Current-driven threshold switching of a small polaron semiconductor to a metastable conductor

David Emin

Phys. Rev. B 74, 035206 – Published 18 July 2006

Abstract

Steady-state current flow through a nonuniform medium generally alters the local carrier density. In particular, driving conventional high-mobility charge carriers through a region in which they collapse into low-mobility small polarons propels the small-polaron density beyond its equilibrium value. There are two contributions to the local augmentation of the small-polaron density. These contributions are proportional to the ratios of the rate governing intersite motion of a conventional high-mobility carrier $R_{f}$ to (i) the (relatively slow) rate governing intersite hopping of a small polaron $R$ , and to (ii) the (even slower) rate governing conversion of a carrier between being quasifree and being a small polaron, $r$ . As a result of the very large values of these ratios, $R_{f} ∕ R$ and $R ∕ r ⪢ 1$ , large increases in the small-polaron density can be obtained with accessible electric-field strengths, $< 10^{6} V ∕ cm$ . However, upon being driven to a sufficiently high density, small polarons become unstable with respect to conversion into nonpolaronic carriers. As a result, currents beyond a threshold value can convert a small-polaron semiconductor to a high-mobility semiconductor. Reducing the current permits the material’s carriers to relax back to being small polarons. This phenomenon may account for the threshold switching that is observed in materials in which equilibrated carriers appear to form small polarons (e.g., amorphous boron, transition-metal oxide glasses, and chalcogenide glasses).

Received 3 May 2006

DOI:https://doi.org/10.1103/PhysRevB.74.035206

Authors & Affiliations

David Emin

Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA

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Issue

Vol. 74, Iss. 3 — 15 July 2006

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Figure 1
(Color online) The model envisions a linear chain of $m$ sites upon which carriers self-trap as small polarons. Quasifree carriers enter and exit this chain through two $n$ -member chains. In the absence of a driving electric field, independent small polarons hop between adjacent sites of their chain with the rate $R$ . The field-free rate characterizing quasifree carriers moving between adjacent sites of their $n$ -member chains is denoted by $R_{f}$ . The rate $r$ characterizes the conversion of a charge carrier between being quasifree and being a small polaron at the boundaries between the two types of chains. The relative magnitudes of these three rates are described by the inequalities: $R_{f} ⪢ R ⪢ r$ .Reuse & Permissions
Figure 2
(Color online) Steady-state flow of carriers from the left drives nonuniform accumulation of small polarons within their $m$ -member chain. With low currents the small-polaron accumulation tends to fall with distance from the site at which carriers enter the small-polaron chain. High currents drive the accumulation to peak where carriers exit the $m$ -member chain. This accumulation typically increases the small-polaron density by a factor $\sim (R_{f} ∕ r)$ beyond its equilibrium value. The function $f_{u}$ denotes the current-driven increase of the small-polaron density at site $u$ on the $m$ -site chain divided by its equilibrium value. This function, as given by Eq. (33), divided by $(1 ∕ B - 1)$ is plotted against $u$ for $\ln A = 0.1$ and 1.0 with $(1 ∕ B C - 1) ∕ (1 ∕ B - 1) = 1 ∕ 4$ for a 10-site chain. The parameters $A$ , $B$ , and $C$ are defined in terms of the three intersite transition rates in the introductory paragraph of Sec. 3.Reuse & Permissions
Figure 3
(Color online) A sufficiently strong current can drive the local density of small polarons high enough to destabilize them with respect to forming quasifree carriers. This effect is modeled as reducing the length of the chain on which carriers self-trap to form small polarons. As depicted above, reducing the chain length from 10 sites (solid lines) to 9 sites (dotted-dashed lines) alters the pattern of small-polaron accumulation. The accumulation driven by a low current is reduced as the chain of sites with small-polaron formation is shortened. By contrast, the small-polaron density driven by a high current rises as the value of $m$ is reduced. These effects are illustrated by plotting $f_{u}$ , the current-driven increase of the small-polaron density at site $u$ divided by its equilibrium value, for $m = 10$ (solid lines) and $m = 9$ (dotted-dashed lines) with both low and high currents. The plot shows $f_{u}$ , as given by Eq. (33), divided by $(1 ∕ B - 1)$ plotted against $u$ for $\ln A = 0.1$ and 1.0 with $(1 ∕ B C - 1) ∕ (1 ∕ B - 1) = 1 ∕ 4$ . In the introductory paragraph of Sec. 3, the parameters $A$ , $B$ , and $C$ are defined in terms of three rates characterizing carriers’ intersite motion.Reuse & Permissions
Figure 4
(Color online) A plot of the current density $J$ versus applied electric field $F$ is used to describe the small-polaron model of threshold switching. An applied electric field drives high-mobility charge carriers into a material in which they form small polarons and move via low-mobility hopping. As the applied field is increased the small-polaron mobility rises exponentially and small polarons increasingly accumulate in the material. Beyond a threshold current, the density of small polarons is driven high enough so that they are destabilized with respect to their forming quasifree carriers. The region with small polarons then shrinks to a small remnant. The resistance of the remnant contributes to that of the high-conductivity state. Reducing current in the high-conductivity state below its “holding” value permits carriers to relax into small polarons thereby reforming the low-conductivity state. Alternatively, current flow in the high-conductivity state may be high enough and prolonged enough to induce a structural phase (e.g., amorphous to crystalline) transition. The switch becomes a memory switch as its maintenance no longer requires a holding current.Reuse & Permissions

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