Nanoscale Advances PAPER 2D pn junctions driven out-of-equilibrium† Cite this: Nanoscale Adv., 2020, 2, Ferney A. Chaves, * Pedro C. Feijoo and David Jiménez 3252 The pn junction is a fundamental electrical component in modern electronics and optoelectronics. Currently, there is a great deal of interest in the two-dimensional (2D) pn junction. Although many experiments have demonstrated the working principle, there is a lack of fundamental understanding of its basic properties and expected performances, in particular when the device is driven out-of- equilibrium. To fill the current gap in understanding, we investigate the electrostatics and electronic transport of 2D lateral pn junctions. To do so we implement a physics-based simulator that self- consistently solves the 2D Poisson's equation coupled to the drift-diffusion and continuity equations. Notably, the simulator takes into account the strong influence of the out-of-plane electric field through the surrounding dielectric, capturing the weak screening of charge carriers. Supported by simulations, we propose a Shockley-like equation for the ideal current–voltage (J–V) characteristics, in full analogy to the bulk junction after defining an effective depletion layer (EDL). We also discuss the impact of recombination–generation processes inside the EDL, which actually produce a significant deviation with respect to the ideal behavior, consistently with experimental data. Moreover, we analyze the capacitances and conductance of the 2D lateral pn junction. Based on its equivalent circuit we investigate its cut-off frequency targeting RF applications. To gain deeper insight into the role played by Received 4th April 2020 Accepted 8th June 2020 material dimensionality, we benchmark the performances of single-layer MoS2 (2D) lateral pn junctions against those of the Si (3D) junction. Finally, a practical discussion on the short length 2D junction case DOI: 10.1039/d0na00267d together with the expected impact of interface states has been provided. Given the available list of 2D rsc.li/nanoscale-advances materials, this work opens the door to a wider exploration of material-dependent performances. enable even more possibilities to create different pn junction 1. Introduction concepts, compared with 3D materials. The need for miniaturization of structures in modern semi- 2D pn junctions can be integrated into a device either by conductor electronics has directed signicant attention toward using an out-of-plane conguration, featuring vertically stacked the fabrication of pn junctions based on atomically thin mate- van der Waals-bonded layers,5,14,21,24 or by a lateral (in-plane) rials such as graphene, phosphorene, silicene, germanene, etc., conguration, spanning adjacent regions of different doping in and the family of transition metal dichalcogenides (TMDs).1–7 the covalently bonded 2D plane.15–18,23,25–27 At the same time, This kind of 2D pn junction takes advantage of the ultra-thin they can be classied as homojunctions (based on a single 2D nature of 2D materials to offer new and exciting possibilities material) and heterojunctions (formed by joining two different which are impossible to achieve by its 3D counterpart. These 2D 2D materials). The 2D homojunctions can be (1) thickness- pn junctions will be an essential part of the new generation of based junctions,22 (2) elemental doping based junctions,5,28 (3) 2D crystal based electronic and optoelectronic devices such as electrostatically doped junctions,11,15,29,30 and (4) chemically photodiodes, transistors, solar cells, photo-detectors, etc.8–23 doped junctions, in which there is surface chemical doping or Some advantages of 2D pn junctions are the thickness-depen- substitutional doping of a region in a 2D material.16,31–35 dency of the semiconductor band gap for some materials, the In particular, 2D lateral pn junctions (2DJ) are useful for gate-tunable doping and rectication by external out-of-plane solar cells and photodetectors because the built-in potential electric elds and the possibility of tuning some electrical created in the extended space charge region separates and properties such as the depletion region width and capacitance drives the photogenerated e–h pairs to generate by changing the surrounding dielectrics. These advantages photocurrent.25–27 While there are many reports of experimental studies on 2D junctions, the same is not true for theoretical studies, where the Departament d'Enginyeria Electrònica, Escola d'Enginyeria, Universitat Autònoma de impact of low dimensionality on their electrical properties and Barcelona, Campus UAB, 08193 Bellaterra, Spain. E-mail: ferneyalveiro.chaves@uab. their differences with 3D junctions (3DJs) are reported. A few cat † Electronic supplementary information (ESI) available. See DOI: theoretical studies have contributed to the quantitative under- 10.1039/d0na00267d standing of the electrostatics in the equilibrium of 2DJs.36–45 3252 | Nanoscale Adv., 2020, 2, 3252–3262 This journal is © The Royal Society of Chemistry 2020 Paper Nanoscale Advances Most of them have been focused on chemically doped edges of the device make perfect ohmic contacts and supply symmetric pn junctions, achieving analytical expressions for a ground reference potential to the right edge and a potential V the electrostatic potential prole and the depletion width with to the le edge. dependency on physical parameters such as doping densities A 2D non-linear Poisson's equation describes the electro- and the dielectric constant of the surrounding media. However, statics of the device, involving the electrostatic potential due to the presence of a very extended transition region between distribution in the xz-plane f(x,z;V), the space charge density s the fully depleted layer and the quasi-neutral region in 2DJs, in ¼ q(p  n + ND  NA) and the effective dielectric permittivity 3eff comparison with 3DJs, inheriting a weak screening of charge of the insulating media surrounding the junction. Strictly, out- carriers, it is truly complex to obtain a closed analytical theory in of-equilibrium electrostatics is coupled with the 1D charge the out-of-equilibrium regime, as exists in 3DJs. Thus, theoret- transport along the lateral dimension x by means of the dri- ical research studies of 2D pn junctions driven out-of-equilib- diffusion and continuity equations. The transport involves non- rium, considering the impact of its low dimensionality, remain linear rst-order differential equations for the electron (hole) to be developed. quasi-Fermi potential Vn(p)(x) and current density Jn(p), and Here, we report a study of the 2D pn junction driven out-of- contains physical parameters such as the electron (hole) equilibrium based on numerical simulations, which sets the mobility mn(p), the minority electron (hole) lifetime sn(p), the basis for gaining understanding of experimental measurements energy gap of the semiconductor Eg and the temperature T.46 and for the assessment of future analytical models. Our study The in-plane potential 4(x) ¼ f(x,0;V) and the quasi-Fermi has been focused, specically, on the chemically doped 2D potentials Vn(p) are related to the local intrinsic energy Ei and lateral pn junction (2DJ), leaving the investigation of electro- electron (hole) Fermi energy as Ei(x) ¼ q4 and EFn(p)(x) ¼ statically doped and vertical 2D pn junctions for a future study. qVn(p), respectively. Processes of generation and recombina- The manuscript has been divided in the following sections: in tion of charge carriers, mediated by capture and emission of Section 2, we have described the numerical model used to carriers at trap centers located in the band gap of the semi- investigate the properties of the 2DJ out-of-equilibrium; and in conductor, are modeled with a net recombination rate U Section 3 we have proposed a simple model describing the ideal obeying the Shockley–Read–Hall model. A complete description electrical characteristics of the 2DJ. Deviations from the ideality of the coupled equations modeling the physics of the 2DJ driven have also been discussed. In Section 4 we have investigated the out-of-equilibrium and details about the algorithm to numeri- impact of dimensionality on the J–V characteristics, specically cally solve them are given in the ESI.† All the simulations in this we have benchmarked the monolayer MoS2 junction against the work have been carried out assuming room temperature to be T silicon junction. In Section 5, we have addressed the calcula- ¼ 300 K. tions of capacitances and conductance of the 2DJ out-of-equi- Fig. 2 shows the band diagram, quasi-Fermi energies and librium. Based on these results, we have investigated its FoM, intrinsic energy proles of a 2DJ, as calculated from our such as the cut-off frequency. In Section 6 the effect of the interface states on the electrical performances is addressed and nally in Section 7 we have discussed the role played by the dielectric environment in dening the J–V characteristics and the possibility of tuning the diode rectication. 2. Model Fig. 1 represents the physical structure of the 2DJ considered in this work. The 2DJ, made of an abrupt junction of p-type and n- type semiconductors at x ¼ 0, lies in the plane z ¼ 0. The device along the y-axis is large enough to consider the independence of its properties in that direction. The metal electrodes at the Fig. 2 Band diagram of a 2D lateral pn junction in thermal equilibrium Fig. 1 Scheme of the 2D lateral pn junction geometry. (a); reverse biased (b); and forward biased (c). This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 3252–3262 | 3253 Nanoscale Advances Paper numerical model. In equilibrium a built-in potential 4bi except for their corresponding DOS. Remarkably, the semi- develops across the depletion layer, while in the out-of-equi- conductor of the 2DJ1 device has an energy gap similar to that librium regime it is reduced/augmented as 4bi–V according to of Si and equivalent doping characterized by the energies q4n the sign of V. and q4p represented in Fig. 2a, i.e. 4n,2D ¼ 4n,3D ¼ |4p,2D| ¼ |4p,3D|. In Fig. 3, the shaded regions represent the transition regions between the fully depleted layer (s/qND  1) and the 3. The ideal 2D diode quasi-neutral region (s/qND  0) for both devices. It is worth It is possible to obtain an analytical model ( equivalent to noting that we have used the notation s and ND for both the 2D Shockley's equation) for the ideal 2DJ, like in conventional 3DJs. case and 3D case, that is, s and ND represent surface (volume) For 3DJs the following approximations are made to develop the charge density and surface (volume) doping density in the 2D ideal model:47,48 (i) a dipole layer fully depleted of carriers with (3D) case, respectively. The length of the transition regions is abrupt boundaries supports the voltage drop 4bi–V and, outside L2D  48 mm and L3D  0.1 mm for the examined 2DJ and 3DJ, those boundaries, the semiconductor is assumed to be neutral. respectively. In the ESI† the possible impact of the edge effects (ii) The carrier statistics is assumed to follow a Boltzmann for several device lengths is reported, concluding it to be low. distribution, implying that the semiconductor is assumed to be The 2D transition region is much wider (500 for the non-degenerate. (iii) The excess of minority carriers injected in examined case) than the 3D transition region because of the quasi-neutral region is much less than the corresponding a weaker screening of charge carriers related to a signicant majority carrier density (low injection approximation). (iv) out-of-plane electric eld, as compared with the strong There is a zero net recombination rate (U ¼ 0) inside the screening exhibited in 3DJs. The strong screening in 3DJs depletion layer such that the total current is only due to the justies the total depletion approximation, giving rise to diffusion component of minority carriers in the quasi-neutral a well-dened depletion region of width: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi regions. In this section, we discuss the possibility of extending 43ð4bi  V Þ the previous assumptions to 2DJs in order to obtain the 2D WD ¼ ; (1) qN equivalent to the Shockley equation. This will be done on the basis of numerical simulations relying on the model described where N ¼ NA ¼ ND for the symmetric junction. For 2DJs it is in the ESI.† still possible to dene an effective depletion layer (EDL) width Fig. 3 shows, from our simulations, the normalized charge analogous to the 3D case, which allows us to obtain an equiv- density along the n-side of symmetric 2DJ and 3DJ devices in alent Shockley model. One of the most relevant theoretical thermal equilibrium. These devices have similar parameters studies reporting an analytical expression for the EDL in both which correspond to 2DJ1 and 3DJ1 described in Table 1, symmetric and asymmetric 2DJs has been carried out by Nipane et al., which uses a method based on an innite number of reections of the image charge.38 The expression for the given EDL is p2 3eff ð4bi  V Þ WD;effð1Þ ¼ (2a) qND f ðNA =ND Þ where h i   pk X N 1  ð1Þk ð1 þ xÞsin 1þx f ðxÞ ¼ (2b) k¼1 k 2 The half of the values given by eqn (2a) corresponds to s/qN ¼ 56.4% for the symmetric case (see Fig. 3). However, eqn (2) is Fig. 3 Charge density profile for symmetric 2D lateral and 3D pn junctions, showing the depletion region and transition region for both inaccurate for asymmetric 2DJs when compared with our cases. Only the n-side region is shown. numerical simulations. We have found that the numerically Table 1 Parameters used in our simulations. SL: single layer Device Description Eg (eV) 3eff/30 ni (cm2) mp (cm2 V1 s1) mn (cm2 V1 s1) sp (s) sn (s) fp (eV) fn (eV) 2DJ1 Sym. pn. Small gap 1.12 11.9 5  103 280 350 1  106 2  107 0.4 0.4 2DJ2 Asym. p+n. Large gap (SL MoS2) 1.8 3.9 9.9  103 50 50 3  1010 3  1010 0.89 0.81 2DJ3 Sym. pn. Large gap (SL MoS2) 1.8 3.9 9.9  103 50 50 3  1010 3  1010 0.81 0.81 3DJ1 Sym. pn. Low doping (Si) 1.12 11.9 1.4  1010 280 350 1  106 2  107 0.4 0.4 3DJ2 Sym. pn. High doping (Si) 1.12 11.9 1.4  1010 280 350 1  106 2  107 0.47 0.47 3DJ3 Asym. p+n (Si) 1.12 11.9 1.4  1010 350 350 2  107 2  107 0.55 0.47 3254 | Nanoscale Adv., 2020, 2, 3252–3262 This journal is © The Royal Society of Chemistry 2020 Paper Nanoscale Advances determined depletion width WD,eff(2) under the condition s/qN is not strictly applicable. On the other hand, Fig. 4b shows the ¼ 56.4% is more correct for the general case,36 guaranteeing an ideal J–V characteristics together with the numerical simulations almost zero eld at the edges of the EDL (see the ESI† for for an asymmetric 2D p+n junction with parameters described for a detailed comparison between WD,eff(1) and WD,eff(2)). device 2DJ2 in Table 1. Again, the black dashed curve has been Once the appropriate EDL is dened, it is possible to calculated with the Shockley-like equation and the others are from consider the 2DJ as consisting of three regions as in 3DJs: the the numerical model with the two different denitions given for quasi-neutral p-side, the quasi-neutral n-side, and the EDL. WD,eff. Importantly, the 2D Shockley-like equation ts the Now, by assuming zero net recombination (U ¼ 0) inside the numerical simulation much better taking WD,eff(2) as the effective EDL, low injection, and the Boltzmann approximation to depletion layer width rather than WD,eff(1). calculate the carrier densities, a Shockley-like equation for the Next, we have analyzed in more detail the electrostatics of the ideal 2DJ case can be obtained. Although not shown here, the asymmetric 2DJ. Fig. 5a shows the relative total charge density 2D Shockley-like equation is obtained, in total analogy with the calculated at forward bias as a function of the position. The 3D case,48 from the solution of the continuity equation to obtain asymmetric 3DJ case is also shown for comparison purposes. the diffusion current of the minority carriers Jn(p) in the quasi- The parameters used here are those of the 2DJ2 and 3DJ3 neutral regions, which allows the determination of the total devices in Table 1. Clearly, the widths WD,eff(1) and WD,eff(2) are current (Jideal) density for a given bias as: quite different. The reason why WD,eff(2) results in a better t of the Shockley model with the numerical model can be under- Jideal ¼ Jp + Jn ¼ Jo[eqV/kT  1], (3) stood by means of Fig. 5b, which shows the electrostatic potential along the device. At WD,eff(2) the electric eld is much where closer to zero compared to WD,eff(1), so the beginning of the   Dp Dn quasi-neutral region is better dened with WD,eff(2). Fig. 5c Jo ¼ qni 2 þ (4) Lp ND Ln NA shows that the electric eld for the 2D case is not linear with position, as in the 3D case, but it is described as ln(x/(x  Xn)) is the reverse saturation current density per unit length. Here, ni near the junction,38 which comes from the contribution of the y g2DkT exp(Eg/2kT) is the intrinsic carrier density, with g2D lines of charge in which the block of sheet charge within the the band-edge density of states (DOS). The electron (hole) space charge region can be split, each of those lines contrib- diffusion coefficient and diffusion length are dened as Dn(p) ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uting an electric eld f1/x. This is in striking contrast with the kTmn(p)/q and LnðpÞ h DnðpÞ snðpÞ , respectively. For 2D semi- 3D case where the contribution of the vertical planes of charge conductor crystals ni is typically much smaller than 1011 cm2 of produces a linear electric eld inside the depletion layer. zero-gap graphene. For example, monolayer MoS2 and MoTe2 Finally, Fig. 5d shows the carrier density proles n(x) and p(x) of semiconductors, with gaps of 1.8 eV and 1.15 eV, respectively, the 2DJ2 device at equilibrium and out-of-equilibrium. In the have intrinsic carrier densities 102 cm2 and 103 cm2, inset, we have repeated the carrier density proles at forward respectively.46 bias (V ¼ 0.5 V) in a wider region, exhibiting a constant slope (in Next, the ideal 2D diode equation has been benchmarked logarithmic scale) in the quasi-neutral region, near the edge of against numerical simulations assuming U ¼ 0 inside the EDL. the EDL, related to the hole diffusion length Lp. The extracted Fig. 4a presents the results for the symmetric case. The parame- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ters used here correspond to the device 2DJ1 described in Table 1, value of Lp is 0.19 mm which coincides with Lp h kTmp sp =q, for which Ln  27 mm and Lp  13.5 mm. As can be observed from as expected from the parameters of the 2DJ2 device in Table 1. the gure, the 2D Shockley-like equation (ideality factor h ¼ 1) ts very well with the numerical simulation at both reverse and 4. Impact of the dimensionality in 2D forward polarizations even when the full depletion approximation lateral pn junctions In order to investigate the impact of the dimensionality in 2DJs, we have carried out simulations of both symmetric 2DJ and 3DJ with similar parameters as given by devices 2DJ1 and 3DJ1, respectively, exceptfor their DOS. We have considered R–G processes inside the EDL. The devices have been simulated up to forward biases such that the series resistance of the quasi-neutral regions can be observed. The resulting current densities are shown in Fig. 6, normalized to the corresponding Jo. In the main panel, marked as “Ideal”, are the J–V characteristics with U ¼ 0 inside the EDL dened by WD and WD,eff in eqn (1) and (2), respectively. The “Ideal” curves t the Shockley equation J ¼ Fig. 4 Ideal J–V characteristics of the 2D lateral pn junction. (a) Jo(eqV/hkT  1), with an ideality factor h ¼ 1, quite well within the Symmetric 2DJ1 device. (b) Asymmetric 2DJ2 device. WD,eff(1) has been calculated according to eqn (2) and WD,eff(2) according to the criterion voltage range where no appreciable inuence of series resistance s/qN  0.56. In both cases a zero net recombination rate (U ¼ 0) has is observed. More specically, as shown in the inset, the ideality been assumed inside the EDL. factor extracted from the numerical results is 1 up to V  0.7 V This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 3252–3262 | 3255 Nanoscale Advances Paper Fig. 5 Electrostatic properties of the asymmetric 2D lateral pn junction. (a) Normalized total charge; (b) electrostatic potential; (c) electric field; (d) carrier density (p and n) in the 2DJ at equilibrium, and forward and reverse bias. Inset: Numerical extraction of the hole diffusion length. The asymmetric 3DJ case is also shown in (a)–(c) for comparison purposes. similar. In the main gure, at reverse bias, the relative current density of the 2DJ is larger than that of the 3DJ. To understand this result, let us consider, rstly, the current driven by R–G processes under reverse-bias. Inside the EDL p  ni, n  ni, so U < 0 is then dominated by generation of electron–hole pairs, and can be expressed as: ni 2 ni Uz  (    ) y  s ; (5) n p g ni sp 1 þ þ sn 1 þ ni ni where sg is the generation lifetime, dened by the expression in brackets. The current due to generation in the effective deple- Fig. 6 J–V characteristics for symmetric 2D lateral and 3D pn junc- tion region can be roughly estimated by assuming a mean value tions showing the different components of the current. Inset: Ideality U ¼ ni =sg of net recombination inside the EDL as factor vs. V. The labels “G”, “R”, “D” and “Rs” correspond to the regions ð Xn dominated by the generation current, recombination current, diffusion qni WD;eff JG ¼ q|U|dx z (6) current and current driven by series resistance, respectively. Xp sg for the “Ideal” case, and then it quickly increases due to the Fig. 7 shows the proles of U for a 2DJ at both reverse and resistance of the quasi-neutral regions. For the calculations we forward bias, obtained from our numerical model, along with have not taken into account any breakdown mechanism such as  in the EDL with sg  sp  sn . Thus, the ratio JG/Jo can be |U| tunneling, thermal instability or avalanche multiplication written as because of the low electric elds at the considered low reverse JG ND WD;eff bias.48,49 On the other hand, the “Non-ideal” curves are obtained f  : (7) JE ni Lp þ Ln aer considering U s 0 inside the EDL. Different regions of J–V characteristics can be observed depending on the bias, namely: generation current (G) at reverse bias, recombination current (R) This result is also valid for 3DJs aer exchanging WD,eff with at low forward bias, diffusion current (D) at moderate forward WD. Typically, WD,eff for 2DJs is larger than WD in 3DJs because bias, and current dominated by the series resistance (Rs) at high of the weaker screening of charge carriers in 2DJs. On the other forward bias. High injection conditions are not present within hand, Lp and Ln are typically smaller in 2DJs as a consequence of the considered voltage range. Clearly, some differences between smaller lifetimes and mobilities. Assuming the same ND/ni for the 2D and 3D cases appear, even though the used parameters are the 2DJ and 3DJ, it does result in a larger JG/Jo for the 2D case. 3256 | Nanoscale Adv., 2020, 2, 3252–3262 This journal is © The Royal Society of Chemistry 2020 Paper Nanoscale Advances 0 can be obtained from our simulation V3D  0:42 V, as illus- trated in the inset in Fig. 6. These effects can also be under- stood through the difference between their EDL in thermal equilibrium with WD,eff ¼ 5.24 mm and WD ¼ 0.17 mm for the 2DJ and 3DJ, respectively. Given the larger transition region L2D exhibited by the 2DJ compared with the 3DJ, as shown in Fig. 3, it might be possible that the physical length of the 2DJ is shorter than the diffusion length. This special case, which is of practical interest, has been discussed in the ESI.† Fig. 7 Net recombination rate profile at reverse bias (V ¼ 0. 3 V) and Now, let's examine the differences between a typical 3DJ forward bias (V ¼ 0.3 V and 0.6 V). made of Si and a 2DJ made of monolayer MoS2. For this, we have simulated symmetric devices 2DJ3 and 3DJ2 described in Table 1. For the sake of making a fair comparison the selected doping On the other hand, at forward bias, the major R–G processes in each device should full (EF  EV)|2D ¼ (EF  EV)|3D ¼ (Ec  in the EDL are the electron and hole capture processes EF)|2D ¼ (Ec  EF)|3D in the quasi-neutral region, far away from (recombination) from the conduction and valence bands, the EDL. The numerical results of the normalized J–V charac- respectively, at the trap centers in the bandgap. Thus, a net teristics are shown in Fig. 8. We can observe that JG/Jo for the recombination rate U > 0 produces a dominant recombination 2DJ is 108 larger than the 3D one, and the crossover voltage 0 0 current (R) over the diffusion current (D) in the quasi-neutral V2D for the 2D pn junction is approx. 1 V larger than V3D . Again, regions. This is valid from zero bias up to the crossover voltage this can be understood because of the drastic reduction of both 0 V2D reached when JR/Jideal  1 (see Fig. 6). By following the same the intrinsic carrier density and diffusion length of minority procedure in the literature for 3D pn junctions, it is possible to carriers (smaller lifetimes and mobilities) together with the nd a simple expression for the recombination current density larger EDL in the 2D case. As for the 2D case the effect of Rs is JR at V > kT/q:50 not visible within the considered voltage range. However, for rffiffiffiffi   the 3DJ the Rs effect starts to be strong at 0.8 V. Remarkably, p kTni qV JR ¼ exp ; (8) the MoS2 2DJ exhibits an ideality factor close to 2 up to 1.3 V, 2 sFo 2kT indicating the dominance of the recombination current over the where Fo is the electric eld at the location of maximum diffusion current. Interestingly, experimental evidence of this recombination and there is a characteristic ideality factor h ¼ 2. phenomenon has been reported by Choi et al. for 7 nm thick To get this expression for JR we have assumed sp ¼ sn ¼ s for chemically doped MoS2 lateral pn junctions and Baugher et al. simplicity. As shown in Fig. 5c, the dependence of the electric for electrically tunable pn diodes.15,16 Their experimental eld for the 2DJ on the position is not as simple as in the 3D evidence reveals that the thinner the crystal semiconductors the case and, in general, numerical approximations are needed to nearer to 2 the ideality factor. nd Fo as a function of the applied voltage. However, by using f V the simple approximation Fo z F ¼ bi WD;eff , which is the 5. Capacitances and the cut-off average electric eld in the EDL, a bias independent formula frequency in 2D lateral pn junctions 4GqND Fo ¼ 2 , valid for symmetric junctions, where G ¼ 0.915 is In this section we address the investigation of the capacitances p 3eff Catalan's constant, can be obtained. Thus, the ratio JR/Jideal at associated with the 2DJ in the out-of-equilibrium regime. An forward bias results in equivalent circuit will be presented and the cut-off frequency of rffiffiffiffi   JR p kT p2 3eff qV z exp  : (9) Jideal 2 q 8Gqni Lp 2kT Again, this ratio is typically larger for 2DJs than for 3DJs due to the lower intrinsic carrier densities, carrier lifetimes and mobilities for 2D semiconductors. In addition, a closed 0 expression for the crossover voltage V2D , can be obtained: rffiffiffiffi  0 2kT p kT p2 3eff V2D z log : (10) q 2 q 8Gqni Lp 0 Using the parameters for the device 2DJ1, V2D  0:57 V agrees with the obtained value in Fig. 6 from our numerical Fig. 8 J–V characteristics of symmetric pn junctions: MoS2 junction 0 model. The value of V3D cannot be analytically obtained, but (2D case) and Si junction (3D case). Inset: Ideality factor vs. voltage. This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 3252–3262 | 3257 Nanoscale Advances Paper the 2DJ will be analyzed. For the sake of simplicity, we have where Dp(x) and Dn(x) are the excess of hole and electron focused the discussion on a symmetric 2DJ. Let us begin dis- surface densities, respectively, in the quasi-neutral regions. The cussing the depletion-layer capacitance per unit length, dened black solid lines refer to the theoretical model, where the Cdp as Cdp ¼ dQD/dV, where dQD is the incremental depletion charge and Cdf capacitances have been calculated from eqn (11) and on each side of the junction aer applying an incremental (14), respectively. In Fig. 9 we have included the corresponding voltage change dV. By assuming the total depletion approxi- 3DJ simulations and theoretical model outcome for comparison mation, the depletion charge on the p-side, for example, can be purposes (brown dashed lines).48 The 2DJ and 3DJ capacitances calculated using QD ¼ qNWD,eff(V)/2. From eqn (2a) we can have units of F/mm and F/mm2, respectively. Importantly, eqn obtain: (11) and (14) t very well to the numerical results, both in 2D and 3D cases. p2 3eff Cdp ¼ : (11) By taking advantage of the Cdp insensitivity to the voltage, 8G a simple formula for the critical bias V* that makes Cdp y Cdf, Cdp does not depend on the applied voltage, in striking contrast can be obtained: to the 3D case. The depletion-layer capacitance accounts for ! most of the junction capacitance when the junction is reverse- kT kT p2 3eff N V ¼ log 2   (18) biased. At forward bias, a diffusion capacitance exists in addi- q q 4Gni 2 Lp þ Ln tion to the depletion capacitance. The diffusion capacitance is the result of the rearrangement of minority surface carriers in For the examined devices 2DJ1 and 2DJ3, the calculated the quasi-neutral regions. Thus, in a similar way to the 3D case, critical voltages are 0.66 V and 1.5 V, respectively. These when a small ac signal, characterized by an angular frequency values agree with the numerical results shown in Fig. 9. In u, is applied to a 2DJ that is forward-biased at a dc voltage V, the general, 2DJs made of large gap semiconductors with both low total voltage and current can be described as carrier lifetimes and mobilities exhibit a relatively high critical V*(t) ¼ V + v* exp(iut) (12) voltage V*, resulting in a total capacitance CT dominated by the depletion component, that is CT ¼ Cdp + Cdf  Cdp that, inter- J*(t) ¼ J + j* exp(iut) (13) estingly, could be modulated by the dielectric permittivity of the surrounding media, according to eqn (11). where v* and j* are the small-signal voltage and current density, The equivalent circuit model of the diode can be obtained by respectively. The real and imaginary parts of the admittance j*/ the total capacitance CT, in parallel with its ac differential v* are related to the diffusion conductance Gd and diffusion resistance rd. The resistance of the quasi-neutral regions and capacitance Cdf, respectively. By applying the same procedure as contact resistance can be modeled by a resistance Rs in series the one used for 3DJs,48 the following expressions for the low with the parallel combination of CT–rd as shown in the inset in frequency diffusion capacitance and conductance, respectively, Fig. 10b. It is worth noting that the units of Rs and rd are U mm are obtained: (U mm2) for 2D (3D) junctions, respectively. Once the diode     equivalent circuit is determined, the cut-off frequency can be q2 ni 2 Lp þ Ln qV Cdf ¼ exp ; (14) dened as the frequency at which v*/j* ¼ rd 2kTN kT 1 and fc ¼ (19) 2pCT ðrd kRs Þ     q ni Dp Dn 2 2 qV Gd ¼ þ exp ; (15) kTN Lp Ln kT Fig. 10a shows fc as a function of the dc voltage for both 2DJ which are the same results as the one that can be obtained by and 3DJ ignoring the extrinsic contact resistance. For the 2DJ differentiating eqn (3), that is Gd ¼ dJideal/dV ¼ rd1, where rd is case, we have considered the devices 2DJ1 (small gap) and 2DJ3 dened as the differential resistance. In Fig. 9 we have shown (large gap), which have been compared with equivalent 3D the behavior of both Cdp and Cdf capacitances as a function of V devices 3DJ1 and 3DJ2, respectively, satisfying (EF  EV)|2D ¼ (EF for 2DJ1 with a small gap (Fig. 9a) and 2DJ3 with a large gap  EV)|3D ¼ (Ec  EF)|2D ¼ (Ec  EF)|3D. At low bias (V < 0.4 V), fc (Fig. 9b). More details of these devices are given in Table 1. The exhibits the highest values for both 2DJ and 3DJ cases. However green solid lines correspond to the capacitances obtained from in this range of voltages rd [ Rs and Cdp [ Cdf, and the cut-off our numerical model, according to frequency can be approximated as fc z 1/2pCdpRs. Our simu- ð lation shows that the 2DJ1 device, which has the same gap as d WD;eff =2 Cdp ¼ qsðxÞdx; (16) the 3DJ1 device, exhibits fc almost two orders of magnitude less dV 0 that of than its 3D counterpart at low bias. By choosing large and gap semiconductors like the monolayer MoS2 the maximum fc ð xmax ð WD;eff =2 can reach the GHz range, although far away from the frequen- d d cies reached by 3DJs, as exemplied by the 3DJ2 device. Cdf ¼ qDpðxÞdx þ qDnðxÞdx; (17) dV WD;eff =2 dV xmin As shown in Fig. 10b we have analyzed the inuence of doping in dening fc. Specically, it shows the zero bias cut-off 3258 | Nanoscale Adv., 2020, 2, 3252–3262 This journal is © The Royal Society of Chemistry 2020 Paper Nanoscale Advances Fig. 9 Depletion-layer and diffusion capacitances of 2D lateral pn junctions and comparison with the 3D case. (a) Small gap case, represented by devices 2DJ1 and 3DJ1. (b) Large gap case, represented by devices 2DJ3 and 3DJ2.   frequency for 2DJ3 and 3DJ2 devices as a function of doping   Vn þ Vp sint 4; Vn ; Vp ¼ q Dit 2 4 þ sint;0 (20) density which has been characterized by (Ec  EF)/kT. Given that 2 we have considered symmetrical junctions, then (EF  EV)|p-side ¼ (Ec  EF)|n-side. The results obtained reveal the benecial where Dit is the density of interface defects (eV1 cm2) and effect of high doping to get the highest possible fc. According to sint,0 is the constant sheet density at the interface (C cm2). The our numerical result, the silicon based 3DJ can reach zero bias details of the model and additional simulations are described in cut-off frequencies over tens of GHz; meanwhile the MoS2 based the ESI.† 2DJ can reach 1 GHz. Further investigation will be needed Fig. 11 shows the impact of the interface states on the current with calibrated parameters to determine the potential of 2DJs. and capacitance characteristics for a symmetric 2DJ with parameters of 2DJ1, from our augmented numerical model. Here we have modied Dit from 0 up to 3  1010 eV1 cm2 and kept sint,0 ¼ 0 in order to get symmetric 2DJs in every case. In the 6. Impact of interface states gure, the arrows indicate the direction of increasing Dit, which Understanding the interface properties between the atomically produces a wider EDL (see the inset in Fig. 11a). However, as thin semiconductors and dielectrics is fundamentally impor- a rectier diode we have predicted a slight reduction of the on/off tant for enhancing the carrier transport properties. The impact ratio with the increasing of Dit (main panel of Fig. 11a). Our of interface states on the electrical properties of 2D material numerical results predict no signicant effects on the capaci- based MOSFETs has been addressed in some studies.51–53 tances within the range of considered Dit, leaving invariant the However, as far as we know, the dependence of properties such cutoff frequency. In the ESI (Section S4†) we have included as diode rectication, capacitance or cutoff frequencies on the numerical results for asymmetric p+n junctions. density of interface states (Dit) has not been reported yet and deserves further investigation. Here, we have included a simple 7. Impact of the surrounding model for the interface states between the 2D semiconductor dielectric permittivity and its surrounding dielectric media, in order to shed light on their effects. We add an interface charge sint to the space charge In this section, we have analyzed the impact of the surrounding density s when solving the Poisson's equation dielectric medium on the J–V characteristics of 2DJs. Because of Fig. 10 (a) Cut-off frequency as a function of the dc bias for 2DJ and 3DJ devices. (b) Zero bias cut-off frequency as a function of the doping for 2DJ (MoS2) and 3DJ (Si) devices, represented by 2DJ3 and 3DJ2, respectively. This journal is © The Royal Society of Chemistry 2020 Nanoscale Adv., 2020, 2, 3252–3262 | 3259 Nanoscale Advances Paper depletion layer on the J–V characteristics. Deviations from ideal J–V characteristics and the impact of the dimensionality have been thoroughly discussed. We have also discussed the possi- bility of tuning the J–V characteristics by using the permittivity of the surrounding environment. Moreover, we have analyzed the capacitances and conductance of 2DJs, which allows us to dene an equivalent circuit. Based on this, we have bench- marked the cut-off frequency of 2DJs for different gaps and doping scenarios, taking the 3DJ case as the reference. Finally, a practical discussion on the short length 2D junction case Fig. 11 Impact of the interface states on electrical characteristics of together with the expected impact of interface states has also the 2DJ1 device. (a) J–V characteristics normalized to the reverse been provided. The present work opens the door to a wider saturation current given by eqn (4). Inset: Normalized total charge. (b) exploration of potential advantages that 2DJs could bring in Depletion and diffusion capacitances. The arrows indicate the direc- tion of increasing Dit. terms of FoMs given the long list of available 2D materials. For this to happen a set of material-dependent calibrated parame- ters is required for any given technology. Conflicts of interest There are no conicts to declare. Acknowledgements This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreements No. GrapheneCore2 785219 and No. Graphe- neCore3 881603, and from Ministerio de Ciencia, Innovación y Universidades under grant agreement RTI 2018-097876-B-C21 Fig. 12 Impact of the surrounding dielectric permittivity 3eff on the J– (MCIU/AEI/FEDER, UE). This article has been partially funded V characteristics of 2DJs. by the European Regional Development Funds (ERDF) allocated to the program Operatiu FEDER de Catalunya 2014–2020, with the support of the Secretaria d'Universitats I Recerca of the the bi-dimensionality 2DJs are quite sensitive to the Departament d'Empresa I Coneixment of the Generalitat de surrounding dielectric, so it is possible to design devices with Catalunya for the Emerging Technology cluster to carry out tailored depletion capacitance (eqn (14)) and current. Eqn (6) valorization and transfer of research results. 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