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The Importance of the Current-voltage Characteristics of FETs, HEMTs and Bipolar Transistors in Contemporary Circuit Design
Peter Ladbrooke and James Bridge
Accent Optical Technologies
Traditionally, when applying computer-aided methods to the design and practical realization of nonlinear circuits, the active device models required for the circuit simulator have been derived from small-signal S-parameter measurements made at multiple bias points. Although this procedure was useful for low specification circuits, it is not adequate for contemporary circuits where linearity, power-added efficiency and insertion phase must all be correctly predicted. The traditional method fails, because under large-signal RF conditions, PHEMTs, FETs and HBTs pass currents at any given RF voltage that are different from the current the device passes if that same voltage is reached slowly, that is, at DC. In other words, the current-voltage characteristics traditionally used for the device model are incorrect. In applications where the maximum frequency of operation is well below the current cut-off frequency fT of the device, the current-voltage characteristics dominate the nonlinear circuit behavior, while the charge or capacitance characteristics have a secondary role. Thus, knowledge of the exact current function is of paramount importance. This article describes the application in nonlinear circuit design of an instrument which measures the dynamic large-signal characteristics of FETs, PHEMTs and bipolar transistors.
The quantities indicated by a lower case symbol with upper case subscripts such as
iG , iGS , vGS , vDG ....
signify total instantaneous (large-signal RF or microwave or pulsed) currents and voltages. The quantities indicated by an upper case symbol with upper case subscripts such as
IG , IGS , VGS , VDG ....
signify DC (or slow-sweep measured) currents and voltages. The symbol I(V) refers to the current-voltage characteristics of devices, irrespective of the way they are measured.
What is Dispersion?
Dispersion is the term used to indicate that the dynamic (or RF) characteristics of a device are different from the static (or DC) characteristics. To one degree or another, all transistor devices are dispersive. Dispersion has two physical origins: changes in charge held in deep levels (or traps) and self-heating.
Charging and discharging of deep levels occurs at DC as the voltages on the device change, but does not occur at RF because the rate of change of the signal voltage is too high for the charge trapped in deep levels to be able to vary. In mature materials technologies there are too few bulk deep levels to affect device behavior; instead, it is the deep levels located at the device surface which give rise to dispersion. The shorthand Dss will be used to signify the density per unit area of charge held in surface states.
The second mechanism giving rise to dispersion is the changing of the lattice temperature T0 , or self-heating. Changes in self-heating occur in an obvious manner at DC as the product of V and I changes, but at RF any changes arise in a far more complicated manner. Suffice it to say that changes in T0 under RF conditions, if they occur at all, are very different from those which occur at DC.
There are two main reasons for studying dispersion: first, from a processing point of view, to reduce the effect as far as possible; secondly, to provide the correct data for the design of nonlinear circuits. This article is concerned with the circuit design application.
Why the Current Function Matters
Considering just the vGS dependence of iD , it can easily be shown that1
- the first derivative of iD with respect to vGS , that is, the transconductance, controls the gain at the fundamental frequency, and the output power compression at high RF input power levels;
- the second derivative of iD with respect to vGS controls the second harmonic generation in the nonlinear regime of operation;
- governs the third harmonic generation, etc...
Consequently, if correct prediction of gain, gain compression and harmonic generation in circuits is desirable, not only must the vGS dependence of the current function be measured with enough accuracy, but it must be preserved when fitting the equations of a large-signal model to the measured current function. The vDS dependence of the current function has an equally important but much less obvious role in nonlinear circuits, particularly in power amplifiers such as the ones used in CDMA. Of reasonably self-evident importance is the "linear" region below the knee in the iD (vDS ,vGS ) characteristics. This region can be loosely characterized by an ON resistance, and has a major bearing on the power-added efficiency of class F type power amplifiers where the load trajectory traverses this linear region. If the linear region is measured inaccurately, or if the measurements are poorly represented by a model, the power-added efficiency will be poorly predicted.
Of more obscure significance is the detailed dependence of iD on vDS , which governs AM-to-PM conversion. It is the shape of iD (vDS ), which, in conjunction with the load impedance at the output of the device, governs AM-to-PM conversion and, with it, all those circuit performance measures which are related to AM-to-PM conversion.
Fig.1 Phase shift for a signal at the fundamental frequency as a function of input signal amplitude.
Figure 1 sketches a typical change in the phase shift of a signal passing through a power amplifier as the input signal amplitude is increased. This nonlinear shift in the insertion phase arises from the dependence of iD on vDS . To understand such a dependence, the key requisite is some mechanism which introduces a phase shift at the fundamental frequency of the excitation. Expressed another way, the RF current iD must depend upon, or at least be in part driven by, a voltage that is delayed, or time-shifted, relative to the excitation input. Careful consideration of the simplified device circuit is shown in Figure 2 , demonstrating that the dependence of iD on vDS , which is shifted in phase relative to vGS as a result of vDS arising from iD flowing through ZL , gives us the delay mechanism sought for. In sequence, changes in vGS initiate changes in iD , which in turn give rise to changes in vDS via the current flowing through ZL . iD is then further altered through its dependence on vDS . Furthermore, as the function f is nonlinear, the phase shift can be expected to be dependent upon the signal amplitude, or equivalently upon the input power, PIN . The phase shift through the amplifier is dependent upon the amplitude of the applied signal, thus amplitude modulation leads to phase modulation. This effect is referred to as AM-to-PM conversion or distortion.
Fig. 2 Simplified device equivalent circuit.
AM-to-PM conversion has the direct effect of leading to phase distortion even at power levels at which there is little amplitude distortion arising from gain compression. Predicting such distortion is important for phase modulation schemes. Additionally, phase distortion gives rise to third-order intermodulation products. The third-order intermodulation products arising from phase distortion are in addition to the third-order intermodulation products that arise directly from amplitude distortion.
Thus, just as the dependence of iD on vGS must be measured exactly and preserved in a model if gain, gain compression and harmonic generation are to be well predicted, so too must the dependence of iD on vDS be known exactly and preserved in a model if power-added efficiency, AM-to-PM conversion and intermodulation distortion are to be well predicted in circuit design. As will be seen in the following sections, the DC I(V) characteristics do not generally qualify for RF use, nor do characteristics resulting from multi-bias small-signal S-parameter measurements. The desired RF large-signal device characteristics must be measured directly.
What is the True Large-signal RF Current Function?
Refer to Appendix A for a definition of the current function and the charge function. The true large-signal set of RF I(V) characteristics of a FET or HEMT are those obtained when the instantaneous voltages and current of the device are moved by the RF well away from the bias point, but the occupied surface state density and the lattice temperature of the device remain as set by the quiescent (or bias) point. This pre-set condition is referred to as the "state" of the device. The principle is illustrated in Figure 3 .
Fig. 3 How the true RF I(V) characteristics differ from the DC I(V) characteristics.
Because the "state" of the device is uniquely set by the bias point, at any bias point, a FET or HEMT has a unique set of dynamic I(V) characteristics, which are the paths the drain current will follow in moving away from that point when driven with large-amplitude time-varying signals. RF signals are just one very important example of such large signals. Similarly, at each bias point, a device possesses a unique Q(V) characteristic set. As shown in the next section, these I(V) and Q(V) characteristics, which change when the bias point is changed, are not what is obtained from bias-dependent S-parameter measurements. Therefore, any device which is dispersive possesses not just one set of large-signal RF characteristics, but as many sets as there are points at which it can be biased (which in theory is infinite).
The state of the device is a hidden parameter of the current (or charge function). A traditional plot of current versus voltage measured at DC for a device (such as is measured on a curve tracer) is misleading. The device state changes with the slowly varying DC voltages applied during the measurements and is different at each point. The resulting curves show only how the device behaves under slowly varying conditions. Under fast varying conditions, the device state does not have time to change, so the characteristics may be very different.
Similarly, though S-parameters are measured at high frequency, they are measurements of very small perturbations about a quiescent state that is set by a DC bias point. Models based on S-parameter measurements made at many different bias points will incorporate data sampled from the device in many different states. For a dispersive device, such a model will not be a good representation of its behavior under large-signal RF conditions.
Models Based on Bias-dependent S-parameter Measurements
A common practice is to derive the voltage dependence of the elements in a large-signal device model from S-parameter measurements made at multiple bias points. The I(V) and Q(V) functions thus derived are distorted representations of the physical behavior of the device when driven with large-amplitude RF signals about any starting, or bias, point. The problem arises because S-parameter measurements about a series of bias points are always measurements of "local" properties. In other words, when bias-dependent S-parameter measurements are made about a bias point, the small-signal properties measured are "local" properties (where "local" means "local to the bias point"). The concept of "localness" can be contrasted with what happens at a point well removed (or "remote") from the bias point under conditions of large-signal excitation.
The dynamic large-signal properties at a point remote from the bias point are those which result when the state of the device remains as set by the bias point, despite large excusions in the values of the device voltages and current away from that point. In direct contrast, the customary bias-dependent small-signal measurements at the remote point perforce have to move the bias to that point, so that the Dss and T0 values, as expressed by the state of the device, are changed by the new bias condition. Expressed another way, bias-dependent small-signal I(V) and Q(V) characteristics are comprised solely of local measurements at multiple bias points, whereas the true large-signal characteristics consist of the I(V) and Q(V) properties at only one local point (the bias point), and all the others are properties at points remote from the bias point. What is different is that the bias-dependent small-signal S-parameter method of characterization makes measurements with the device biased into many different states, whereas in the true large-signal behavior the device remains in a single state as set by the bias point.
Thus the I(V) and Q(V) functions derived from bias-dependent small-signal S-parameters are distortions. For example, if measurements at one hundred bias points have been made, these functions are made up of one point (the local point properties) from each of one hundred different full large-signal I(V) and Q(V) functions. The distortion can be very easily detected by forming the integrals of the extracted transconductance and output conductance as
Both should give the same result for the drain current iD . In dispersive devices they do not, and neither integral gives the correct drain current function. To obtain the true RF large-signal current function it must be measured directly. Accent's DiVA makes measurements of the desired I(V) function in the time domain.
True Large-signal Dynamic I(V) Measurements
Figure 4 shows the characteristics of a microwave FET measured under DC conditions. Figure 5 shows the characteristics of the same device measured using fast pulses (200 ns) about the bias point VDS = 6 V, VGS = -2.4 V. The pulsed characteristics are very different from the DC characteristics and any model based on the DC behavior would produce erroneous results if used to model large-signal RF behavior.
Fig. 4 DC characteristics measured at 1 V/sec for a sample FET.
Fig. 5 Pulsed I(V) characteristics about the bias point VDC = 6.0 V and VGS = -2.4 V.
To illustrate how large-signal characteristics will alter with bias point, Figure 6 shows the pulsed characteristics plotted about the bias point VDS = 2 V, VGS = 0 V. The two sets of characteristics for the two different bias points are very different from each other.
The pulsed characteristics shown were measured using 200 ns pulses. Is this fast enough to give the correct characteristics for microwave signals at several gigahertz? Figure 7 shows the characteristics about the bias point VDS = 6 V, VGS = -2.4 V measured using both 200 ns pulses and 100 ns pulses. The characteristics are identical, thus illustrating that the pulse lengths are short enough to give the true RF characteristics.
Figure 8 shows the same characteristics compared with measurements made using 2 ms pulses. There are now some differences illustrating that for this device the state time constants are in the tens of microsecond range.
Fig. 6 Pulsed characteristics measured about the bias point vDS = 2.0 V, vGS = 0.0 V.
Fig. 7 Pulsed characteristics about the bias point vDS = 6.0 V and vGS = -2.4 V with pulse lengths of 200 and 100 ns.
Fig. 8 Pulsed characteristics about the bias point vDS = 6.0 V and vGS = -2.4 V for two different pulse lengths.
Given the importance of maintaining the accuracy of the measured current function in fitting a model, it is useful to have a model fitting program built into the software interface that drives the instrument. By including a pre-optimization algorithm tailored specially for the equations of each separate model, a complete fit takes no more than a few seconds. That way, various models can be tried in quick succession to see which provides the best quality of fit to any given set of I(V) measurements. The results are surprising: generally there is no "best model" for any given device, but different models produce the most acceptable fits to the dynamic I(V) characteristics measured about different bias points.
Figure 9 shows the Triquint Own Model (TOM) fitted to the pulsed characteristics measured about the bias point VDS = 6 V, VGS = -2.4 V for the example previously shown. The overall idea is to make the best use of the standard nonlinear device models offered in commercial circuit simulators. Although much improvement in the design of nonlinear circuits is made possible through the procedures suggested in this article, it may be that none of the standard large-signal modelling approaches suffice for the most demanding contemporary design tasks. Part of the reason is that none of the standard model equations can fit the I(V) data with sufficient accuracy: they do not preserve the differentials of iD with respect to vGS , and they do not model iD (vDS ) accurately enough. When the customary forms fail in this way, a more elaborate approach is needed, an example of which is given in Parker.2 Excellent results have been achieved for a number of high specification circuits with what is referred to as a separated function approach, using separate functional forms to represent the vDS and vGS dependences of iD .3
Fig. 9 An example of a model fit to the pulsed characteristics measured about the bias point vDS = 6.0 V, vGS = -2.4 V.
Obtaining the Charge Function Q(V)
The current function, I(V), can be measured directly under pulsed conditions. The charge function, Q(V), cannot be measured in the same way. S-parameter measurements may be used to obtain differential changes in the charge function about a particular bias point but moving to a different point alters the device state thus changing the charge function. Using generic device physics, it is possible to obtain the charge function indirectly by using information from the directly measured current function.
The differential quantities gm0 and g0 from the current function provide a measure of the depletion edge movement with voltage changes. The same depletion edge movements lead to changes in charge stored, so it is possible, using generic physics, to make a good estimate of the charge function. Any inaccuracy in estimating the charge function from the current function rather than using direct measurement is less significant than the inaccuracy inherent in fitting the equations of standard models to the charge function (or equivalently to the model capacitances).
1. O. Pitzalis, "Computer-aided Design of GaAs FET Power Amplifiers," in High Power GaAs FET Amplifiers , J.L.B. Walker Ed., Artech House Inc., Norwood, MA 1993.
2. A.E. Parker, "Implementing High Order Continuity and Rate Dependence in SPICE Models," IEE Proceedings of Ccts Dev Syst ., Vol. 141, No. 4, August 1994.
3. Application Note PMI-AN1 available from Accent Optical Technologies.
4. I.W. Smith, H. Statz, H.A. Haus and R.A. Pucel, "On Charge Nonconservation in FETs," IEEE Transactions on Electron Devices , Vol. ED-34, No. 12, December 1987.
Peter Ladbrooke earned his PhD degree from Cambridge University in 1971 for work on GaAs devices. Since then he has worked in various organizations in England including Plessey Radar Research Centre, Allen Clark Research Centre and Hirst Research Centre. He spent nine years as a lecturer at the University of New South Wales, Australia, and was a visiting scientist at Allen Clark Research Centre in the UK and Cornell University, NY. He co-founded GaAs Code Ltd. in 1988, applying FET and HEMT physics to large-signal modeling, yield-driven circuit design and MMIC process design. In April of 2001, he joined Accent Optical Technologies as a device operation specialist.
James Bridge obtained his degree in engineering science from Oxford University in 1982. From 1982 to 1986, he worked for the GEC Hirst Research Centre on GaAs MMICs. From 1986 to 1988, he worked for Analytic Associates on radar tracking and signal processing algorithms. He co-founded GaAs Code Ltd. in 1988, and in June of 2001 joined Accent Optical Technologies as project manager, device measurements.
The purpose of this appendix is to define the charge and current functions, Q(V) and I(V), which are referred to in the body of the article. Reference will be made to FETs but the behavior of HEMTs is very similar.
At the core of a FET is a positively charged depletion region under the gate electrode that restricts the current passing between the source and drain electrodes. To maintain charge neutrality the positive depletion charge is matched by a negative charge on the gate electrode. The size and shape of the depletion charge depends on two voltages - VGS between gate and source and VDS between drain and source. (Equivalently it can be considered to depend on any other combination of independent voltages such as VGS and VGD between gate and drain.)
The size and shape of the depletion region is also affected by charges trapped in surface states or other deep levels that, under DC conditions, may change with applied voltage. In this appendix current and charge functions are considered that are applicable at RF where the changes in voltage are too rapid for the trapped charges to respond.
The current passing between drain and source is thus dependent on the two voltages VGS and VDS . The current can be expressed as a function ID (VGS ,VDS ), which is simplified to I(V). The partial differential of the current with respect to VDS is a direct conductance referred to as g0 (the voltage is between the drain and source and the current flow is between drain and source). The partial differential of the current with respect to VGS is an indirect transconductance referred to as gm0 (the voltage controlling the current is across different electrodes to those between which the current flows). There is a single current with two differentials, gm0 and g0 , which form a conservative pair.
The charge function is less straightforward. This is because charge flow occurs to and from all three electrodes, while in most circumstances the current flow is restricted to the drain and source electrodes. The negative gate charge flows on and off the gate electrode but changes in the positive depletion charge can result in charge flow in either the source or drain electrodes depending on circumstances.
Consider the magnitude of the gate charge, which equals the magnitude of the depletion charge as a function Q of the voltages VGS and VDS , that is Q(VGS ,VDS ), which can be simplified to Q(V). Care is needed in considering the meaning of the partial differentials of Q(V). In addition, how both the charge changes with voltage needs and where the change of charge flows as a capacitive current need to be considered.
In simple standard models there are two capacitances - Cgs and Cgd . Cgs is the partial differential of the charge with respect to gate source voltage and Cgd is the partial differential of the charge with respect to the gate drain voltage (the two capacitances can equally be expressed in terms of partial differentials with respect to VGS and VDS but the one-to-one correspondence is lost).
Such a model, containing only two gate capacitances Cgs and Cgd , imposes non-physical charge flow on the device. In the model, fixing VGD results in no charge change in Cgd , so all charge change occurs between gate and source. Similarly, fixing VGS causes all charge change to occur between gate and drain. In the real device some charge change occurs between gate and drain even when it is only VGS varying and some charge change occurs between gate and source even when only VGD varies.
A better model of charge changes includes transcapacitances in addition to Cgs and Cgd . The transcapacitances model the charge change between gate and drain when VGS varies and the charge change between gate and source when VGD varies.
Four capacitances and transcapacitances correspond to four partial differentials but there are only two independent voltages. Therefore, there should be two associated charges, one corresponding to charge flow between gate and source and one corresponding to charge flow between gate and drain.
Unfortunately, the capacitance and transcapacitance pairs are not conservative. Physically there is only one depletion charge and the value assigned to branch charges obtained by integrating capacitance and transcapacitance will depend on the path of integration selected (see Smith, et al.,4 for example).
To implement a more sophisticated model including transcapacitance requires a circuit simulator which works on the capacitance rather than charge level and does not impose charge conservation as a requirement.