An Analytical Approach to Understanding Mushroom-Type Phase Change Memories
Mushroom-type phase change memories have been investigated for memory and computing-in-memory applications due to their non-volatility, good retention properties and compatibility to CMOS processing . Phase change memory device can be programmed to analog resistance states, which makes the highly interesting for analog computing-in-memory applications. The resistance states are encoded in the shape and size of the amorphous volume within the crystalline phase change material (PCM). The size of this amorphous volume determines further device characteristics such as the threshold voltages, the resistance drift, and the retention characteristics. To capture these properties accurately a suitable model should also contain the information about the size and shape of the phase-configuration. So far, most groups used physical continuum simulations, which accounts for the exact shape. Moreover, often the investigations were limited to large amorphous volumes (i.e the full RESET states). The drawback of this method, however, is the long computation time, which makes it unsuitable for circuit simulation. In this work, an analytical model is presented for mushroom-type PCM cells that is based on the shape and size of the amorphous mark under different programming conditions. The model is framed for both projecting and non-projecting devices . To this end, analytical equations for the differently shaped amorphous and crystalline regions are derived. One of the main features of the model is the inclusion of a current ‘leakage’ path which injects the current directly at the outer edge of the heater. It is shown that the model reproduces well the experimental programming curve and drift data of a PCM mushroom cell. The state-dependent drift observed in experiment can be well explained within the model due to the current flowing through amorphous and crystalline regions. The drift coefficient for the amorphous and crystalline regions remains constant. The model accounts also for the asymmetry of the current conduction with respect to voltage  and includes the field-dependent conductivity of the amorphous material. To simulate the threshold switching, the resistance model is coupled with two ordinary differential equations describing the temperature in the leaky path and the main current path. To this end, all resistance elements of the model are modeled as temperature dependent. The simulation results show that the initial threshold switching is triggered in the leaky path and then extends to the main current path. The model allows us to simulate the threshold switching behavior and the retention behavior as a function of the phase configuration. For example, the typical trends of the threshold switching such as switching delay, sweep rate-dependence or state dependence can be well reproduced. Our work provides a simple framework for understanding the characteristics of PCM devices and lays the basis for a fully dynamic compact model.  M. Le Gallo and A. Sebastian, “An overview of phase-change memory device physics,” J. Phys. D Appl. Phys., vol. 53, pp. 213002, 2020.  S. G. Sarwat, T. M. Philip, C.-T. Chen, B. Kersting, R. L. Bruce, C.-W. Cheng, N. Li, N. Saulnier, M. BrightSky and A. Sebastian, “Projected Mushroom Type Phase-Change Memory,” Adv. Funct. Mater., vol. 31, pp. 2106547, 2021.  S. G. Sarwat, M. Le Gallo, R. L Bruce, K. Brew, B. Kersting, V. P. Jonnalagadda, I. Ok, N. Saulnier, M. BrightSky and A. Sebastian, “Mechanism and Impact of Bipolar Current Voltage Asymmetry in Computational Phase-Change Memory (early view),” Adv. Mater., vol., pp. 2201238, 2022.