Definitions

• the present invention relates to a method for modeling semiconductor devices and more particularly to a method for modeling semiconductor devices, such as field effect transistors (FET) and high electron mobility transistors (HEMT) for relatively accurately determining the physical device characteristics and small-signal characteristics to enable the high frequency performance of the device to be forecasted.
• FET field effect transistors
• HEMT high electron mobility transistors
• HEMT technology provides unparalleled, high-performance characteristics at high frequencies (microwave to millimeter wave) .
• HEMTs are used in various RF applications.
• In order to accurately forecast the performance of such devices it is necessary accurately model the effect of the components physical structure on its high frequency small signal characteristic.
• it is necessary to know how physical changes to the device will effect device performance in order to determine what process changes may be acceptable to improve RF yield product and which may be unacceptable which decrease yield.
• HEMT devices are known: equivalent circuit modeling; and physical device simulation.
• Equivalent circuit modeling utilizes networks of linear electrical elements to model the small signal performance of the device.
• a typical equivalent circuit topology is shown in FIG. 1. This equivalent circuit model is known to accurately model measured S-parameters (small signal characteristics) of HEMT devices up to 120 GHz.
• Such physical device simulators utilize comprehensive data about material characteristics and the basic device physics to simulate the actual physical location and structure of HEMT devices. Such simulators are known to be based upon finite element and Monte Carlo approaches. Such analytical tools are adapted to accept input in the form of the device physical structure, as generally shown in FIGS. 3, 4 and 5. In particular, these figures show the typical cross section and the "epi" stack used for physical simulation of specific device structures.
• FIG. 3 illustrates a rough scale drawing of a cross section of an exemplary HEMT device.
• FIG. 4 illustrates how the cross section of information regarding device structure is input into a known physical device simulator tool, such as
• FIG. 5 illustrates how the epi stack information is input into the physical device simulator.
• FIG. 6 illustrates where the epi stack physically resides within the total device structure.
• the present invention relates to a semi-physical device model that can represent known physical device characteristics and measured small signal characteristics relatively accurately.
• the semi-physical device model in accordance with the present invention uses analytical expressions to model the fundamental electric charge and field structure of a HEMT internal structure. These expressions are based on the device physics but are in empirical form. In this way, the model is able to maintain physical dependency with good fidelity while retaining accurate measured-to- modeled small signal characteristics.
• the model in accordance with the present invention provides model elements for a standard small signal equivalent circuit model of FET.
• the model elements are derived from small signal excitation analysis of intrinsic charge and electric field as modeled within the device by the semi-physical HEMT model. As such, the RF performance can be predicted at arbitrary bias points. DESCRIPTION OF THE DRAWINGS
• FIG. 1 is schematic diagram of an exemplary small signal equivalent circuit model for a HEMT device.
• FIG. 2 is a sectional view of an exemplary HEMT illustrating the rough translation of the physical origins for each of the equivalent circuit elements illustrated in the small signal circuit model in FIG. 1.
• FIG.3 is a cross-sectional view of a HEMT illustrating the regions in the HEMT which correspond to the various circuit elements in the small signal equivalent circuit model illustrated in FIG. 1.
• FIG.4 is an example of a cross-sectional descriptive input of a physical HEMT device structure by a conventional physical device simulation tool.
• FIG. 5 is an example of an epi stack descriptive input of a physical HEMT device structure for a known physical device simulation tool.
• FIG. 6 is an example illustrating the location of the epi stack within the device structures cross-sectional view.
• FIG. 7 is an example of a relatively accurate measured-to-model I-V characteristics using the semi-physical modeling method in accordance with the present invention.
• FIG. 8 is a elevational view illustrating an epi stack for an exemplary HEMT.
• FIG. 9 is a cross-sectional view of a HEMT for the exemplary epi stack illustrated in FIG. 8.
• FIG. 10 is a blown up diagram of the cross-sectional parameters pertaining to the T-gate geometry for the exemplary epi stack illustrated in FIG. 8.
• FIG. 11 is a diagram of an electric conductance model used in the semi-physical example.
• FIG. 12 is a Smith chart illustrating the measured vs modeled S-parameters S 11 , S12 and S22 simulated in accordance with the method in accordance with the present invention.
• FIG. 13 is similar to FIG. 12 and illustrates the measured vs. modeled values for the S21 parameter.
• FIG. 14 is similar to FIG. 12 but for the S12 S-parameter.
• FIG. 15 is a graphical illustration of the semi-physically modeled vs measured small signal Gm.
• FIG. 16 is a graphical illustration of the semi-physically simulated bias dependence of the small-signal output conductance Rds.
• FIG. 17 is a graphical illustration of the semi-physically simulated bias dependence of the small signal gate-source and gate-drain capacitance Cgs and Cgd.
• FIG. 18 is a graphical illustration of the semi-physically simulated bias- dependence of the small signal gate source charging resistance Ri.
• FIG. 19 is a graphical illustration of the semi-physically bias dependence of the small signal source and drain resistance Rs and Rd.
• FIG. 20 is a graphical illustration of the measured vs modeled bias dependent gain at 23.5 Ghz for a K-band MMIC amplifier.
• FIG. 21 A and 21B are graphical illustrations of the extracted parameters from measured device I-V's for process control monitor testing.
• FIG.22 is a graphical illustration of the measured vs semi-physically simulated process variation for Gmpk and Vgspk.
• FIG.23 is a graphical illustration of the measured vs semi-physically simulated process variation for Idpk and Gmpk.
• FIG.24 is a graphical illustration of the measured vs semi-physically simulated process variation for Imax and Vpo.
• FIG. 25 is a graphical illustration of the measured/extracted vs semi-physically simulated process variation for the small signal equivalent model Rds and Gm.
• FIG.26 is a graphical illustration of the measured/extracted vs semi-physically simulated process variation for the small signal equivalent model Cgs and Gm.
• FIG.27 is a graphical illustration of the measured vs semi-physically simulated physical dependence for Imax as a function of physical gate length.
• FIG. 28 is a graphical illustration of the measured/extracted model vs semi- physically simulated physical dependence for Rds as a function of physical recess undercut width.
• the present invention relates to a semi-physical device model which canbe used to simulate RF performance through physically-based device models.
• the semi- physical model is an analytical model based upon empirical expressions that model the physics of HEMT operation, hence the terminology "semi-physical".
• the model incorporates real process parameters, such as gate length recess, etch depth, recess undercut dimensions, passivation nitrite thickness, and the like.
• the semi-physical model is able to maintain relatively good measured to modeled accuracy while accounting for the effects of process variations on the device performance.
• the semi-physical model in accordance with the present invention provides model elements for the standard small signal equivalent circuit model or FET as illustrated in FIG. 1.
• the model elements are derived from small signal excitation analysis of the intrinsic charge and electric fields within the device.
• the simulated small signal model elements represent a relatively accurate physical equivalent circuit description of a physical FET.
• the general methodology for the semi-physical modeling of intrinsic charge, electric conductance and electrical field is as set forth below.
• the relationships between the conduction band offsets and electrical permitivities and material composition for the various materials in the epi stack are determined. These relationships can be performed analytically or by fitting simulated data from physical simulators.
• the basic electron transport characteristics in any of the applicable bulk materials in the epi stack are determined. Once the electron transport characteristics are determined, the undepleted linear channel mobility is determined either through material characterization or physical simulation. Subsequently, the
• Schottky barrier height value or expressions are determined. Once the Schottky barrier height value is determined, the semi-physical equations are constructed modeling the following characteristics:
• the empirical terms of the semi-physical modeling equations are adjusted to fit the model I-V (current voltage) characteristics against measured values. Subsequently, the empirical terms are interactively readjusted to achieve a simultaneous fit of measured C-V (capacitance- voltage) and I-V characteristics. Lastly, the empirical modeling terms are fixed for future use.
• FIG. 7 illustrates a set of relatively accurate measured-to-modeled I-V characteristics for a HEMT using the semi-physical modeling discussed herein.
• FIG. 7 illustrates the drain-to- source current I ds as a function of the drain-to-source voltage V ⁇ for various gate biases, for example, from 0.4V to - 1.0V.
• solid lines are used to represent the semi-physical model while the Xs are used to represent measured values.
• a close relationship exists between the measured values and the modeled parameters.
• FIG. 10 relates to a blown up T-gate characteristic which is correlated to the parameters identified in Table 2.
• the semi-physical modeling of the intrinsic charge and electric field within the HEMT device is initiated by determining the relationships between the conduction band offset, electrical permitivities and material composition for the various materials in the epi stack.
• Material composition related band offset and electrical permitivity relationships may be obtained from various references, such as "Physics of Semiconductor Devices," by Michael Shur, Prentice Hall, Englewood Cliffs, New Jersey 1990.
• the basic electron transport characteristics, for example, for the linear mobility of electron carriers in the bulk GaAs cap layer may be determined to be 1350cm 2 /Vs, available from "Physics of Semiconductor Devices", supra.
• the linear mobility of electron carriers in the undepleted channels is assumed to be 5500cm 2 /Vs.
• This value may be measured by Hall effect samples which have epi stacks grown identically to the stack in the example, except for some differences in the GaAs cap layer.
• the Schottky barrier height is assumed to be 1.051 volts, which is typical of platinum metal on a AlGaAs material.
• Vgt M V gs - ⁇ b - ⁇ E c - V T0 - ⁇ V d5
• Threshold Voltage V TO M ⁇ b - ⁇ E 0 -V ⁇
• V th [1+V ⁇ l /2V th + Effective Gate Voltage V g ,e m sqrt( ⁇ 2 + (V at /2V th - 1) 2 ]
• Ns represents the model sheet carrier concentration within the active channel.
• Ns' represents the ideal charge control law and is modeled as a semi- physical representative of the actual density of state filling rate for energy states within the channel v. gate voltage.
• the gate-to-channel voltage used for the charge control, Vgt is a function of the Schottky barrier height, conduction band offsets and doping in the epi stack as is known in the art.
• the following equations represent the semi- physical expression used to model the position of regional charge boundaries within the HEMT device. These expressions govern how to partition the model charge between the influence of different terminals.
• Gate-Source Control Region 1 gs [ ⁇ m] L ! 2+ ⁇ L, +X 01
• Region 1 denotes the linear region
• Region 2 denotes the saturated region of the channel
• FIG. 11 is a schematically illustrates how electrical conductance in the source and drain access regions are modeled in the example.
• Equivalent circuit element Gm delta(Ids)/delta (Vgs') where delta (Vgs 5 ) is mostly the applied voltage deltas, but also subtracting out that voltage which is dropped across the gate source access region, shown as RsCont, RsundepCap, RsundepRec, ResdepRec, and RsBoundary in FIG. 11, above.
• Equivalent circuit element Cgs and Cgd takes the form of delta(Nsn)/delta(Vgs)*Lgn, where delta (Nsn) is the appropriate charge control expression, and Lgn is the gate source or gate drain charge partitioning boundary length.
• Equivalent circuit element Ri Lgs/(Cgschannel * vs) where Cgs channel is the portion of gate source capacitance attributed to the channel only, and vs is the saturated electron velocity.
• Equivalent circuit element Rds l/ ⁇ delta(Ids)/delta(Vds') ⁇ where Vds' is mostly the applied voltage deltas, but also subtracting out voltage which is dropped over both the gate source and gate drain access regions.
• Equivalent circuit element Cds is taken to be the sum of the appropriate fringing capacitance Semi-Physical models, or can take the form of delta(Nsd)/delta(Vds')*Xsat, were Nsd is the charge control expression for charge accumulation between the appropriate source and drain charge boundaries, and Xsat is the length of the saturated region, if in saturation. 3)
• On-mesa Parasitic Elements The equivalent circuit elements, Rs and Rd are expressed by the appropriate electrical conduction models of the source and drain access regions.
• the RF performance can be predicted at an arbitrary bias point.
• Table 4 represents a comparison of the values for a high frequency equivalent circuit model derived from equivalent circuit model extraction from and semi-physical modeling for the sample illustrated in Table 2.
• results of the semi-physical modeling method produce a small-signal equivalent circuit values which are relatively more accurate than the physical device simulator in this case. Furthermore, given the differences in the parasitic embedding, treatment of the two approaches, the results given in Table 3 yield much closer results than a comparison of equivalent circuit values.
• Table 4 lists the values of parasitic elements used in the model derivations.
• FIG. 15. illustrates the semi-physically simulated bias equations of the small signal Gm compared to measured data.
• FIG. 16 illustrates the semi-physically simulated bias-dependence of the small-signal Rds.
• V DSAT m 'dsW* ( R DSaturat ⁇ d )
• FIG. 17 illustrates the semi-physically simulated bias-dependence of the small-signal Cgs and Cgd.
• C gdf [fF/ ⁇ m] Cg S ⁇ rf C HQrm1 SiNF + Cgdf Cap + Cgdf Pad
• V-s M V ds / [1+(V ds /V saton ) m ] 1 ""
• FIG. 18 shows the semi-physically simulated bias-dependence of the small-signal Ri.
• FIG. 19 shows the semi-physically simulated bias-dependence of the on- mesa parasitic access resistances, Rs and Rd.
• the following example verifies how the semi-physical small-signal device model is able to provide accurate projections for bias-dependent small-signal performance.
• the same semi-physical device model as used in the previous examples was used because the example MMIC circuit was fabricated utilizing the same HEMT device technology.
• the bias-dependence small-signal gain and noise performance of a two-stage balanced K-band MMIC LNA amplifier is replicated through microwave circuit simulation using small signal and noise equivalent circuits that were generated by the semi-physical model.
• the results of the measured and modeled results are shown below in Table 5. As seen from these results, the semi-physical device model was able to accurately simulate the measured bias-dependent performance, even though the bias variation was quite wide.
• FIG. 20 A plot of measured vs. modeled gain for the values listed in Table 5, above, is shown in FIG. 20.
• the following example verifies how the semi-physical small-signal device model is able to provide accurate projections for physically dependent small-signal performance.
• the same semi-physical device model as used in the previous examples was used.
• FIGs. 21 A and 2 IB show schematically the kind of data that is extracted and recorded from measured device I-V's during PCM testing. Since the semi-physical device model is able to simulate I-V's, it was able to simulate the variation of I-V's due to physical process variation. These I-V's were analyzed in the same fashion to extract the same parameters that are recorded for PCM testing. Figures 22, 23, and 24 show how accurately the simulated results match with measured process variation.
• Figure 19 shows how the semi-physically simulated Vgpk and Gmpk match with actual production measurements.
• Figure 20 shows how simulated Idpk and Gmpk match, also.
• FIGs. 21 A and 2 IB show how the simulated Imax and Vpo also match very well.
• Small-signal S-parameter measurements are also taken in process for process control monitoring. These measurements are used to extract simple equivalent circuit models that fit the measured S-parameters. Since the semi-physical device model is able to simulate these equivalent circuit models, it was able to simulate the variation of model parameters due to physical process variation.
• FIGs 25, and 26 show how accurately the simulated results match with measured/extracted process variation for the small-signal model parameters.
• FIG. 25 shows how the semi-physically simulated Rds and Gm match very well with actual extracted model process variation
• the semi-physical model is able to very accurately reproduce the dependence of Imax upon gate length.
• the semi-physical model is also able to replicate physical dependence for high-frequency small-signal equivalent circuits. This is shown in FIG. 28, which shows that it is able to reproduce the dependence of Rds with Recess undercut width.

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