Models for the self-heating evaluation of a gallium nitride-based high electron mobility transistor

The investigated In_0.185Al_0.815N/AlN/GaN HEMT structure including 6.3 nm In_0.185Al_0.815N/1.5 nm AlN/1830 nm GaN heterostructure was grown by MOVPE on 350 μm thick 4 H-SiC substrate. The backside Au contact is soldered to 1 mm thick CuMo leadframe by 60 μm thick AuSn solder. Top ohmic drain/source and gate contacts were created by standard Au-based metallization.

Gated transmission line model (GTLM) HEMT of width w ≈ 100 μm with a gate of length d_G ≈ 0.15 μm situated in the middle of the source to drain gap of length d_SD ≈ 2.5 μm was investigated. As a temperature sensor, a parallel TLM structure of length d_TLM ≈ 2.5 μm and width w ≈ 100 μm located in the distance of d_GT ≈ 120 μm from GTLM HEMT was utilized as shown in figure 1. The device is set in the open package placed on the Al thermal chuck preserved at a constant temperature.

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Semiconductor parameter analyzer Agilent 4155C and controlled thermal chuck were utilized to acquire output characteristics at zero gate-source voltage V_GS and drain-source voltage V_DS varied from 0 V up to 20 V together with average TLM temperature T_S obtained as the current temperature dependence at constant voltage ∼0.5 V in the low-power operating regime. The chuck temperature was set in the range 25 °C–85 °C to demonstrate and to compare methods based on ambient temperature and dissipated power variation. The device was recovered by white LED illumination for 1 min between measurements. The three-dimensional (3D) thermal FEM simulations of the device were performed by Synopsys TCAD Sentaurus [8, 9]. The 3D model was created according to device geometry, layout and thickness of individual layers. Material thermal conductivity values were obtained from the previous work and calibrated utilizing the measurements [10]. The boundary condition of the constant ambient temperature is set to the structure backside based on ideal heat transfer to the heatsink. The structure self-heating is simulated by three thermal contacts placed along 2DEG between drain and source. The individual heat sources represent heat contribution from drain to source the access region, the region under the gate electrode and the pinch-off region located at the drain side gate edge [11].

The drain–source current I_DS dependence on the drain–source voltage V_DS for gate–source voltage V_GS = 0 V at varying ambient temperature T₀ in the range of 25 °C–85 °C and sensor temperature T_S dependence on total dissipated power P were acquired as shown in figures 2 and 3, respectively. Low temperature-dependent threshold voltage V_TH ≈ −3.8 V acquired from semi-logarithmic transfer characteristics of the transistor, V_GS = 0 V and drain to gate voltage drop V_DG ≈ 1.5 V allows us to estimate saturation voltage ${V_{{\text{DS,SAT}}}} = {V_{{\text{GS}}}} - {V_{{\text{TH}}}} + {V_{{\text{DG}}}}$ corresponding to the measured value V_DS,SAT ≈ 6 V.

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The electro-thermal model offers an approximate pinch-off area length increase under the gate d_PO/V_PO ≈ 0.4 nm V⁻¹ and gate to drain area d_PD/V_PO ≈ 2.2 nm V⁻¹ utilizing pinch-off voltage V_PO in the saturation regime. 3D thermal FEM simulations are keeping up with HEMT dimensions such as d_PO and d_PD. Assuming ${\text{d}}{V_{{\text{PO}}}} \approx {\text{d}}{V_{{\text{DS}}}}$ the channel length modulation caused by d_PO results in the constant ratio ${g_{\text{D}}}/{I_{{\text{DS}}}} \approx {\left( {{d_{\text{G}}} - {d_{{\text{PO}}}}} \right)^{ - 1}}{\text{d}}{d_{{\text{PO}}}}/{\text{d}}{V_{{\text{PO}}}} \approx {2.7.10^{ - 3}}{{\text{V}}^{ - 1}}$ [8]. Small d_G brings notable g_D in comparison with measured output conductance dI_DS/dV_DS including self-heating. Various methods are advisable to compare and obtain proper g_D value having a significant impact on average channel temperature T_A determination for short gated HEMT.

Differential thermal resistance R_A(T₀) vs. P and R_A(T₀) vs. T₀ at different T₀ calculated by equation (11) (see appendix) is shown in figures 4 and 5, respectively. The increase of R_A with P at defined T₀ is partially caused by spatially distributed electrical parameters of the active device area such as g_D variation which is caused mainly by non-linear pinch-off length dependence on V_DS insufficiently described in the electro-thermal model. Therefore, R_A approximate value at defined T₀ is depicted in figure 5.

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Directly acquired ${R_{{\text{S00}}}} = {\text{d}}{T_{{\text{SM0}}}}/{\text{d}}{P_0}$ from sensor temperature T_S dependence on power dissipation P and R_S00 calculated by equations (11) and (22) (see appendix) are similar due to the negligible influence of dissipated power spatial distribution in linear and saturation regime on the far location of the sensing element despite TLM width. To include the device topology the distance s/l ratio is defined where s is the distance of the sensor device (TLM), and l is the channel distance (GTLM) from the surrounding area with the ambient temperature. In other words, s/l close to one represents sensor device nearby the channel region, whereas s/l ratio approaching zero stands for sensor device far away from the channel region. Utilizing equations (14) and (22) and assuming a_M ≈ 1 dissipated power variation method allows us to plot s/l(T_S) vs. P, R_A(T_S) vs. P and R_A(T_S) vs. T_S dependence at different T₀ as shown in figures 6–8, respectively. The dissipated power variation method based on T_S acquisition across the whole T_A range requires much higher V_DS as well as g_D range than the ambient temperature variation method. Therefore, R_A and s/l deviation are more significant especially for R_A being slightly dependent on T_A. Measured data approximation eliminates missing overlap of R_A(T_S) vs. T_S for different T₀ caused by g_D variation and spatially distributed electrical parameters as shown in figure 8.

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Approximate thermal resistance R_A dependence on ambient temperature T₀ and sensor temperature T_S, respectively, depicted in figures 5 and 8 with consecutive linear fit gives us the opportunity to plot average temperature T_A dependence on dissipated power P depicted in figure 9 utilizing equations (13) and (16) (see appendix). For constant V_GS applied to the InAlN/AlN/GaN HEMT self-heating results in I_DS decrease caused mainly by thermally induced serial source area resistance increase [8]. Therefore, the average temperature of the source to the gate area is related to T_A. Simulated and calculated results are in good agreement as depicted in figure 9.

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3D thermal FEM simulations were utilized to acquire HEMT channel temperature T_C vs. distance x_s from the drain side of the gate edge perpendicular to the gate in the middle of the HEMT width as shown in figure 10. Due to different dissipated power spatial distribution in the pinch-off area and source to drain region, the condition ${x_{\text{s}}} \gg {d_{{\text{SD}}}}$ needed for low V_DS is gradually turned into ${x_{\text{s}}} \gg {d_{{\text{PO}}}} + {d_{{\text{PD}}}}$ for high V_OUT; these conditions are required for regular s/l determination.

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These conditions also allow acquisition of x_s and T_C vs. s/l ratio dependencies for different dissipated power P at T₀ = 25 °C utilizing equations (23) and (24) (see appendix) and the quadratic fit of simulated average temperature T_A vs. P dependence, as shown in figure 11. Fulfilled condition $\left( {{\text{d}}{s_P}{\text{/}}s} \right) \ll \left( {{\text{d}}P{\text{/}}P} \right)$ results in x_s vs. s/l ratio plot independent of dissipated power.

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The procedure mentioned above applied for P = 2 W at different T₀ makes it possible to obtain x_s and T_C vs. s/l ratio dependencies as depicted in figure 12. Condition $\left( {{\text{d}}{s_T}{\text{/}}s} \right) \ll {\text{d}}T_0^*{\text{/}}\left( {P{R_{\text{S}}}} \right)$ fulfilled for P = 2 W is assumed as well as for lower dissipated power resulting in a_M ≈ 1 across the operating range.

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In general under the quasi-static operating condition and dissipated power free space heat transfer equation describes heat flux vector $\overrightarrow {{q_s}} = - {\lambda _s}\nabla T$ dependent on temperature gradient T and thermal conductivity ${\lambda _s} = \left( {{\lambda _{sx}},{\lambda _{sy}},{\lambda _{sz}}} \right)$ in 3D orthogonal space using cartesian vectors $\overrightarrow {{i_{sx}}}$, $\overrightarrow {{i_{sy}}}$, $\overrightarrow {{i_{sz}}}$ and coordinates ${X_s} = \left( {{x_s},{y_s},{z_s}} \right)$ [12]:

$$\begin{equation}\overrightarrow {{q_s}} = - {\lambda _{sx}}\frac{{{\text{d}}T}}{{{\text{d}}{x_s}}}\overrightarrow {{i_{sx}}} - {\lambda _{sy}}\frac{{{\text{d}}T}}{{{\text{d}}{y_s}}}\overrightarrow {{i_{sy}}} - {\lambda _{sz}}\frac{{{\text{d}}T}}{{{\text{d}}{z_s}}}\overrightarrow {{i_{sz}}} .\end{equation} \tag{ 1 }$$

Heat flux from device active area to defined ambient temperature T₀ boundary forms equithermal surface consisting of elementary surfaces $\overrightarrow {{\text{d}}{S_S}} = {\text{d}}{S_S}\overrightarrow {{i_{sl}}}$ utilizing perpendicular $\overrightarrow {{i_{sl}}}$ vector. Equithermal elementary surfaces shifted by $\overrightarrow {{\text{d}}l} = {\text{d}}l\overrightarrow {{i_{sl}}}$ exhibit dT in $\overrightarrow {{q_s}} = {q_s}\overrightarrow {{i_{sl}}}$ direction and equation (1) result in $\overrightarrow {{q_s}} = - {\lambda _l}\left( {{\text{d}}T/{\text{d}}l} \right)\overrightarrow {{i_{sl}}}$ which is possible to be integrated across the equithermal surface utilizing spatially dependent $\overrightarrow {{\text{d}}l}$ and λ_l :

$$\begin{equation}\mathop{\unicode{x222F}}\limits_{{S_S}} {\overrightarrow {{q_s}} .\overrightarrow {{\text{d}}{S_S}} } = - \mathop{\unicode{x222F}}\limits_{{S_S}} {{\lambda _l}\left[ {{\text{d}}T\left( {{X_s}} \right)/{\text{d}}l} \right]\overrightarrow {.{i_{sl}}} .{\text{d}}{S_S}.\overrightarrow {{i_{sl}}} } .\end{equation} \tag{ 2 }$$

The left side of equation (2) represents total dissipated power P of the active area for all power sources situated inside the space bounded by equithermal surface. The right side of equation (2) results in scalar form and makes transformation of 3D heat propagation into 1D possible by application $\mathop{\unicode{x222F}}\limits_{{S_S}} {{\lambda _l}\left[ {{\text{d}}T\left( {{X_s}} \right)/{\text{d}}l} \right].{\text{d}}{S_S}} = {\lambda _{\text{T}}}\left( T \right)S{\text{d}}T\left( x \right)/{\text{d}}x$ utilizing position x of equithermal surface in the space of constant area S, constant heat flux $q = P/S$ and scalar thermal conductivity λ_T as shown in figure 13. This reduces equation (2) into 1D equation (9):

$$\begin{equation}{\text{d}}T\left( x \right)/{\text{d}}x = \lambda _{\text{T}}^{ - 1}\left( T \right)q = {r_{\text{T}}}\left( T \right)q.\end{equation} \tag{ 3 }$$

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Constant heat flux q evenly produced at $x = l > 0$ results in temperature profile T(x) assuming zero λ_T for $x > l$ and $T\left( 0 \right) = {T_0}$ set at zero x. Thermal resistance ${r_{\text{T}}}\left( T \right) = \lambda _{\text{T}}^{ - 1}\left( T \right)$ utilized in equation (3) for $0 < x < l$ is temperature-dependent only.

If a block of fixed length and heatflux is prolonged by dx₀ then ${\text{d}}{T_{\text{T}}}\left( 0 \right) = {\text{d}}{T_0} = {r_0}q{\text{d}}{x_0}$ using ${r_{\text{T}}}\left( 0 \right) = {r_0}$ and subsequently ${\text{d}}{T_{\text{T}}}\left( x \right) = r\left( x \right)q{\text{d}}{x_0}$ as shown in figure 14(a). Length shift dx₀ here is possible to be caused by ambient temperature shift dT₀ and the ratio between dT_T(x) and dT(0) dT_T(x) is determined:

$$\begin{equation}{\text{d}}{T_{\text{T}}}\left( x \right)/{\text{d}}{T_0} = {r_{\text{T}}}\left( x \right)/{r_0}.\end{equation} \tag{ 4 }$$

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Two different heat flux intensities q_A, q_B with corresponding length differences dx_A , dx_B result in similar temperature difference ${\text{d}}T\left( T \right) = {r_{\text{T}}}\left( T \right){q_A}{\text{d}}{x_A} = {r_{\text{T}}}\left( T \right){q_B}{\text{d}}{x_B}$ for the same temperature. Subsequent integration with boundaries 0, x_A and 0, x_B supposing $T\left( {{x_A}} \right) = T\left( {{x_B}} \right)$ yields:

$$\begin{equation}{q_A}{x_A} = {q_B}{x_B}.\end{equation} \tag{ 5 }$$

Heat flux increase ${\text{d}}q = {\text{d}}P/S$ results in temperature increase ${\text{d}}{T_{\text{P}}}\left( x \right) = {r_{\text{T}}}\left( x \right)q{\text{d}}{x_0}$ for x < l as depicted in figure 14(b). Substitution ${q_A} = q$, ${q_B} = q + {\text{d}}q$, ${x_A} = x$, ${x_B} = x + {\text{d}}{x_0}$ in equation (5) resulting in ${\text{d}}{x_0}{\text{/}}x = - {\text{d}}q{\text{/}}q$ makes it possible to determine dT_P(x):

$$\begin{equation}{\text{d}}{T_{\text{P}}}\left( x \right) = {r_{\text{T}}}\left( x \right)x{\text{d}}q.\end{equation} \tag{ 6 }$$

Differential thermal resistance $R\left( x \right) = {\text{d}}{T_{\text{P}}}\left( x \right){\text{/d}}P$ brings $R\left( x \right) = {r_{\text{T}}}\left( x \right)x{\text{/}}S$. In similar way ${R_0}\left( x \right) = {r_0}x{\text{/}}S$ is defined, therefore equation (4) yields:

$$\begin{equation}{\text{d}}{T_{\text{T}}}\left( x \right){\text{/d}}{T_0} = R\left( x \right){\text{/}}{R_0}\left( x \right).\end{equation} \tag{ 7 }$$

It is possible for ${\text{d}}{T_{\text{P}}}\left( x \right) = R\left( x \right){\text{d}}P$ to be invoked by dissipated power increase dP produced at l. On the other hand at constant P ambient temperature increase, dT₀ ^* results in dT^* (x) consisting of ${\text{d}}{T_{\text{T}}}\left( x \right) = \left[ {R\left( x \right){\text{/}}{R_0}\left( x \right)} \right]{\text{d}}T_0^*$ at constant dissipated power and ${\text{d}}T_{\text{P}}^*\left( x \right) = R\left( x \right){\text{d}}{P^*}$ caused by dissipated power difference dP^* at constant T₀.

Subsequently subtitution $k\left( x \right) = {\text{d}}T\left( x \right){\text{/d}}{T^*}\left( x \right)$ in ${\text{d}}{T^*}\left( x \right) \approx {\text{d}}{T_{\text{T}}}\left( x \right) + {\text{d}}{T_{\text{P}}}\left( x \right)$ yields:

$$\begin{equation}{R_0}\left( x \right) = \left( {{\text{d}}T_0^*/{\text{d}}P} \right)/\left[ {{k^{ - 1}}\left( x \right) - \left( {\frac{{{\text{d}}{P^*}}}{{{\text{d}}P}}} \right)} \right].\end{equation} \tag{ 8 }$$

As depicted in figure 1 average temperature T_A assumed as a constant value across the device active area defines the boundary conditions T(0) = T₀ at x = 0 and T(l) = T_A at x = l and dT^* (l) = dT_A ^* , dT_T(l) = dT_A ^T, dT_P(l) = dT_A ^P, R(l) = R_A, R₀(l) = R_A0, k(l) = k_A. Thermal parameters T_S, dT_S ^* , dT_S ^T, dT_S ^P, R_S, R_S0, k_S of spot temperature sensor placed at x = s utilizing transformation shown in figure 1 are assigned in the similar way.

Various spatial power density contribution for dP and dP^* has negligible influence on heat flux distribution relatively far from the active area. On the other hand, due to various temperature dependence of thermal conductivity a heat flux change across the structure caused by dP, dT₀ ^* , dP^* is unequal resulting in equithermal surface deformation causing x shift in 1D thermal model. In general, the ratio between l and s varies with P and T₀.

Active device at operating point characterized by output, namely drain–source voltage and drain–source current (V_DS, I_DS), input, namely gate-source voltage V_GS and zero input, namely gate current at average active area temperature T_A is inspected under quasi-static operating conditions. Output current difference dI_DS is defined by device transconductance ${g_{\text{M}}} = {\text{d}}{I_{{\text{DS}}}}/{\text{d}}{V_{{\text{GS}}}}$ and output conductance ${g_{\text{D}}} = {\text{d}}{I_{{\text{DS}}}}/{\text{d}}{V_{{\text{DS}}}}$ at constant T_A and thermal coefficient ${k_{{\text{TH}}}} = {\text{d}}{I_{{\text{DS}}}}{\text{/d}}{T_{\text{A}}}$ determined at constant V_GS and V_DS [13]:

$$\begin{equation}{\text{d}}{I_{{\text{DS}}}} = {g_{\text{M}}}{\text{d}}{V_{{\text{GS}}}} + {g_{\text{D}}}{\text{d}}{V_{{\text{DS}}}} + {k_{{\text{TH}}}}{\text{d}}{T_{\text{A}}}.\end{equation} \tag{ 9 }$$

Coefficients g_M, g_D, k_TH are determined from electrical parameters at operating point (V_GS, V_DS, T_A) and are required to be independent of th entire device thermal parameters [14].

At constant V_GS and V_DS small ambient temperature increase dT₀ ^* results in dT_A ^* inside the active area and current change dI_DS ^* is supposed and dT_A * is also influenced by dissipated power variation ${\text{d}}{P^*} = {V_{{\text{DS}}}}{\text{d}}I_{{\text{DS}}}^*$.

Increasing V_DS at constant V_GS results in dT_A, caused by dV_DS and dI_DS and in dissipated power change ${\text{d}}P = {V_{{\text{DS}}}}{\text{d}}{I_{{\text{DS}}}} + {I_{{\text{DS}}}}{\text{d}}{V_{{\text{DS}}}}$. Whereas equation (9) results in ${\text{d}}{I_{{\text{DS}}}} = {g_{\text{D}}}{\text{d}}{V_{{\text{DS}}}} + {k_{{\text{TH}}}}{\text{d}}{T_{\text{A}}}$ and ${\text{d}}I_{{\text{DS}}}^* = {k_{{\text{TH}}}}{\text{d}}T_{\text{A}}^*$, consecutive substitution yields:

$$\begin{equation}{k_{\text{A}}} = {\text{d}}{T_{\text{A}}}{\text{/d}}T_{\text{A}}^* = \left( {{\text{d}}{I_{{\text{DS}}}} - {g_{\text{D}}}{\text{d}}{V_{{\text{DS}}}}} \right){\text{/d}}I_{{\text{DS}}}^*.\end{equation} \tag{ 10 }$$

Rising V_DS at constant V_GS brings an increase of dissipated power mainly in the pinch-off area for field-effect transistor (FET in saturation regime, though dissipated power contribution in the entire active device area is evident especially for low V_DS. The different mode of dP and dP^* influence on heat flux distribution across the spot temperature sensor is negligible for $\left| {{\text{d}}P{\text{/d}}{P^*}} \right| \gg {k_{\text{S}}}$ or sensor location far from the active area in comparison with a source to drain distance. In the case of TLM structure the dP and dP^* spatial distribution plays a lesser role. In the case of Schottky diode or multi-fingered device, an appropriate location of the spot temperature sensor is coming out from the principles mentioned above.

Proper device simulations such as pulse measurements allow g_D acquisition taking leakage paths, trapping and other secondary effects into account [15]. Correct T_A determination is influenced by dissipated power produced by temperature sensor as well.

Although this method has been introduced [16], the theoretical background and requirements are mentioned here. Utilizing equation (8) in the active area yields:

$$\begin{equation}{R_{{\text{A0}}}} = \left( {{\text{d}}T_0^*{\text{/d}}P} \right)/\left[ {k_{\text{A}}^{ - 1} - \left( {\frac{{{\text{d}}{P^*}}}{{{\text{d}}P}}} \right)} \right].\end{equation} \tag{ 11 }$$

Coming out from equations (10) and (11) zero g_D results in:

$$\begin{equation}{R_{{\text{A0}}}} = \left( {{\text{d}}T_0^*{\text{d}}{I_{{\text{DS}}}}} \right)/\left( {{I_{{\text{DS}}}}{\text{d}}{V_{{\text{DS}}}}{\text{d}}I_{\text{D}}^*} \right).\end{equation} \tag{ 12 }$$

Variation of T₀ across the assumed operating temperature range allows us to obtain ${R_{{\text{A0}}}} = {r_{\text{T}}}\left( l \right)l{\text{/}}S = {r_{\text{T}}}\left( 0 \right)l{\text{/}}S$ resulting in ${\text{d}}{T_{\text{A}}} = {R_{{\text{A0}}}}{\text{d}}P = {R_{\text{A}}}\left( {{T_{\text{A}}}} \right){\text{d}}P$ for T_A ≈ T₀. Recurrent calculations of T_A are processed in this way:

$$\begin{equation}{T_{\text{A}}} = {T_{00}} + \sum {R_{\text{A}}}\left( {{T_0}} \right){\text{d}}P.\end{equation} \tag{ 13 }$$

In equation (13) T₀₀ represents the starting ambient temperature whereas ${R_{\text{A}}}\left( {{T_0}} \right)$ is calculated using equation (11) at increased T₀. This method requires accurate T₀ determination across the whole T_A range independent on dissipated power. Otherwise R_A0 variation with changing V_DS at defined T_A ≈ T₀ points on improper g_D assumption or phenomena caused by spatially distributed electro-thermal parameters, temperature gradient and different allocation of dissipated power change dP and dP^* along the active device area.

This method is advisable to acquire average temperature T_A if a small-sized temperature sensor is placed relatively far from an active device area because of the negligible influence of different dissipated power spatial distribution on sensor temperature T_S. For FET and temperature sensor placed nearby the active device supposing $\left| {{\text{d}}P/{\text{d}}{P^*}} \right| \gg {k_{\text{S}}}$ the T_A deviation is expected due to different dissipated power spatial distribution in the pinch-off area and source to drain area playing a lesser role with increasing V_DS.

The ratio ${R_{{\text{A0}}}}{\text{/}}{R_{{\text{S0}}}} = l{\text{/}}s$ coming out from constant r_T(x) at T₀ ambient temperature and comparison of equation (11) applied for R_A0 and R_S0 yields:

$$\begin{equation}s{\text{/}}l = \left[ {k_{\text{A}}^{ - 1} - \left( {\frac{{{\text{d}}{P^*}}}{{{\text{d}}P}}} \right)} \right]/\left[ {k_{\text{S}}^{ - 1} - \left( {\frac{{{\text{d}}{P^*}}}{{{\text{d}}P}}} \right)} \right].\end{equation} \tag{ 14 }$$

Similarly the same temperature-dependent r_T(x) at T_S utilizing $R\left( x \right) = {r_{\text{T}}}\left( x \right)x{\text{/}}S$ applied for R_A and ${R_{\text{S}}} = {\text{d}}{T_{\text{S}}}{\text{/d}}P$ results in:

$$\begin{equation}{R_{\text{A}}}\left( {{T_{\text{S}}}} \right) = {R_{\text{S}}}\left( {{T_{\text{S}}}} \right){\left( {s{\text{/}}l} \right)^{ - 1}} = \left( {{\text{d}}{T_{\text{S}}}{\text{/d}}P} \right){\left( {s{\text{/}}l} \right)^{ - 1}}.\end{equation} \tag{ 15 }$$

Consecutive R_A utilization for T_S ≈ T_A brings:

$$\begin{equation}{T_{\text{A}}} = {T_{00}} + \sum {R_{\text{A}}}\left( {{T_{\text{S}}}} \right){\text{d}}P.\end{equation} \tag{ 16 }$$

Results from equations (14)–(16) are applicable in a straight way for the structure consisting of materials exhibiting a constant ratio of particular thermal conductivities within the operating temperature range. In this case, spatial heat flux distribution left intact for dT₀ ^* and evenly increased for dP and dP^* results in unchanged equithermal surface shape corresponding to particular spatial position.

However, various thermal conductivity dependence on temperature requires us to be aware of s position shift caused by unequal heat flux change invoked by dT₀ ^* or dP and dP^* . For temperature sensor the difference between measured temperature shift dT_SM, dT_SM ^* and temperature shift dT_S, dT_S ^* mentioned above exhibit difference of $\left( {{R_{\text{S}}}P{\text{/}}s} \right)\left( {{\text{d}}{s_{\text{P}}} + {\text{d}}{s_{\text{T}}}} \right)$ where ds_T is caused by dT₀ and dP results in ds_P and dP^* proportionally in ${\text{d}}{s_{\text{P}}}{\text{d}}{P^*}{\text{/d}}P$:

$$\begin{equation}{\text{d}}{T_{\text{S}}} = {\text{d}}{T_{{\text{SM}}}} - {R_{\text{S}}}P\left( {{\text{d}}{s_{\text{P}}}{\text{/}}s} \right)\end{equation} \tag{ 17 }$$

$$\begin{equation}{\text{d}}T_{\text{S}}^* = {\text{d}}T_{{\text{SM}}}^* - {R_{\text{S}}}P\left[ {\left( {{\text{d}}{s_{\text{T}}}{\text{/}}s} \right) - \left( {{\text{d}}{P^*}{\text{/d}}P} \right)\left( {{\text{d}}{s_{\text{P}}}{\text{/}}s} \right)} \right].\end{equation} \tag{ 18 }$$

Measured thermal resistance ${R_{{\text{SM}}}} = {\text{d}}{T_{{\text{SM}}}}{\text{/d}}P$ exhibits:

$$\begin{equation}{R_{{\text{SM}}}}{\text{/}}{R_{\text{S}}} - 1 = \left( {{\text{d}}{s_{\text{P}}}{\text{/}}s} \right)/\left( {{\text{d}}P{\text{/}}P} \right).\end{equation} \tag{ 19 }$$

Substitution ${k_{\text{S}}} = {\text{d}}{T_{\text{S}}}{\text{/d}}T_{\text{S}}^*$ and ${R_{\text{S}}} = {\text{d}}{T_{\text{S}}}{\text{/d}}P$ in ${\text{d}}T_{\text{S}}^* = {\text{d}}T_0^*\left( {{R_{\text{S}}}{\text{/}}{R_{{\text{S0}}}}} \right) + {R_{\text{S}}}{\text{d}}{P^*}$ brings:

$$\begin{equation}{k_{\text{S}}} = {\text{d}}{T_{\text{S}}}{\text{/d}}T_{\text{S}}^* = {\text{d}}P{\text{/}}\left[ {\left( {{\text{d}}T_0^*{\text{/}}{R_{{\text{S0}}}}} \right) + {\text{d}}{P^*}} \right]\end{equation} \tag{ 20 }$$

$$\begin{equation}s{\text{/}}l = {R_{{\text{S0}}}}{\text{/}}{R_{{\text{A0}}}} = \left[ {k_{\text{A}}^{ - 1} - \left( {\frac{{{\text{d}}{P^*}}}{{{\text{d}}P}}} \right)} \right]{R_{{\text{S0}}}}{\text{d}}P{\text{/d}}T_0^*.\end{equation} \tag{ 21 }$$

For ${T_{\text{S}}} \approx {T_0}$ the conditions ${\text{d}}P \approx P$ and $\left( {{\text{d}}{s_{{\text{P0}}}}{\text{/}}{s_0}} \right) \ll 1$ result in ${R_{{\text{SM0}}}} = {R_{{\text{SM}}}}\left( {{T_0}} \right) = {R_{{\text{S00}}}}$, ${R_{{\text{S0}}}}{\text{/}}{R_{{\text{S00}}}} = \left( {s{\text{/}}l} \right)/\left( {{s_0}{\text{/}}l} \right)$ giving an opportunity to s₀/l acquisition.

The shift ratio ${a_{\text{M}}} = \left( {{\text{d}}{T_{\text{S}}}{\text{/d}}T_{\text{S}}^*} \right)/\left( {{\text{d}}{T_{{\text{SM}}}}{\text{/d}}T_{{\text{SM}}}^*} \right)$ is required to substitute ${k_{\text{S}}} = {a_{\text{M}}}{\text{d}}{T_{{\text{SM}}}}{\text{/d}}T_{{\text{SM}}}^*$ in equations (14) and (21) due to different spatial heat flux redistribution caused by dT₀ ^* or dP and dP^* . Supposing thin top layers in comparison with a relatively thick substrate and conventional temperature sensor location nearby the active device area on the top of the structure, then ${\text{d}}{T_{\text{S}}}{\text{/d}}T_{\text{S}}^* \approx {\text{d}}{T_{{\text{SM}}}}{\text{/d}}T_{{\text{SM}}}^*$ assumption resulting in ${a_{\text{M}}} \approx 1$ brings sufficient accuracy. Otherwise, the thermal model based on partial heat flux distribution is requisite. Substitution ${\text{d}}{s_{\text{P}}}{\text{/}}s = \left( {{\text{d}}{s_{\text{P}}}{\text{/}}l} \right){\left( {s{\text{/}}l} \right)^{ - 1}}$ coming out from s/l ratio dependence on P utilized in equations (15), (17) and (19) allows R_A calculation:

$$\begin{equation}{R_{\text{A}}}\left( {{T_{\text{S}}}} \right) = {P^{ - 1}}{\left( {s{\text{/}}l} \right)^{ - 1}}{\text{d}}{T_{{\text{SM}}}}{\text{/}}\left[ {\left( {\frac{{{\text{d}}P}}{P}} \right) + \left( {\frac{{{\text{d}}{s_{\text{P}}}}}{s}} \right)} \right].\end{equation} \tag{ 22 }$$

The starting operating point is commonly set at ${T_{\text{A}}} \approx {T_{\text{S}}} \approx {T_0}$ for small P resulting in R_S00 and s₀/l. In the FET case, the initial operating point is shifted to the saturation voltage V_DS,SAT with active area average temperature T_A,SAT whereas relatively low g_D for V_DS > V_DS,SAT exhibits minor impact on T_A determination especially for long gated FET [12]. Initial s/l ratio at ${T_{\text{A}}} \approx {T_{{\text{A,SAT}}}}$ is calculated utilizing equation (14).

Dissipated power variation method requires T_S to be swept across the whole T_A range being aware of dissipated power limitation. The solution in equation (11) is consistent with equation (22) for s/l ratio close to zero; therefore, both methods mentioned above are allowed to be combined if measurement setup is limited by T₀ or P operating conditions.

This part deals with s/l ratio acquisition if thermal parameters of the active device area and temperature sensor P, T₀, T_A, T_S are obtained from simulations or spatial calculations. The equithermal surface of temperature T_S and s/l ratio meets the requirements mentioned above. In a simple way equation (3) is able to be rewritten as:

$$\begin{equation}{\text{d}}x = {\text{d}}T\left( x \right)/\left[ {{r_{\text{T}}}\left( T \right)\left( {P{\text{/}}S} \right)} \right].\end{equation} \tag{ 23 }$$

At defined T₀, the dissipated power P_A needed for reaching T_S in the active device area at l position is applied. Subsequently the dissipated power P_S at l position needed for reaching T_S nearby temperature sensor at s position is set. Substitution ${x_A} = l$ and ${x_B} = s$ in equation (5) yields:

$$\begin{equation}s/l = {P_{\text{A}}}/{P_{\text{S}}}.\end{equation} \tag{ 24 }$$

The procedure mentioned above allows us to acquire T_S dependence on s/l ratio for various P and T₀ and subsequently calculate $\left( {{\text{d}}{s_{\text{P}}}{\text{/}}s} \right){\text{/d}}P$ and $\left( {{\text{d}}{s_{\text{T}}}{\text{/}}s} \right)/{\text{d}}T_0^*$ for defined s/l. Utilizing thermal resistance ${R_{\text{S}}}\left( {{T_{\text{S}}}} \right) = {\text{d}}{T_{\text{S}}}{\text{/d}}P$, $R_{\text{S}}^*\left( {{T_{\text{S}}}} \right) = {\text{d}}T_{\text{S}}^*{\text{/d}}T_0^*$ at defined s/l and ${R_{{\text{SM}}}}\left( {{T_{\text{S}}}} \right) = {\text{d}}{T_{{\text{SM}}}}{\text{/d}}P$, $R_{{\text{SM}}}^*\left( {{T_{\text{S}}}} \right) = {\text{d}}T_{{\text{SM}}}^*{\text{/d}}T_0^*$ at a spatial position corresponding to s/l ratio allows us to verify them coming out from equations (17)–(19):

$$\begin{equation}\left( {{\text{d}}{s_{\text{P}}}{\text{/}}s} \right){\text{/d}}P = \left( {{R_{{\text{SM}}}}{\text{/}}{R_{\text{S}}} - 1} \right){\text{/}}P\end{equation} \tag{ 25 }$$

$$\begin{equation}\left( {{\text{d}}{s_{\text{T}}}{\text{/}}s} \right){\text{/d}}T_0^* = \left( {R_{{\text{SM}}}^* - R_{\text{S}}^*} \right){\text{/}}\left( {P{R_{\text{S}}}} \right).\end{equation} \tag{ 26 }$$

The shift ratio calculated as ${a_{\text{M}}} = {R_{\text{S}}}R_{{\text{SM}}}^*{\text{/}}{R_{{\text{SM}}}}R_{\text{S}}^*$ brings correction of s/l ratio calculations equation (14) from experimental data.