Analytical and Physical Investigation on Source Resistance in In_xGa_1−xAs Quantum-Well High-Electron-Mobility Transistors

Abstract

We present a fully analytical model and physical investigation on the source resistance (R_S) in In_xGa_1−xAs quantum-well high-electron mobility transistors based on a three-layer TLM system. The R_S model in this work was derived by solving the coupled quadratic differential equations for each current component with appropriate boundary conditions, requiring only six physical and geometrical parameters, including ohmic contact resistivity (ρ_c), barrier tunneling resistivity (ρ_barrier), sheet resistances of the cap and channel regions (R_{sh_cap} and R_{sh_ch}), side-recessed length (L_side) and gate-to-source length (L_gs). To extract each model parameter, we fabricated two different TLM structures, such as cap-TLM and recessed-TLM. The developed R_S model in this work was in excellent agreement with the R_S values measured from the two TLM devices and previously reported short-L_g HEMT devices. The findings in this work revealed that barrier tunneling resistivity already played a critical role in reducing the value of R_S in state-of-the-art HEMTs. Unless the barrier tunneling resistivity is reduced considerably, innovative engineering on the ohmic contact characteristics and gate-to-source spacing would only marginally improve the device performance.

1. Introduction

The evolving sixth-generation (6G) wireless communication technologies demand higher operating frequencies of approximately 300 GHz with data rates approaching 0.1 Tbps [1,2]. To meet this urgent requirement, transistor technologies must be engineered to sustain the evolution of digital communication systems, guided by Edholm’s law [3]. Among various transistor technologies, indium-rich In_xGa_1−xAs quantum-well (QW) high-electron-mobility transistors (HEMTs) on InP substrates have offered the best balance of current-gain cutoff frequency (f_T) and maximum oscillation frequency (f_max), and the lowest noise figure characteristics in the sub-millimeter-wave region [4,5,6,7,8]. These transistors adopt a combination of L_g scaling down to sub-30 nm, enhancement of the channel carrier transport by incorporating the indium-rich channel design, and reduction of all parasitic components.

Among various parasitic components, it is imperative to minimize the source resistance (R_S) itself to fully benefit from the superior intrinsic performance of the In_xGa_1−xAs QW channel [9,10], demanding an analytical and physical model for the source resistance. Considering state-of-the-art In_xGa_1−xAs HEMT technologies [11,12,13,14], source and drain contacts have been created with a non-alloyed metal stack of Ti/Pt/Au with a source-to-drain spacing (L_ds) between 1 μm and 0.5 μm. Historically, R_S is minimized by reducing the ohmic contact resistivity (ρ_c) [15] and shrinking the gate-to-source spacing (L_gs) using da self-aligned gate architecture [16,17]. However, it is very challenging to reduce R_S to below 100 Ω·μm, because of the tunneling resistance component between the heavily doped In_0.53Ga_0.47As capping layer and the In_xGa_1−xAs QW channel layer. To understand the limit of R_S in HEMTs in an effort to further reduce R_S, a sophisticated and comprehensive model must be developed for R_S in state-of-the art HEMTs, rather than the simple lumped-elements-based one-layer model [18,19].

Previously, two-layer system-based R_S model was developed by Feuer [20], which was applicable to alloyed ohmic contact structures with two different contact resistances: one was associated with a heavily doped GaAs capping layer and the other with a undoped GaAs QW channel layer. In this letter, we present a fully analytical and physical model for R_S in advanced HEMTs, requiring only six physical and geometrical parameters. The model considers three different regions: (i) a one-layer transmission-line model (TLM) for the side-recess region, (ii) an analytical TLM for the access region and (iii) an analytical three-layer TLM for the source electrode region, to accurately predict a value of R_S in a given HEMT structure and identify dominant components to further minimize R_S. To do so, we proposed and fabricated two different types of TLM structures to experimentally extract each component of R_S. The analytical model proposed in this work is in excellent agreement with the measured values of R_S from the fabricated r-TLMs, as well as recently reported advanced HEMTs. Most importantly, findings in this work reveal that the ρ_barrier is a bottleneck for further reductions of R_S in advanced HEMTs.

2. Analytical Model for R_S

Figure 1a–c show the cross-sectional schematic and TEM images of advanced In_xGa_1−xAs QW HEMTs on an InP substrate [4]. They adopt non-alloyed S/D ohmic contacts such as a metal stack of Ti/Pt/Au with contact resistance (R_C) values between 10 Ω·μm and 20 Ω·μm. Carrier transfer from the cap to channel replies on a tunneling mechanism via an In_0.52Al_0.48As barrier layer. To model R_S, a comprehensive transport mechanism from the source ohmic electrode to the In_xGa_1−xAs QW channel via the In_0.53Ga_0.47As cap and In_0.52Al_0.48As barrier layers must be considered in a distributed manner.

Figure 2a illustrates a complete distributed equivalent circuit model for R_S, comprising three regions. One is the source ohmic electrode region (Region-I), where the electrons are injected from the ohmic metal to the In_0.53Ga_0.47As cap and then to the In_xGa_1−xAs QW channel through the In_0.52Al_0.48As barrier, which is governed by a three-layer TLM system. Another is the source access region (Region-II), where the electron transfer mechanism is governed by a cap-to-channel two-layer TLM system with transfer length (L_{T_barrier}) given by $\sqrt{{\rho }_{barrier}/\left(R_{sh\_ch}+R_{sh\_cap}\right)}$. The other is the side-recessed region (Region-III), where a simple one-layer model works. In comparison, lumped-elements based one-layer model is shown in Figure 2c [18,19].

Next, let us derive a fully analytical and physical expression for R_S. Given the coordinate system in Figure 3a, R_S can be determined by V_ch(x = −L_gs)/I_O from Ohm’s law, and then the problem is how to express each current component as a function of x such as I_ch(x), I_cap(x), and I_met(x). In a given segment as highlighted in Figure 3b, we can define a differential contact conductance as dg_c = (W_g/ρ_c) dx, a differential barrier conductance as dg_barrier = (W_g/ρ_barrier) dx, a differential lateral cap resistance as dr_{s_cap} = (R_{sh_cap}/W_g) dx and a differential lateral channel resistance as dr_{s_ch} = (R_{sh_ch}/W_g) dx. At location x, Kirchhoff’s current and voltage laws yield, respectively

$$\frac{d^2{I}_{met}\left(x\right)}{d{x}^2}=\left[R_{met}·I_{met}\left(x\right)-R_{sh\_ch}·I_{cap}\left(x\right)\right]{\rho }_c^{-1}$$

(1)

$$\frac{d^2{I}_{ch}\left(x\right)}{d{x}^2}=\left[R_{sh\_ch}·I_{ch}\left(x\right)-R_{sh\_cap}·I_{cap}\left(x\right)\right]{\rho }_{barrier}^{-1}$$

(2)

$$I_{cap}=I_O-I_{met}-I_{ch}$$

(3)

These are coupled quadratic differential equations for three current components (I_ch(x), I_cap(x) and I_met(x)). From the general solution for these differential equations with existing six boundary conditions (listed in

Table 1. Six boundary conditions, the general solution for three current components, and their corresponding eigenvalues and eigenvectors.

BCs	$I_{ch}$$=I_O$ for x = −Lgs	$I_{met}$= 0 for −Lgs < x < 0	$I_{met}$ = $I_O$ for x = ∞
	$I_{cap}$$(x=$0−$)=I_{cap}$$(x=0^{+}$)	$I_{ch}$$(x=$0−$)=I_{ch}$$(0^{+}$)	$\frac{d{I}_{ch}(x=0^{-}) }{dx}=-\frac{d{I}_{ch}\left(x=0^{+}\right)}{dx}$
Region-I	$\left[\begin{array}{c}{I}_{met}\\ I_{ch}\end{array}\right]=\left[\begin{array}{c}{v}_{11}\\ v_{12}\end{array}\right]C_{1}exp\left(-\sqrt{{\lambda }_1}x\right)+\left[\begin{array}{c}{v}_{21}\\ v_{22}\end{array}\right]C_{2}exp\left(-\sqrt{{\lambda }_2}x\right)+\left[\begin{array}{c}{I}_O\\ 0\end{array}\right]$ $I_{cap}=I_O-I_{met}-I_{ch}$	Region-II	$I_{cap}=C_{3}exp\left(\frac{-x}{L_{T\_barrier}}\right)+C_{4}exp\left(\frac{x}{L_{T\_barrier}}\right)+\frac{R_{sh\_ch}}{R_{sh\_cap}+R_{sh\_ch}}{I}_O$ $I_{ch}=I_O-I_{cap}$
−${\lambda }_1$ & ${\lambda }_2$ are eigenvalues of $A=\left[\begin{array}{cc}\left(\frac{R_{sh\_cap}}{{\rho }_c}\right)& \left(\frac{R_{sh\_cap}}{{\rho }_c}\right)\\ \left(\frac{R_{sh\_cap}}{{\rho }_{barrier}}\right)& \left(\frac{R_{sh\_cap}+R_{sh\_cap}}{{\rho }_{barrier}}\right)\end{array}\right]$ −$\left[\begin{array}{c}{v}_{11}\\ v_{12}\end{array}\right] \&$ $\left[\begin{array}{c}{v}_{21}\\ v_{22}\end{array}\right]$ are their corresponding eigenvectors		$\left[\begin{array}{c}{C}_1\\ C_2\\ \begin{array}{c}{C}_3\\ C_4\end{array}\end{array}\right]={\left[\begin{array}{cccc}0& 0& exp\left(\frac{L_{gs}}{L_{T\_barrier}}\right)& exp\left(\frac{-L_{gs}}{L_{T\_barrier}}\right)\\ v_{12}& v_{22}& 1& 1\\ v_{11}& v_{21}& 0& 0\\ -\sqrt{{\lambda }_1}{v}_{12}& -\sqrt{{\lambda }_2}{v}_{22}& -\frac{1}{L_{T\_barrier}}& \frac{1}{L_{T\_barrier}}\end{array}\right]}^{-I}\left[\begin{array}{c}-\frac{R_{sh\_ch}\times I_O}{R_{sh\_cap}+R_{sh\_ch}}\\ I_O\left(1-\frac{R_{sh\_ch}}{R_{sh\_cap}+R_{sh\_ch}}\right) \\ \begin{array}{c}-I_O\\ 0\end{array}\end{array}\right]$

), we obtain an analytical expression for I_ch(x), I_cap(x), and I_met(x) for both regions, as written in Table 1. The expression for V_ch(x = −L_gs) can then be derived. Although there are several ways to express V_ch(x = −L_gs), it is useful to focus on the total voltage drop across the In_xGa_1−xAs QW channel from x = −L_gs to x = ∞ in this work. From this,

$$V_{ch}(x=-L_{gs})={\displaystyle \int }{I}_{ch}\left(x\right)·d{r}_{S\_ch} dx$$

(4)

The source resistance, defined as V_ch(x = 0)/I_O, is

$$R_S=\frac{V_{ch}\left(x=-L_{gs}\right)}{I_O}=\frac{R_{sh\_ch}}{W_g{I}_O}{{\displaystyle \int }}_{-L_{gs}}^{\infty }{I}_{ch}\left(x\right)dx$$

(5)

$$\begin{gathered} R_S·W_g=[\frac{\sqrt{2}·C_1·\sqrt{{\rho }_C·{\rho }_{barrier}}}{{\rho }_C·\left(R_{sh\_ch}+R_{sh\_cap}\right)+{\rho }_{barrier}·R_{sh\_cap}-\eta }+\frac{\sqrt{2}·C_1·\sqrt{{\rho }_C·{\rho }_{barrier}}}{{\rho }_C·\left(R_{sh\_ch}+R_{sh\_cap}\right)+{\rho }_{barrier}·R_{sh\_cap}+\eta } \\ +\frac{\left(C_3-C_4\right)·\left(1-e^{-L_{gs}/L_{T\_barrier}}\right)}{L_{T\_barrier}}+\frac{R_{sh\_cap}·L_{gs}}{{\rho }_{barrier}·L_{T\_barrier}{}^2}]+R_{sh\_ch}·L_{side} \end{gathered}$$

(6)

$$\eta =\sqrt{{\rho }_c{}^2(R_{sh\_ch}{}^2+R_{sh\_cap}{}^2)+2R_{sh\_ch} R_{sh\_cap}{\rho }_c\left({\rho }_c-{\rho }_{barrier}\right)+R_{sh\_cap}{}^2{\rho }_{barrier}\left(2{\rho }_c+{\rho }_{barrier}\right)}$$

(7)

$$L_{T\_barrier}=\sqrt{\frac{{\rho }_{barrier}}{R_{sh\_ch}+R_{sh\_cap}}}$$

(8)

Overall, R_S depends on the ohmic contact resistivity, the sheet resistances of the cap and QW channel layers, the barrier tunneling resistivity, and the length of the gate-to-source region and side-recessed regions.

3. Experimental Results and Discussion

Two types of TLM structures were fabricated, as shown in Figure 3: the cap-only TLM structure (cap-TLM, (a)) to evaluate the contact characteristics of the non-alloyed ohmic metal stack, and the recessed TLM structure (r-TLM, (b)) which is identical to the real device without a Schottky gate electrode. Details on the epitaxial layer design and device processing were reported in our previous paper [4]. All device processing was conducted on a full 3-inch wafer with an i-line stepper to ensure fine alignment accuracy within 0.05 μm. In the r-TLM, we varied L_g from 40 μm to 0.5 μm and L_gs from 10 μm to 0.2 μm. In this way, the split of L_g yielded the sheet resistance of the QW channel (R_{sh_ch}) from the linear dependence, and the source resistance (R_S) from the y-intercept at a given L_gs. Lastly, we investigated the dependence of R_S on L_gs in detail.

Figure 4 plots the measured total resistance (R_T) against L_ds, which corresponds to the length between the edge of source and the edge of drain. for the fabricated cap-TLM structures. This yielded values of R_{sh_cap} = 131 Ω/sq, R_C = 32 Ω·μm, L_{T_cap} = 0.34 μm and ρ_c = 15 Ω·μm², with an excellent correlation coefficient of 0.99999. Figure 5a plots the measured R_T against L_g for the r-TLM structures with various dimensions of L_gs from 10 μm to 0.2 μm. When L_g was long enough, each r-TLM device yielded approximately the same slope for all L_gs with excellent correlation coefficient. This is plotted in Figure 5b with averaged R_{sh_ch} = 145 Ω/sq and excellent ∆(R_{sh_ch}) = 1.56 Ω/sq, confirming that the In_0.8Ga_0.2As QW channel sheet resistance was independent of L_gs. Because we designed the symmetrical L_gs and L_gd, half of the y-intercept from Figure 6a corresponded exactly to R_S. In analyzing r-TLM structures with various L_gs, values of the correlation coefficient were also greater than 0.999, increasing the credibility of the overall TLM analysis.

Figure 6 plots the measured R_S (filled symbols) from the r-TLM analysis against L_gs, as well as the projected R_S (line) from Equation (5) with the model parameters of ρ_barrier = 91 Ω·μm² and others directly from the cap-TLMs and r-TLMs. Additionally, the open symbols in Figure 6 came from the R_S extracted directly from the reported HEMTs [4] using the gate-current injection technique [21]. There are two points to identify in Figure 6. First, all of the measured R_S characteristics were explained by the modeled R_S. Second, R_S was linearly proportional to L_gs for L_gs > 1 μm, where its slope was 69 Ω/sq. Interestingly, this was similar to the parallel connection of R_{sh_cap} and R_{sh_ch}. However, this linear dependence of R_S on L_gs was no longer valid for L_gs < 1 μm and, most importantly, the measured R_S eventually saturated to approximately 123 Ω·μm even with L_gs approaching 0. Our model clearly indicated that this was because of the barrier tunneling resistivity. The saturation of R_S in L_gs = 0 was because the necessary lateral length for the cap-to-channel tunneling was supplied by its equivalent transfer length from the leading edge of the source metal contact (−L_{T_barrier} < x < 0) in Region-I.

Finally, let us discuss how to further reduce R_S with the R_S model proposed in this work. The three solid lines in Figure 6 are the model projections of R_S with the ohmic contact resistivity improve from 15 Ω·μm² (present) to 1 Ω·μm². Surprisingly, R_S would not be minimized even with a significant reduction in ρ_c and L_gs because of the ρ_barrier. Alternatively, the three dashed lines in Figure 6 are from the same model projection, but with ρ_barrier = 20 Ω·μm². Note that a reduction in the ρ_barrier is important; in consequence, the projected R_S would be significantly scaled down to 70 Ω·μm and below. Under this circumstance, R_S could then be further reduced by the improved ohmic contact characteristics and the reduction of L_gs.

4. Conclusions

A fully analytical and physical investigation on R_S in advanced In_xGa_1−xAs QW HEMTs was carried out with a three-layer TLM system. Analytical solutions to the three current components (source metal, cap, and channel) along the selected coordinate system with appropriate boundary conditions were produced. The proposed R_S model in this work required only six physical and geometrical parameters (ρ_c, ρ_barrier, R_{sh_cap}, R_{sh_ch}, L_side and L_gs), yielding excellent agreement with the R_S values measured from the two TLM devices and previously reported In_xGa_1−xAs QW HEMTs. The developed model in this work was capable of explaining the saturation behavior of R_S for L_gs < 1 μm, which was due to the ρ_barrier. Therefore, one must pay a more careful attention to cut down the ρ_barrier to further minimize R_S in future HEMTs.