This work is also discussed in our archival publication:
M. S. Islam, E. Tuncer, and D. P. Neikirk, "Accurate Model for Schottky-Contacted Coplanar Waveguide Including Finite Epilayer Resistance Effects," Electronics Letters, vol. 30, pp. 712-713, 1994.
Accurate Quasi-Static Model for Schottky-Contacted Voltage-Controlled
Coplanar Waveguide Phase Shifters
M. Saiful Islam, Emre Tuncer and Dean P. Neikirk
Department of Electrical & Computer Engineering
The University of Texas at Austin
Austin, Texas 78712
A new quasi-static model for Schottky-contacted coplanar waveguide (CPW) on a semiconductor substrate is shown. Comparison between experimental measurements for a CPW on a GaAs epilayer and calculations show excellent agreement. The new model includes the effect of the finite resistance of the undepleted epilayers under the CPW electrodes [1,2].
Introduction: Coplanar waveguide (CPW) on semiconducting substrates has been extensively studied for possible use in phase shifting applications. The most common structure consists of CPW electrodes Schottky-contacted to a doped semiconducting layer on a semi-insulating substrate . Two main techniques for the control of the propagation constant in such CPWs have been used: optical control  and voltage control [5-8]. For both cases, models of the behavior of the CPW are usually based on slow-wave effects. Here we report a quasi-static model for Schottky-contacted CPWs that is in excellent agreement with experimentally measured data over a wide range of frequency and bias conditions. While past models have emphasized the impact of depletion layer capacitance on the slow wave effect, we show that significant sensitivity to bias voltage is induced by resistance in the epi layer.
Model: A cross-sectional drawing of a Schottky-contacted CPW is shown in Fig. 1. The CPW center conductor is 2a wide, the gap between center conductor and ground plane is , and the ground plane is w wide. The depleted and undepleted layers under the CPW electrodes contribute a distributed capacitance and resistance, respectively, to the admittance per unit length Ytot for the transmission line. Figure 2 shows the equivalent circuit for Ytot. The normal CPW capacitance per unit length consists of air-side ( ) and substrate-side ( ) contributions, found using conformal mapping. The distributed RC circuits formed by the CPW electrodes and the epi layer under them produce Zgp and Zcntr. The finite resistance of the undepleted epi layer has been ignored previously; however, it can be of critical importance in capacitance-voltage measurements on doped epitaxial layers [9,10]. The sheet resistances shown in Fig. 1 are
where the superscript n is either gp, gap, or cntr, for the region under the ground plane, in the gap between electrodes, or under the center conductor, respectively. The epi layer is tepi thick, with conductivity [[sigma]]epi, and the bias-dependent depletion layer is hn thick. For simplicity, we have assumed a single, uniformly doped epitaxial layer, on a high resistivity substrate with residual sheet resistance , in parallel with the epi layer. A simple depletion approximation is used to find hn, given the bias condition and doping level in the epi layer. For the geometry used here, we then find
where [[omega]] is the angular frequency, [[epsilon]] is the dielectric constant of the epi layer, n is either gp or cntr, and wn is the width of the region ( and ). If is small enough is just the parallel plate capacitance across the depletion region. However, if becomes large the series resistance prevents the capacitance from contributing fully. Again referring to Fig. 2, Ytot for the Schottky-contacted CPW is
where Rgap is the total gap region resistance, . Finally, the complex propagation constant is given by , where the series impedance per unit length Ztot is found using a new, highly accurate quasi-static technique .
The propagation constant can be very sensitive to bias voltage when either hgp or hcntr are close to the total epi layer thickness tepi (i.e., the bias voltage has nearly depleted the epi layer) since small changes in bias induce very large changes in RS (even though h, and therefore the depletion layer capacitance, changes very little), inducing large changes in Ytot. This also explains the extreme sensitivity to optical illumination when biased near full depletion , since very low levels of illumination can induce large changes in RS.
Results and Conclusions: To verify the accuracy of this model, measurements have been made on coplanar waveguide fabricated on doped GaAs layers grown by molecular beam epitaxy on semi-insulating (SI) GaAs wafers. The epi layer consisted of a 1.5 um thick n-type (4 x 1015 cm-3) GaAs layer, a 0.1 um thick AlAs layer, and finally a 0.2 um thick GaAs undoped buffer layer. Accounting for Fermi level pinning at the SI substrate interface, the effective epi layer thickness used in eq. 1 is tepi = 1 um. In the gap region, Fermi level pinning also depletes 0.5 um from the front surface, so hgap = 0.5 um. The center conductor half-width a is 5 um, the gap is 7 um, the ground plane width w is 500 um, and the CPW metal (silver) thickness is 1 um. Figures 3 (attenuation constant) and 4 (effective refractive index, neff = [[beta]]/[[beta]]o) show a comparison between the experimental results (measured using an HP 8510B Network Analyzer) and our new model for the center conductor reverse biased. Excellent agreement between the experimental results and the calculations over the full frequency range (45 MHz - 40 GHz) is shown; excellent agreement between measurement and calculation is also obtained when the ground planes are reverse biased. Again, no fitting factors are used; only the dimensions of the CPW and the characteristics of the semiconducting substrate are required for the calculation.
Acknowledgments: This work was sponsored in part by the Joint Services Electronics Program under grant number AFOSR 49620-92-C-0027 and the Advanced Research Projects Agency ASEM program.
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Figure 1: Schottky-contacted coplanar waveguide on an epitaxial layer. Depletion regions under the electrodes produce distributed capacitance, while the undepleted epi layer produces a distributed resistance.
Figure 2: Equivalent circuit for the shunt admittance per unit length of the coplanar waveguide.
Figure 3: Measured (solid lines) and modeled results (dashed lines) for attenuation constant at 0 V, 2 V reverse bias on center conductor, and 7 V reverse bias on center conductor; at 7 V the epilayer is fully depleted, so hcntr is taken to be the full SI substrate thickness.
Fig. 4: Measured (solid lines) and modeled results (dashed lines) for the effective index of refraction [[beta]]/[[beta]]o (slow wave factor) at 0 V, 2 V reverse bias on center conductor, and 7 V reverse bias on center conductor.