14 C. Modeling of PCM physics and devices at high temperature. Extensive materials and device re- search continues to decrease crystallization times below these values[117, 118], offering hope of a bright future for PCM technology in application niches that call for rapid switching. that hold the most attraction for applications. The best case scenario would be to have a finite-difference simu- lation tool capable of handling large and arbitrary 3-D structures, which could potentially be matched against fast electrical SET and RESET experiments, slow thin- film crystallization experiments, and optical pulse exper- iments performed with the same material. From our experience with such a simulation tool[42], the RESET condition is not dictated by the maximum temperature at the cell-center, but by what happens at the edge of the cell. Typically, a voltage pulse just below the RESET condition leaves a small portion of the limit- ing cross-sectional aperture remaining in the crystalline state, usually at the extreme edges of the cell[42]. In general, besides the obvious choice of reducing the diam- eter of this limiting aperture, the best way to reduce the RESET current is to improve the efficiency with which injected electrical power heats the cell. In the best case scenario, this power would heat just the portion of the cell needed to block all of the cross-sectional aperture and produce a high resistance state. However, in any practical case, the surrounding material is also heated to some degree. Optimization can be performed by en- suring that the heated volume is minimized and by re- ducing the heat-loss through the thermally-conductive electrodes as much as possible. Another popular way to decrease RESET current is to increase the overall resis- tance of the cell by increasing the series resistance of the contact electrode[133], although it is not clear how much of this benefit may be due to associated changes in ther- mal resistance. As with any simulation, care must be taken to establish the boundary conditions correctly, because the computer memory available for simulation is inevitably finite. For instance, Dirichlet boundary conditions, which call for the edge of the cell to be held at room temperature, are frequently used[128] and are easy to program. However, in a simulation where the hot central region of the PCM cell is not very far from this boundary, then the effective heat transfer over this boundary can become unphysi- cally large, skewing the results. In contrast, Neumann boundary conditions, in which the spatial derivative of temperature (or equivalently, outgoing heat flow) is held constant at the boundaries, allow a truncated simulation to act as if it is embedded within a large expanse of sur- rounding material. Another important consideration is the optimization metric. It is conventional in the PCM community to discuss the importance of RESET current. However, it is the dissipated power, not current, which leads to the heating of the cell. The focus on current comes from the assumption that a co-located access transistor will have a current-voltage characteristic which saturates. If the applied current needs to be higher than this saturation value, there is no way for that particular transistor to supply it. This in turn implies that density will need to be sacrificed in order to provide this current. In ac- tuality, the transistor and PCM device will interact in In these equations, temperature T and voltage V are each computed as a function of time t . Even inside each ma- terial of density d , parameters such as specific heat ( C$_{p}$ ), thermal conductivity ( κ ), and electrical resistivity and conductivity ( ρ and σ ) are frequently functions of both position and temperature. The current density J and the temperature dependence of the electrical conductiv- ity serve to intimately cross-couple these two equations. A number of studies have used analytical equations[49, 108, 119–122], finite-element techniques[123–131], and finite-difference techniques[42, 132] to analyze either PCM cells or phase change material. Pirovano et al. studied the RESET current and the thermal prox- imity effect of scaled PCM by both simulation and experiment[41]. Although analytical techniques are at- tractively simple, and work well for explaining the incubation of new crystalline nuclei[108] or threshold switching[49], it is difficult to include the effects of inho- mogeneous temperature distributions and temperature- dependent resistivity, which critically affect the RESET current through their effect on the dynamic resistance of the cell. Finite element techniques can include these effects, and work well for cylindrically symmetric cell de- signs such as the conventional “mushroom” cell, since such structures can be reduced to a single ( r, z ) plane. However, because of the inherent computational difficulty in inverting matrices as they grow very large, these tech- niques are difficult to extend to three-dimensional cell designs. And finally, even though nucleation is unlikely to play an effect during the fast RESET pulse, recrys- tallization at the end of a RESET pulse does play an important role in the value of the RESET current, espe- cially for the fastest-crystallizing phase change materials Because of the large number of factors influencing the performance of PCM devices, a number of groups have begun to perform predictive numerical simulations. Par- ticularly for the consideration of reducing the RESET current that can limit density by requiring a overly large access device, even straightforward electrothermal mod- eling of the temperature produced by a particular in- jected current can be highly revealing. Such electrother- mal studies typically need to simultaneously solve the heat diffusion equation, and Laplace’s equation, dC$_{p}$ dT dt = ∇ · ( κ ∇ T ) + ∣ $_{∣}$ρJ 2 ∣ $_{∣}$, (2) ∇ · ( σ ∇ V ) = 0 . (3)