AlGaN/GaN HEMT channel temperature determination utilizing external heater

Lateral FET consists of the gated core FET area, ungated and ohmic contact source/drain area as shown in figure 1. The total FET resistance is the sum of the core FET channel resistance R_CH in the length d_G, source to gate channel resistance R_S in the length d_S, drain to gate channel resistance R_D in the length d_D and source/drain ohmic contact resistances R_CS, R_CD in the transfer length d_CS, d_CD. In saturation regime pinch-off area of voltage drop V_PO is formed nearby the drain-side gate edge at x ∈ (–d_PO, 0) under the gate as well as at x ∈ (0, d_PD) in the drain to gate gap. For quasi-static operation the difference between investigated FET gate voltage V_GS, drain voltage V_DS and core FET gate voltage V_GO, drain voltage V_DO, respectively, is given by the voltage drop caused by resistances R_S, R_D, R_CS, R_CD at drain current I_D:

$$\begin{eqnarray}&&{V}_{GS}-{V}_{GO}=\left({R}_{S}+{R}_{CS}\right){I}_{D},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{V}_{DS}-{V}_{DO}=\left({R}_{S}+{R}_{D}+{R}_{CS}+{R}_{CD}\right){I}_{D}.\end{eqnarray} \tag{ 2 }$$

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Core FET behavioural model [8] coming out from core FET functional approximation ${I}_{D}\sim f\left({V}_{GO}-{V}_{TH}\right)$ [9] results in core FET transconductance g_GO = $d{I}_{D}/\left(d{V}_{GO}-d{V}_{TH}\right).$ Further substitution $d{I}_{A}=d{I}_{D}-{g}_{GO}\left(d{V}_{GO}-d{V}_{TH}\right)$ and ${R}_{SX}={R}_{S}+{R}_{CS},$ ${R}_{DX}={R}_{D}+{R}_{CD}$ leads to:

$$\begin{eqnarray}&&d{I}_{D}-d{I}_{A}={g}_{GO}\left(d{V}_{GS}-d{V}_{TH}-{I}_{D}d{R}_{SX}-{R}_{SX}d{I}_{D}\right).\end{eqnarray} \tag{ 3 }$$

Utilizing [8] dI_A involves output conductance g_DO = dI_D/dV_DO and thermal coefficient determined partially by threshold voltage V_TH change, channel concentration n shift, carrier velocity v and carrier mobility μ = v/E_X change caused by temperature T change having an impact on horizontal and vertical electric field (E_X and E_Z).

External heater utilization causes small temperature increase dT*(x) across the structure resulting in current change dI_D* and dI_A* addicted to dR_SX* and dV_TH*:

$$\begin{eqnarray}&&d{I}_{D}^{* }-d{I}_{A}^{* }={g}_{GO}\left(d{V}_{GS}^{* }-d{V}_{TH}^{* }-{I}_{D}d{R}_{SX}^{* }-{R}_{SX}d{I}_{D}^{* }\right).\end{eqnarray} \tag{ 4 }$$

If both (3) and (4) are linearly independent and constant V_GS, V_GS* are supposed then (4) divided by (3) results in:

$$\begin{eqnarray}&&\begin{array}{l}\left(d{I}_{D}^{* }-d{I}_{A}^{* }\right)\left(d{V}_{TH}+{I}_{D}d{R}_{SX}\right)-\left(d{I}_{D}-d{I}_{A}\right)\left(d{V}_{TH}^{* }+{I}_{D}d{R}_{SX}^{* }\right)\\ \,=\,{R}_{SX}\left(d{I}_{D}d{I}_{A}^{* }-d{I}_{D}^{* }d{I}_{A}\right).\end{array}\end{eqnarray} \tag{ 5 }$$

Leakage current path allocation with respect to thermal gradient is requisite for proper calculations mentioned in [8].

Temperature dependence of threshold voltage V_TH and particular resistances R_S, R_D, R_CS, R_CD are crucial for FET channel temperature determination [10]. R_S(0), R_D(0), R_CS(0), R_CD(0) at low I_D are lower than R_S(I_D), R_D(I_D), R_CS(I_D), R_CD(I_D) used in (1), (2) or (5) due to finite current saturation level [11]. Therefore particular correction factors r_S = R_S(I_D)/R_S(0), r_D = R_D(I_D)/R_D(0), r_CS = R_CS(I_D)/R_CS(0), r_CD = R_CD(I_D)/R_CD(0) at I_D and T are requisite.

Relative small r_S, r_D, r_CS, r_CD change allows correction of R_S, R_D, R_CS, R_CD and dR_S, dR_D, dR_CS, dR_CD in simple multiplication way and usually r_S ≈ r_D, r_CS ≈ r_CD. In general R_S(I_D), R_D(I_D), R_CS(I_D), R_CD(I_D) and I_D change cause relative resistance drop demonstrated for R_S(I_D):

$$\begin{eqnarray}&&d{R}_{S}\left({I}_{D}\right)/{R}_{S}\left({I}_{D}\right)=d{R}_{S}\left(0\right)/{R}_{S}\left(0\right)+d{r}_{S}/{r}_{S},\end{eqnarray} \tag{ 6 }$$

whereas r_S depends on both I_D and T and R_S(0) on T only dR_S(I_D) is a linear combination of dI_D and dT.

Thermal dependence of V_TH(0) and particular thermal coefficient k_VTH(0) = dV_TH(0)/dT are typically acquired from transfer low power I–V characteristics [12–14]. Under quasi-static condition dV_TH(I_D) = r_VTHdV_TH(0) is supposed I_D dependent only using correction factor r_VTH in core FET functional approximation ${I}_{D}\sim f\left({V}_{GO}-{V}_{TH}\right).$

Low power output I–V characteristics at defined T allow to determine R_S(0), R_D(0), R_CS(0), R_CD(0) temperature dependencies and calculate particular thermal coefficients k_RS(0) = dR_S(0)/dT, k_RD(0) = dR_D(0)/dT, k_RCS(0) = dR_CS(0)/dT, k_RCD(0) = dR_CD(0)/dT [8]. In the case of channel thermal gradient those coefficients in symbiosis with correction factors determine the temperature dependence of channel resistance per length.

Under the quasi-static condition an average temperature T_A along the whole channel is assumed. This hypothetic situation is demonstrated by infinite channel thermal conductance in the FET of width w and length d_S + d_G + d_D. Therefore power density distribution in the channel plays no role. Differential FET behavioral model offers T_A determination utilizing core FET thermal coefficient ${k}_{GC}=d{I}_{A}/d{T}_{A}$ and substitution $d{I}_{A}={k}_{GC}d{T}_{GO}+{g}_{DO}d{V}_{DO}.$ Dissipated power change $dP={V}_{DS}d{I}_{D}+{I}_{D}d{V}_{DS}$ results in dT_A. Ambient temperature T₀ increase dT₀* caused by the external heater is set before measurement. Therefore active area temperature increase $d{T}_{A}^{* }\approx d{T}_{A}^{T}+d{T}_{A}^{P}$ at the device operation consists of the active area temperature change $d{T}_{A}^{T}$ at constant dissipated power and $d{T}_{A}^{P}$ caused by dissipated power difference $d{P}^{* }={V}_{DS}d{I}_{D}^{* }$ at constant V_DS utilizing differential thermal resistance ${r}_{TH}\,\approx d{T}_{A}/dP\approx d{T}_{A}^{P}/dP* .$ Substitution ${k}_{A}=d{T}_{A}/d{T}_{A}^{* }$ yields:

$$\begin{eqnarray}&&d{T}_{A}=d{T}_{A}^{T}{\left[{k}_{A}^{-1}-\left(d{P}^{* }/dP\right)\right]}^{-1}.\end{eqnarray} \tag{ 7 }$$

It is advisable to calculate dT_A for each operating point, however $d{T}_{A}^{T}$ invoked by dT₀* depends on T_A. Therefore dT_A caused by defined dP is possible to be obtained for varying T₀ across the operating T_A range assuming $d{T}_{A}^{T}$ ≈ dT₀* [13].

Dissipated power change and thermal parameters are used in (7) whereas electrical parameters utilizing I_D and T_A dependence of R_SX and V_TH substituted in (5) lead to k_A determination and the following solutions:

• (1)  
  If $d{I}_{A}\approx {k}_{GC}d{T}_{A}$ and $d{I}_{A}^{* }\approx {k}_{GC}d{T}_{A}^{* }$ are thermally dependent only then ${k}_{A}=d{I}_{D}/d{I}_{D}^{* }$ in (7) yields:

  $$\begin{eqnarray}&&d{T}_{A}=d{T}_{A}^{T}\left(d{I}_{D}/d{I}_{D}^{* }\right)\left[\left(d{I}_{D}/{I}_{D}\right){\left(d{V}_{DS}/{V}_{DS}\right)}^{-1}+1\right].\end{eqnarray} \tag{ 8 }$$

  This approximation is suitable for small g_DO when d_PO and leakage current modulation with V_DO are negligible. Moreover r_S, r_D, r_CS, r_CD, r_VTH variation with I_D and T_A has no influence on (8) if low g_DO condition is satisfied.
• (2)  
  Thermal coefficients k_RSX(I_D) = r_Sk_RS(0) + r_CSk_RCS(0), k_RDX(I_D) = r_Dk_RD(0) + r_CDk_RCD(0) are possible to be obtained at operating I_D and T_A for relatively small r_S, r_D, r_CS, r_CD, r_VTH variation. Considering g_DO determined by pinch-off area and leakage current modulation caused by dV_DO, substitution $a={I}_{D}{k}_{RSX}\left({I}_{D}\right)\,+{k}_{VTH}\left({I}_{D}\right)$ in (5) yields:

$$\begin{eqnarray}&&{k}_{A}=\displaystyle \frac{\left(a+{R}_{SX}\left({I}_{D}\right){k}_{GC}\right)d{I}_{D}-a{g}_{DO}d{V}_{DO}}{\left[\left(a+{R}_{SX}\left({I}_{D}\right){k}_{GC}\right)+\left({R}_{SX}\left({I}_{D}\right){g}_{DO}d{V}_{DO}/\left({r}_{TH}dP\right)\right)\right]d{I}_{D}^{* }}.\end{eqnarray} \tag{ 9 }$$

Recurrent calculations coming out from (7) and (9) allow to obtain dT_A incrementing T_A and utilizing ${r}_{TH}\approx d{T}_{A}/dP$ from the previous operating point. Zero a and R_SX causing zero dV_GO result in solution similar to (8) where the term $d{I}_{D}/d{I}_{D}^{* }$ is replaced by $\left(d{I}_{D}-{g}_{DO}d{V}_{DO}\right)/d{I}_{D}^{* }.$ Substitution of dV_DO by dV_PO and dV_DS using a differential form of (2) and leakage current are recommended to be incorporated in (9).

Negligible leakage current and dV_TH ≪ dV_PO yields dV_PO ≈ dV_DO and ${g}_{DO}={I}_{D}{\left({d}_{G}-{d}_{PO}\right)}^{-1}d{d}_{PO}/d{V}_{PO}$ where dd_PO is d_PO change with V_PO whereas thermal d_PO change is included in k_GC [8]. If $d[{R}_{SX}({I}_{D}){I}_{D}]+d[{R}_{DX}({I}_{D}){I}_{D}]\langle \langle d{V}_{DS}$ then condition dV_PO ≈ dV_DO ≈ dV_DS is satisfied. The condition ${r}_{TH}\approx \left({T}_{A}-{T}_{0}\right)/P\approx d{T}_{A}/dP$ for T_A close to T₀ is used for the temperature assumption in the linear regime area [8].

Temperature profile evaluation for a device under quasi-static condition takes thermal gradient along the channel and temperature dependent thermal conductance into account. An uniform power dissipation density is assumed along z ∈ (d_CH–h_CH/2, d_CH + h_CH/2) using average channel depth d_CH and vertical width h_CH. Constant power density and temperature consideration along y ∈ (–w/2, +w/2) for x-position are requisite to define channel temperature T(x) and resistance element per length R_i(x) [8]. Power density per length P_i(x) = R_i(x)I_D² = I_DE_X(x) = I_DdV_CH(x)/dx where E_X(x) and V_CH(x) = E_X(x)dx are the channel electric field and potential, respectively. In the ungated source and drain area E_X(x) is distorted by R_i(x) modulation due to self-heating. Because of thermal gradient in the structure R_S and R_D at I_D are declared in a way shown for R_S:

$$\begin{eqnarray}&&\begin{array}{l}{R}_{S}=\displaystyle {\int }_{-{d}_{G}-{d}_{S}}^{-{d}_{G}}{R}_{i}\left(x\right)dx\\ \,=\displaystyle {\int }_{-{d}_{G}-{d}_{S}}^{-{d}_{G}}{r}_{S}\left({I}_{D},T\left(x\right)\right){R}_{iS}\left(0,T\left(x\right)\right)dx,\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\begin{array}{l}d{R}_{S}=\displaystyle {\int }_{-{d}_{G}-{d}_{S}}^{-{d}_{G}}\left[{r}_{S}\left({I}_{D},T\left(x\right)\right){k}_{RS}\left(0,T\left(x\right)\right)dT\left(x\right)\right.\\ \,\,\left.+\,{R}_{i}\left(x\right)d{r}_{S}\left({I}_{D},T\left(x\right)\right)\right]dx.\end{array}\end{eqnarray} \tag{ 11 }$$

Assuming small d_CS and d_CD both dR_CS and dR_CD at I_D are defined in a way shown for dR_CD:

$$\begin{eqnarray}&&\begin{array}{l}d{R}_{CD}={r}_{CD}\left({I}_{D},T\left({d}_{D}\right)\right){k}_{RCD}\left(0,T\left({d}_{D}\right)\right)dT\left({d}_{D}\right)\\ \,+\,{R}_{C}d{r}_{CD}\left({I}_{D},T\left({d}_{D}\right)\right).\end{array}\end{eqnarray} \tag{ 12 }$$

The core FET acts as the sum of elementary FETs of transconductance g_GO(x) and length dx. Current continuity equation is partially satisfied for I_D change caused by dV_TH(x) and dd_PO resulting in V_CH(x) bending. Similarly k_GC(x) is advised to be calculated at appropriate x ∈ (–d_G, –d_PO) using Poisson and current continuity equation and taking n, v, E_X and E_Z variation into account. Those calculations specific for each FET type allow to obtain core FET average dV_TH. Although simple dV_TH approximation is advised to be applied for short channel FET or for ${k}_{VTH}({I}_{D})\langle \langle {I}_{D}{k}_{RSX}({I}_{D}):$

$$\begin{eqnarray}&&d{V}_{TH}={\left({d}_{G}-{d}_{PO}\right)}^{-1}\displaystyle {\int }_{-{d}_{G}}^{-{d}_{PO}}{k}_{VTH}\left({I}_{D},T\left(x\right)\right)dT\left(x\right)dx.\end{eqnarray} \tag{ 13 }$$

Temperature profile evaluation based on simple iterations is described in [8]. For simple temperature estimation in the saturation regime to avoid the iteration process total dissipated power increase $dP\approx d\left({V}_{PO}{I}_{D}\right)$ is supposed as uniform pinch-off area power dissipation contribution with respect to rising d_PO and d_PD. Pinch-off area dimensions d_PO, d_PD, h_CH affect T(x) significantly inside this area whereas farther those parameters become less dominant.

Temperature change $d{T}_{A}^{T}\left(x\right)$ caused by dT₀* such as $d{T}_{A}^{P}\left(x\right)$ invoked by uniform dissipated power change along the active area $d{P}^{* }={V}_{DS}d{I}_{D}^{* }$ are possible to be acquired by thermal simulations giving opportunity to assume dT*(x) in a simple way:

$$\begin{eqnarray}&&d{T}^{* }\left(x\right)\approx d{T}_{A}^{T}\left(x\right)+d{T}_{A}^{P}\left(x\right).\end{eqnarray} \tag{ 14 }$$

Differential correction factor k_T as ratio between simulated and calculated dT(x) gives opportunity to compare results obtained by simulations and calculated from experimentally acquired electrical FET attributes using differential FET behavioral model. Inaccurate assumption of particular thermal conductance, leakage current paths or power density cause k_T deviation from one. In that case self-heating in entire channel area is advised to be considered or more proper electrical and thermal model to be employed. Differential equations are suitable to be replaced by difference formulas in numerical recurrent calculations.

Mentioned procedure is allowed to be used for for multi-gated FET as well. In the case of symmetric double gated structure drain current under each gate is ~I_D/2. Self-heating plays role in a different non-uniform I_D distribution for multi-gated FET, V_CH(x) boundary conditions are useful here.

The investigated Al_0.29Ga_0.71N/GaN HEMT structure including 1.5 nm GaN/14.5 nm Al_0.29Ga_0.71N/50 nm undoped GaN/1650 nm GaN heterostructure was grown by MOVPE on 500 μm thick 4H-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 prepared by standard Au-based metallization. Horizontal view of investigated GTLM HEMT of w ≈ 125 μm, d_G ≈ 1 μm, d_S ≈ d_D ≈ 5 μm and source/drain ohmic contact length ∼100 μm is shown in figure 2. The device is set in the package open from top, placed on the Al thermal chuck preserved at a constant temperature.

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The output and transfer I–V characteristics were measured using semiconductor parameter analyzer Agilent 4155C and controlled thermal chuck. To obtain I_D dependence zero gate voltage V_GS was kept while drain voltage V_DS varied from 0 up to 40 V. The parameters R_S and V_TH were acquired from the output and transfer I–V characteristics measured in low power regime at temperature range 25 °C–120 °C, respectively. The device was recovered by one minute white LED illumination between measurements. 3D thermal finite element method (FEM) simulations of the device were performed by Synopsys TCAD Sentaurus coming out from thermal conductivity values shown in table 1 [15, 16].

Linearization of R_S(0) and V_TH(0) temperature dependence depicted in figure 3 results in R_iS(0) ≈ R_iD(0) ≈ 4.14 Ω μm⁻¹ at 25 °C, k_RS(0) ≈ k_RD(0) ≈ 29.4 mΩ K⁻¹ μm⁻¹, k_VTH(0) ≈ 1.46 mV K⁻¹ assuming r_VTH ≈ 1. From transmission line measurements (TLM) correction coefficients r_S ≈ r_D ≈ 1+k_DI_D, k_D ≈ 6 A⁻¹ for R_iS, R_iD at operating temperature and the current range was obtained using FEM simulations to eliminate self-heating part. Constant R_CS ≈ R_CD ≈1.2 Ω and r_CS ≈ r_CD ≈ 1 were verified. Transfer I–V characteristics point on low leakage current.

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The measured I_D dependence on V_DS for V_GS = 0 V at T₀ = 25 °C and 35 °C exhibiting ΔT* = 10 °C and ΔI_D* are shown in figure 4. Additionally, nearly constant d_PO/V_PO ≈ 0.6 nm V⁻¹ and d_PD/V_PO ≈ 3 nm V⁻¹ were obtained from electro-thermal model. HEMT channel temperature profile shown in figure 5 was obtained using heat flow injector [8] in 3D thermal FEM simulations. Utilizing A_n ≈ 0.15 the relative saturation velocity change ${v}_{SAT}(T)/{v}_{SAT}\,=-T/[T+1700\,{\rm{K}}]$ was assumed.

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Simulated average temperature of the ungated source area, gate area and peripheral temperature of the pinch-off area exhibit linear dependence on dissipated power therefore the condition ${\rm{\Delta }}{T}_{A}^{T}\approx {\rm{\Delta }}{T}_{0}^{\ast }$ is satisfied reducing the external heater requirements on the small temperature increase only utilizing ΔT₀* stabilization unaffected by ΔP and ΔP*.

An average temperature T_A shown in figure 6 was calculated using (8) neglecting d_PO. Considering d_PO slightly higher T_A was obtained solving (7) and (9) incorporating ${g}_{DO}={I}_{D}{\left({d}_{G}-{d}_{PO}\right)}^{-1}{\rm{\Delta }}{d}_{PO}/{\rm{\Delta }}{V}_{PO}$ and ${k}_{GC}\,=\left({I}_{D}/{v}_{SAT}\right)\left({\rm{\Delta }}{v}_{SAT}/{\rm{\Delta }}{T}_{A}\right)$ and. Calculated T_A is close to simulated average temperature of ungated source area pointing on major thermal R_S and minor V_TH and v_SAT change.

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Finite small ΔI_D* and ΔT₀* caused by external heater takes effect on finite point differential analysis deviation resulting in correct T_A determination.

Simulated channel temperature profile T_SIM(x) at T₀ and ${T}_{0}+{\rm{\Delta }}{T}_{0}^{* }$ for various dissipated power was obtained. In the case of temperature independent k_VTH(0), k_RS(0) ≈ k_RD(0) and zero k_RCS(0) ≈ k_RCD(0) average temperature difference values of ungated source area ΔT_ARS and gated area ΔT_ACH are requisite for ΔR_S determination and for simple ΔV_TH approximation (13):

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{ARS}={d}_{S}^{-1}\displaystyle {\int }_{-{d}_{G}-{d}_{S}}^{-{d}_{G}}{\rm{\Delta }}T\left(x\right)dx,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{ACH}={\left({d}_{G}-{d}_{PO}\right)}^{-1}\displaystyle {\int }_{-{d}_{G}}^{-{d}_{PO}}{\rm{\Delta }}T\left(x\right)dx.\end{eqnarray} \tag{ 16 }$$

Utilization of (14), (15) and (16) leads to ΔT*_ARS, ΔT*_ACH and ΔT*(–d_PO) acquisition. Polynomial approximation of ΔT_ARS, ΔT_ACH, ΔT(–d_PO) and ΔT*_ARS, ΔT*_ACH, ΔT*(–d_PO) allows to acquire those parameters for each operating point.

Accurate k_GC(x) and g_DO(x) determination requires calculations based on Poisson and current continuity equation taking channel thermal gradient into account. To avoid this V_CH(x) bending caused by Δd_PO and ${\rm{\Delta }}{V}_{DO}\approx {\rm{\Delta }}{V}_{PO}$ due to ${\rm{\Delta }}{V}_{DO}\gg {\rm{\Delta }}{V}_{TH}$ are supposed resulting in ${g}_{DO}\,={I}_{D}{\left({d}_{G}-{d}_{PO}\right)}^{-1}{\rm{\Delta }}{d}_{PO}/{\rm{\Delta }}{V}_{PO}.$ Because of negligible thermal 2DEG concentration change in AlGaN/GaN channel carrier velocity variation is the determining factor. For small V_CH(x) thermal bending excluding V_TH(x) thermal change current continuity equation results in ${\rm{\Delta }}{I}_{A}=qn\left(x\right){\rm{\Delta }}v\left(x\right)$ for x ∈ (–d_G, –d_PO), n(x) includes built-in such as E_Z induced carrier concentration, gate boundaries exhibit ΔT(–d_PO) > ΔT(–d_G), E_X(–d_PO) ≫ E_X(–d_G), ΔT(–d_PO) > ΔT(–d_G). Assuming acceptable deviation in $n\left(-{d}_{PO}\right){\rm{\Delta }}{v}_{SAT}\left(-{d}_{PO}\right)\approx n\left(-{d}_{G}\right){E}_{X}\left(-{d}_{G}\right){\rm{\Delta }}\mu \left(-{d}_{G}\right)$ and Δv_SAT minor impact on ΔI_D in comparison with ΔR_SX the formula ${k}_{GC}\left(-{d}_{PO}\right)=\left({I}_{D}/{v}_{SAT}\right)\left({\rm{\Delta }}{v}_{SAT}/{\rm{\Delta }}T\left(-{d}_{PO}\right)\right)$ is found to be sufficient approximation [8].

The ratio between calculated ΔT(x) and simulated ΔT_SIM(x) temperature change ${k}_{T}={\rm{\Delta }}T\left(x\right)/{\rm{\Delta }}{T}_{SIM}\left(x\right)$ applied in (5) allows to compare simulated and calculated thermal profile in a multiplication way for ΔP applied for simulated ΔT_ARS, ΔT_ACH, ΔT(–d_PO). Values ΔT*_ARS, ΔT**_ACH, ΔT*(–d_PO) are supposed as simulated using (14). Preliminarily calculated parameters R_SX(I_D), g_DO, k_RS(I_D), k_RD(I_D), k_VTH(I_D), k_GC(–d_PO) for defined operating point and relatively small r_S, r_D, r_VTH variation allow (5) to be transformed into the following difference equation to calculate k_T:

$$\begin{eqnarray}&&\begin{array}{l}\left[{\rm{\Delta }}{I}_{D}^{* }-{k}_{GC}\left(-{d}_{PO}\right){\rm{\Delta }}{T}^{* }\left(-{d}_{PO}\right)\right]\\ \,\times \,\left[{I}_{D}{r}_{S}{k}_{RS}\left(0\right){k}_{T}{\rm{\Delta }}{T}_{ARS}+{r}_{VTH}{k}_{VTH}\left(0\right){k}_{T}{\rm{\Delta }}{T}_{ACH}\right]\\ \,+\,\left[{k}_{GC}\left(-{d}_{PO}\right){k}_{T}{\rm{\Delta }}T\left(-{d}_{PO}\right)\right]\left[{I}_{D}{r}_{S}{k}_{RS}\left(0\right){\rm{\Delta }}{T}_{ARS}^{* }\right.\\ \,+\,\left.{r}_{VTH}{k}_{VTH}\left(0\right){\rm{\Delta }}{T}_{ACH}^{* }+{R}_{SX}\left({I}_{D}\right){\rm{\Delta }}{I}_{D}^{* }\right]\\ =\,{\rm{\Delta }}{I}_{D}\left[{I}_{D}{r}_{S}{k}_{RS}\left(0\right){\rm{\Delta }}{T}_{ARS}^{* }+{r}_{VTH}{k}_{VTH}\left(0\right){\rm{\Delta }}{T}_{ACH}^{* }\right.\\ \,+\,\left.{R}_{SX}\left({I}_{D}\right){k}_{GC}\left(-{d}_{PO}\right){\rm{\Delta }}{T}^{* }\left(-{d}_{PO}\right)\right]\\ \,-\,{g}_{DO}{\rm{\Delta }}{V}_{DO}\left[{I}_{D}{r}_{S}{k}_{RS}\left(0\right){\rm{\Delta }}{T}_{ARS}^{* }+{r}_{VTH}{k}_{VTH}\left(0\right){\rm{\Delta }}{T}_{ACH}^{* }\right.\\ \,+\,\left.{R}_{SX}\left({I}_{D}\right){\rm{\Delta }}{I}_{D}^{* }\right].\end{array}\end{eqnarray} \tag{ 17 }$$

Channel temperature T_SAT(x) obtained from simulations at saturation voltage is used as initial parameter for recurrent calculations. Therefore the resultant temperature in the saturation regime is yielded as:

$$\begin{eqnarray}&&T\left(x\right)={T}_{SAT}\left(x\right)+{\sum }^{}{k}_{T}{\rm{\Delta }}{T}_{SIM}\left(x\right).\end{eqnarray} \tag{ 18 }$$

Obtained k_T is shown in figure 7, particular simulated and calculated average temperature T_ARS, T_ACH such as T(−d_PO) are compared in figure 8.

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Maximal peripheral temperature T ≈ 130 °C was reached for P = 1.5 W, however channel temperature is higher inside pinch-off area depending on d_PO, d_PD, and h_CH. Although k_T deviation is ∼12% a good correspondence between thermal simulations and calculations using (17) was achieved.