythe 2DEHP from the absorbed radiation and the energytransmitted to the optical phonons when Ω /ω ≃
1. Asfollows from Eqs. (10) and (11), at the latter relation, µ and T − T become equal to zero, that leads to the zerovalues of the photocurrent. IV. MODULATED QUASI-FERMI ENERGY,EFFECTIVE TEMPERATURE, AND ACPHOTOCURRENT
When the incident radiation comprises the ac modu-lation component I ω Ω ( t ), we arrive at the following equa-tions for the ac components of the quasi-Fermi energyand effective temperature, µ ω and T ω :2(1 − iωt (1) ) µ ω T + (1 − iωt (2) ) (cid:18) ~ ω T (cid:19) T ω T = β tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I , (14)2(1 − iωt (2) ) µ ω T + (1 + a )(1 − iωt (3) ) (cid:18) ~ ω T (cid:19) T ω T = β (cid:18) Ω ω (cid:19) tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I , (15)which in the limit ω → t (1) , t (2) ,and t (3) are given in the Tables I and II and are be-ing proportional to the time t = Σ /I related to theelectron-hole pair generation rate associated with the op-tical phonon absorption in GLs. Assuming that I =(1 − × cm − s − at T = 25 meV, for b ≪ t ≃ (84 − c ≃ .
58 [seeAppendix B, Eq. B(6)]The times t (1) and t (3) are the effective carrier recom-bination and cooling/heating times, respectively. Theinclusion of the Auger recombination leads to a decreasein the recombination time t (1) . One can see that t (3) isproportional to the product of the heat capacitance perone carrier c and the carrier energy relaxation time dueto the interaction with optical phonons t ( T / ~ ω ) (thatis in line with the previous calculation of this time in the2D systems [4]). Despite a relatively short time t (3) , it islonger than one could assume in the case of the classicalvalue c = 1 (see the Appendix B). TABLE I: GL, PGL parameters (definitions) t (1) /t t (2) /t t (3) /t GL, PGL 6 ln 2 π (1 + b ) 2 T ~ ω (cid:18) T ~ ω (cid:19) c (1 + a )TABLE II: GL, PGL parameters (numerical values) t (1) (ps) t (2) (ps) t (3) (ps)GL, PGL 35 – 70 21 – 42 7 – 14 For the 2DEHP in the GL, the hierarchy of the charac-teristic times is as follows: t (1) > t (2) > t (3) . One can seefrom Eqs. (14) and (15) that µ ω substantially depends on T ω . In particular, an increase in T ω leads to a decreasein µ ω . Equations (14) and (15) result in µ ω T = β Z ω Ω (1 − iωt (1) ) tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I , (16) T ω T = β (cid:18) T ~ ω (cid:19) (1 − Z ω Ω )(1 − iωt (2) ) tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I , (17)where Z ω Ω = 1 + a − (cid:18) Ω ω (cid:19) (1 − iω t (2) )(1 − iω t (3) a − (1 − iω t (2) ) (1 − iωt (1) )(1 − iω t (3) ) . (18)Calculating the ac components of the ac photocurrentcurrent ∆ J ωGL as in the previous section, we arrive at thefollowing equation:∆ J ωGL J darkGL = β Z ω Ω (1 − iωt (1) ) tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I . (19)Analogously, invoking Eq. (9), for the ac photocurrent inthe PGL photodiodes we obtain∆ J ωP GL J darkP GL = β (cid:18) ∆ GL T (cid:19) (cid:20) Z ω Ω (1 − iωt (1) )+ (cid:18) T ~ ω (cid:19) (1 − Z ω Ω )(1 − iωt (2) ) (cid:21) tanh (cid:18) ~ Ω4 T (cid:19) I ω Ω I . (20)
V. MODULATION DEPTH
The modulation depths normalized by the dark cur-rents (in units µ m / µ W) can be defined as M ωGL = (cid:18) ∆ J ωGL J darkGL (cid:19) ~ Ω I ω Ω , M ωP GL = (cid:18) J ωP GL J darkP GL (cid:19) ~ Ω I ω Ω , (21) for the GL photoresistors and the PGL photodiodes, re-spectively.Figure 3 shows the absolute value (modulus) of themodulation depths | M ωGL | and | M ωP GL | as functions ofthe modulation frequency ω calculated using Eqs. (21)with Eqs. (19) and (20) for different carrier frequenciesΩ / π (different photon energies ~ Ω). It is assumed that I = 2 × cm − s − and, hence, t = 84 ps, t (1) =35 ps, t (2) = 21 ps , and t (3) = 7 ps (see Table II). Otherparameters are the same, as in Fig. 2, namely, κ = 4, a = 0 . < ω . the modulation depth decreases withthe modulation frequencies exceeding several GHz. AtΩ > ω , a marked decrease in | M ωGL | and | M ωP GL | takesplace starting at more than 10 GHz.It is interesting that at Ω > ω , the dependences of | M ωGL | and | M ωP GL | on the modulation frequency ω/ π exhibit maxima at ω/ π about several MHz at which | M ωGL | and | M ωP GL | exceed | M GL | and | M P GL | . Thiscan be explained by a change in the phase shift betweenthe quasi-Fermi and the effective temperature oscillationswith a change of the modulation frequency. Modulation frequency, ω /2 π (GHz) | M ω P G L | ( µ m / µ W ) | M ω G L | ( µ m / µ W ) (a)(b) h _ Ω = 75 meV100175325250 h _ Ω = 75 meV100175250 325 FIG. 3: Modulation depth (a) in GL photoresistor | M ωGL | and(b) in PGL photodiode | M ωPGL | versus modulation frequency ω/ π for different photon energies ~ Ω. VI. COMMENTSA. Comparison of GL and PGL photodetectors
Using Eqs. (8), (9), (13), (14), (19), and (20), we canarrive at the following ratios of the photocurrents nor-malized by the dark currents in the GL photoresistorsand the PGL photodiodes:∆ J P GL /J darkP GL ∆ J GL /J darkGL ∼ δJ ωP GL /J darkP GL ∆ J ωGL /J darkGL ∼ (cid:18) ∆ GL T (cid:19) ≫ . (22) B. Comparison of GL and PGL photodetectorswith reverse-biased GLD photodiodes
The photocurrents ∆ J GL = J GL − J darkGL in the GL pho-todetectors under consideration and ∆ J GLD = J GLD − J darkGLD in the p-i-n photodiodes [41] at a reverse strongbias providing the depletion of the GL with not too longabsorbing GL (with the spacing between the side contact2 L . L D , where L D is the drift length) can be estimatedas follows [41]:∆ J GLD LH = 2 eβ tanh (cid:18) ~ Ω − µ T (cid:19) I Ω . (23)The dark current associated with the carrier generationdue to the interband absorption of optical phonons in theGLDs is given by J darkGLD LH = 2 eI. (24)Using Eq. (12), for the ratios of the dc photocurrents(detector responsivities) normalized by the dark currentswe obtain η GL = ∆ J GL /J darkGL ∆ J GLD /J darkGLD ≃ a (cid:18) a − Ω ω (cid:19) , (25) η P GL = ∆ J P GL /J darkP GL ∆ J GLD /J darkGLD ≃ a (cid:18) ∆2 T (cid:19) (cid:20) a − Ω ω + (cid:18) T ~ ω (cid:19) (cid:18) Ω ω − (cid:19)(cid:21) . (26)One can see from Eq. (25) that at Ω < ω , i.e., in theTHz and far-IR spectral ranges η GL ∼
1. If Ω > ω (mid- and near-IR ranges), | η GL | can markedly exceedunity. Due to a large factor (∆ / T ), | η P GL | can beparticularly large in a wide spectral range. Indeed, if,for example, ∆ = 400 meV and ~ Ω = 40 −
100 meV,Eq. (26) yields η P GL ≃ −
18. The GL-photoresistor and PGL photodiode responsiv-ities R GL = ∆ J GL / LHI Ω and R P GL = ∆ J P GL / LHI Ω can be derived using Eqs. (12) and (13): R GL ≃ β e π ~ a (2 L ) I (cid:18) eV τ ~ (cid:19)(cid:18) a − Ω ω (cid:19) × tanh( ~ Ω / T )( ~ Ω / T ) , (27) R P GL ≃ β eN π ~ a (2 LH ) I (cid:18) eVT (cid:19)(cid:18) ∆2 T (cid:19) exp (cid:18) − ∆2 T (cid:19) × (cid:20) a − Ω ω + (cid:18) T ~ ω (cid:19) (cid:18) Ω ω − (cid:19)(cid:21) tanh( ~ Ω / T )( ~ Ω / T ) . (28)Assuming I = 2 × cm − s − , τ = 1 ps, 2 L =10 µ m, and V = 100 mV for ~ Ω = 100 meV, we find R GL ≃ .
04 A/W. For this photon energy, one obtains R GLD ≃ .
17 A/W. Setting 2 L = H = 10 µ m and N =100, for the same voltage and photon energy, we obtain R P GL ≃ . (cid:18) ∆2 T (cid:19) exp (cid:18) − ∆2 T (cid:19) A/W. If ∆ = 125 meV(corresponding to a smaller ∆ / T than in the above es-timate of η P GL ), one obtains R P GL ≃ .
19 A/W. Theseestimates show that the PGL photodiodes can surpassthe GLD photodiodes in the parameter η P GL because oflow values of the dark current, exhibiting, however, theresponsivities of the same order of magnitude.
C. Role of Auger processes
Due to a smallness of the parameter b , the Auger pro-cesses weakly affect the characteristics of the photodetec-tors under consideration. This in particular, implies thatdespite a weak temperature dependence of the uniformGL conductivity, its photoresponse can be marked dueto the deviation of the quasi-Fermi levels from the Diracpoint cased by the 2DEHP heating or cooling. This iscontrast to the situation when a relatively dense 2DEHPis hot [42–50], so that the Auger processes in GLs arerather effective [34]. The latter forces the quasi-Fermilevels be very close to the Dirac point ( µ ≃
0) [51].
D. Assumptions
The screening lengths in GLs with µ ≃ l GL = κ ~ e v W T . (29)For κ = 4, from Eq. (22) we obtain l GL ≃ . λ GL = 2 π ~ v W /T ≃
150 nm.This justifies the assumption of our model that theionized impurities are effectively screened, so that thecarrier scattering on such impurities, point defects, andacoustic phonons is a short-range scattering.In Eq. (2) we used the quantity G given by Eq. (4). Thelatter ignores the contribution of the intraband photonabsorption by the carriers (the Drude absorption). Thisis justified if (see, for example, [9])tanh (cid:18) ~ Ω4 T (cid:19) ≫ D τ /π , (30)where D = (4 T τ /π ~ ) stands for the Drude factor. In-equality (30) yields the following condition: ~ Ω ≫ r π ~ T τ = ~ Ω . (31)For T = 25 meV and τ = 0 . − . ~ Ω ≃ −
25 meV. The photon energies assumed in theabove calculations are much larger. The generalizationof our model for smaller ~ Ω is rather simple.The biased voltage can lead to some 2DEHP Jouleheating. Considering that the Joule power in the GLper unit of its area Q J ≃ σ E , one can find the condi-tion that the pertinent variation of the 2DEHP temper-ature ( T J − T ) /T ≪ E ≪ E J = p π a I/τ ( ~ /e )( ~ ω /T ). For the parametersused in the above estimates E J ≃ (175 − r P GL can beexpressed as r P GL = 2( L − l ) Hσ GL + r NC . (32)Here σ GL is given by Eq. (5) and r NC = π ~ e N exp (cid:18) µ + ∆ / T (cid:19) is the resistance of the perforated area for a parabolicpotential distribution along the nanoconstrictions [seeEq. (13) in [22]] with N being the number of the nanocon-strictions. Hence, the variation of the nanoconstrictionresistance (assumed above) is much larger than that ofthe uniform parts of the GL ifexp (cid:18) ∆2 T (cid:19) ≫ N LH ~ T τ , (33)i.e., ∆ > T ln (cid:20) N ( L − l ) H ~ T τ (cid:21) . (34) Setting 2( L − l ) = 10 µ m, H = 10 µ m, N = 10( h = H/N = 1 µ m) and τ = 0 . N = 100 and τ = 1 . T = 25 meV we find that, according toinequality (33), a marked distinction in the GL and PGLphotodetectors requires ∆ >
115 meV.
VII. CONCLUSIONS
We developed analytical models for the uncooled GLand PGL photodetectors. Their operation is linked tothe correlated variations of the carrier density (i.e., quasi-Fermi energies) and effective temperature associated withthe interband absorption of the incident modulated ra-diation. We derived the modulation characteristic asa function of the carrier and modulation frequencies,demonstrated that the variation of both the carrier den-sity and the effective temperature are essential for theGL- and PGL-based photodetectors, and showed thatthe value 2DEHP capacity (markedly different form theclassical value) affects the response time. The compar-ison of photodetectors under consideration showed thatthe presence of the energy barrier for the carriers in thenanoconstrictions in the PGL photodiodes promotes thehigher photodetectors performance. We also comparedthe the GL- and PGL-based photodetectors with the in-terband photodiodes with the reverse-bias depleted GL.As shown the former can surpass the latter somewhatslower devices.
ACKNOWLEDGMENTS
The Japan Society for Promotion of Science (KAK-ENHI
DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
Appendix A. The density and energy relaxationrates
The rates of the interband and intraband processes in-volving the optical phonons can be presented as: R interGL = I (cid:20) exp (cid:18) ~ ω T − ~ ω − µT (cid:19) − (cid:21) , (A1)and R intraGL = aI (cid:20) exp (cid:18) ~ ω T − ~ ω T (cid:19)(cid:21) − . (A2)Here I is the rate of the electron-hole pair generation dueto the absorption of equilibrium optical phonons in theGLs, the quantities a is the ratio of the rates of the intra-band and interband transitions in the GLs accompaniedby the absorption of an optical phonon. This quantityis mainly determined by the energy dependences of thedensities of states around the Dirac point. For GLs, at | µ | ≪ T , a ≃ ( π T / ~ ω ) (1 + 2 . T / ~ ω ) ≃ .
196 [4].According to the calculation [30], at T = 25 meV,for the intra-valley and inter-valley optical phonons inGLs, I ≃ (1 − × cm − s − [21]. The characteristictime of the interband transitions with absorption of anoptical phonon can be defined as τ = Σ /I , where Σ =( π/ T ~ v W .The rates of the Auger recombination-generation pro-cesses are assumed in the following form: R AGL = A GL (cid:20) exp (cid:18) µT (cid:19) − (cid:21) , (A3)where A GL is the rate of the electron-hole pair genera-tion due to the impact Auger processes in the 2DEHPwith the equilibrium carrier density. In GLs these pro-cesses are characterized by t A ∝ A − GL so that b = t /t A = A GL /I . Since the time of the carrier recom-bination due to the spontaneous optical phonon emission t R ≃ t exp( − ~ ω /T ) ≪ t . The characteristic time t A can be estimated as t A ∼ t RA exp( ~ ω /T ). Setting t RA & (0 . − .
0) ps [34], we find that the parameter t A ( ≃ . − × ps, so that b ≃ . − . Appendix B. GL carrier and energy densities versuscarrier quasi-Fermi energy and effective temperature
The net carrier (electrons and holes) densities Σ in theGL in line with Eq. (2) is given byΣ = 2 π ~ v W Z ∞ dεε (cid:20)
11 + exp (cid:18) ε − µ e T (cid:19) + 11 + exp (cid:18) ε − µ h T (cid:19) (cid:21) . (B1) The density of the carrier energy in the 2DEHP canbe calculated as E = 2 π ~ v W Z ∞ dεε (cid:20)
11 + exp (cid:18) ε − µ e T (cid:19) + 11 + exp (cid:18) ε − µ h T (cid:19) (cid:21) . (B2)In the undoped GLs, the electron and hole densities areequal to each other. In this case, due to a symmetry ofthe valence and conduction bands, µ e = µ h = µ . Exceptthe situations when GLs and GBLs are strongly opticallyor injection pumped with the generation of cold or warmcarriers, | µ e + µ h | = 2 | µ | ≪ T . In such cases, Eqs. (B1)and (B2) yield [40]Σ = (cid:18) T ~ v W (cid:19) (cid:18) π π µT (cid:19) , (B3) E ≃ T π ~ v W (cid:20) ζ (3) + π µT (cid:21) , (B4)where ζ ( x ) is the Riemann zeta function: ζ (3) ≃ . / C GL = ∂ E GL /∂T , from Eq. (B4) we obtain C ≃ T π ~ v W (cid:20) ζ (3) + 2 π µT (cid:21) , (B5)so that the 2DEHP capacity, c = C/ Σ (in units of theBoltzmann constant k B ), per one carrier is given by c = 54 ζ (3) π ≃ . . (B6)This value is markedly different from the classical valuefor nondegenerate 2D systems c D = 1 [40]. [1] V. Ryzhii, M. Ryzhii, and T. Otsuji, “Negative dynamicconductivity of graphene with optical pumping,”J. Appl.Phys. , 083114 (2007).[2] F. T. Vasko and V. Ryzhii, “Photoconductivity of intrin- sic graphene,”Phys. Rev. B , 195433 (2008).[3] A. Satou, F. T. Vasko, and V. Ryzhii, “Nonequilibriumcarriers in intrinsic graphene under interband photoexci-tation,”Phys. Rev. B , 115431 (2008). [4] V. Ryzhii, M. Ryzhii,V. Mitin, A. Satou, and T. Ot-suji, “Effect of heating and cooling of photogeneratedelectron-hole plasma in optically pumped graphene onpopulation inversion,”Jpn. J. Appl. Phys. , 094001(2011).[5] S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou,V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics andterahertz emission in optically pumped graphene at roomtemperature,”Phys. Rev. B , 035443 (2012).[6] T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J.Schmalian, and J.Wang, “Femtosecond population inver-sion and stimulated emission of dense Dirac fermions ingraphene,”Phys. Rev. Lett. , 167401 (2012).[7] I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. E.Turcu, E. Springate, A. St¨ohr, A. K¨ohler, U. Starke, andA. Cavalleri, “Snapshots of non-equilibrium Dirac carrierdistributions in graphene,”Nat. Mater. , 1119 (2013).[8] J. N. Heyman, J. D. Stein, Z. S. Kaminski, A. R. Ban-man, A. M. Massari, and J. T. Robinson, “Carrier heat-ing and negative photoconductivity in graphene,”J. Appl.Phys. , 015101 (2015).[9] V. Ryzhii, D. S. Ponomarev, M. Ryzhii, V. Mitin, M. S.Shur, and T. Otsuji, “Negative and positive terahertz andinfrared photoconductivity in uncooled graphene,”Opt.Mat. Express , 585 - 597 (2019).[10] M. Shur, A.V. Muraviev, S. L. Rumyantsev, W. Knap,G. Liu, and A. A. Balandin, “Plasmonic and bolometricterahertz graphene sensors,”Proc. of 2013 IEEE SensorsConf., 978-1- 4673-4642-9/13/ 2013 IEEE pp. 1688-1690(2013).[11] J. Yan, M.-H. Kim, J. A. Elle, A.B. Sushkov, G.S.Jenkins, H.M. Milchberg, 3, M.S. Fuhrer, and H.D.Drew, “Dual-gated bilayer graphene hot electron bolome-ter,”Nat. Nanotech. , 472–478 (2012).[12] V. Ryzhii, T. Otsuji, M. Ryzhii, N. Ryabova, S. O.Yurchenko, V. Mitin, and M. S. Shur, “Graphene ter-ahertz uncooled bolometers,”J. Phys. D: Appl. Phys. ,065102 (2013).[13] Q Han, T. Gao, R. Zhang, Y. Chen, J.Chen, G. Liu, Y.Zhang, Z. Liu, X. Wu, and D. Yu, “Highly sensitive hotelectron bolometer based on disordered graphene,”Sci.Reports , 3533 (2013).[14] X. Cai, A. B. Sushkov, R J. Suess, M. M. Jadidi, G.S. Jenkins, L. O. Nyakiti, R. L. Myers-Ward, S. Li, J.Yan, D. K. Gaskill, T. E. Murphy, H. D. Drew, and M.S. Fuhrer, “Sensitive room-temperature terahertz detec-tion via the photothermoelectric effect in graphene,”Nat.Nanotech. , 814-819 (2014).[15] X. Du, D. E. Prober, H. Vora, and Ch. B. Mckitterick,“Graphene-based bolometers,”Graphene 2D Mater. , 1-22 (2014).[16] A. El Fatimy, R. L. Myers-Ward, A. K. Boyd, K.M. Daniels, D. K. Gaskill, and P. Barbara, “Epitaxialgraphene quantum dots for high-performance terahertzbolometers,”Nat. Nanotechnol. , 335-338 (2016).[17] S. Yuan, R. Yu, C. Ma, B. Deng, Q. Guo, X. Chen, C.Li, C. Chen, K. Watanabe, T. Taniguchi, F. J. Garc´ıa deAbajo, and F. Xia, “Room temperature graphene mid-infrared bolometer with a broad operational wavelengthrange,”ACS Photonics , 1206-1215 (2020).[18] V. Ryzhii, M. Ryzhii, D. Ponomarev, V. G. Leiman, V.Mitin, M. Shur, and T. Otsuji, “Negative photoconduc-tivity and hot-carrier bolometric detection of terahertzradiation in graphene-phosphorene hybrid structures,”J. Appl. Phys. 125, 151608 (2019).[19] A. Blaikie, D. Miller, and B. J. Alem´an, “A fastand sensitive room-temperature graphene nanomechani-cal bolometer,”Nat. Comm. , 4726 (2019).[20] G.-H. Lee, D. K. Efetov, L. Ranzani, E. Walsh, J.Crossno, T. A. Ohki, T. Taniguchi, K. Watanabe, P.Kim, D. Englund, and K. C. Fong, “Graphene-basedJosephson junction microwave bolometer,”Nature ,42 – 46 (2020).[21] G. S. Simin, M. Islam, M. Gaevski, J. Deng, R. Gaska,and M. S. Shur, “Low RC-constant perforated-channelHFET,”IEEE Electron Device Lett. , 452-454 (2014).[22] V. Ryzhii, M. Ryzhii, M. S. Shur, V. Mitin, A. Satou, andT. Otsuji, “Resonant plasmonic terahertz detection ingraphene split-gate field-effect transistors with lateral p-n junctions, ”J. Phys. D: Appl. Phys. , 315103 (2016).[23] R. P. Panmand, P.Patil, Y Sethi, S. R. Kadam, M.V. Kulkarni,S. W. Gosavi, N. R. Munirathran, and B.B. Kale, “Unique perforated graphene derived from-Bougainvilleaflowers for high-powersupercapacitors: agreen approach,”Nanoscale , 4801 (2017).[24] A. Guirguisa, J. W. Maina, L. Kong, L. C. Henderson, A.Rana, L. H. Li, M. Majumder, L. F. Dumee, “Perforationroutes towards practical nano-porous graphene andanal-ogous materials engineering ”Carbon , 660 (2019).[25] R. A. Suris and V. A. Fedirko, “Heating photoconduc-tivity in a semiconductor with a superlattice,”Sov. Phys.Semicond. , 629 (1978).[26] E. A. Shaner, A. D. Grine, M. C. Wanke, M. Lee, J.R. Reno, and S. J. Allen, “Far-infrared spectrum anal-ysis using plasmon modes in a quantum-well transistor,”IEEE Photonics Technol. Lett. , 1925 (2006).[27] V. Ryzhii, A. Satou, T. Otsuji, and M. S. Shur, “Plasmamechanism of resonant terahertz detection in a two-dimensional electron channel with split gates,”J. Appl.Phys. , 014504 (2008).[28] V. Ryzhii, A. Satou, T. Otsuji, M. Ryzhii, V. Mitin, andM. S. Shur, “Graphene vertical hot-electron terahertz de-tectors,”J. Appl. Phys. , 114504 (2014).[29] M. Ryzhii, V. Ryzhii, V. Mitin, M. S. Shur, andT. Otsuji, “Vertical hot-electron terahertz detectorsbased on black-As − x P x /graphene/black-As − x P x het-erostructures,”Sensors and Materials , 2271-2279(2019).[30] F. Rana, P. A. George , J. H. Strait, S. Sharavaraman, M.Charasheyhar, and M. G. Spencer, “Carrier recombina-tion and generation rates for intravalley and intervalleyphonon scattering in graphene,”Phys. Rev. B , 115447(2009).[31] H. Wang, J. H. Strait, P. A. George, S. Shivaraman,V. D. Shields, M. Chandrashekhar, J. Hwang, F. Rana,M. G. Spencer, C. S. Ruiz-Vargas, and J. Park, “Ul-trafast relaxation dynamics of hot optical phonons ingraphene,”Appl. Phys. Lett. , 081917 (2010).[32] J. M. Iglesias, M. J. Mart´ın, E. Pascual, and R. Rengel,“Hot carrier and hot phonon coupling during ultrafastrelaxation of photoexcited electrons in graphene,”Appl.Phys. Lett. , 043105 (2016).[33] M. S. Foster and I. L. Aleiner, “Slow imbalance relaxationand thermoelectric transport in graphene,”Phys. Rev. B , 085415 (2009).[34] G. Alymov, V. Vyurkov, V. Ryzhii, A. Satou, and D.Svintsov, “Auger recombination in Dirac materials: Atangle of many-body effects,”Phys. Rev. B , 205411 (2018).[35] V. Ryzhii, T. Otsuji, M. Ryzhii, V. E. Karasik, and M.S. Shur, “Negative terahertz conductivity and amplifi-cation of surface plasmons in graphene–black phospho-rus injection laser heterostructures,”Phys. Rev. B Quantum Wells: Physics and Electronics ofTwo-Dimensional systems (World Scientific Singapore,1997)[37] V. V. Mitin, D. I. Sementsov, and N. Vagidov,
QuantumMechanics for Nanostructures (Cambridge Univ. Press,Cambridge, 2010).[38] G. Liang, N. Neophytou, M. S. Lundstrom, and D E.Nikonov, “Contact effects in graphene nanoribbon tran-sistors,”Nano Lett. , 1819-1824 (2008).[39] K.-T. Lam and G. Liang, “Electronic structure of bi-layer graphene nanoribbon and its device application: Acomputational study,”NanoSci. and Technol. , 509-527(2012).[40] V. Ryzhii, M. Ryzhii, T. Otsuji, V. Mitin, and M. S. Shur,“Heat capacity of nonequilibrium electron-hole plasma ingraphene layers and graphene bilayers,”arXiv:2011.03739[cond-mat.mes-hall].[41] V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Tera-hertz and infrared photodetection using p-i-n multiple-graphene layer structures,”J. Appl. Phys. , 054512(2010).[42] M. Freitag, H.-Y. Chiu, M. Steiner, V. Perebeinos,and P. Avouris, ”Thermal infrared emission from biasedgraphene,” Nat. Nanotech. , 497–501 (2010).[43] Y. D. Kim, H. Kim, Y. Cho, Ji 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.-Ho Bae, and Y. D.Park, “Bright visible light emission from graphene,”Nat.Nanotech. , 676–681 (2015).[44] H. R.Barnard, E. Zossimova, N. H. Mahlmeister, L.M. Lawton, I. J. Luxmoore, and G. R. Nash, “Boronnitride encapsulated graphene infrared emitters,”Appl. Phys. Lett. , 131110 (2016).[45] S.-K. Son, M. ˘S i ˘s kins, C. Mullan, J. Yin, V. G.Kravets, A. Kozikov, S. Ozdemir, M. Alhazmi, M. Hol-will, K. Watanabe, T. Taniguchi, D. Ghazaryan, K. S.Novoselov, V. I. Fal’ko, and A. Mishchenko, “Graphenehot-electron light bulb: incandescence from hBN encap-sulated graphene in air, ”2D Materials , 011006 (2017).[46] H. M. Dong, W. Xu, and F. M. Peeters, “Electri-cal generation of terahertz blackbody radiation fromgraphene,”Opt. Express (19), 24621–24626 (2018).[47] R.-J. Shiue, Y. Gao, C. Tan, C. Peng, J. Zheng, D. K.Efetov, Y. D. Kim, J. Hone, and D. Englund, “Thermalradiation control from hot graphene electrons coupled toa photonic crystal nanocavity,”Nat. Commun. , 109(2019).[48] F. Luo, Y. Fan, G. Peng, S. Xu, Y. Yang, K. Yuan,J. Liu, W. Ma, W. Xu, Z. H. Zhu, X.-Ao Zhango, A.Mishchenko, Yu Ye, H. Huang, Z. Han, W. Ren, K. S.Novoselov, M. Zhu, and S. Qin, “Graphene thermal emit-ter with enhanced Joule heating and localized light emis-sion in air,”ACS Photonics , 2117–2125 (2019).[49] Y. D. Kim, Y. Gao, R.-J. Shiue, L. Wang, O. B. Aslan,M.-Ho Bae, H. Kim, D. Seo, H.-J. Choi, S. H. Kim, A.Nemilentsau, T. Low, C. Tan, D. K. Efetov, T. Taniguchi,K. Watanabe, K. L. Shepard, T. F. Heinz, D. Englund,and J. Hone, “Ultrafast graphene light emitters,”NanoLett. , 934–940 (2018).[50] V. Ryzhii, T. Otsuji, M. Ryzhii, V. Leiman, P. P. Malt-sev, V. E. Karasik, V. Mitin, and M. S. Shur, “Theo-retical analysis of injection driven thermal light emittersbased on graphene encapsulated by hexagonal boron ni-tride,”Opt. Mat. Express , 468 (2021).[51] V. Ryzhii, M. Ryzhii, D. Svintsov, V. Leiman, P. P.Maltsev, D. S. Ponomarev, V. Mitin, M. S. Shur, andT. Otsuji, “Real-space-transfer mechanism of negativedifferential conductivity in gated graphene-phosphorenehybrid structures: Phenomenological heating model,”J.Appl. Phys.124