Microwave Analysis for Two-Dimensional CV and Noise Model of AlGaN / GaN MODFET

A new two-dimensional analytical model for the capacitance-voltage and noise characteristics of a AlGaN/GaN MODFET is developed. The two-dimensional electron gas density is calculated as a function of device dimensions. The model includes the spontaneous and polarization effects.The contribution of various capacitances to the performance of the device is shown.Themodel further predicts the transconductance, drain conductance, and frequency of operation. A high transconductance of 160mS/mm and a cut-off frequency of 11.6 GHz are obtained for a device of 50 nm gate length. The effect of gate length on the gate length behaviour of the noise coefficients P, R, and C is also studied.The effect of parasitic source and gate resistance has also been studied to evaluate the minimum noise figure. The excellent agreement with the previously simulated results confirms the validity of the proposed model to optimize the device performance at high frequencies.


Introduction
HEMT is very useful in designing Microwave Monolithic Integrated Circuits (MMIC) and also for designing the circuits at higher frequency range of operation.The superior performance of HEMTs over conventional MESFET structures has been demonstrated in both analog and digital ICs.In recent years, AlGaN/GaN pHEMTs (Pseudomorphic High Electron Mobility Transistors) have emerged as a very promising device for high frequency, high power density, and high temperature applications [1,2].The pHEMTs on GaN are rapidly replacing conventional MESFET technology in military and commercial applications.AlGaN/GaN pHEMTs have emerged strongly as attractive candidate for high power, high temperature, and high speed applications in the microwave regions.The presence of strong spontaneous and piezoelectric polarization fields between AlGaN and GaN in addition to the large band-gap energy, high saturation velocity, large breakdown fields, large conduction band discontinuities, and high thermal stability are some of the characteristic features of the device which leads to outstanding performance on GaN-based HEMTs.Capacitancevoltage characteristics are widely used as a diagnostic tool for semiconductor.The C-V characteristics also provide more information about the nonlinearity of the device for the large signal models [3][4][5][6].Low noise HEMTs are promising devices for millimetre wave due to their excellent high frequency and low noise performances.Because of the high carrier density and high breakdown fields, AlGaN/GaN pHEMTs have low noise performance over conventional GaAs-based HEMTs.The HEMT appears to be an ideal device for low noise application [7,8].The earliest study of HEMTs consisted of a one-dimensional analytical approach, which is not sufficient to describe the behaviour of these devices in the submicrometer regime.A number of analytical models [9][10][11] have been developed so far to characterize the 2-DEG sheet carrier concentration.But these models do not explain the behaviour of the device in the subthreshold region.Therefore to provide a better physical understanding of the device operation, an analytical approach to model the 2-DEG density of AlGaN/GaN pHEMTs is highly desirable.
In the present paper, a new analytical model for an AlGaN/GaN pHEMT is realised by solving the 2D Poisson's equation in the depleted AlGaN region to characterize the dc performance of the device considering the effect of parasitic resistances.A detailed analysis of various polarization effects associated with it is to be performed.The model includes the effects of various fringing capacitances due to the small geometry of the device.These device capacitances are then used to evaluate the microwave performance of the model.The model evaluates the transconductance, cut-off frequency, transit time, minimum noise figure, and noise coefficients of the device.Such models are helpful as they provide good insight into the physical operation of the device.The device capacitances, both gate, drain, and gate, source, play a vital role in determining the microwave performance of the pHEMT.The results so-obtained are compared with the previously published experimental and simulated results, which prove the validity of the model.

Model Formulation
2.1.Charge-Control Model.The basic structure of AlGaN/ GaN pHEMT [12] is as shown in Figure 1.
The device structure consists of GaN quantum well channel separated by undoped AlGaN spacer layer from thick n-type AlGaN supply layer.The function of the undoped spacer layer is to reduce impurity scattering.The structure of the device is shown with drain and source electrodes as ohmic contacts.HEMTs or AlGaN/GaN MODFETs are high performance devices for their potential use at higher frequencies because of the GaN as substrate, which offers thermal stability and high electron velocity.The buffer layer is epitaxially grown on the substrate.Many pHEMTs contain a super lattice structure to further inhibit substrate conduction.A super lattice structure is a periodic arrangement of undoped epitaxial layers used to realize a thicker epitaxial layer of a given property.The most important point about the channel layer in the HEMT and pHEMT devices is the twodimensional electron gas (2DEG) that results from the bandgap difference between AlGaN and GaN.
Other typical models of pseudomorphic HEMT (pHEMT) are rapidly replacing conventional MESFET technology in military and commercial applications requiring low noise figures and high gain particularly at millimeter wave frequencies.The application of pHEMTs for high efficiency power amplifications is gaining popularity.
The 2DEG sheet charge density can be obtained by simplifying and solving Poisson's equation [5] as where  is the separation between the gate and the channel,  is the electronic charge,  is the Al mole fraction,  gs is the applied gate-source voltage, () is the AlGaN/GaN dielectric constant, and   () is the channel potential at  due to the drain voltage. th is the threshold voltage and is given as where The total polarization induced sheet charge density is given [3] as where  SP and  PZ are the spontaneous and piezoelectric polarization of AlGaN and GaN layers, respectively.The total amount of polarization induced sheet charge density for AlGaN/GaN heterostructure field effect transistor is obtained as In the above expression, it has been assumed that the GaN layer is fully relaxed.This is a reasonable assumption since the thickness of GaN layer is much larger than that of strained AlGaN layer.Thus,  PZ (GaN) ≈ 0. And ( The piezoelectric polarization induced charge density depends upon Al mole fraction which is given as where () and (0) are lattice constants,  31 () and  33 () are piezoelectric constants, and  13 () and  33 () are elastic constants, respectively.
For a given aluminum mole fraction, there is a certain critical thickness of barrier layer.Equation (6) shows that the device remains completely strained for Al content up to 38% and fully relaxed for Al content greater than 67% for thickness range of 200-400 Å.

Current-Voltage Characteristics.
The drain-source current in the channel is obtained from the current density equation and is given [13] as where  is the gate width,   is Boltzmann constant,  is temperature, and () is the field dependent electron mobility and is given [2] as By substituting ( 1) and ( 8) in ( 7) and on integrating using boundary conditions where   and   are the parasitic source and drain resistances, respectively.
The drain to source current in the channel is given as where The drain saturation voltage  dsat is obtained as where with The detailed analysis for (10) and ( 12) is reported in [6].

Capacitance-Voltage Characteristics.
The equation for channel current in terms of position dependent sheet carrier concentration and electron velocity is given as [10]  ds =   () V  () , where  is the gate width,  is the electronic charge,   () is the electron concentration in the channel, and   () is the electron drift velocity and is given as where and  0 is the low field mobility Total channel charge is given as Rearranging (15) and using (1), we get Integrating ( 19) from 0 to , using [10] By substituting (21) into (20) and using boundary conditions in ( 9), the total channel charge is obtained as where It is defined as the rate of change of charge on the gate electrode with respect to the drain bias when the source and the gate potentials are kept constant: 3. Small-Signal Parameters ] . (27)

Cut-Off Frequency (𝑓 𝑡
).The unity gain cut-off frequency is an important figure of merit of the MODFET performance at microwave frequencies.The high velocity of electrons in the channel at the drain side results in a high transconductance.The cut-off frequency is given as the ratio of the transconductance to the sum of capacitances (both  gs and  gd ); that is, And by substituting the value of   , which has already been derived in (27) and  gd and  gs from (24) and (25), respectively, the cut-off frequency can be evaluated.The cutoff frequency   determines the ultimate speed of a switching device.

Transit Time.
The transit time effect is the result of a finite time being required for carriers to traverse from source to drain.Smaller transit times are desirable to attain a high frequency response from a device.The transit time for the AlGaN/GaN pHEMT is evaluated as By substituting (28) in (29), transit time can be obtained.

Power Added Efficiency
In certain applications such as mobile and satellite communications [14], DC power is at a premium.In modern communication system and radar applications, the RF power amplifier may be the subsystem that consumes most of the DC power.For this reason, the efficiency of the RF power amplifier can be critical to the design and performance of the overall system.

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The basic efficiency is expressed as When input RF power is included in the efficiency calculations, the modified expression is referred to as Power Added Efficiency (PAE) and is expressed [15] as where  is the power gain and is given as

Noise Analysis
In the present work, the noise is produced in the ohmic section of the channel in region I termed as thermal or Johnson noise.And in region II, the noise is produced in the velocity saturated section of the channel region termed as diffusion noise.The analytic expressions for the mean square values  2  and  2  are obtained in this section.

Drain Noise Source.
The drain voltage fluctuation due to an elementary noise produced at  is obtained as where Δ is the frequency range,  is the noise temperature parameter, and  0 is the lattice temperature.The mean square drain noise voltage is given as Open circuit drain voltage fluctuation due to diffusion noise can be given as where  is the effective channel thickness and is given as

Gate Circuit Noise.
The noise current flowing may be obtained by Δ 2 2 = ΔV 2 2 / 2  , where   is the drain resistance.The expression for short circuit gate noise current can be obtained as where (38)

Noise Coefficients.
The correlation between the drain noise source and the gate noise source is given as The details of (39) are reported in [16].
The drain noise coefficient associated with the drain noise current is given as The gate noise coefficient associated with the gate noise current is given as where  2  is the equivalent noise voltage at the drain region and  2  is the equivalent noise induced gate current.The noise coefficients  1 and  2 and  1 and  2 are of the same form as  and , but with subscripts 1 and 2 on different noise sources in the ohmic and saturation region, respectively.
The correlation coefficient between different noise sources can be written as The correlation coefficient in the linear region can be given in terms of  and  as The correlation coefficient in the saturation region can be given in terms of  and  as The correlation coefficient between   1 and   1 can be calculated as There is full correlation between   2 and   2 with a capacitive phase shift of 90 ∘ .Therefore, Therefore, there is no correlation between   1 and   2 or   2 and   1 .

Noise Figure.
The minimum noise figure in terms of source resistance and gate resistance can be calculated as where Experimental [17] Calculated where   ,   , and   are the functions of , , and  and are given as (49) 5.5.Noise Temperature.The minimum noise temperature can be obtained as where  is absolute temperature and   is the correlation resistance.

Results and Discussion
To prove the validity of the model, the calculated/obtained results are plotted and compared with the previously published experimental and simulated results.
Figure 2 shows the current-voltage characteristics of AlGaN/GaN MODFET of 50 nm gate length for different values of gate voltages.It can be seen that current increases with the increase in drain-source voltage.A peak drain current of 501 mA/mm is achieved for gate bias of 1 V.The model exhibits high current driving capabilities and good pinch-off characteristics.In saturation region, the current continues to increase slightly due to short-channel effects.The calculated results are in good agreement with the experimental results [17].

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Transconductance, g (mS/mm)  The variation of transconductance with gate voltage is depicted in Figure 3.The high value of transconductance can be explained due to high saturation velocity and better carrier mobility.The calculated transconductance has a peak value of 160 mS/mm at a gate bias of −1.2 V.As  gs increases, the pHEMT initially operates in the saturation region.As  gs continues to increase, the transistor enters in the linear region Figure 6 shows the variation of cut-off frequency with drain current.For low values of drain current, transconductance is low and hence cut-off frequency is also low.At a drain current of 203 mA/mm, a cut-off frequency of 11.6 GHz is obtained.The present results are in close proximity with the experimental results [18].
Figure 7 shows the dependence of gate-source capacitance on drain-source voltage.It is observed from the figure that, for small drain voltages, the dependence is small and for higher values of drain voltage there is rapid rise in gate-source capacitance due to the carrier saturation in the device.
Figure 8 shows the variation of  gs with  gs for different values of  ds .It is evident from the figure that  gs decreases with decrease in  gs and increases with increase in  gs .Figure 9 shows the variation of  gd with  dg for different values of  gs .It can be seen from the figure that there is sharp increase in  gd at  dg of −3 V.The calculated results are in good agreement with experimental results [13].
The variation of PAE, gain, and output power with input power is shown in Figure 10.A PAE of 54% with an associated power gain of 14 dB and 31 dBm output power with a frequency of 20 GHz is obtained.It is evident from the figure that the input power varies linearly with output power.These plots are consistent with those reported in [19].Figures 11,12   Figure 14 shows the dependence of noise temperature on drain current for 50 nm gate length which is consistent with those reported in [20].It can be seen from the graph that the transistor achieves lower noise temperature due to better confinement and effective channel width.
The variation of minimum noise figure with frequency is presented in Figure 15.It is clear from the figure that, with increase in frequency, minimum noise figure also increases.The results are in close agreement with the previously published results [20].

Conclusion
The proposed model is developed for the 2D MODFET in characterizing and evaluating the dc and microwave performance for a gate length of 50 nm device.The model is developed for the I-V characteristics and small signal parameters of an AlGaN/GaN MODFET considering the strong polarization effects.Including the analysis of the physical meaning of the capacitances, the C-V model reflects the variations in the capacitance with  gs and  ds .The dependencies of noise performance on drain current and frequency were characterized.The noise analysis is quiet good in linear and saturation region.All the noise coefficients like drain noise coefficient (), gate noise coefficient (), and

Figure 1 :
Figure 1: (a) Schematic of AlGaN/GaN MODFET.(b) Diagram for showing linear and saturation region used for 2D analysis.

Figure 3 :
Figure 3: Variation of transconductance with gate voltage.

Figure 5 :
Figure 5: Variation of transit time with gate voltage for different mole fraction.

ForFigure 8 :
Figure9shows the variation of  gd with  dg for different values of  gs .It can be seen from the figure that there is sharp increase in  gd at  dg of −3 V.The calculated results are in good agreement with experimental results[13].The variation of PAE, gain, and output power with input power is shown in Figure10.A PAE of 54% with an associated power gain of 14 dB and 31 dBm output power with a frequency of 20 GHz is obtained.It is evident from the figure that the input power varies linearly with output power.These plots are consistent with those reported in[19].Figures 11,12, and 13 show the dependence of noise coefficients (, , and ) on drain current.The correlation coefficient  shows the symmetric behaviour.It is almost constant for all values of drain currents.It decreases with shorter gate lengths.The gate noise coefficient  is very

Figure 10 :Figure 11 :
Figure 10: Variation of PAE, gain, and output power with input power.

Figure 12 :Figure 13 :
Figure 12: Variation of gate noise coefficient  with drain current.
( 0 ) is the time-dependent Schottky barrier height, Δ im is the dielectric image force, Δ  is the interface dipole bond energy,   and   are the electron concentration in the valence and conduction band, respectively, Δ  () is conduction-band discontinuity of AlGaN/GaN interface,   is the doping concentration of the AlGaN barrier, () is the net polarization induced sheet charge density at the AlGaN/GaN interface, Δ exc is the electron exchangecorrelation energy, and   () is the Fermi potential.
the gain mechanism is embodied in the active channel transconductance, which is evaluated as   () =  ds ()  gs           ds , ) 3.2.Transconductance.It is the most important parameter for optimization of FET high frequency behaviour.The major part of Variation of  gd with  dg .
[20] at low drain currents due to decrease in gate to source capacitance.It increases for shorter gate lengths.The drain noise coefficient  decreases with lower values of drain currents.It also decreases for shorter gate lengths.The calculated/obtained results are in close proximity with the previously published results[20].