RADIOENGINEERING, VOL. 21, NO. 1, APRIL 2012 213 A New Model of Multiphonon Excitation Trap-Assisted Band-to-Band Tunneling Juraj RACKO, Miroslav MIKOLÁŠEK, Alena GRMANOVÁ, Juraj BREZA, Peter BENKO, Ondrej GALLO, Ladislav HARMATHA Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19 Bratislava, Slovakia juraj.racko@stuba.sk Abstract. The paper describes a new approach to calcu- lating the currents in a pn-diode based on the extension of the Shockley-Read-Hall recombination-generation model. The presented theory is an alternative to Schenk’s model of trap-assisted tunneling. The new approach takes into ac- count generation and recombination as well as tunneling processes in pn-junctions. Using this model, the real “soft” I-V curve usually observed in the case of switching diodes and transistors was modeled as a result of the high con- centration of traps that assist in the process of tunneling. Keywords Shockley-Read-Hall model, Schenk model, trap assisted tunneling, pn-junction, I-V curve. 1. Introduction Electrical characteristics of gate dielectric stacks strongly affect the reliability and non-volatility of memory devices. Leakage currents in these stacks are adversely influenced by defects present in the dielectrics that act as deep energy levels and bring about indirect, trap-assisted tunneling (TAT) of charge carriers. Under certain densities and distributions of the traps in the dielectrics, the TAT leakage current may assume dominant magnitudes. In the last decade a number of models of TAT have been proposed [1 - 5]. In common, they allow to retrieve the energy levels and the densities of trapping centers by fitting the parameters of the models to experimentally obtained data. More recently, also a simple graphical method was developed for extraction of TAT parameters in thin n-Si/SiO 2 structures [6]. Attempts have been made to formulate compact uni- fied models of TAT [7 - 9], nevertheless, some questions remain still open and new approaches to the issue can be expected. Trap-assisted tunneling results in a reduction of the Shockley-Read-Hall (SRH) recombination lifetimes in the regions of strong electric fields [10, 11]. I-V characteristics of a reverse biased pn-junction are extremely sensitive to defect-assisted tunneling. The classical SRH model as- sumes that intermediate trap centers with concentration N t lie on a discrete energy level E t . In our model we assume that the discrete energy level E t is broadened due to inter- actions of intermediate trap centers with multiphonon lat- tice vibrations, which gives rise to a band of multiphonon excitation traps. 2. Theory In our model, the distribution function D t i of the den- sity of traps (index i denotes a particular donor or acceptor band of traps) in the band gap satisfies the normalizing condition     ) ( ) ( t t C V d ) , ( x E x E i i x D N (1) where i N t are donor or acceptor trap concentrations, and   C V d ) ( ) ( ) , ( t E E i i i t i M M N x D     . (2) Here, M() is the multiphonon non-radiative transition probability for electron and hole capture [10] where S is the Huang-Rhys factor representing the elec- tron-phonon coupling, ħ 0 is the effective phonon energy,  r = S ħ 0 is the lattice relaxation energy,  = i t /(ħ 0 ),  i t =  - E C (x) + E i t , f B is the Bose distribution function 1 0 1 exp                   kT f B  and ) 1 ( 2 B B f f S z   . The sign inside the bracket in the nominator is negative for ε > E C (x) – E t and positive for ε < E C (x) – E t .                                          kT f S z z z z S x M i B r i 2 1 2 1 ln exp 2 1 ) , ( t 2 2 2 4 1 2 2 2            (3)