## 3 Experimental Results
The average change in resistance versus applied voltage plot of these resistors is shown in Figure 5. The resistors with thinner GeTe (0.09 and 0.15 \(\upmu\)m) show amorphous to crystalline transition at a small voltage pulse of magnitude \(\sim\)0.50 V. However, a large voltage pulse (\(\sim\)3.25 V) was required to induce a phase transition in the thicker resistor (0.40 \(\upmu\)m). As the phase transition in PCM devices is quite sharp, a voltage drop across the resistors was measured using a current source for accurate explanation of such transition. However, this method was not found adequate to capture the abrupt transition. Therefore, we plan to use high speed oscilloscope to capture the precise transition times exhibited by these novel devices in our future work. Furthermore, similar to [PERSON] et al., we also used \(E_{t}=\frac{V_{t}}{T}=\frac{\rho V_{t}}{RA}\) to obtain threshold field values from threshold voltage [16]. Here, it is observed that \(E_{t}\) increases from 3.30 V/\(\upmu\)m to 8.13 V/\(\upmu\)m as GeTe layer thickness increases from 90 nm to 400 nm. On the other hand, horizontal devices show such phase transition at relatively higher threshold voltage (30-100 V) due to the presence of larger inter-electrode distance (5 and 10 \(\upmu\)m). Here, \(E_{t}\) varies from \(\sim\)6.0 V/\(\upmu\)m to 10 V/\(\upmu\)m as channel length increases. An important relationship between threshold voltage, threshold field and inter-electrode distance can be established; resistors with large inter-electrode distance require higher threshold voltage, hence the larger threshold field for the phase transition. Figure 4: Temperature induced amorphous to crystalline phase transition in a horizontal GeTe resistor. After establishing the above relationship, the impact of device cross-sectional area on threshold voltage was also investigated. Various resistors of the different area but same inter-electrode distance were tested using a probe station. Figure 6 shows the average resistance change versus threshold voltage plot of tested resistors. It was observed that all the resistors (except one) demonstrate amorphous to crystalline phase transition at \(\sim\)3.20 V. The independence of threshold voltage on the resistor's cross-sectional area can be attributed to bottom conduction layer, which assures a uniform Joule heating in GeTe resistors. Therefore, using a relationship between heat intensity (\(I\)) and power (\(P\)) i.e., \(I=P/V\) (where \(V\) is the effective conduction volume and can be given as product of area A and channel length \(L\)), one can obtain \(=V^{2}/\rho L^{2}\), which is independent of the cross-sectional area. This explanation suggests that the fluctuating resistance of device of size 4 \(\times\) 4 \(\times\) 0.40 \(\upmu\)m\({}^{3}\) can be attributed to unstable contacts. Similar behavior was observed for horizontal devices (results not shown here). Here, it is noteworthy that the Figure 4 shows the indirect heating (i.e., hot plate)-induced phase transition in a horizontal PCM resistor (Temperature on X-axis), whereas Figures 5 and 6 corresponds  
Figure 5: Effect of inter-electrode distance on Joule heating induced crystallization in GeTe vertical resistors (same cross-sectional area). Figure 6: Effect of cross-sectional area on Joule heating-induced crystallization in GeTe vertical resistors (same inter-electrode distance). to Joule heating (i.e., current flow)-induced phase transition in vertical resistors. As in the Joule heating case, device temperature is increased by applying external voltage, and the measured resistance is given as a function of applied voltage not applied temperature.