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Tunnel Junction I ( V ) Characteristics: Review and a New Model for P-N Homojunctions N

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Tunnel Junction I ( V ) Characteristics: Review and a New Model for P-N Homojunctions N Tunnel junction I ( V ) characteristics: Review and a new model for p-n homojunctions N. Moulin, Mohamed Amara, F. Mandorlo, M. Lemiti To cite this version: N. Moulin, Mohamed Amara, F. Mandorlo, M. Lemiti. Tunnel junction I ( V ) characteristics: Review and a new model for p-n homojunctions. Journal of Applied Physics, American Institute of Physics, 2019, 126 (3), pp.033105. 10.1063/1.5104314. hal-03035269 HAL Id: hal-03035269 https://hal.archives-ouvertes.fr/hal-03035269 Submitted on 2 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tunnel junction model Tunnel junction I(V) characteristics: review and a new model for p-n homojunctions N. Moulin,1 M. Amara,1, a) F. Mandorlo,1, b) and M. Lemiti1, c) University of Lyon, Lyon Institute of Nanotechnology (INL) UMR CNRS 5270, INSA de Lyon, Villeurbanne, F-69621, FRANCE (Dated: 1 December 2020) Despite the widespread use of tunnel junctions in high-efficiency devices (e.g., multijunction solar cells, tunnel field effect transistors, and resonant tunneling diodes), simulating their behavior still remains a challenge. This paper presents a new model to complete that of Karlovsky and simulate an I(V ) characteristic of an Esaki tunnel junction. A review of different analytical models of band-to-band tunneling models is first presented. As a complement to previous work on tunnel junction simulation, the transmission coefficient is precisely determined and incorporated, the valley current between the tunneling and drift regimes is included, and calculations of physical parameters are updated. It is found that the model works for a broad range of values of the forward bias. Keywords: tunnel junction, simulation, model Eg Band gap Et Carrier transverse energy NOMENCLATURE EV Energy of valence band Acronyms EF n, EF p Fermi levels on n and p sides BT BT Band-to-band tunneling F Electric field T AT Trap-assisted tunneling IP Peak tunnel current TC Transmission Coefficient IV Valley current TM Transfer Matrix method IBT BT Tunneling current WKB Wentzel Kramers Brillouin method ICh Excess tunnel current Physical Constants k1 Wave vector of an electron of energy E ~ Planck constant=2π mC , mV Effective mass of the electron on the conduc- tion, valence band c Speed of light in vacuum me Effective mass of the electron h Planck constant ND, NA Density of donor, acceptor impurity atoms kB Boltzmann constant ni Intrinsic carriers density q Charge of the electron S Junction surface Variables T Temperature θ Parameter in Chynoweth's model V Applied bias Dv Volume density of occupied levels above EV V0 Barrier height E Energy of an electron VP Voltage at I = IP EC Energy of conduction band VV Voltage at I = IV Vbi Built-in voltage a)Electronic mail: [email protected] W Free charge space b)Electronic mail: [email protected] c)Electronic mail: [email protected] WN , WP Depletion zone on n,p side Tunnel junction model 2 I. INTRODUCTION veloped for TAT, since he considered this to be dominant over direct tunneling at forward-bias polarization. Hurkx's work was tested in 2008 by Baudrit and The tunneling effect is a quantum phenomenon that 16 allows carriers to cross a potential barrier without jump- Algora, who underlined the fact that Hurkx's model does not work at forward bias. Currently, the reference ing over it. If the barrier is thin enough and empty sites 17 4 are available in the right range of energies, carriers from model remains that of Tsu and Esaki. Hermle et al. one side can tunnel to these empty sites. It should be presented a method to simulate an isolated GaAs tunnel noted that although the word \tunnel" carries the im- junction at forward bias. age of going through, it is rather the case that carriers The first part of the present paper compares these analyt- disappear and then reappear on the other side of the ical models in the context of simulation of direct BTBT. barrier. The probability for this phenomenon to occur is As far as the simulation method is concerned, a review highly correlated with the width of the barrier and the of different analytical models and simulations leads to 1 the conclusion that the nonlocal approach (see Fig. 1 of energy of the carrier. There are many devices (TFETs , 4,5,18 RTDs2...) that rely on this effect for their properties, Ref. 5) is the most precise, since it considers effective with specific materials and doping levels being chosen to carrier transport. Also, only BTBT need be considered as the tunneling regime, since it is sufficient for correct create a tunnel effect. For example, in the field of pho- 4,5 tovoltaics, tandem solar cells score the highest efficiency simulation of an I(V ) curve. The tunneling probabil- ity can be calculated using the transfer matrix method, on both laboratory and industrial scales by using tunnel 5 junctions as key parts of their structures linking different since this method has proven to be accurate and is com- subcells.3 putationally less expensive than the Wentzel{Kramers{ Brillouin (WKB) method.19 Simulations are run under However, there have been few studies focusing on the Matlab software. behavior of this particular component of these cells, and indeed detailed simulations conducted by Hermle et al.4 The main goal of this work is to construct an analyt- and Liu et al.5 have highlighted the difficulty of such a ical model that is able to calculate I(V ) curves to aid task. A reliable tunneling model should be able to sim- in the design of the doping levels and dimensions of a ulate three different regimes: the peak current, where tunnel diode. As the experimental work of the present tunneling is dominant; the valley current, where the tun- authors is mainly concerned with silicon-based diodes, neling probability is low and drifting starts to occur;6,7 the review will focus on models that can be applied to and the diode regime, where drifting is dominant. Yajima this material. and Esaki discovered the tunneling effect experimentally in a highly doped germanium diode8 in 1958, and Esaki9 proposed a model to describe it. In 1960, an extensive theory was developed by Kane,6 following a suggestion II. REVIEW OF TUNNEL JUNCTION MODELS from Zener about a tunneling phenomenon. Soon af- ter, Karlovsky10 proposed a simpler version of the Esaki A. Tsu{Esaki based models model that was valid as long as the difference between the gap and the Fermi levels was small enough. In 1969, the 1. Tsu{Esaki model first review of tunneling models was presented by Duke.11 In his book, he compared several models and approaches This model17 is based on a previous one-dimensional to establish which of these was the most accurate at that (1D) superlattice model published in 197120. The 1973 time. He considered, among others, the previous work model incorporates a finite number of periods and a short by Kane and Esaki, as well as that by Keldysh,12 who electron mean free path. It also applies to multibarrier had found the same expression as Kane, although they tunneling. The effective mass is calculated for unper- had worked independently because of the lack of scien- turbed structures, and the 3D Schr¨odingerequation is tific communication between East and West during the solved for a 1D periodic potential V . Making the sim- Cold War. plifying assumption that the transmission coefficient TC As research on semiconductors progressed, several mod- is a function of kl only (the wave vector lies along the els were proposed (p{n diode, CMOS, SOI, III{V, etc.). barrier dimension), we obtain In 1989, at a conference in Berlin, Hurkx13 presented a new model of the recombination rate based on the J = 1952 work of Shockley and Read14 on Shockley{Read{ 0 1 Hall (SRH) recombination. This model was based on Z 1 1 + exp EF n−El qmekBT kB T TC(El)·ln dEl; trap-assisted tunneling (TAT) associated with SRH re- 2 3 @ E −E −qV A 2π ~ 0 1 + exp F p l combination and band-to-band tunneling (BTBT) at re- kB T verse bias. In his paper, Hurkx calculated the contribu- (1) tion of the tunneling effect as a recombination rate in- stead of a current density. Along the same lines, in 1991, where me is the electron effective mass and El is the 15 Klaassen presented a model that was also mainly de- electron energy along kl. This expression is general and Tunnel junction model 3 holds regardless of the type of semiconductor (direct or 4. Duke model indirect bandgap). In 1969, Duke11 presented what was then the state of art in tunneling theory, describing all the main tunneling 2. Kane model models, their weak points, and their advantages. The model simulated in this review is the one described as the most accurate at that period. In this model, the In 1961, Kane developed a model21 similar to Esaki barrier profile is corrected by a coefficient introduced into and the later Karlovsky models, but using a different ap- the expression for the maximum electric field.
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