Thermal Crosstalk Analysis of Phase Change Memory Considering Thermoelectric Effect and Thermal Boundary Resistance

Phase change memory (PCM) has emerged as a promising memory for next-generation applications due to its high-speed read and write capabilities as well as non-volatility. However, as PCM scales down to smaller feature sizes, it faces the challenge of thermal crosstalk. During the reset operation, a large amount of heat is generated and dissipated in the PCM array, potentially affecting adjacent memory cells, compromising device stability, and limiting high-density integration. To accurately investigate the thermal crosstalk in the PCM array, the conventional finite element model of the PCM array is improved by incorporating the thermoelectric effect and thermal boundary resistance. Under the 65, 45, 32, and 22-nm process nodes, the improved model reveals the occurrence of thermal crosstalk within the PCM array, whereas the conventional model is unable to detect this phenomenon at the 65-nm node. The improved model proposed in this paper incorporates more comprehensive considerations, providing a more precise analysis of the thermal crosstalk phenomenon in the PCM array, and thereby offering theoretical reference for high-density integration of the PCM.


Introduction
Phase change memory (PCM) is a promising new generation of semiconductor memory devices, boasting fast read and write speeds as well as non-volatility [1].However, the continuous reduction in device feature size has led to an increasingly severe thermal crosstalk problem in high-density integrated PCM arrays [2][3][4].Thermal crosstalk occurs when a PCM cell being reset dissipates heat to neighboring cells that are already in a reset state, resulting in compromised data stability [5][6].To ensure the stability of PCM cells under advanced technology nodes and promote their high-density integration, it is crucial to study and analyze this problem through precise thermal simulation.
Currently, extensive research has been conducted in both academic and industrial domains on the thermal crosstalk problem of PCM.Kim et al. investigated the thermal crosstalk of PCM with different structures through simulation [7].While PCM with confined structures is capable of mitigating thermal crosstalk, the cost of mass production for such structures is relatively high, thus making it difficult to replace traditional T-type PCM [8]. Lee et al. experimentally demonstrated that parasitic heat affects PCM, leading to partial crystallization of amorphous GST.They also employed finite element simulations to mitigate thermal crosstalk in PCM arrays by adjusting reset pulse and inter-cell spacing [9].Russo et al. utilized finite element simulations to investigate thermal crosstalk within the PCM array.Their research focused on assessing the impact of isotropic or non-isotropic scaling methods on heat transfer in the array.They also illustrated the relationship between thermal crosstalk and device structural parameters [10].Although these studies are constructive and inspiring, there remain some deficiencies in their simulation methodology.Specifically, they have not considered the thermoelectric effect (TE) and thermal boundary resistance (TBR).The exclusion of these factors can potentially impair the thermal simulations' results and analysis, which, in turn, can restrict their capability to offer effective theoretical guidance for optimizing device structure and processes during the scaling of PCM.
Thermoelectric effect (TE) is an electrical heating or thermoelectricity phenomenon that occurs in thermoelectric materials (such as phase change material GST), and it affects the electrical-thermal coupling within the PCM array.Thermal boundary resistance (TBR) refers to the resistance of heat diffusion across the boundary between adjacent materials, and it limits the dissipation of heat generated within PCM during the reset process.The TBR improves the thermal confinement ability of PCM.Accordingly, in pursuit of a more precise simulation model to enhance the investigation of thermal crosstalk in the PCM array, this study incorporates the TE into the conventional governing equations for the PCM's reset process, while also introducing TBR into the conventional structural modeling.The improved model, which incorporates the TE and TBR, more accurately characterizes the thermal effects of the PCM array compared to the conventional model.Utilizing the improved model, this study employs computational simulations across distinct process nodes to analyze the impact of TE and TBR on the heat transfer within the PCM array.

Methodology
When a cell being programmed generates heat, it can transfer to the adjacent cell and may cause thermal crosstalk.As a result, the temperature in the adjacent cell's phase transition region increases, which may exceed the crystallization temperature and lead to the re-crystallization of phase change material.Consequently, during the reset operation of a cell within the PCM array, the temperature distribution across the array can be analyzed to determine whether the adjacent cells are affected by thermal crosstalk.This study utilizes the finite element method to simulate the reset process of a PCM cell.The temperature distribution is obtained through the computation of multi-physics, and subsequent analysis is conducted to assess the impact of heat transfer on neighboring cells.

Structural model of adjacent PCM cell
Figure 1 depicts the finite element structural model of adjacent PCM cells.It shows the cross-section of two T-shaped PCM cells located on the same bit line within the PCM array.Table 1 lists the corresponding values of adjacent PCM cell pitch (Lp) and bottom electrode heater width (φ) for different process nodes (F).This set of relationships is directed by the trends in the scaling roadmap of PCM [10].Using conventional structural modeling as a foundation, this study introduces the concept of TBR at the interface of each material.To approximate the TBR, a narrow rectangle with a thickness of 0.1 nm is employed, as depicted by the thick line delineating the material contact region in Figure 1.Additionally, the TE is integrated into the multi-physics simulation of PCM, which enables an accurate simulation of the PCM cell's reset process.

Governing equations of the PCM reset process
The reset process of PCM is described by a set of governing equations.The comprehensive TE is incorporated in the conventional governing equations in the paper, and the improved components are indicated with bold markings, as expressed in Equations ( 1) and (2).Equation ( 1 The reset operation generates a temperature difference between the GST and the TiN heater, which produces an electromotive force, also known as the Seebeck effect.This results in the thermally driven diffusion current  , as defined by Equation (1).When the current flows from the GST to the TiN heater, the overall reset current decreases.The Thomson effect, which arises in isotropic conductors like GST, is characterized by the Thomson term that is proportional to the Seebeck coefficient gradient S ∇ .When the current in GST flows from the crystalline GST region with a small Seebeck coefficient gradient to the programming area with a large Seebeck coefficient gradient, the Thomson term is positive and acts as a heat source, releasing a substantial amount of heat.
On the other hand, when the current flows from the TiN heater to the GST, it moves from the programming area with a high Seebeck coefficient gradient to the area with a low Seebeck coefficient gradient in the GST, resulting in a negative Thomson term, which absorbs heat.The Peltier effect arises at the junction of two conductors, and when the current flows through materials that have different Seebeck coefficients (S), such as GST and TiN heater, the Peltier term releases or absorbs heat at the interface.When the reset current flows from GST to the TiN heater, the Seebeck coefficient changes from positive to negative (S GST >0, S TiN <0).As a result, the Peltier term and there is a significant release of heat at the GST-TiN heater interface.Conversely, when the reset current flows from the TiN heater to the GST, the Peltier term becomes negative, acting as a heat sink at the interface, and absorbing heat.
In summary, the TE can induce strong coupling between electricity and heat within the PCM array, leading to notable influence in the distribution of electric and temperature fields.By incorporating the comprehensive TE in the conventional governing equations, we can more realistically simulate the complex multi-physics inside the PCM array.This provides a more accurate temperature distribution than conventional governing equations, allowing us to effectively analyze the thermal crosstalk phenomenon in the PCM array.

PCM thermal boundary resistance modeling
In a PCM array, the heat generated by a cell being reset is conducted through the material, causing thermal crosstalk to affect surrounding cells.During heat conduction, the thermal resistance of the boundary is orders of magnitude higher than that of the material [11].To reflect the thermal confinement capability of the boundary and enhance simulation accuracy, this study incorporates the TBR into the conventional structural model, as depicted by the thick line in the material contact area in Figure 1.In this structural model, we introduce a virtual material layer with a thickness of 0.1 nm to approximate TBR and define a boundary thermal conductivity value of 0.05 W/m•K in this region [12].As the interface between materials affects heat transfer within the array, the improved structural model in this study enables a more accurate simulation of heat transfer between adjacent cells compared to the studies that do not consider TBR.

Conditions and Settings for Simulation
The material parameters employed for the PCM adjacent cell model are presented in Table 2.The Seebeck coefficient of GST is obtained from [13], making it temperature-dependent with an expression 30 ) 300 , which more accurately characterizes the influence of TE on the temperature field of PCM.The melting temperature of GST is set to 873 K [12].During the reset operation of the PCM cell, the temperature in the programming area gradually increases.Once the molten GST completely covers the TiN heater, the current path is blocked by the high-resistance amorphous GST, and the PCM cell is reset, at which point the simulation terminates.The reset voltage pulse width is 50 ns, and the simulation step is 1/100 of the pulse, i.e., 0.5 ns.
To investigate the impact of TE and TBR on the array temperature distribution in the improved model, this study establishes three models for comparative analysis.The first model is the conventional model, which does not consider TE and TBR.The second model is the improved model with TE, which is compared to the first model to assess the effect of TE on the array temperature.The multiphysics of this model is described by the improved Equations ( 1) and ( 2), but the TBR is not introduced in the structural model.The third model is the improved model with TE and TBR, which is compared to the second model to reflect the influence of TBR on the array temperature.Furthermore, this article conducts simulations under different process nodes in Table 1 to analyze the thermal crosstalk problem under the corresponding nodes.

Result and Discussion
Initially, simulations were conducted at the 90-nm node, and the results are presented in Figure 2. Notably, when comparing Figures 2(a) and (b), it is observed that the TE has an impact on the reset process of cell-1 in Figure 2(b) due to the application of a forward bias voltage, causing a significant drop in the temperature of the programming area.In the PCM array, this effect leads to a reduction in the radial heat flux, and as a result, the 350 K isotherm in Figure 2(b) shifts to the left as compared to Figure 2(a).Therefore, the simulations reveal that cell-2 is less affected by dissipated heat in the improved model with TE.Further analysis comparing Figures 2(b) and (c) highlights the role of TBR.The inclusion of TBR in Figure 2(c) prevents heat dissipation from the programming area to the electrode, thereby improving the thermal efficiency of cell-1.However, this effect is accompanied by an increase in radial heat transfer in the GST layer.As a consequence, the 350 K isotherm in Figure 2(c) shifts significantly to the right as compared to Figure 2(b).
Figure 3 depicts the temperature profiles obtained along the marked dotted line in Figure 2.This dotted line crosses the region with the highest temperature inside PCM and is crucial to evaluating the thermal crosstalk problem in the PCM array.The T 0 at the measurement point is illustrated in Figure 3. Compared with the conventional model, the T 0 value for the model consisting of only TE is the lowest, because TE reduces the reset current, leading to a decrease in the average temperature inside the array.In contrast, the model incorporates both the TE and TBR showing the highest T 0 values, which attribute to the confine of heat dissipation caused by TBR.Notably, the T 0 value in all three models is lower than the critical temperature of 470 K. Consequently, our simulation results show that thermal crosstalk is not a concern in the PCM array under the 90-nm node.
To further validate the improved model's applicability across various process nodes, the T 0 at 90, 65, 45, 32 and 22-nm nodes are presented in Table 3.At the 65-nm process node, both the conventional model and the improved model incorporating TE exhibit a T 0 lower than the critical temperature of 470 K.However, in the improved model that considers both TE and TBR, T 0 reaches 486 K, indicating the occurrence of thermal crosstalk.Table 3 presents a comparative analysis of T 0 under four process nodes.Specifically, T 0 initially decreases with the inclusion of TE and subsequently increases with the inclusion of TBR.The application of a forward bias voltage to the PCM causes a decrease of T 0 due to the influence of TE, whereas TBR limits heat dissipation, thereby causing the increase of T 0 .Notably, the improved model that considers TE shows a remarkable decrease in T 0 as the feature size scales down, and the percentage reduction of T 0 becomes progressively greater.For instance, we observe a decrease of 28 K ((370-342)/370=7.6%) at 90 nm and a decrease of 219 K ((937-718)/937=23.4%) at 22 nm.This trend is due to the TE exacerbating the reduction in PCM's reset current as the device feature size decreases, in line with the conclusion in [14].
To sum up, the improved model proposed in this paper enhances the thermal analysis capability compared to the conventional model.By utilizing the temperature distribution calculated by this model, researchers and engineers can assess the extent of thermal crosstalk in the PCM array at the corresponding process node and devise appropriate solutions to mitigate its influence.These solutions may include dynamically adjusting the spacing of adjacent cells or leveraging TE by selecting materials with a larger Seebeck coefficient to achieve better cooling effects.The improved model offers valuable insights into the complex thermal behavior of PCM arrays.

Conclusion
This paper proposes an improved finite element model of the PCM array that incorporates TE and TBR into the conventional model, enabling accurate analysis of heat conduct within the array.The temperature distribution in the PCM array is then simulated and analyzed across different process nodes.The research in this paper makes the following contributions.
1) It has enhanced the simulation environment for PCM thermal analysis with an improved model that considers more comprehensive physical factors and produces calculation results that are more accurate.
2) It has revealed the influence of TE and TBR on PCM thermal crosstalk, providing a theoretical reference for the physical analysis in the PCM array.
Our results indicate that there is no risk of thermal crosstalk in the PCM array under the 90-nm process node.However, under the 65, 45, 32 and 22-nm process nodes, thermal crosstalk occurs within the PCM array, while the conventional model fails to demonstrate this phenomenon at 65-nm node.Thus, our research provides a more comprehensive analysis of the thermal crosstalk problem in PCM arrays using the improved model, which can serve as a theoretical reference for the high-density integration and optimization of PCM arrays.

C
), which denotes the electrical continuity equation, is augmented by incorporating the Seebeck term ) σ , V , and S correspond to the electrical conductivity of the material, electrical potential in the array, and the Seebeck coefficient of the material, respectively.While Equation (2) represents the heat conduction equation and is improved by adding the Thomson term ) represents the heat capacity of the material, d represents the mass density of the material, T represents the temperature, κ represents the thermal conductivity of the material, and J  represents the current density. 0

Figure 2 .
Figure 2. Comparison of the temperature between the conventional model and the improved model.

Figure 3 .
Figure 3. Temperature distribution of the adjacent PCM at 90-nm node.

Table 1 .
Relationship between the adjacent cell pitch Lp, bottom heater width φ with process node F.
Figure 1.Structural model of adjacent PCM cell

Table 2 .
PCM material parameters for simulation