Recombination in Semiconductors: Appearance of Nonequilibrium Carriers due to Injection or Redistribution in the Sample

Abstract

It is shown that the traditional approach for consideration of recombination under condition of steady-state current in the absence of external carrier generation is internally contradictory. Sometimes the approach leads to obviously incorrect results. Such situations are demonstrated and a new method for consideration of recombination is proposed. In the present paper we would like to consider some internal inconsistencies in the conventional description when we have carrier recombination under condition of steady-state current. Practically all text-books 1,2,3,4,5 present technique to solve the given problem based on the solution of a set of continuity equations:where n and p are the electron and hole concentration, j n,p are the current of electrons and holes, R n,p are the electron and hole recombination rate respectively. Here we consider the absencens of external generation of carriers (by light, etc.). Thus the nonequilibrium carriers are a result of injection or accumulation of carriers near potential barriers at interfaces. For small concentration of non-equilibrium carriers δn ≡ n − n 0 ≪ n 0 , δp ≡ p − p 0 ≪ p 0 (where n 0 , p 0 are concentration of electrons and holes without the current) the recombination rates are widely assumed to be of the following form:where τ n , τ p are life times of electrons and holes respectively. Obviously, by virtue of charge preservation law an extra conditionshould hold. Although it is not mentioned in the literature that Eq. (2) makes the system overdetermined. Some authors 3 use the latter expression as an equation to find the carrier concentration or to reduce by one the number of unknowns in the problem. However, such approach seems to be incorrect because the Eq. (2) is not a new condition for the concentration of non-equilibrium carriers, rather it is the criterion for correctness of the recombination description, and should fulfill identically at any concentration of non-equilibrium carriers. Probably due to this reason other approach frequently is used, 1,2,3,4,5,6,7 assumingwhere δp means the non-equilibrium concentration of minority carriers. Just this approach is classical and is used widely both in text-books, and in papers devoted to kinetic phenomena in semiconductors. This description possesses a serious inconsistency. It becomes especially obvious if we consider injection of majority carriers. From physical reasons it is evident: injected non-equilibrium majority carriers should recombine. While from a formal point of view, as far as non-equilibrium minority carriers do not occur (δp = 0) the recombination rates are

Effect of electron–phonon energy exchange on thermal wave propagation in semiconductors considering carrier diffusion and recombination

“…This term takes into account the change in the rate of carrier thermal generation, where it is proportional to the square of the intrinsic carrier concentration in the semiconductor. Therefore, it is straightforward to show in a similar manner that in the case for different electron and phonon temperature distributions, the electron-hole pair recombination rate is proportional to the heating of carriers, i.e., [20] …”

Current-Voltage Characteristic of a p-n junction: Problems and solutions

“…14, that if the current flows from the p into the n-region the p-n junction is cooled, [18], this means that from Eqs. 8 in the p-region and in the n-region. When we reverse the electrical current, the sign of changes too and the p-n junction is heated.…”

“…However, the above recombination model for recombination, as it was shown in [8], is basically incorrect. The correct expression for the recombination rates is [9][10][11]:…”

Transport boundary condition for semiconductor structures

“…If we consider a different model of ''cross-recombination", we will arrive to the same structure of the expression for R r . We write down it by analogy with the interband recombination rate [23]: R r ¼ aðT e ; TÞnp À aðT; TÞn 0 p 0 ; ð30Þ where n ¼ n 0 ¼ n m ; p is the hole concentration in the semiconductor (p 0 is the equilibrium hole concentration at the uniform equilibrium temperature T 0 ); T e ¼ T 0 is the electron temperature in the metal, and T ¼ T p ðÀ0Þ is the hole temperature in the semiconductor.…”

“…At the same time, the rate of thermal generation of conduction electrons (the second term) is controlled only by the temperature of the crystal lattice (for more details, see Ref. [23]). …”