Currents under high driving forces Ilan Riess a, ⁎, Dima Kalaev a , Joachim Maier b a Physics Department, Technion-IIT, Haifa 32000, Israel b Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany abstract article info Article history: Received 6 October 2012 Accepted 19 November 2012 Available online 23 December 2012 Keywords: Current–voltage relations High driving force Transfer at interfaces Transfer in thin layer The current density vs. driving force relation is discussed for the hopping of localized charges between two neighboring sites under high driving force and high charge carrier concentration. First the bulk situation is treated, then the situation at the interface between two phases. To demonstrate the significance of these equations numerical simulations are presented for two cases of thin samples on which high driving forces are imposed. The results are compared with those obtained assuming a linear relation between the current density and the driving force. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Usually the flux-driving force relations for electronic conductors, ionic conductors or mixed conductors are considered to be linear, [1] the simplest example being Ohm's law. The linear relations between the current density and the gradient of the corresponding electro- chemical potential may lose their validity for very large overpotentials or very small systems. Whether a driving force is low or high is deter- mined by the difference in the corresponding chemical or electro- chemical potential for a single hop of the propagating particle. When discussing charged particles it is the difference in the corresponding electrochemical potential, Δ ˜ μ , across the hopping distance, Δx, while for neutral mobile particles it is the corresponding difference in their chemical potential across the distance Δx. Thus for electrons a high driving force occurs for, Δ ˜ μ e k B T > 1 ð1:1Þ while for ions, Δ ˜ μ ion k B T > 1 ð1:2Þ and for neutral particles, Δμ X k B T > 1 ð1:3Þ where k B is the Boltzmann constant and T the temperature. Under what experimental condition do we expect to encounter high driving forces? Note that the applied voltages, V, may be raised to the mega Volt level. A typical jump distance is Δx ~ 0.25 nm while k B T at room temperature is 0.026 eV. Voltages in the regime of MV are necessary to bring the electrical situation into the nonlinear re- gime if the sample thickness is about 1 cm, while for 1–10 nm layers which are becoming a standard in modern solid state devices voltages less than 1 V are already critical. This assumes uniform voltage distri- bution. In the case of non-uniformities regions may exist with much greater voltage gradients in a narrow regime (see Fig. 1). This is par- ticularly relevant for interfaces, where quite usually the major part of the voltage drops over a very narrow zone, hence making nonlinear- ities much more abundant. Nonlinearities can also be induced by chemical potential differ- ences. As they are limited to the order of ~1 eV, thicknesses smaller than 10 nm are required in the case of uniform gradients. Thus non- linearities may occur at interfaces and in the smaller samples. If the gradient is not uniform then thicker samples may include regions with high chemical driving forces. These considerations are far from being academic and various ex- amples can be found where high driving forces appear. As one exam- ple let us consider electrical insulation barriers. In solid state devices thin, 1–10 nm, layers are used to insulate between two conducting layers. Since the prevailing voltage is of the order of 1 V the driving force is high. A second example is diffusion barriers used to separate two phases and limit the inter-diffusion between the two phases. Let us consider phases A and B being separated by a thin diffusion barrier. This results in chemical potential differences Δμ A and Δμ B of materials A and B across the barrier, as shown schematically in Fig. 2. These chemical potential differences are of the order of 1 eV and the corresponding driving forces acting on the ions of A and B within the barrier 1–10 nm thick, are high. The third example refers Solid State Ionics 251 (2013) 2–8 ⁎ Corresponding author. E-mail address: riess@tx.technion.ac.il (I. Riess). 0167-2738/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2012.11.019 Contents lists available at ScienceDirect Solid State Ionics journal homepage: www.elsevier.com/locate/ssi