Influence of Thomson effect on amorphization in phase-change memory: dimensional analysis based on Buckingham's П theorem for Ge₂Sb₂Te₅

Our analyses were performed on a simplified two-dimensional PCRAM cell (see figure 1). The PCM lay on a SiO₂ layer enclosing a TiN heater. Tungsten (W) layers formed the top and bottom electrodes. The calculation domain was a rectangle with dimensions of 1500 nm × 450 nm (width × height). The W top electrode, PCM, SiO₂ and TiN layers were 150, 100, 50, and 50 nm thick, respectively, and the TiN heater was 34 nm wide. The current flowed from the top to the bottom electrode.

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To investigate the balance between the Thomson effect and Joule heating, we simulated the following governing equations:

$$\begin{eqnarray}&&\rho {c}_{p}\displaystyle \frac{\partial T}{\partial t}={\rm{\nabla }}\cdot \left(k{\rm{\nabla }}T\right)+\displaystyle \frac{1}{\sigma }({\boldsymbol{j}}\cdot {\boldsymbol{j}})-{\mu }_{T}({\rm{\nabla }}T\cdot {\boldsymbol{j}}),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left({\rm{\sigma }}{\rm{\nabla }}\phi \right)=0,\end{eqnarray} \tag{ 4 }$$

where ρ and c_p are the density and heat capacity, t denotes time, k and σ are the thermal and electrical conductivities, respectively, and ϕ is the electric potential. The current is given by

$$\begin{eqnarray}&&{\boldsymbol{j}}=-{\rm{\sigma }}{\rm{\nabla }}\phi .\end{eqnarray} \tag{ 5 }$$

We solved the above equations for the composed materials of the PCRAM cell. The used physical properties are shown in table 1.

Table 1. Physical properties of the materials in the present calculation [24, 29–33].

	σ [Sm−1]	k [Wm−1K−1]	ρc [Ωcm2]	ρ [kgm−3]	cp [Jkg−1K−1]	μT [VK−1]
A-GST	$1.0\,\times \,{10}^{-1}$	$1.9\,\times \,{10}^{-1}$	$6.2\,\times \,{10}^{-5}$	$5.9\,\times \,{10}^{3}$	$2.1\,\times \,{10}^{2}$	$-4.0\,\times \,{10}^{-4}$
C-GST	$2.3\,\times \,{10}^{5}$	$1.6\,\times \,{10}^{0}$	$6.0\,\times \,{10}^{-8}$	$6.4\,\times \,{10}^{3}$	$2.1\,\times \,{10}^{2}$	$3.5\,\times \,{10}^{-5}$
TiN	$1.0\,\times \,{10}^{6}$	$1.0\,\times \,{10}^{1}$	—	$5.3\,\times \,{10}^{3}$	$7.9\,\times \,{10}^{2}$	—
SiO2	$1.0\,\times \,{10}^{-14}$	$1.4\,\times \,{10}^{0}$	—	$2.7\,\times \,{10}^{3}$	$7.5\,\times \,{10}^{2}$	—
W	$2.0\,\times \,{10}^{7}$	$1.7\,\times \,{10}^{2}$	—	$1.9\,\times \,{10}^{4}$	$1.4\,\times \,{10}^{2}$	—

At the side boundaries, an adiabatic temperature condition was imposed, and the normal gradient of the electric potential was set to zero. At the top and bottom boundaries, the electrical potential was set as

$$\begin{eqnarray}&&{\phi }_{top}={\phi }_{h},\,{\phi }_{bottom}=0,\end{eqnarray} \tag{ 6 }$$

and the temperature was fixed at 298 K. The thermal boundary resistance was neglected in the present simulation. The heat flux was balanced at all boundaries between the materials:

$$\begin{eqnarray}&&{k}_{a}{({\rm{\nabla }}T\cdot {\boldsymbol{n}})}_{a}={k}_{b}{\left({\rm{\nabla }}T\cdot {\boldsymbol{n}}\right)}_{b},\end{eqnarray} \tag{ 7 }$$

where the subscripts a and b indicate the adjacent materials at the interface, and n is the normal unit vector to the boundary. The boundary conditions of the electrical potential at the W/PCM, PCM/SiO₂, TiN/SiO₂, SiO₂/W, and TiN/W interfaces were set as

$$\begin{eqnarray}&&{\sigma }_{a}{({\rm{\nabla }}\phi \cdot {\boldsymbol{n}})}_{a}={\sigma }_{b}{\left({\rm{\nabla }}\phi \cdot {\boldsymbol{n}}\right)}_{b}.\end{eqnarray} \tag{ 8 }$$

At the TiN/PCM interface, the contact electrical resistance was imposed as

$$\begin{eqnarray}&&{\sigma }_{a}{({\rm{\nabla }}\phi \cdot {\boldsymbol{n}})}_{a}=\displaystyle \frac{{\rm{\Delta }}{\phi }_{i}}{{\rho }_{c}},\end{eqnarray} \tag{ 9 }$$

where Δϕ_i is the contact electrical potential and ρ_c is the contact resistance.

The physical properties of the materials in the present numerical simulation are listed in table 1. All properties were assumed at room temperature, and the crystalline and amorphous phases of GST were designated as C-GST and A-GST, respectively. In order to be consistent with the experimental results (Supplementary Information), the physical properties of the hexagonal-type crystalline GST were used as C-GST parameters. Note that we ignored the temperature dependence of thermal conductivity (k), pseudo-contact resistivity (ρ_c), density (ρ), and heat capacity (c_p). The ρ_c incorporates the contact resistance into the numerical analysis. This parameter will be explained in the following section and the supplementary information.

The governing equations were discretized by the finite volume method. The spatial and temporal derivatives were discretized by the second-order linear interpolation scheme and the first-order implicit Euler scheme, respectively. The discretized algebraic equations were weakly coupled by an outer iteration of each component. At each time step, the Laplace equation for the electrical potential was solved until it converged. The calculation time step and duration time were set to 1 ns and 100 ns, respectively. A constant voltage was applied at t = 0 s and the temperature variation was evaluated at 1-ns intervals. The phenomena were evaluated at 100 ns. At this time, the system was assumed as steady state because the temperature distribution was stabilized at approximately 50 ns. The computational grid was structured and uniform. The side-length of each grid cell was 2 nm. These calculation models were incorporated into the open-source software, OpenFOAM (v1812). As the standard OpenFOAM solver cannot solve the above thermoelectric phenomena, we created a new solver based on the multi-region function of OpenFOAM.

3.1.1. Buckingham's Π theorem

We first classified the phenomena into two cases: (I) without contact resistance, and (II) with contact resistance. In the case without contact resistance, one can define seven independent physical variables as

$$\begin{eqnarray}&&f({\rm{\sigma }},{\rm{\Delta }}\phi ,\rho ,{c}_{{\rm{p}}},k,{\rm{\Delta }}x,{\mu }_{{\rm{T}}})=0,\end{eqnarray} \tag{ 10 }$$

where Δϕ is the applied voltage and Δx is the PCM thickness. In this system, the physical variables can be described by five independent physical units:

$$\begin{eqnarray}&&{f}^{{\prime} }({\rm{m}},{\rm{kg}},{\rm{s}},{\rm{A}},{\rm{K}})=0.\end{eqnarray} \tag{ 11 }$$

Applying Buckingham's П theorem, the phenomena in this system can be represented by two dimensionless numbers.

In the case with contact resistance, one can define eight independent physical variables as

$$\begin{eqnarray}&&F({\rm{\sigma }},{\rm{\Delta }}\phi ,\rho ,{c}_{{\rm{p}}},k,{\rm{\Delta }}x,{\mu }_{{\rm{T}}},{\rho }_{{\rm{c}}})=0.\end{eqnarray} \tag{ 12 }$$

These variables are described by five independent physical units as shown in equation (11). Again applying Buckingham's П theorem, the phenomena in the case of contact resistance can be described by three dimensionless numbers.

3.1.2. Dimensionless numbers without contact resistance

To non-dimensionalize the governing equations (3) and (4), we define the following dimensionless numbers:

$$\begin{eqnarray}&&{T}^{* }=\displaystyle \frac{T}{({\alpha }^{2}/{\rm{\Delta }}{x}^{2}{c}_{p})},\,{\phi }^{* }=\displaystyle \frac{\phi }{{\rm{\Delta }}\phi },{{\boldsymbol{j}}}^{* }=\displaystyle \frac{{\boldsymbol{j}}}{\displaystyle \frac{\sigma {\rm{\Delta }}\phi }{{\rm{\Delta }}x}},{{\boldsymbol{x}}}^{* }=\displaystyle \frac{{\boldsymbol{x}}}{{\rm{\Delta }}x},{t}^{* }=\displaystyle \frac{t}{{\rm{\Delta }}{x}^{2}/\alpha },\end{eqnarray} \tag{ 13 }$$

where the asterisk indicates a dimensionless variable. Substituting expressions (13) into equations (3) and (4), the dimensionless governing equations are respectively obtained as

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\phi }^{* }=0,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {T}^{* }}{\partial {t}^{* }}+{{\rm{\nabla }}}^{2}{T}^{* }+\displaystyle \frac{\sigma {\rm{\Delta }}{\phi }^{2}{\rm{\Delta }}{x}^{2}}{\rho {\alpha }^{3}}\left({{\boldsymbol{j}}}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)-\displaystyle \frac{{\mu }_{T}\sigma {\rm{\Delta }}\phi }{k}\left({\rm{\nabla }}{T}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)=0.\end{eqnarray} \tag{ 15 }$$

In these dimensionless governing equations, we define the dimensionless numbers as follows:

$$\begin{eqnarray}&&A=\displaystyle \frac{\sigma {\rm{\Delta }}{\phi }^{2}{\rm{\Delta }}{x}^{2}}{\rho {\alpha }^{3}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&B=\displaystyle \frac{{\mu }_{T}\sigma {\rm{\Delta }}\phi }{k}.\end{eqnarray} \tag{ 17 }$$

As predicted by Buckingham's П theorem, we could define two dimensionless numbers in this system.

To evaluate the influence of the Thomson effect on the temperature increase in PCRAM, we require another normalized temperature. Considering the thermal diffusion and Joule heating terms, the governing equation in steady state is written as

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left(k{\rm{\nabla }}T\right)+\displaystyle \frac{1}{\sigma }\left({\boldsymbol{j}}\cdot {\boldsymbol{j}}\right)=0.\end{eqnarray} \tag{ 18 }$$

Here, the dimensionless governing equations were normalized under the following assumptions:

• The steady-state temperature is linearly distributed along the vertical direction
• The steady-state electrical potential is linearly distributed along the vertical direction
• The contact resistances except at the PCM/TiN interface are sufficiently small

Substituting equation (5) into equation (18), we obtain the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{k{\rm{\Delta }}T}{{\rm{\Delta }}{x}^{2}}\sim \displaystyle \frac{\sigma {\rm{\Delta }}{\phi }^{2}}{{\rm{\Delta }}{x}^{2}},\end{eqnarray} \tag{ 19 }$$

Finally, the temperature increase is given by

$$\begin{eqnarray}&&{\rm{\Delta }}T\,\sim \displaystyle \frac{\sigma {\rm{\Delta }}{\phi }^{2}}{k}.\end{eqnarray} \tag{ 20 }$$

The dimensionless temperature can be normalized by equation (20). Taking the dimensionless temperature as ${T}^{* }=\tfrac{T}{(\sigma {\rm{\Delta }}{\phi }^{2}/k)},$ the steady-state dimensionless energy equation transforms into

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{T}^{* }+\left({{\boldsymbol{j}}}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)-B\left({\rm{\nabla }}{T}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)=0.\end{eqnarray} \tag{ 21 }$$

As seen in this equation, the influence of the Thomson effect on the temperature variation in PCRAM can be evaluated by a single dimensionless number, B.

3.1.3. Dimensionless numbers with contact resistance

In the case with contact resistance, we must consider three dimensionless numbers as discussed in section 3.1.1 The contact resistance is non-dimensionalized as

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{i}^{* }=\displaystyle \frac{{\rm{\Delta }}{\phi }_{i}}{{\rm{\Delta }}\phi }.\end{eqnarray} \tag{ 22 }$$

Substituting equation (9) into equation (22), we obtain

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{i}^{* }=\displaystyle \frac{{\rho }_{c}\sigma {\rm{\nabla }}\phi }{{\rm{\Delta }}\phi }=\displaystyle \frac{{\rho }_{c}\sigma }{{\rm{\Delta }}x}{{\boldsymbol{j}}}^{* }.\end{eqnarray} \tag{ 23 }$$

An additional dimensionless number is defined as

$$\begin{eqnarray}&&C=\displaystyle \frac{{\rho }_{c}}{{\rm{\Delta }}x/\sigma }.\end{eqnarray} \tag{ 24 }$$

Equation (23) can be physically interpreted as

$$\begin{eqnarray}&&C=\displaystyle \frac{Contact\,\,resistance}{Volumetric\,\,resistance\,\,of\,\,PCM}.\end{eqnarray} \tag{ 25 }$$

We decided that contact resistance dominates the effect when C is higher than O(1) (i.e., order 1). In this case, the magnitude of the electric current is determined not by the volumetric resistance of PCM but by the contact resistance. Under this condition, the effective electrical voltage (Δϕ_eff) in the PCM is calculated as

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{eff}={\rm{\Delta }}\phi -{\rm{\Delta }}{\phi }_{i}.\end{eqnarray} \tag{ 26 }$$

The electric potential applied at the PCM–electrode interface is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{\phi }_{i}}{{\rho }_{c}}=\displaystyle \frac{{\rm{\Delta }}\phi -{\rm{\Delta }}{\phi }_{eff}}{{\rho }_{c}}={\boldsymbol{j}}\sim \sigma \displaystyle \frac{{\rm{\Delta }}{\phi }_{eff}}{{\rm{\Delta }}x}.\end{eqnarray} \tag{ 27 }$$

Equation (27) is finally transformed as

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{eff}\sim \displaystyle \frac{{\rm{\Delta }}\phi }{1+C}.\end{eqnarray} \tag{ 28 }$$

The current density and temperature increase are respectively given by

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \frac{{\rm{\Delta }}{\phi }_{i}}{{\rho }_{c}}\sim \displaystyle \frac{C}{1+C}\displaystyle \frac{{\rm{\Delta }}\phi }{{\rho }_{c}},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}T\,\sim \displaystyle \frac{\sigma {\rm{\Delta }}{\phi }_{eff}^{2}}{k}\sim \displaystyle \frac{1}{{\left(1+C\right)}^{2}}\displaystyle \frac{\sigma {\rm{\Delta }}{\phi }^{2}}{k}.\end{eqnarray} \tag{ 30 }$$

Using equations (28)–(30), the dimensionless variables are defined as

$$\begin{eqnarray}&&{T}^{* }=\displaystyle \frac{T}{\tfrac{1}{{\left(1+C\right)}^{2}}\tfrac{\sigma {\rm{\Delta }}{\phi }^{2}}{k}},\,{\phi }^{* }=\displaystyle \frac{\phi }{\tfrac{{\rm{\Delta }}\phi }{1+C}},{{\boldsymbol{j}}}^{* }=\displaystyle \frac{{\boldsymbol{j}}}{\tfrac{C}{1+C}\tfrac{{\rm{\Delta }}\phi }{{\rho }_{c}}\,},{{\boldsymbol{x}}}^{* }=\displaystyle \frac{{\boldsymbol{x}}}{{\rm{\Delta }}x}.\end{eqnarray} \tag{ 31 }$$

Substituting equation (31) into equations (3) and (4) in steady-state, the dimensionless governing equations are given as

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{\phi }^{* }=0,\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{T}^{* }+\left({{\boldsymbol{j}}}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)-\displaystyle \frac{1}{1+C}\displaystyle \frac{{\mu }_{T}{\rm{\Delta }}\phi {\rm{\Delta }}x}{k{\rho }_{c}}\left({\rm{\nabla }}{T}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)=0.\end{eqnarray} \tag{ 33 }$$

Finally, the equation is described in terms of the dimensionless numbers:

$$\begin{eqnarray}&&{{\rm{\nabla }}}^{2}{T}^{* }+\left({{\boldsymbol{j}}}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)-\displaystyle \frac{B}{1+C}\left({\rm{\nabla }}{T}^{* }\cdot {{\boldsymbol{j}}}^{* }\right)=0.\end{eqnarray} \tag{ 34 }$$

In the case with contact resistance, the contribution of the Thomson effect on the temperature distribution can be evaluated by the ratio of two dimensionless numbers, B/(1 + C).

To validate the above-defined dimensionless numbers, we conducted numerical simulations with and without contact resistance while varying the electric potential (voltage). First, the influence of Thomson effect on the temperature increase was evaluated in the absence of contact resistance. The dimensionless number B under the different applied potentials is listed in table 2. Amorphization in C-GST was confirmed under applied voltages of 0.2–0.3 V in a preliminary numerical simulation. Therefore, the B value in this case is shown only at 0.1 and 0.5 V. In A-GST, the dimensionless number was much smaller than 1, so the Thomson effect on the temperature increase was assumed negligibly small. Conversely, in C-GST, the B approached 1 and increased with voltage.

Table 2. Dimensionless number B at different applied voltages.

	A-GST	C-GST
Δϕ = 0.1 V	$2.11\,\times \,{10}^{-5}$	$5.08\,\times \,{10}^{-1}$
Δϕ = 0.5 V	$1.05\,\times \,{10}^{-4}$	$2.54\,\times \,{10}^{0}$
Δϕ = 1.0 V	$2.11\,\times \,{10}^{-4}$	—
Δϕ = 2.0 V	$4.21\,\times \,{10}^{-4}$	—
Δϕ = 4.0 V	$8.42\,\times \,{10}^{-4}$	—

To confirm the relationship between the B and temperature increase, we simulated the temperature distribution of C-GST for various B values. The Thomson coefficient was varied from 0 to 10 μ_T. Figure 2 shows the simulated temperature distribution at 100 ns and 0.2 V in the C-GST without contact resistance. Increasing B decreased the maximum temperature. When B exceeded O(1), the Thomson effect largely affected the temperature increase. According to the simulation results, the influence of the Thomson effect on the temperature increase can be evaluated through B, as derived in section 3.1.2.

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Second, we validated the classification of dimensionless numbers in the case of contact resistance. Prior to the numerical simulation, we investigated the incorporation of contact resistance into the system. In classical theory, the contact resistance is calculated by dividing the contact resistivity by the contact area between the PCM and electrode. However, when the contact resistance was defined in this way, the temperature was hardly increased under the applied voltage because the contact resistance was large and blocked the current flow (Supplementary Information). In a highly scaled PCRAM cell, the device resistance was reportedly dominated by the contact resistance [8, 27, 28]. However, the device resistance calculated using the contact resistance is known to exceed the experimentally obtained device resistance because the latter is lowered by leakage current [34, 35]. Therefore, when including the contact resistance, we should also consider the leakage current. To extract the influence of contact resistance and apply it to the numerical simulation, we introduced and experimentally determined a new parameter called the pseudo-contact resistivity, ρ_c, which brings the contact resistance into the numerical simulation. The details are given in the Supplementary Information.

When contact resistance is present, the current flow requires a larger voltage than in the case without contact resistance. Therefore, in the numerical simulations with contact resistance, the voltage in the C-GST was set to 4.0 V. To evaluate the contribution of the Thomson effect on the temperature increase, we set four Thomson coefficients: 0, 0.1μ_T, 1.0μ_T, and 10 μ_T, giving dimensionless number ratios B/(1 + C) of 0, 1.37 × 10⁻¹, 1.37, and 1.37 × 10¹, respectively. Figure 3 shows the temperature distributions at 100 ns in the contact resistance cases with the four values of B/(1 + C). Through the parameter ratio B/(1 + C), we can estimate the contribution of the Thomson effect under the influence of contact resistance. When the ratio was smaller than O(1) (figure 3 (a) and (b)), the temperature increase was independent of the Thomson coefficient because the Thomson effect was much smaller than Joule heating. Conversely, when the ratio exceeded O(1), the Thomson effect noticeably lowered the maximum temperature (see figure 3 (d)).

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In addition, the dimensionless parameters can locate the maximum temperature in the GST. Figure 4 shows the temperature distributions along the central vertical line of the C-GST for different evaluation parameters in the cases without contact resistance (panel (a)) and with contact resistance (panel (b)). The origin of the x coordinate was set at the bottom surface of the C-GST. The maximum temperature shifted downward as B or B/(1 + C) increased, reflecting an increased cooling effect by the Thomson effect in the PCM component.

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Figure 5 is a flowchart of the method for evaluating the contribution of the Thomson effect in PCM. The first step calculates the dimensionless number C, which determines whether the evaluation method changes. When C is much smaller than 1, the contribution of the Thomson effect on the temperature increase can be evaluated using B = μ_T σΔϕ/k, but when C is closer to 1, the contribution of Thomson effect can be evaluated by the ratio of dimensionless numbers, B/(1 + C). In both cases, the Thomson effect alters the temperature increase when the evaluation parameter exceeds O(1), and its contribution increases with increasing dimensionless evaluation parameter.

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Following the flowchart in figure 5, we evaluated the characteristics of C-GST in PCRAM; especially, the influence of the Thomson effect on the temperature increase. We first evaluated the dominant resistance factor through C. The temperature-dependent electrical conductivity and Thomson coefficient [33] were respectively determined as follows:

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{6.789\,\times \,{10}^{7}}{T},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{\mu }_{T}=1.162\,\times \,{10}^{-7}T.\end{eqnarray} \tag{ 36 }$$

The temperature-dependent resistivity is positive (equation (35)), indicating metallic characteristics of C-GST[33]. That is, the temperature-elicited change in the electrical characteristics originates from a mobility change rather than a carrier-density change. As the temperature dependence of carrier density is assumed negligible in C-GST, the temperature dependence of the contact resistivity was ignored in the calculation. This assumption is based on the contact resistivity measurements of Roy et al [36] and Deshmukh et al [37], who demonstrated a near-negligible temperature dependence of the contact resistivity in both face-centered cubic and hexagonal closest packed crystalline forms of GST. The temperature dependence of C in the C-GST is shown in figure 6. The C was a decreasing function of temperature. The contact resistance exceeded the volumetric resistance of C-GST across the temperature range, but (as shown in figure 6) its contribution reduced with increasing temperature.

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To establish the relationship between B/(1 + C) and temperature, we must understand the relationship between the temperature change and voltage, which is roughly estimated by equation (30). We thus simulated the temperature distribution at different voltages. Figure 7 plots the maximum temperature increase versus voltage in the C-GST with contact resistance. As the approximation curve differed from that given by equation (20), it was multiplied by a constant and fitted. When weighted by 0.02, the temperature increase calculated by equation (30) well agreed with the simulated curve. The analytical solution (equation (30)) was derived in a one-dimensional configuration, so its results differed from the simulated results and the value of weighting factor was meaningless. This result also indicates that the temperature increase can be roughly estimated from the physical properties using the dimensionless numbers. From the temperature change versus voltage relationship, the B/(1 + C)–temperature relationship was calculated and is plotted in figure 8. In C-GST at low temperature, the Thomson effect was negligibly small because the B/(1 + C) was lower than O(1). In contrast, the B/(1 + C) exceeded 1 in the higher temperature range, and the Thomson effect in the C-GST dominated the temperature increase. As estimated from figure 4, the Thomson effect suppressed the temperature increase by approximately 10%–20% in the high-temperature range. The present study confirmed that the Thomson effect increased the operating energy in C-GST, especially in the high-temperature range.

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In the present study, we investigated the contribution of the Thomson effect on the thermal efficiency using dimensionless numbers. Thus far, evaluating the contribution of Thomson effect on the thermal efficiency has been difficult, because this effect depends on the relationship between the electric potential and temperature fields. Using the proposed dimensionless numbers, we can easily evaluate the contribution of Thomson effect on the temperature increase and hence understand the phenomenon.