The family of these materials are often referred to as chalcogenides (although strictly speaking they don't always include only metals from that periodic group). They've been investigated quite a bit for use in phase change memory if you have access to academic journals (I could probably root around later and see what's not behind a paywall).
The strange looking curve is really kind of two parts forward biasing/reverse biasing. Assuming you're starting from a virgin device you move along the curve where you're sweeping the voltage upwards but not really getting much increase in current (since there's no real channel yet). In these materials the mechanism is the establishment of little "fingers" of metal migrating and completing the circuit. As the "fingers" move forward you suddenly start seeing more and more current without increasing voltage (your effective resistance is dropping) and then jumps up as the connection is made. Eventually you sweep the voltage backwards. Something similar happens in the part of the AC curve in reverse.
The strange looking curve is really kind of two parts forward biasing/reverse biasing. Assuming you're starting from a virgin device you move along the curve where you're sweeping the voltage upwards but not really getting much increase in current (since there's no real channel yet). In these materials the mechanism is the establishment of little "fingers" of metal migrating and completing the circuit. As the "fingers" move forward you suddenly start seeing more and more current without increasing voltage (your effective resistance is dropping) and then jumps up as the connection is made. Eventually you sweep the voltage backwards. Something similar happens in the part of the AC curve in reverse.
There is a really good diagram in this paper: "Mechanism for resistive switching in chalcogenide-based electrochemical metallization memory cells" from Zhuge et al http://scitation.aip.org/content/aip/journal/adva/5/5/10.106...