# Microwave Analysis for Two-Dimensional C-V and Noise Model of AlGaN/GaN MODFET

Authors: Ramnish Kumar; Sandeep K. Arya; Anil Ahlawat
Journal: Advances in Materials Science and Engineering
 (2014)

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## Abstract

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. The model further predicts the transconductance, drain conductance, and frequency of operation. A high transconductance of 160 mS/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 coefficientsP, 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.


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## Body

## 1. 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. Capacitance-voltage 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 [3–6]. 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 [9–11] 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.


## 2. Model Formulation

### 2.1. Charge-Control Model

The basic structure of AlGaN/GaN pHEMT [12] is as shown in Figure 1.(a) Schematic of AlGaN/GaN MODFET. (b) Diagram for showing linear and saturation region used for 2D analysis.
(a)
(b)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 two-dimensional electron gas (2DEG) that results from the band-gap 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
(1)ns(m,x)=ε(m)q·D(Vgs-Vc(x)-Vth(m)),
where D is the separation between the gate and the channel, q is the electronic charge, m is the Al mole fraction, Vgs is the applied gate-source voltage, ε(m) is the AlGaN/GaN dielectric constant, and Vc(x) is the channel potential at x due to the drain voltage. Vth is the threshold voltage and is given as
(2)Vth(m)=ϕm(T0)+m⌊εgap⁡AIGaN(T)-εgap⁡AIGaN(T0)⌋-ΔEg2-ET2ln⁡[NcAIGaNNVGaNNVAIGaNNCGaN]+ΔEexc-ΔEim+ΔEb-qNddd22ε(m)-qNbcb+Ef(m)q,
where ϕm(T0) is the time-dependent Schottky barrier height, ΔEim is the dielectric image force, ΔEb is the interface dipole bond energy, NV and NC are the electron concentration in the valence and conduction band, respectively, ΔEc(m) is conduction-band discontinuity of AlGaN/GaN interface, Nd is the doping concentration of the AlGaN barrier, σ(m) is the net polarization induced sheet charge density at the AlGaN/GaN interface, ΔEexc is the electron exchange-correlation energy, and Ef(m) is the Fermi potential.The total polarization induced sheet charge density is given [3] as
(3)σ(m)=Psp(AlmGa1-mN)-Psp(GaN)+PPZ(AlmGa1-mN)-PPZ(GaN),
where PSP and PPZ 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
(4)|σ(m)|=|Psp(AlmGa1-mN)-Psp(GaN)+PPZ(AlmGa1-mN)|.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,PPZ(GaN)≈0.And(5)PPZ(AlmGa1-mN)=2(a(0)-a(m)a(m))×(e31(m)-e33(m)c13(m)c33(m)+Psp(m)we×-Psp(0)c13(m)c33(m))C/m2PSP(AlmGa1-mN)=-0.052m-0.029c/m2PSP(GaN)=-0.029c/m2.
The piezoelectric polarization induced charge density depends upon Al mole fraction which is given as
(6)PPZ(AlmGa1-mN)={(2(a(0)-a(m))a(m))(e31(m)-e33(m)C13(m)C33(m))wwwwfor0≤m≤0.38(2(2.33-3.5m)a(m)/(a(0)-a(m)))×(e31(m)-e33(m)C13(m)C33(m))wwwwfor0.38≤m≤0.670,wwwwfor0.67≤m≤1,
where a(m) and a(0) are lattice constants, e31(m) and e33(m) are piezoelectric constants, and c13(m) and c33(m) 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.


### 2.2. Current-Voltage Characteristics

The drain-source current in the channel is obtained from the current density equation and is given [13] as
(7)Ids(m,x)=zqμ(x,m)(ns(m,x)dvc(x)dx+KBTqdns(m,x)dx),
where z is the gate width, KB is Boltzmann constant, T is temperature, and μ(x) is the field dependent electron mobility and is given [2] as
(8)μ(m)=μ0(m)1+((μ0(m)Ec-Vsat⁡)/EcVsat⁡)(dvcx/dx).
By substituting (1) and (8) in (7) and on integrating using boundary conditions
(9)vc(x)|x=0=IdsRs,vc(x)|x=L=Vds-Ids(Rs+Rd),
where Rs and Rd are the parasitic source and drain resistances, respectively.The drain to source current in the channel is given as(10)Ids=-α1+α12-4α2α32α2,
where
(11)α1=E1(2Rs+Rd)-E2Rd2(Rd+2Rs),α2=-[L+E1Vds+E2(Vgs-Vth-KBTq)hhhhhh×(2Rs+Rd)-E2Vds(Rs+Rd)(Vgs-Vth-KBTq)],α3=E2(Vgs-Vth-KBTq)Vds-E22Vds2.
The drain saturation voltage Vdsat is obtained as
(12)Vdsat=-δ1+δ12-4δ2δ32δ2,
where
(13)δ1=μ0Zε(m)Ecd[α1μ0zε(m)Ecd+E1-E2(Rs+Rd)]-E2E,δ2=μ0Zε(m)Ecd[-2Qα1μ0zε(m)Ecd+E1Qμ0Zε(m)Ecdw-E2Q(Rs+Rd)+L+E2Q(2Rs+Rd)-2Qα1μ0zε(m)Ecd]+E2Q,δ3=Qμ0Zε(m)Ecd×[α1μ0zε(m)E2Qd-L-E2(2RS+Rd)d]
with
(14)Q=Vgs-Vth-KBTq.
The detailed analysis for (10) and (12) is reported in [6].


### 2.3. Capacitance-Voltage Characteristics

The equation for channel current in terms of position dependent sheet carrier concentration and electron velocity is given as [10]
(15)Ids=Zqns(x)vd(x),
where Z is the gate width, q is the electronic charge, ns(x) is the electron concentration in the channel, and Vd(x) is the electron drift velocity and is given as
(16)Vd(x)=μ0E(x)1+(1/E3)E(x),
where
(17)E3=EcVsat⁡μ0Ec-Vsat⁡,E(x)=dVc(x)dx
and μ0 is the low field mobilityTotal channel charge is given as(18)Q=∫0tIdsdt=∫0LIdsdxvd(x).
Rearranging (15) and using (1), we get
(19)Idsvd(x)=Zε(m)D{Vgs-Vth(m)-Vc(x)}.
Integrating (19) from 0 to L,
(20)∫0LIdsvd(x)dx=Zε(m)D∫0L{Vgs-Vth(m)-Vc(x)}dx
using [10]
(21)dx=Zqnsμ0-(Ids/E3)IdsdVc(x).
By substituting (21) into (20) and using boundary conditions in (9), the total channel charge is obtained as
(22)Q=-A13Ids[B-{Vds-IdsR}]3-[B-IdsRs]3-A2[B{Vds-IdsR}-BIdsRs-12{Vds-1dsR}2hhhh+12Ids2Rs2],
where
(23)A1=zε(m)2zμ0D2,B=Vgs-Vth,R=Rs+Rd,A2=Zε(m)E3D.


#### 2.3.1. Gate-Drain Capacitance

It is also called feedback capacitance. 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:(24)Cgd=∂Q∂Vds=-A13Ids[3{B-(Vds-1dsR)}2{1-gdR}]+[{B-(Vds-IdsR)}]3[A13Ids2gd]+A1Ids·(B-IdsRs)2(-gdRs)+(B-IdsRs)3(-A13Ids2gd)-A2[B(1-gdR)-BRsgd-(Vds-IdsR)(1-gdR)hhhhhh+IdsRs2gd].


#### 2.3.2. Gate-Source Capacitance

The gate-source capacitance (Cgs) is defined as the change in total charge with the change in gate voltage. It is mainly due to the 2DEG in the normal operating region and is given as
(25)Cgs=∂Q∂Vgs=-A13Ids[3{B-(Vds-IdsR)}2{1-gmR}]+{B-(Vds-IdsR)}3A13Ids2gm+A1Ids·(B-IdsRs)2(1-gmRs)+(B-IdsRs)3(-A13Ids2gm)-A2[IdsRs2gm{B(-gmR)+(Vds-IdsR)}-{BgmRs+IdsRs}-A2w-(Vds-IdsR)(-gmR)+IdsRs2gm].


## 2.1. Charge-Control Model

The basic structure of AlGaN/GaN pHEMT [12] is as shown in Figure 1.(a) Schematic of AlGaN/GaN MODFET. (b) Diagram for showing linear and saturation region used for 2D analysis.
(a)
(b)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 two-dimensional electron gas (2DEG) that results from the band-gap 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
(1)ns(m,x)=ε(m)q·D(Vgs-Vc(x)-Vth(m)),
where D is the separation between the gate and the channel, q is the electronic charge, m is the Al mole fraction, Vgs is the applied gate-source voltage, ε(m) is the AlGaN/GaN dielectric constant, and Vc(x) is the channel potential at x due to the drain voltage. Vth is the threshold voltage and is given as
(2)Vth(m)=ϕm(T0)+m⌊εgap⁡AIGaN(T)-εgap⁡AIGaN(T0)⌋-ΔEg2-ET2ln⁡[NcAIGaNNVGaNNVAIGaNNCGaN]+ΔEexc-ΔEim+ΔEb-qNddd22ε(m)-qNbcb+Ef(m)q,
where ϕm(T0) is the time-dependent Schottky barrier height, ΔEim is the dielectric image force, ΔEb is the interface dipole bond energy, NV and NC are the electron concentration in the valence and conduction band, respectively, ΔEc(m) is conduction-band discontinuity of AlGaN/GaN interface, Nd is the doping concentration of the AlGaN barrier, σ(m) is the net polarization induced sheet charge density at the AlGaN/GaN interface, ΔEexc is the electron exchange-correlation energy, and Ef(m) is the Fermi potential.The total polarization induced sheet charge density is given [3] as
(3)σ(m)=Psp(AlmGa1-mN)-Psp(GaN)+PPZ(AlmGa1-mN)-PPZ(GaN),
where PSP and PPZ 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
(4)|σ(m)|=|Psp(AlmGa1-mN)-Psp(GaN)+PPZ(AlmGa1-mN)|.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,PPZ(GaN)≈0.And(5)PPZ(AlmGa1-mN)=2(a(0)-a(m)a(m))×(e31(m)-e33(m)c13(m)c33(m)+Psp(m)we×-Psp(0)c13(m)c33(m))C/m2PSP(AlmGa1-mN)=-0.052m-0.029c/m2PSP(GaN)=-0.029c/m2.
The piezoelectric polarization induced charge density depends upon Al mole fraction which is given as
(6)PPZ(AlmGa1-mN)={(2(a(0)-a(m))a(m))(e31(m)-e33(m)C13(m)C33(m))wwwwfor0≤m≤0.38(2(2.33-3.5m)a(m)/(a(0)-a(m)))×(e31(m)-e33(m)C13(m)C33(m))wwwwfor0.38≤m≤0.670,wwwwfor0.67≤m≤1,
where a(m) and a(0) are lattice constants, e31(m) and e33(m) are piezoelectric constants, and c13(m) and c33(m) 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.


## 2.2. Current-Voltage Characteristics

The drain-source current in the channel is obtained from the current density equation and is given [13] as
(7)Ids(m,x)=zqμ(x,m)(ns(m,x)dvc(x)dx+KBTqdns(m,x)dx),
where z is the gate width, KB is Boltzmann constant, T is temperature, and μ(x) is the field dependent electron mobility and is given [2] as
(8)μ(m)=μ0(m)1+((μ0(m)Ec-Vsat⁡)/EcVsat⁡)(dvcx/dx).
By substituting (1) and (8) in (7) and on integrating using boundary conditions
(9)vc(x)|x=0=IdsRs,vc(x)|x=L=Vds-Ids(Rs+Rd),
where Rs and Rd are the parasitic source and drain resistances, respectively.The drain to source current in the channel is given as(10)Ids=-α1+α12-4α2α32α2,
where
(11)α1=E1(2Rs+Rd)-E2Rd2(Rd+2Rs),α2=-[L+E1Vds+E2(Vgs-Vth-KBTq)hhhhhh×(2Rs+Rd)-E2Vds(Rs+Rd)(Vgs-Vth-KBTq)],α3=E2(Vgs-Vth-KBTq)Vds-E22Vds2.
The drain saturation voltage Vdsat is obtained as
(12)Vdsat=-δ1+δ12-4δ2δ32δ2,
where
(13)δ1=μ0Zε(m)Ecd[α1μ0zε(m)Ecd+E1-E2(Rs+Rd)]-E2E,δ2=μ0Zε(m)Ecd[-2Qα1μ0zε(m)Ecd+E1Qμ0Zε(m)Ecdw-E2Q(Rs+Rd)+L+E2Q(2Rs+Rd)-2Qα1μ0zε(m)Ecd]+E2Q,δ3=Qμ0Zε(m)Ecd×[α1μ0zε(m)E2Qd-L-E2(2RS+Rd)d]
with
(14)Q=Vgs-Vth-KBTq.
The detailed analysis for (10) and (12) is reported in [6].


## 2.3. Capacitance-Voltage Characteristics

The equation for channel current in terms of position dependent sheet carrier concentration and electron velocity is given as [10]
(15)Ids=Zqns(x)vd(x),
where Z is the gate width, q is the electronic charge, ns(x) is the electron concentration in the channel, and Vd(x) is the electron drift velocity and is given as
(16)Vd(x)=μ0E(x)1+(1/E3)E(x),
where
(17)E3=EcVsat⁡μ0Ec-Vsat⁡,E(x)=dVc(x)dx
and μ0 is the low field mobilityTotal channel charge is given as(18)Q=∫0tIdsdt=∫0LIdsdxvd(x).
Rearranging (15) and using (1), we get
(19)Idsvd(x)=Zε(m)D{Vgs-Vth(m)-Vc(x)}.
Integrating (19) from 0 to L,
(20)∫0LIdsvd(x)dx=Zε(m)D∫0L{Vgs-Vth(m)-Vc(x)}dx
using [10]
(21)dx=Zqnsμ0-(Ids/E3)IdsdVc(x).
By substituting (21) into (20) and using boundary conditions in (9), the total channel charge is obtained as
(22)Q=-A13Ids[B-{Vds-IdsR}]3-[B-IdsRs]3-A2[B{Vds-IdsR}-BIdsRs-12{Vds-1dsR}2hhhh+12Ids2Rs2],
where
(23)A1=zε(m)2zμ0D2,B=Vgs-Vth,R=Rs+Rd,A2=Zε(m)E3D.


### 2.3.1. Gate-Drain Capacitance

It is also called feedback capacitance. 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:(24)Cgd=∂Q∂Vds=-A13Ids[3{B-(Vds-1dsR)}2{1-gdR}]+[{B-(Vds-IdsR)}]3[A13Ids2gd]+A1Ids·(B-IdsRs)2(-gdRs)+(B-IdsRs)3(-A13Ids2gd)-A2[B(1-gdR)-BRsgd-(Vds-IdsR)(1-gdR)hhhhhh+IdsRs2gd].


### 2.3.2. Gate-Source Capacitance

The gate-source capacitance (Cgs) is defined as the change in total charge with the change in gate voltage. It is mainly due to the 2DEG in the normal operating region and is given as
(25)Cgs=∂Q∂Vgs=-A13Ids[3{B-(Vds-IdsR)}2{1-gmR}]+{B-(Vds-IdsR)}3A13Ids2gm+A1Ids·(B-IdsRs)2(1-gmRs)+(B-IdsRs)3(-A13Ids2gm)-A2[IdsRs2gm{B(-gmR)+(Vds-IdsR)}-{BgmRs+IdsRs}-A2w-(Vds-IdsR)(-gmR)+IdsRs2gm].


## 2.3.1. Gate-Drain Capacitance

It is also called feedback capacitance. 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:(24)Cgd=∂Q∂Vds=-A13Ids[3{B-(Vds-1dsR)}2{1-gdR}]+[{B-(Vds-IdsR)}]3[A13Ids2gd]+A1Ids·(B-IdsRs)2(-gdRs)+(B-IdsRs)3(-A13Ids2gd)-A2[B(1-gdR)-BRsgd-(Vds-IdsR)(1-gdR)hhhhhh+IdsRs2gd].


## 2.3.2. Gate-Source Capacitance

The gate-source capacitance (Cgs) is defined as the change in total charge with the change in gate voltage. It is mainly due to the 2DEG in the normal operating region and is given as
(25)Cgs=∂Q∂Vgs=-A13Ids[3{B-(Vds-IdsR)}2{1-gmR}]+{B-(Vds-IdsR)}3A13Ids2gm+A1Ids·(B-IdsRs)2(1-gmRs)+(B-IdsRs)3(-A13Ids2gm)-A2[IdsRs2gm{B(-gmR)+(Vds-IdsR)}-{BgmRs+IdsRs}-A2w-(Vds-IdsR)(-gmR)+IdsRs2gm].


## 3. Small-Signal Parameters

### 3.1. Drain/Output Conductance

It is an important microwave parameter that determines the maximum voltage gain attainable from a device. The drain conductance of the AlGaN/GaN pHEMT is evaluated as(26)gd(m)=∂Ids(m)∂Vds|Vgs,gd(m)=12α1[-E1+E2(Rd+Rs)+12α12-4α2α3=12α1ew×{2α2(-E1+E2(Rd+Rs))-4α1E2(Q-Vds)}12α12-4α2α3].


### 3.2. Transconductance

It is the most important parameter for optimization of FET high frequency behaviour. The major part of the gain mechanism is embodied in the active channel transconductance, which is evaluated as(27)gm(m)=∂Ids(m)∂Vgs|Vds,gm(m)=12α1[-E2(2Rs+Rd)+12α12-4α2α3=12α1ew×{2α2(-E2(2Rs+Rd))-4α1E2Vds}12α12-4α2α3].


### 3.3. Cut-Off Frequency (ft)

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 (bothCgs and Cgd); that is,
(28)ft=gm2π(Cgs+Cgd).
And by substituting the value of gm, which has already been derived in (27) and Cgd and Cgs from (24) and (25), respectively, the cut-off frequency can be evaluated. The cut-off frequency ft determines the ultimate speed of a switching device.


### 3.4. 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(29)Tt(m)=gm2πft(m).
By substituting (28) in (29), transit time can be obtained.


## 3.1. Drain/Output Conductance

It is an important microwave parameter that determines the maximum voltage gain attainable from a device. The drain conductance of the AlGaN/GaN pHEMT is evaluated as(26)gd(m)=∂Ids(m)∂Vds|Vgs,gd(m)=12α1[-E1+E2(Rd+Rs)+12α12-4α2α3=12α1ew×{2α2(-E1+E2(Rd+Rs))-4α1E2(Q-Vds)}12α12-4α2α3].


## 3.2. Transconductance

It is the most important parameter for optimization of FET high frequency behaviour. The major part of the gain mechanism is embodied in the active channel transconductance, which is evaluated as(27)gm(m)=∂Ids(m)∂Vgs|Vds,gm(m)=12α1[-E2(2Rs+Rd)+12α12-4α2α3=12α1ew×{2α2(-E2(2Rs+Rd))-4α1E2Vds}12α12-4α2α3].


## 3.3. Cut-Off Frequency (ft)

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 (bothCgs and Cgd); that is,
(28)ft=gm2π(Cgs+Cgd).
And by substituting the value of gm, which has already been derived in (27) and Cgd and Cgs from (24) and (25), respectively, the cut-off frequency can be evaluated. The cut-off frequency ft determines the ultimate speed of a switching device.


## 3.4. 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(29)Tt(m)=gm2πft(m).
By substituting (28) in (29), transit time can be obtained.


## 4. 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.The basic efficiency is expressed as(30)η=PoutPdc×100%.
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
(31)PAE=Pout-PinPdc=PoutPdc(1-PinPout)=PoutIdsVds(1-1G)×100%,
where G is the power gain and is given as
(32)G=PoutPin.


## 5. 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 valuesig2¯ and id2¯ are obtained in this section.


### 5.1. Drain Noise Source

The drain voltage fluctuation due to an elementary noise produced atx is obtained as
(33)Δvd12¯=4kT0ΔfIdcosh⁡2(πL22d)((WP)2+δPW)dvx,
where Δf is the frequency range, δ is the noise temperature parameter, and T0 is the lattice temperature.The mean square drain noise voltage is given as(34)vd12¯=16kTΔfK42|Voff⁡|qzvsat⁡(P1-K2)2cosh⁡2(πL22d)×((1P2)(s3-p33)+δpln⁡(SP)).
Open circuit drain voltage fluctuation due to diffusion noise can be given as
(35)vd22-=64qDΔfId(d+b)3π5ε5z2vsat⁡3b5sin2⁡(πb2(d+b))×(3+exp⁡(πL2d+b)-4exp⁡(πL22(d+b))+πL2(d+b)),
where b is the effective channel thickness and is given as
(36)b=εdqdQ{Ef(L1)}=εq(K2K4P1-K2+K3).


### 5.2. Gate Circuit Noise

The noise current flowing may be obtained byΔlg22¯=Δvd22¯/rd2, where rd is the drain resistance.The expression for short circuit gate noise current can be obtained as(37)Δlg22¯=(16q3Δfw2(d+b)3DL12k02π5ε2b2vsat⁡3k44rd2)sin2⁡(πb2(d+b))×{3+exp⁡(πL2(d+b))-4exp⁡(πL22(d+b))+(πL2(d+b))},
where
(38)K0=K+L22L1(P1-K2)2.


### 5.3. Noise Coefficients

The correlation between the drain noise source and the gate noise source is given as(39)Lg1*Ld1¯=-jw2kT0Δfqz|Voff⁡|L1K42Id2rd2cosh⁡(πL22d)(s0+s1+s2+s3).
The details of (39) are reported in [16].The drain noise coefficient associated with the drain noise current is given as(40)P=Vd2¯4KT0Δfgm=Vd2¯4KT0Δfgmrd2=ld12¯+ld22¯4KT0Δfgm=ld12¯+ld22¯4KT0Δfgmrd2=P1+P2.
The gate noise coefficient associated with the gate noise current is given as
(41)R=lg2¯gm4KT0Δfw2Cgs2=(lg12-+lg22-)gm4KT0Δfw2Cgs2=R1+R2,
where Vd2¯ is the equivalent noise voltage at the drain region and ig2¯ is the equivalent noise induced gate current.The noise coefficientsP1 and P2 and R1 and R2 are of the same form as P and R, 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(42)C=lg*ld¯lg2ld2j=C1+C1.
The correlation coefficient in the linear region can be given in terms of P and R as
(43)C1=C11·P1R1PR.
The correlation coefficient in the saturation region can be given in terms of P and R as
(44)C2=C22·P2R2PR.
The correlation coefficient between lg1 and ld1 can be calculated as(45)C11=lg1*ld1¯lg12ld12j.
There is full correlation between id2 and ig2 with a capacitive phase shift of 90°. Therefore,
(46)C22=lg2*ld2¯lg22ld22j=1.
Therefore, there is no correlation between id1 and ig2 or id2 and ig1.


### 5.4. Noise Figure

The minimum noise figure in terms of source resistance and gate resistance can be calculated as(47)Fmin⁡=1+2gn(Rc+Rc2+rngn),
where
(48)gn=noiseconductance=w2Cgs2gm(P+R-2CPR)=Kgw2Cgs2gm,rn=noiseresistance=Rs+Rg+1+w2Cgs2Ri2gm·PR(1-C2)P+R-2CPRRs+Rg+Kr(1+w2Cgs2Ri2gm),Zc=correlationimpedance=Rc+jXc=Rs+Rg+(Ri-jwCgs)·P-CPRP+R-2CPR=Rs+Rg+Kc(Ri-jwCgs),
where Kg, Kr, and Kc are the functions of P, R, and C and are given as(49)Kg=P+R-2CPR,Kr=PR(1-C2)P+R-2CPR,Kc=P-CPRP+R-2CPR.


### 5.5. Noise Temperature

The minimum noise temperature can be obtained as(50)Tmin⁡=2Tgn(Rc+Zc),
where T is absolute temperature and Rc is the correlation resistance.


## 5.1. Drain Noise Source

The drain voltage fluctuation due to an elementary noise produced atx is obtained as
(33)Δvd12¯=4kT0ΔfIdcosh⁡2(πL22d)((WP)2+δPW)dvx,
where Δf is the frequency range, δ is the noise temperature parameter, and T0 is the lattice temperature.The mean square drain noise voltage is given as(34)vd12¯=16kTΔfK42|Voff⁡|qzvsat⁡(P1-K2)2cosh⁡2(πL22d)×((1P2)(s3-p33)+δpln⁡(SP)).
Open circuit drain voltage fluctuation due to diffusion noise can be given as
(35)vd22-=64qDΔfId(d+b)3π5ε5z2vsat⁡3b5sin2⁡(πb2(d+b))×(3+exp⁡(πL2d+b)-4exp⁡(πL22(d+b))+πL2(d+b)),
where b is the effective channel thickness and is given as
(36)b=εdqdQ{Ef(L1)}=εq(K2K4P1-K2+K3).


## 5.2. Gate Circuit Noise

The noise current flowing may be obtained byΔlg22¯=Δvd22¯/rd2, where rd is the drain resistance.The expression for short circuit gate noise current can be obtained as(37)Δlg22¯=(16q3Δfw2(d+b)3DL12k02π5ε2b2vsat⁡3k44rd2)sin2⁡(πb2(d+b))×{3+exp⁡(πL2(d+b))-4exp⁡(πL22(d+b))+(πL2(d+b))},
where
(38)K0=K+L22L1(P1-K2)2.


## 5.3. Noise Coefficients

The correlation between the drain noise source and the gate noise source is given as(39)Lg1*Ld1¯=-jw2kT0Δfqz|Voff⁡|L1K42Id2rd2cosh⁡(πL22d)(s0+s1+s2+s3).
The details of (39) are reported in [16].The drain noise coefficient associated with the drain noise current is given as(40)P=Vd2¯4KT0Δfgm=Vd2¯4KT0Δfgmrd2=ld12¯+ld22¯4KT0Δfgm=ld12¯+ld22¯4KT0Δfgmrd2=P1+P2.
The gate noise coefficient associated with the gate noise current is given as
(41)R=lg2¯gm4KT0Δfw2Cgs2=(lg12-+lg22-)gm4KT0Δfw2Cgs2=R1+R2,
where Vd2¯ is the equivalent noise voltage at the drain region and ig2¯ is the equivalent noise induced gate current.The noise coefficientsP1 and P2 and R1 and R2 are of the same form as P and R, 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(42)C=lg*ld¯lg2ld2j=C1+C1.
The correlation coefficient in the linear region can be given in terms of P and R as
(43)C1=C11·P1R1PR.
The correlation coefficient in the saturation region can be given in terms of P and R as
(44)C2=C22·P2R2PR.
The correlation coefficient between lg1 and ld1 can be calculated as(45)C11=lg1*ld1¯lg12ld12j.
There is full correlation between id2 and ig2 with a capacitive phase shift of 90°. Therefore,
(46)C22=lg2*ld2¯lg22ld22j=1.
Therefore, there is no correlation between id1 and ig2 or id2 and ig1.


## 5.4. Noise Figure

The minimum noise figure in terms of source resistance and gate resistance can be calculated as(47)Fmin⁡=1+2gn(Rc+Rc2+rngn),
where
(48)gn=noiseconductance=w2Cgs2gm(P+R-2CPR)=Kgw2Cgs2gm,rn=noiseresistance=Rs+Rg+1+w2Cgs2Ri2gm·PR(1-C2)P+R-2CPRRs+Rg+Kr(1+w2Cgs2Ri2gm),Zc=correlationimpedance=Rc+jXc=Rs+Rg+(Ri-jwCgs)·P-CPRP+R-2CPR=Rs+Rg+Kc(Ri-jwCgs),
where Kg, Kr, and Kc are the functions of P, R, and C and are given as(49)Kg=P+R-2CPR,Kr=PR(1-C2)P+R-2CPR,Kc=P-CPRP+R-2CPR.


## 5.5. Noise Temperature

The minimum noise temperature can be obtained as(50)Tmin⁡=2Tgn(Rc+Zc),
where T is absolute temperature and Rc is the correlation resistance.


## 6. 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.Figure2 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].Figure 2
I-V characteristics of AlGaN/GaN MODFET.The variation of transconductance with gate voltage is depicted in Figure3. 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 Vgs increases, the pHEMT initially operates in the saturation region. As Vgs continues to increase, the transistor enters in the linear region of operation and the transconductance decreases. The present results are in close agreement with the experimental results [17].Figure 3
Variation of transconductance with gate voltage.The plot of output/drain conductance for different gate voltages is shown in Figure4. As shown in the figure, the output conductance increases gradually with the increases in gate voltage for different values of drain bias. Starting from left to right, the transistor starts in saturation and slowly approaches the linear region as gate bias increases.Figure 4
Variation of drain conductance with gate voltage.The variation of transit time with gate voltage is depicted in Figure5. The figure shows the effect of Al mole fraction composition on transit time variation with gate bias. Minimum value of transit time is achieved at the gate voltage corresponding to the peak transconductance. Increase in Al mole content shifts the gate voltage to a minimum transit time.Figure 5
Variation of transit time with gate voltage for different mole fraction.Figure6 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 6
Variation of cut-off frequency with drain current.Figure7 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 7
Dependence ofCgs on Vds.Figure8 shows the variation of Cgs with Vgs for different values of Vds. It is evident from the figure that Cgs decreases with decrease in Vgs and increases with increase in Vgs.Figure 8
Variation ofCgs with gate voltage.Figure9 shows the variation of Cgd with Vdg for different values of Vgs. It can be seen from the figure that there is sharp increase in Cgd at Vdg of −3 V. The calculated results are in good agreement with experimental results [13].Figure 9
Variation ofCgd with Vdg.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].Figure 10
Variation of PAE, gain, and output power with input power.Figures11, 12, and 13 show the dependence of noise coefficients (P, R, and C) on drain current. The correlation coefficient C shows the symmetric behaviour. It is almost constant for all values of drain currents. It decreases with shorter gate lengths. The gate noise coefficient R is very high at low drain currents due to decrease in gate to source capacitance. It increases for shorter gate lengths. The drain noise coefficient P 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].Figure 11
Dependence of noise correlation coefficientC on drain current.Figure 12
Variation of gate noise coefficientR with drain current.Figure 13
Variation of drain noise coefficientP with drain current.Figure14 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.Figure 14
Variation of noise temperature with drain current.The variation of minimum noise figure with frequency is presented in Figure15. 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].Figure 15
Dependence of minimum noise figure on frequency.


## 7. 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 withVgs and Vds. 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 (P), gate noise coefficient (R), and correlation coefficient (C) and noise figure are also evaluated. The comparisons between the power performance and the model simulation validate the approximations of the present capacitance and noise model.



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*Source: 197937-2014-09-10.xml*