Device model for pixelless infrared image up-converters based on polycrystalline graphene heterostructures
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Device model for pixelless infrared image up-converters based on polycrystallinegraphene heterostructures
V. Ryzhii , , , M. S. Shur , M. Ryzhii , V. E. Karasik , and T. Otsuji Research Institute of Electrical Communication,Tohoku University, Sendai 980-8577, Japan Institute of Ultra High Frequency Semiconductor Electronics of RAS,Moscow 117105, Russia Center for Photonics and Infrared Engineering,Bauman Moscow State Technical University,Moscow 111005, Russia Department of Electrical,Computer, and Systems Engineering,Rensselaer Polytechnic Institute,Troy, New York 12180, USA Department of Computer Science and Engineering,University of Aizu, Aizu-Wakamatsu 965-8580, Japan
We develop a device model for pixelless converters of far/mid-infrared radiation (FIR/MIR)images into near-infrared/visible (NIR/VIR) images. These converters use polycrystalline graphenelayers (PGLs) immersed in the van der Waals (vdW) materials integrated with light emitting diode(LED). The PGL serves as an element of the PGL infrared photodetector (PGLIP) sensitive tothe incoming FIR/MIR due to the interband absorption. The spatially non-uniform photocurrentgenerated in the PGLIP repeats (mimics) the non-uniform distribution (image) created by theincident FIR/MIR. The injection of the nonuniform photocurrent into the LED active layer resultsin the nonuniform NIR/VIR image reproducing the FIR/MIR image. The PGL and the entirelayer structure are not deliberately partitioned into pixels. We analyze the characteristics of suchpixelless PGLIP-LED up-converters and show that their image contrast transfer function and theup-conversion efficiency depend on the PGL lateral resistivity. The up-converter exhibits highphotoconductive gain and conversion efficiency when the lateral resistivity is sufficiently high.Several teams have successfully demonstrated the large area PGLs with the resistivities varyingin a wide range. Such layers can be used in the pixelless PGLIP-LED image up-converters. ThePGLIP-LED image up-converters can substantially surpass the image up-converters based on thequantum-well infrared photodetector (QWIP) integrated with the LED. These advantages are dueto the use of the interband FIR/NIR absorption and a high photoconductive gain in the GLIPs.
Keywords : graphene; van der Waals heterostructure; infrared photodetector; image up-conversion.
I. INTRODUCTION
The main problem in the transformation of far-infraredradiation (FIR), mid-infrared radiation (MIR), or near-infrared radiation (NIR) images, into visible (VIR) oreven ultraviolet images is the availability of the pertinentdetector technology. Despite tremendous success associ-ated with CCD and CMOS digital technology, imagingat relatively long wavelengths where silicon is ”blind”,is very complicated and expensive [1, 2]. Therefore, thedemand for practical devices effectively converting FIR,MIR, and NIR images to VIR images is very strong.Different approaches have been explored, including ther-mal imaging, nonlinear up-conversion and photochemicalup-conversion based on sensitized triplet-triplet annihi-lation, and others [2–4]. In particular, the integrationof the quantum-well infrared photodetectors (QWIPs)with the light-emitting diodes (LEDs) for the image up-conversion was proposed and implemented almost twodecades ago [5–11]. However, despite reasonable char-acteristics of the QWIP-LED image up-converters, theyhave not found wide applications because of the incline incidence requirement (or the necessity to use special ra-diation couplers), a relatively low conversion efficiency(due to a relatively low intersubband radiation absorp-tion and the absence of the photoelectric gain resultingin a modest contrast transfer). Technologically, the real-ization of effective QWIP-LED up-converter requires theformation of large area multiple-QW heterostructures.Some drawbacks of the QWIP-LED image up-convertersmight be eliminated in the image up-converters based onthe integration of quantum-dot infrared photodetectors(QDIPs) [16] and QD- or QW-LEDs as was proposed inRef. [17]. But this idea was not realized yet, althoughthe lamp (pixell) QD-based up-converters were recentlyreported [18–20].Recently, we proposed to use the graphene-layer in-frared photodetectors (GLIPs) integrated with the light-emitting diodes (LEDs) for the photon energy up-conversion leading to the transformation of far/mid-infrared (FIR/NIR) signals into near-infrared/visible(NIR/VIR) signals [12]. In such GLIP-LED up-converters, the photocurrent produced in the GLIP partof the device due to the FIR/MIR interband absorp- P G L V h w p + h W N I R / V I R p h o t o n s ` PL E DG L I P NN n h W N I R / V I R p h o t o n sF I R / M I R p h o t o n s h w F I R / M I R p h o t o n sE m i t t e r
FIG. 1: Schematic view of (a) the PGLIP-LED up-converter structure. Wavy arrows correspond to the incident photons(with the energy ~ ω ) and the photons generated in the LED part (with the energy ~ Ω). Arrows indicate passes of the electronsinjected from the emitter n-emitter layer and those excited from the GL by FIR/MIR photons. ( b ) E E = E C W h h w E E < E C P G L W h h w E E > E C W h h w ( a ) ( c ) FIG. 2: Schematic view of the PGLIP-LED up-converter band diagrams with the emitter electric field E E (a) smaller thanthe collector field E C , (b) E E = E C , and (c) E E > E C , respectively. Dashed arrows correspond to the electron capture intothe GLs. tion [13–15] is injected into a LED resulting in the emis-sion of NIR/VIR. The GLIP-LED elements can form pix-els of the system (which consists of an array of such pix-els) up-converting the FIR/MIR images. In this paper,we show that the GLIPs with the large area macroscop-ically uniform and sufficiently resistive polycrystallineGLs (PGLs) (not intentially partitioned into pixels) in-tegrated with the large area LEDs can up-convert theFIR/MIR images. We develop the device model for thepixelless PGLIP-LED image up-converters and evaluatetheir characteristics. The operation of the PGLIP-LEDup-converters is associated with the injection of the spa-tially nonuniform photocurrent produced in the GLIPpart of the device by the spatially nonuniform FIR/MIR(FIR/MIR image) into its LED part resulting in the emis-sion of the spatially nonuniform NIR/VIR (i.e.,NIR/VIRimage). The pixelless PGLIP-LED image up-converterscan be implemented in the heterostructures with thePGL and the barrier layers made of different materials,in particular, the so-called van der Waals (vdW) materi-als [21–27] (hBN, WS , WSe , and many others) and us-ing these materials for the LED part of such devices [28–31]. A weak inter-layer bonding enables effective stackingof these layers with different lattice constants. The pix-elless PGLIP-LED image up-converters can surpass the pixeless QWIP-LED image up-converters due to:(i) the GL (and PGL) sensitivity to the normally inci-dent input FIR/MIR [32] because of the use of the inter-band transitions (avoiding the need for FIR/MIR cou-pling structures);(ii) a higher probability of the direct or followed by tun-neling electron photoexcitation from the GLs (than thatfrom QWs) into the continuum states above the inter-GLbarriers [32–35];(iii) the photoconductive gain due to the possibility ofnonuniform lateral potential distribution formation inthe PGLs with relatively high lateral resistivity under thenonuniform incident radiation (such a gain occurs due toa low probability of the capture of the electrons into theGL [36] and can provide substantially higher contrast ofthe output images and elevated up-conversion efficiency);(iv) easy fabrication due to the robust technology of largesize formation of PGLs [37–41] with relatively low con-ductivity due to their polycrystalline nature (with thescattering of charge carriers at grain boundaries degrad-ing their performance relative to exfoliated, single-crystalgraphene) and as well as due to other types of disor-der [41–47].In contrast to the QWIP-LED image up-converters inwhich multiple-QW structures are indispencible [5, 8],the PGLIP-LED devices can comprise a single PGL.These advantages of the pixelless PGLIP-LED imageup-converters should stimulate their implementation anduse in different applications. II. DEVICE STRUCTURE AND MODEL
Figure 1 shows the PGLIP-LED device structure witha single PGL sandwiched by the N-barrier layers and withthe top emitter n-layer. The structure comprises also thep + -layer (on the P-type substrate), which serves as theactive region of the LED part of the device. Figures 2(a)- 2(c) show the band diagrams corresponding to differentelectric field in the emitter barrier layer E E and in thecollector barrier layer E C . The PGLIP part of the devicestructure under consideration is somewhat different fromthose studied in Refs. [12–15], where the emitter n-layeris assumed to be a GL.Under the bias voltage applied between the n-emitterand p + -collector layer (serving as the LED active re-gion), the electron tunneling through the triangular bar-rier provides the electron injection from the emitter tothe barrier layer between the n-emitter region and thePGL. A portion of the injected electrons crosses the PGLand enters to the collector barrier layer and than to thep + -layer. The electron tunneling or thermionic emis-sion from the PGL also contribute to the net currentcollected by the p+-layer. The incident FIR/MIR spa-tially nonuniform in the device plane generates the spa-tially nonuniform electron photocurrent from the PGL.It is associated with the electrons photoexcited in thePGL from its valence band into the conduction band (seeFigs. 1 and 2) which go to the barrier layer either directlyor after the tunneling through the triangular barrier be-tween the PGL and the collector barrier layer (dependingon the FIR/MIR photon energy ~ ω and the height ∆ GL of the barrier between the PGL and the collector bar-rier layer). The spatial distribution of the photocurrentfrom the PGL repeats the spatial distribution of the in-cident FIR/MIR. The photoexcitation of the PGL leadsto the deviation of its potential Φ GL from its dark value Φ dark . If the PGL lateral resistivity is relatively low (asin sufficiently perfect and/or doped GLs), Φ GL is virtu-ally independent of the lateral coordinates, so that thevariation of the photocurrent injected from the emitteris uniform as well. Thus, in such a case, the nonuni-form irradiation leads to the nonuniform current gener-ated solely from the PGL, whereas the net photocurrentis produced by both the PGL and the emitter. Simi-lar situation occurs in the pixelless QWIP-LED imageup-converters due to low QW lateral resistivity. The lat-ter can not normally be made sufficiently high becauseof the necessity to maintain relatively high electron con-centration (doping) in the near emitter QW to providea sufficient intersubband absorption and photoemission.In contrast, in the PGLIP-LED devices, the lateral re-sistivity of the PGL can be so high that the nonuniform distributions of the photogenerated holes (left in the PGLafter the escape of the photoelectons) do not manage torelax. Hence in this case, the PGL electric potential spa-tial distribution becomes similar to that of the incidentradiation. This results in the nonuniform density of thephotocurrent emitted not only from the PGL but alsoinjected from the n-emitter. As a result, the spatiallynonuniform component of the net current stimulated bythe incident FIR/MIR can be larger than the componentassociated with the photoemission from PGL solely. Inother words, the effect of photoconductive gain amplifiesnot only spatially uniform currents (the dark current andthe current generated by the averaged component of theincident FIR/MIR intensity I ω, = h I ω i ) but the ”image”component as well.The PGLIP-LED image up-converter model accountsfor the main processes responsible for the device opera-tion, namely, the electron photoemission from the PGL(direct and followed by tunneling), capture of the elec-trons injected from the emitter into the PGL, processes ofthe PGL lateral conductivity, injection of the photocur-rent to the LED active layer, and the lateral electronpropagation due to the diffusion and the reabsorption (re-cycling) of the NIR/VIR photons trapped in this layer.The main feature of the device under consideration is theuse of large area polycrytalline GL as a photosensitive el-ement with decreased dc conductivity.In the absence of irradiation, the densities ofthe electron tunneling current from the emitterand the current of the electrons photoescaped fromthe PGL, j E and j GL , respectively, can be pre-sented as j E = j maxE exp( − E tunnE /E E ) and j GL = j maxGL exp( − E tunnGL /E C ). Here j maxE and j maxGL are themaximum electron current densities, which can be ex-tracted from the emitter n-layer and the PGL. Thesequantities are determined by the doping and the elec-tron try-to-escape times. The characteristic tunnel-ing fields for the near-equilibrium electrons in the n-emitter and for the photoexcided electronsin the PGLare equal to E tunnE = 4 √ m ∆ / E / e ~ and E tunnGL =4 √ m ∆ / GL / e ~ [48], respectively, where ∆ E and ∆ GL are the electron activation energies in the n-emitter layerand the GL, m is the electron effective mass in the bar-rier layers, e is the electron charge, and ~ is the Planckconstant. The emitter and collector fields E E and E C satisfy the equation E E W E + E C W C = V , were W E and W C are the thicknesses of the barrier layers and V is thebias voltage. In the following, to avoid to cumbersomeformulas we, for simplicity, set W E = W C = W and j maxE = j maxGL = j max .Equalizing the capture rate of the injected electronscrossing the PGL into the latter j E p/e , where p < p ≪ j GL /e , one can find in the case of the undoped GL,which will be primarily considered in the following, thecondition E E = E C = V W (1)is achieved at V = V with V = 2 W ( E tunnGL − E tunnE )ln( j maxGL /pj maxE ) . (2)In the situation under consideration, the surface chargein the PGL Σ = 0, so that the PGL Fermi level coincideswith the Dirac point, carrier density is minimized, thatpromotes an elevated GL resistivity. Such a situationcan take place when E tunnGL > E tunnE , i.e., when ∆ E < ∆ GL . The latter inequality implies that ∆ E = χ E − χ B − ε F < ∆ GL = χ GL − χ B , where ε F is the electronFermi in the emitter. Here χ E − χ B and χ GL − χ B are thedifferences between the electron affinities of the emittermaterial ( χ E ) and of the PGL ( χ GL ) and that of thebarrier material χ B . Hence, the structure materials andthe emitter doping should be chosen in a such a way that χ E > χ GL > χ B and ε F > χ E > χ GL .The deviation of V from V leads to E E > E C or E E < E C and to the formation of the excess electronor hole charges in the undoped GL. Usually the lattercan result in a marked drop of the GL resistivity. If thePGL is doped, the appropriate choice of the bias voltage V = V doped = V , can decrease the carrier density and,hence, increase the GL resistivity. In this case, E E = E C ,although the PGLIP-LED characteristics can be foundanalogously.The consequences of the departure of the GL electron-hole system from the Dirac point will be discussed below.To provide an effective injection of the electrons fromthe GLIP part to the p-layer in the LED part andNIR/VIR emission from the latter, the following two con-ditions should be fulfilled: (1) absence of the barrier atthe p-layer and (2) sufficiently large band gap in the lat-ter layer (to secure emission of the NIR/VIR photons).The first condition requires χ B ≤ χ LED . III. CURRENT OUTPUT FROM THE GLIP
The intensity I inω of the incident FIR/MIR with thefrequency ω and the variation of the GL potential causedby irradiation Φ GL comprise the spatially averaged andspatially nonuniform (in the in-plane x -direction) com-ponents: I inω = I inω, + I inω,q cos qx, Φ GL = Φ GL, + Φ GL,q cos qx. (3)Here q is the wavenumber characterizing the scale of theimage details. The components o of the injected currentdensity induced by the incident FIR/MIR (photocurrentdensity) are given by j E, = σ E ϕ /W, j E,q = σ E ϕ q /W, (4)where σ E = dj E /dE | E = E E = j maxE exp( − E tunnE /E E )( E tunnE /E E ) is the differen-tial conductance of the emitter. The spatially uniformcomponents ϕ and j E, can be found accounting for thebalance of the electron captured into and photoescapedfrom the GL. As a result, j E, = 4 π eαθ ω p ( √ κ + 1) I inω, (5)with the quantity θ ω = 11 + τ esc τ relax exp (cid:18) η / ω E tunnGL E C (cid:19) (6)describing the dependence of the electron photoescapeon the FIR/IR photon energy ~ ω [12–15], η ω = (∆ GL − ~ ω/ / ∆, α ≃ /
137 is the fine structure constant and √ κ is the barrier material refractive index.Taking into account the spreading of the holes photo-generated in the PGL due to the lateral conductivity ofthe latter, the spatially nonuniform components of thePGL potential Φ GL,q can be derived using the followingequation (the continuity equation): d Φ GL,q dx − Q GL Φ GL,q = 4 παθ ω ρ GL ( √ κ + 1) I inω,q cos qx, (7)Here Q GL = p pσ E ρ GL /W is the parameter characteriz-ing the lateral spreading of the GL potential and ρ GL isthe GL resistivity.Equations (4) and (7) yield Φ GL,q = − παθ ω ρ GL ( √ κ + 1) I inω,q cos qx ( q + Q GL ) , (8)so that the spatially nonuniform component of the elec-tron current density from the emitter reads j E,q = σ E ρ GL W παθ ω ( √ κ + 1) I inω,q cos qx ( q + Q GL ) , (9)Considering that the fraction of the electrons injectedfrom the emitter and crossed the PGL is equal (1 − p ) andthat the spatially uniform and nonuniform componentsof the electron current density emitted from the PGL are,respectively, given by j GL, = 4 π e αθ ω ( √ κ + 1) I inω, , (10) j GL,q = 4 π e αθ ω ( √ κ + 1) I inω,q cos qx, (11)for the components of the electron photocurrent densityinjected to the p + -layer, one can obtain j C, = 4 π e αθ ω ( √ κ + 1) (cid:18) − pp + 1 (cid:19) I inω, , (12) j C,q = 4 π e αθ ω ( √ κ + 1) (cid:20) − pp Q GL ( q + Q GL ) + 1 (cid:21) I inω,q cos qx. (13)The first term in the brackets in Eqs. (12) and (13) aredue to the contribution of the photoelectric gain effect.When the GL lateral conductivity increased, the param-eter Q GL tends to zero, so that the photoelectric gaineffect for the nonuniform current vanishes.If Q GL tends to zero, Eqs. (12) and (13) become similarto the pertinent equation in Ref. [12]. Some distinctionsare associated with different photosensitivity of the emit-ter contacts.The effect of photoconductive gain becomes substan-tial when Q GL /q ≫
1. Depending on the emitter differ-ential conductance, capture probability, GL lateral mo-bility, the parameter Q GL can vary in a wide range.Let us estimate the ratio Q GL /q for q max = 2 π/λ ω =6 π × cm − , corresponding to the FIR/MIR with thewavelength λ ω = 10 µ m.Using Eqs. (12) and (13), the photocurrent densities j C, and j C,q can be expressed via the PGLIP character-istic responsivity R GLIPω = 4 π e αθ ω ~ ω ( √ κ + 1) . (14)This yields j C, = R GLIPω ~ ω I inω, p , (15) j C,q = R GLIPω (cid:20) − pp Q GL ( q + Q GL ) + 1 (cid:21) ~ ω I inω,q cos qx. (16)If ∆ E = 0 . GL = 0 . j maxE = j maxGL =1 . × A/cm , m = 0 . m ( m is the mass of bare elec-tron), and p = 10 − , one obtains E tunnE ≃ × V/cm, E tunnGL ≃ . × V/cm, E E = E C = V / W ≃ . × V/cm, and σ E ≃ .
41 A/V · cm. At ∆ E = 0 . GL = 0 . E tunnE ≃ . × V/cm, E tunnGL ≃ × V/cm, E E = E C = V / W ≃ . × V/cm, and σ E ≃ .
12 A/V · cm. Using thesedata, setting W = 10 − cm and ρ GL > Q GL & . × cm − and Q GL & . × cm − , respec-tively. This implies that to achieve the ratio Q GL /q ≫ q , corresponding to the incident FIR with the wave-length λ ω = 10 µ m, one needs to use the GLs with theresistivity much larger than 5 kΩ.The radiative recombination of the electrons injectedto the LED p-layer with the holes provides the emissionof NIR/VIR photons with the energy ~ Ω > ∆ G , where∆ G is the energy gap of the p-layer. The intensity of theoutput NIR/VIR stimulated by the incident FIR/MIR I outω = I out Ω , + I Ω ,q cos qx is determined by the internalquantum efficiency τ n / ( τ n + τ rad ) (where τ n and τ rad are the times of nonradiative and radiative recombina-tion, respectively) and by the fraction of the generatedNIR/VIR photons trapped in the LED p-layer due tototal internal reflection η .Considering the electron diffusion in the p + -LED layerand the effect of recycling of the NIR/VIR photons [49–53] in this layers, the density of the electrons producedby the photocurrent Σ LED , which comprises the uniformand spatially nonuniform components, can be found asin Refs. [8, 52, 53]:Σ
LED, = j C, e (cid:18) τ n + 1 − ητ rad (cid:19) , (17)Σ LED,q = j C,q e (cid:20) τ n + 1 − ητ rad + q τ rad (cid:18) ηq + æ + L D (cid:19)(cid:21) . (18)Here æ and L D = √ Dτ rad are the interband absorptioncoefficient of the NIR/VIR photons and the electron dif-fusion length in the LED p + -layer. Using Eqs. (17) -(18), we obtain I out Ω , = j C, e (cid:18) τ n + 1 − ητ rad (cid:19) Θ out (1 − η ) τ rad , (19) I out Ω ,q = j C,q e (cid:20) τ n + 1 − ητ r + q τ r (cid:18) ηq + æ + L D (cid:19)(cid:21) × Θ out (1 − η ) τ rad . (20)Here Θ out ≤ out depends on the ratios of the refractiveindices of the LED p-layer and the surrounding layers. IV. DERIVATION OF UP-CONVERSIONCHARACTERISTICS
Substituting j C, and j C,q from Eqs. (15) and (16) toEqs. (19) and (20), for the pixelless PGLIP-LED aver-age up-conversion and image up-conversion efficienciesdefined as C GLIP − LEDω → Ω , = Ω ω I out Ω , I inω, , C GLIP − LEDω → Ω ,q = Ω ω I out Ω ,q I inω,q , respectively, we arrive at the following formulas: C GLIP − LEDω → Ω , = ~ ΩΓ e R GLIPω p , (21) C GLIP − LEDω → Ω ,q = ~ ΩΓ e R GLIPω F LEDq P GLIPq . (22)Here Γ = Θ out (1 − η )( τ rad /τ n + 1 − η ) , (23)1 P GLIPq = 1 + 1 − pp Q GL ( q + Q GL ) , (24) F LEDq = 11 + q ( τ rad /τ n + 1 − η ) (cid:18) ηq + æ + L D (cid:19) . (25)The function 1 /P GLIPq describes the role of the GL lat-eral potential spreading with decreasing of the nonuni-formity scale.The image contrast transfer function, i.e., the ratio ofthe conversion efficiencies of the image nonuniform com-ponent to its averaged value K GLIP − LEDω → Ω ,q = C P GLIP − LEDω → Ω ,q C P GLIP − LEDω → Ω , , which characterizes the output NIR/VIR image contrast,as follows from Eqs. (21) and (22) is described by K GLIP − LEDω → Ω ,q = pP GLIPq F LEDq = p + (1 − p ) Q GL ( q + Q GL )1 + q ( τ rad /τ n + 1 − η ) (cid:18) ηq + æ + L D (cid:19) . (26) V. RESULTS
The plots based on the above calculations of the up-conversion characteristics are shown in Figs. 3 - 6. Forthe definiteness, the following device parameters are as-sumed: and σ E = 0 .
41 A/V · cm, κ = 5, W = 10 − cm, q ( µ m -1 ) / P q G L I P p = 0.01 p = 0.02 p = 0.05 FIG. 3: PGLIP potential spreading factor 1 /P GLIPq versusimage nonuniformity wave number q for different captureprobabilities p and PGL resistivity ρ GL = 20 kΩ (solid lines)and 10 kΩ (dashed lines). q ( µ m -1 ) C on t r a s t t r an s f e r f un c t i on , K (a) (b) L D = 1 µ m ρ GL = 20 k Ω ρ GL = 20 k Ω p = 0.01 10 k Ω Ω p = 0.01 L D = 0.2 µ m1 µ m2 µ m FIG. 4: Image contrast transfer function K GLIP − LEDω → Ω ,q versusimage nonuniformity wave number q for (a) different PGLresitivities ρ GL and electron diffusion length in the LED p-layer L D = 1 . µ m and (b) different L D and ρ GL = 20 kΩ. τ rad /τ n = 0 .
1, and η = 0 .
5. Other parameters are indi-cated below.Figure 3 shows the dependence of the the PGL lat-eral potential spreading factor 1 /P GLIPq on the imagewavenumber q calculated using Eq. (24) for different val-ues of the capture probability p and the PGL resistivity ρ GL . One can see that a decrease in the capture prob-ability p (leading to an increase of the photoconductivegain and, therefore, in the enhancement of the role ofthe nonuniform injection from the emitter), can markedlysuppress the lateral potential spreading. An increase inthe PGL resistivity also promotes the latter (compare thesolid and dashed lines in Fig. 3).Figure 4 shows the image contract transfer function K GLIP − LEDω → Ω ,q versus the image wavenumber q calculatedfor different PGL resitivities ρ GL and electron diffusionlength in the LED p-layer L D . The pertinent calculationsare based on Eqs. (22) -(26). The following parametersare assumed: p = 0 . τ rad /τ n = 0 .
1, and æ = 1 µ m − .In particular, Fig. 4(a) indicates an improvement of thecontrast of the NIR/VIR image when the PGL resistiv-ity rises. This is due to the pertinent suppression of the Resistivity, ρ (k Ω ) C on t r a s t t r an s f e r f un c t i on , K λ = 12.5 µ m p = 0.01 L D = 1 µ m 10.0 µ m7.5 µ m5.0 µ m FIG. 5: Image contrast transfer function K GLIP − LEDω → Ω ,q as afunction of the PGL resistivity for different NIR/MIR wave-length λ = 2 π/q (5, 7.5, 10, and 12.5 µ m). Resistivity, ρ (k Ω ) C on v e r s i on e ff i c i en cy , C p = 0.010.020.10.05 L D = 1 µ m q = 0.5333 µ m -1 FIG. 6: Image conversion efficiency C GLIP − LEDω → Ω ,q as a func-tion of the PGL resistivity for different capture probabilities p , q = q max = ω/c , ~ ω = 0 . ~ Ω = 1 . PGL potential spreading shown in Fig. 3. As followsfrom Fig. 4(b), a weaker lateral diffusion of the electronsinjected into the LED p + -layer also promotes better con-trast.Plots of K GLIP − LEDω → Ω ,q and C GLIP − LEDω → Ω ,q shown inFigs. 5 and 6 indicate that using the PGLs with higherresistivity improves the output image contrast and in-creases the energy conversion efficiency. Diminishing thephoton recycling effect in the LED (characterized by pa-rameters η and æ in Eqs. (25) and (26)), which attenu-ates the effective electron diffusion,also leads to a higherquality of the output image.Figure 6 corresponds to ~ ω = 0 . ~ Ω = 1 . q = q max = ω/c = 0 . µ m − , Γ = 0 . θ ω = 0 .
5, andother parameters as in the above figures. As seen fromFig. 6, the energy conversion efficiency can markedly ex-ceed unity. The optimization of the parameters, first ofall the LED parameters (increase in the internal and ex-ternal LED efficiencies), might provide even higher val-ues of the PGLIP-LED energy conversion efficience thanthose in Fig. 6. As follows from Figs. 4(a), 5, and 6, the PGLIP-LED up-converter characteristics improve with increas-ing PGL lateral resistivity ρ GL . This is attributed to theincreasing role of the photoconductive gain when the re-sistivity rises (see below). Using PGLs with small grainsizes a [41, 42] (see also Refs. [43–47]), one can realizethe resistivities much higher than those considered in theabove figures (say, ρ G L >
500 kΩ at a ∼ | V − V | fromzero leads to violation of condition (1) and, hence, to adecrease in the GL responsivity ρ GL due to the deviationof the Fermi level in the GL from the Dirac point. A de-crease in the resistivity with increasing carrier density inthe PGL is complicated by the mobility density depen-dence and the specifics of the inter-grain transport. De-viation of the voltage from the that corresponding to theDirac (neutrality) point in rather wide range might leadto a decrease of the PGL resistivity by several times[41].This, can result in a marked decrease in the contrasttransfer function and the conversion efficiency (see Figs. 5and 6, respectively). VI. MATERIALS FOR PGLIP-LED DEVICES
Different materials can be used for the PGLIP-LEDlayered structures, providing their proper relations be-tween the electron affinities (see, in particular, Refs.[54,55]).For example, the PGLIP section can include:(a) n-Si emitter, SiO or hBN -emitter barrier, WS col-lector barrier (as in GL-based vertical-field effect transis-tors [22]);(b) the n-Si emitter, Si0 emitter barrier layer (as in GL-based hot electron transistors [57–59]) and Si collectorbarrier layer;(c) the Ti-base emitter, Al O emitter barrier layer, andSi collector barrier layer (the material of the LED activelayer should have the electron affinity and energy gaplarger than that in Si);(d) As an option, the emitter layer can also be an n-typeGL (as considered in Refs. [12–14]).The LED active (emitting NIR/VIR) layer can, inparticular, be made of such a direct bandgap materialas WS , WSe , MoSe , MoS [28–30]. In particular,in the case of the WS and MoSe LED active layers,the energy of the output image photons is in the range ~ Ω ∼ . − . VII. DISCUSSIONA. Effect of photoconductive gain
The electron photoemission from the PGL leads notonly to the photocurrent generation but also (due to thePGL charging and the consequent variation of its poten-tial) to the injection of extra electrons from the emitter.The latter results in a higher net photocurrent in compar-ison with the photocurrent provided solely by the pho-toemission from the PGL that constitutes what is usuallycalled as the effect of the photoconductive gain. The pho-toconductive gain is described by the factor 1 /P GLIPq inEqs. (22) and (26). As follows from this factor defini-tion given by Eq. (24), at ρ GL tending to zero the factor1 /P GLIPq tends to unity. This is because at small val-ues of ρ GL , the PGL is virtually equipotential. This im-plies that in the limit ρ GL = 0, the photoconductive gainof the nonuniform photocurrent component vanishes, sothat the nonuniformity of the photocurrent injected tothe LED p + layer and, hence, the nonuniformity of theoutput radiation intensity are associated only with theelectrons photoexcited from the PGL. Simultaneously,the uniform (averaged) component still can exhibit a sub-stantial gain, i.e., such a component comprises not onlythe photocurrent created by the electrons photoexcitedfrom the PGL but also by the photocurrent associatedwith the extra electrons injected from the emitter. This,in particular, seen from Eqs. (21), (22), and (24), where C GLIP − LEDω → Ω , ∝ /p ≫
1, while C GLIP − LEDω → Ω ,q does notcontain a large factor 1 /p (in the limit ρ GL = 0). As aconsequence, at small values of ρ GL , the contrast trans-fer function becomes very small ( K GLIP − LEDω → Ω , ≃ p ≪ ρ QW ) to provide the carrierdensity sufficient for a reasonable photosensitivity.In the multiple-PGL devices, the spatially uniform andnonuniform components of the photocurrent output fromthe PGLIP with the resistive PGLs are virtually inde-pendent of the number of the PGLs N . Hence, in suchmultiple-PGL devices with all resistive PGLs, the pho-toelectric gain and almost all PGLIP-LED characteris-tics are close to those of the PGLIPs with a single PGL.However, the GLIP-LED image up-converters can ex-hibit lower noise (by a factor of 1 / √ N (see, for example,Refs. [34]). B. PGLIP-LED versus QWIP-LED
Comparing the image up-conversion efficiency of thePGLIP-LEDs with a single PGL, given by Eqs. (21) -(24), with that of the QWIP-LED imagers [5, 8] (assum-ing the same properties of the LED sections), we find C GLIP − LEDω → Ω ,q C QW IP − LEDω → Ω ,q = (cid:18) ασ GL Σ GL (cid:19) p QW [1 − (1 − p QW ) N ] × (cid:20) − pp Q GL ( q + Q GL ) (cid:21) ≃ (cid:18) ασ GL Σ GL (cid:19) pN . (27)Here σ GL , Σ QW , p QW , and N are the cross-section of thephoton absorption and the electron density in the QW,the electron capture probability onto the QW, and the number of the QWs in the QWIP. It is assumed for sim-plicity that the LED sections of both image up-convertershave the same characteristics, the number of the QWs isnot too large (say, several thens or less), the GL resistiv-ity and the scale of the image nonuniformities are suffi-ciently large q < Q GL , /l D , where l D = p W D B /v B , D B and v B are the electron diffusion length, electron dif-fusion coefficient, and drift velocity in the barrier layers.Both factors in the right-hand side of Eq. (27) are largeor very large.Analogously, for the ratio of the image contrast trans-fer functions of the PGLIP-LED and QWIP-LED oneobtains K GLIP − LEDω → Ω ,q K QW IP − LEDω → Ω ,q ≃ p QW N . (28)Due to small values of the capture probability p QW even at a relatively large but practical number ofthe QWs, the ratio K GLIP − LEDω → Ω ,q /K QW IP − LEDω → Ω ,q exceedsunity. C. Role of lateral diffusion of the injected electronsin the barrier layers
In Eqs. (8) and (27) we disregarded the lateral diffu-sion of the electrons propagating above the barriers (incontrast to Ref. [5, 8]). This is justified because the lat-eral displacement of these electrons during their rathershort flight across the barrier layers is very small. Indeed,such a displacement ∆ x ≃ L D . Setting W = 10 − cm, D B = (10 − /s, and v B = 10 cm/s, we obtain∆ x ≃ (1 . − . × − cm, i.e., the value negligibly smallin comparison with the incident radiation wavelength. D. Role of the GL and barriers doping (electricaland chemical)
When the bias voltage V deviates from the character-istic voltage V , the Fermi energy in the PGL shifts withrespect to the Dirac point. This leads to the followingconsequences. First, an increase in the electron or holedensity Σ GL results in the increase in the GL lateral con-ductivity and, hence, in the smoothening of the lateralpotential distribution and the suppression of the photo-electric gain. The same occurs when the GL is chemicallydoped.Second, the Fermi energy shift affects the PGL absorp-tion spectrum due the Pauli principle (toward higher en-ergies of the FIR/MIR photons [15]). This might be usedfor a voltage control of the spectral characteristics (say,for the ”filtering”) of the incident FIR/MIR.Third, the deviation of V from V as well as chemicaldoping (giving rise to the formation of the hole gas inthe GLIP) can be used for a lowering of the GLIP darkcurrent and, therefore for decrease in the background uni-form component of the output NIR/VIR.Forth, the selective dipole doping of the barrier layerscan markedly modify the PGLIP characteristics [15] af-fecting the operation of both the lamp GLIP-LED andPGL-LED up-converters [12] and the pixelless PGLIP-LED imagers. E. Optical feedback
If a substantial portion of the NIR/VIR photons gener-ated in the LED active p + -layer (and not reflected by thePGLIP collector barrier) enters the PGLIP (the pertinentwavy arrows are not shown in Figs. 1 and 2), the inter-band absorption of these photons in the PGLIP (in theGL) leads to an extra photocurrent, which further rein-forces the emission of the NIR/VIR photons. Such a pos-itive optical feedback can reinforce not only the averageup-conversion efficiency [12] but the image up-conversionas well. VIII. CONCLUSIONS
We reported on the proposal of the pixelless FIR/MIRto NIR/VIR up-converter based on the vdW heterostruc-tures with the highly resistive (polycrystalline) PGLs -PGLIP-LED upconverter. Using the developed devicemodel which accounts for generation of nonuniform pho-tocurrent in the GLIP section by FIR/MIR, the pho- tocurrent injection to the LED section, and emissionof NIR/VIR from the latter section, we calculated thePGLIP-LED characteristics (the image contrast transferfunction and the conversion efficiency). The photocur-rent lateral spreading was considered taking into accountthe PGL lateral conductivity and the effective diffusionof the electrons injected into LED (combining their stan-dard diffusion and the lateral spreading due to the pho-ton recycling). We showed that the pixelless PGLIP-LEDup-converters can be effective imaging devices exhibitingthe power image up-conversion efficiency substantiallyexceeding unity. Recent publications [54–57] and oth-ers support the feasibility of realization of the PGLIP-LED devices with elevated performance. The proposedand evaluated pixelless PGLIP-LED up-converters canmarkedly surpass the pixelless QWIP-LED imagers.
Acknowledgments
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