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НАУЧНО-ТЕХНИЧЕСКИИ ВЕСТНИК ИНФОРМАЦИОННЫХ ТЕХНОЛОГИИ, МЕХАНИКИ И ОПТИКИ январь-февраль 2022 Том 22 № 1 http://ntv.ifmo.ru/
SCIENTIFIC AND TECHNICAL JOURNAL OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS January-February 2022 Vol. 22 No 1 http://ntv.ifmo.ru/en/
ISSN 2226-1494 (print) ISSN 2500-0373 (online)
ИНФОРМАЦИОННЫХ ТЕХНОЛОГИЙ. МЕХАНИКИ И йПТИКИ
doi: 10.17586/2226-1494-2022-22-1-147-154
A new analytical model of drain current and small signal parameters for AlGaN-GaN high-electron-mobility transistors Azzeddine Farti1H, Abdelkader Touhami2
!>2 Hassan II University of Casablanca, Casablanca, 20190, Morocco
1 azzeddine.farti-etu@etu.univh2c.mas, https://orcid.org/0000-0002-8694-1471
2 abdelkadertouhami2016@gmail.com, https://orcid.org/0000-0001-8582-1884
Abstract
The paper proposes a new analytical model of the drain current in AlGaN-GaN high-electron-mobility transistors (HEMT) on the basis of a polynomial expression for the Fermi level as a function of the concentration of charge carriers. The study investigated the influence of parasitic resistances (source and drain sides), high-speed saturation, the amount of aluminum in the AlGaN barrier, and low field mobility. To isolate the output characteristics, cut-off frequency and steepness, the parameters of the hyper frequency signal were developed. Comparison of analytical calculations with experimental measurements confirmed the validity of the proposed model. Keywords
AlGaN-GaN, HEMTs, 2-DEG two-dimensional electron gas, current-voltage characteristics, cut-off frequencies,
Transconductance
Acknowledgements
This work carried out at the Faculty of Sciences Ain Chok Km 8, Hassan II University, Casablanca, Morocco. For citation: Farti A., Touhami A. A new analytical model of drain current and small signal parameters for AlGaN-GaN high-electron-mobility transistors. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2022, vol. 22, no. 1, pp. 147-154. doi: 10.17586/2226-1494-2022-22-1-147-154
УДК 621.382.3
Новая аналитическая модель тока стока и параметров малых сигналов AlGaN-GaN транзисторов с высокой подвижностью электронов
Аззеддин Фарти1Н, Абделькадер Тухами2
!>2 Университет Хасана II, Факультет наук Касабланки, Касабланка, 20190, Марокко
1 azzeddine.farti-etu@etu.univh2c.mas, https://orcid.org/0000-0002-8694-1471
2 abdelkadertouhami2016@gmail.com, https://orcid.org/0000-0001-8582-1884
Аннотация
Предложена новая аналитическая модель тока стока в устройствах AlGaN-GaN транзисторов с высокой подвижностью электронов (HEMT) на основе полиномиального выражения для уровня Ферми как функции концентрации носителей заряда. В ходе исследования изучено влияние паразитных сопротивлений (стороны истока и стока), высокоскоростного насыщения, количества алюминия в барьере AlGaN и низкой подвижности поля. Для выделения выходных характеристик, частоты среза и крутизны разработаны параметры гиперчастотного сигнала. Сравнение аналитических расчетов с экспериментальными измерениями подтвердило справедливость предложенной модели. Ключевые слова
AlGaN-GaN, транзистор с высокой подвижностью электронов, HEMT, двумерный электронный газ,
вольтамперная характеристика, частота среза, крутизна характеристики
Благодарности
Работа выполнена на факультете наук Айн Чок Km 8, Университет Хасана II, Касабланка, Марокко.
© Farti A., Touhami A., 2022
Ссылка для цитирования: Фарти А., Тухами А. Новая аналитическая модель тока стока и параметров малых сигналов AlGaN-GaN транзисторов с высокой подвижностью электронов // Научно-технический вестник информационных технологий, механики и оптики. 2022. Т. 22, № 1. С. 147-154 (на англ. яз.). doi: 10.17586/2226-1494-2022-22-1-147-154
Introduction
High-electron-mobility transistors (HEMT) based on the AlGaN-GaN hetero-structure have rapidly emerged in the fabrication of very high-speed integrated circuits [1]. These transistors have taken an important place in the realization of high-power, high-voltage, and high-temperature devices because of their very tall switching speed, poor power consumption and relatively easy manufacturing technology.
Due to their considerable gap energy and significant discontinuity of conduction bands between the AlGaN and GaN layers, HEMT transistors can create a two-dimensional electron gas (2-DEG), with a concentration in the order of 1013 cm-2 under the spontaneous polarization effect and the piezoelectric effect [2].
To develop a reliable model for HEMT transistors, an accurate estimate of the 2-DEG density at the heterointerface is significant. A number of charge control models for HEMTs and several theoretical models of drain current in HEMT transistors have been developed to characterize the concentration of the two-dimensional electron's gas (2-DEG) at the AlGaN-GaN interface [2-6].
The previous work on theoretical analysis of HEMT AlGaN-GaN [5] has been done by solving simultaneously the Schrodinger and Poisson equations. According to the assumption's authors, the polarization in nitride alloys interpolates linearly [7]; meanwhile, Fiorentini et al [8] have argued that there is a significant variety in the theoretical results when nonlinear polarization is applied instead of linear polarization. This can be explained by the fact that nonlinearity modifies the interface charge densities induced by the polarization. Analyzing the models involving polarization nonlinearity [9-12], we found that theoretical models are not suitable with the experimental result (the current-voltage characteristic and the transconductance).
In the paper, we present a new model of charge control using the 2-DEG channel current equation developed by Rashmi et al [5], and considering the nonlinearity of the
polarization which will produce a good optimization with the experimental result. In this model we also introduce the Fermi level Ep{T, m) in its simple polynomial form as a function of the electron gas density ns(T, m).
Based on the proposed analytical model, we will extract the current-voltage characteristics and describe the variations of the channel transconductance of the device. To highlight the performance and operating limits of HEMT transistors at microwave frequencies, first, we study the variation of cut-off frequencies fT and the transcanductance gm as a function of the drain current at fixed drain voltage, and secondly, we analyze the variation of the cut-off frequency as a function of the bias voltages Vds and Vgs.
Model formulation
Charge-control model
The structure of HEMT mainly consists of three different materials: substrate, material with a wide bandgap, and material with a low bandgap. The junction of these two materials leads to the formation of two-dimensional electron gas (2-DEG) at the interface. It can be modulated by the gate-source voltage, while the formation of the electron gas is performed between the spacer and the small gap layer (Fig. 1). The sheet carrier concentration of the two-dimensional electron gas (2-DEG) constituted at the AlGaN-GaN heterojunction is given by [9]:
8 (m)
n(T, m) = — (VgS - Vth(T, m) - ЕДТ, m)),
(1)
where T is the temperature; m is the Al mole fraction in AlGaN-GaN; q is the electron charge; e(m) is the dielectric constant of AlmGa1-mN; d is the summation of the doped AlGaN layer thickness (dd) and the undoped AlGaN spacer layer thickness (d). Vth(m) is the threshold voltage, Ep(T, m) is the position of the Fermi level which is expressed as a function of ns(T, m) and it is given by the form of a simple polynomial [13].
a L
Ad 1
S ourse4) f Gate ^ f Drain ^
N-AlGaN
У■ UID-AlGaN Spacer
С X UID-GaN - 2-DEG
Substrate
Fig. 1. HEMT transistor AlGaN-GaN in cross-section (a) and the band diagram of the AlGaN-GaN structure (b)
Ef(T, m) = Kl(T) + K2(TMT m) + K3(T)ns(T, m), (2)
where Kb K2 and K3 are three different parameters depending on the temperature T.
Where L is the Gate length, x present the position along the transistor channel. Vb presents the Schottky-barrier height, ^ and are work functions of AlGaN and GaN, respectively. x1 (X2) is electron affinity of AlGaN (GaN). Evl, Ec and EV show the vacuum level, the energy of the conduction band, the energy of the valence band, respectively.
By introducing the equation (2) into (1), we obtained a quadratic equation, to demonstrate the concentration of 2-DEG charge carriers at the heterojunction [14].
ns(T, m) = (3)
¡-K2(T) + imT) + 4K4(T, m)(Vgs - Vth(T, 7))f
where
2Ka(T, m)
K4(T, m) = K3(T) +
qd e(m)
The threshold voltage Vth(T, m) depends on the aluminum mole fraction of MOSFET, and its expression given by [15].
qNjdj oP2(m)
Vth(T, m) = ^(m) - AEc(T, m) - - d,
2z{m) E{m)
where 9(m), AEC(m) are the Schottky barrier height and conduction band discontinuity between GaN and AlGaN, respectively. Nd is the doping density of the Al-Ga-N layer and oPZ(m) is the charge density induced by the total polarization, can be written as follows [8]:
°PZ(m) = |PPP(m) + PAmGa1-mN(m) - PAlmGa1-mN(0)1,
where PPP(m) is the piezo-electric polarization with its expression and it varies as a function of the value of m.
PAlmGa1-mN(m) and PAlmGa1-mN(0) are the spontaneous
polarizations of AlmGa1-mN and GaN, respectively [8].
nPz
Ppp(m) =
For 0 < m < 0.38
PAlmGai_mN(m)
PAmGa1-mN(2.33-3.5m) For 0.38 < m < 0.67, 0 For 0.67 < m < 1
The expressions of spontaneous and piezo-electric polarizations are given in Table 1 [8].
Current-voltage characteristics (Ids - Vds)
By fixing the value of voltage Vds superior of the threshold voltage, the electron movement at the AlGaN/ GaN interface is ensured by the application of the source-drain voltage Vds superior to zero. The generated drain current Ids is given by the following equation [5].
i ,t ^ 7 , J st ^TO , KBTdns(T, m)\
Ids(T, m, x) = Zq^(x)ns(T, m)-+----, (4)
\ dx q dx I
where Z, VC(x) and KB are the gate width, the channel potential at position x, and Boltzmann's constant, respectively. ^(x) is the field dependent of the mobility given by:
Kx) = "-—-——, (5)
1
(\ioEc -vsat)dVc(x)
EcvSi
dx
where EC is the saturation electric field, ^ is the low-field mobility, and vsat is the saturation drift velocity.
Depending on the variation of the bias voltage Vds, the HEMT transistor can be function under two different regimes (linear regime and saturation regime). The linear region (Vds < Vda)
To describe the variation of potential at each position x in the 2-DEG channel, the voltage Vgs is replaced by Vgs - VC(x) in the expression (3) of sheet carrier concentration of two-dimensional electron gas formed at the AlGaN/GaN heterojunction.
ns(T, m, x) = fe^ + (6)
+ V^KT) + AUT, m)iVgs - Vth(T, m) - V<jx) - ^(7))\2 IK^T, m) I .
Using the following approximation [16] in equation (6) we get:
4K4(T, m)VC(x) << << Kl(T) + 4K4(T, m)(VgS - Vth(T, m) - Ki(T)).
After using the above given approximation in equation (6), a simple expression of the concentration ns(T, m) have the following form:
ns(T, m) = (7)
-K2(T) + iKMT) + 4K4(T, m)(Vgs - Vth{T, m) - UT))\2 2KlT,m) r
Table 1. Parameters of the AlmGa1-mN/GaN heterostructure
Description Parameter Expression
Spontaneous polarization of AlGaN P^Ga^m C/m2 -0.09m - 0.034(1 - m) + 0.019m(1 - m)
Piezo-electric polarization of AlGaN PAÎmGa i-mN(m X C/m2 mPArn[8b(m)] + (1 - m)Pc;aN[86(m)]
Piezo-electric polarization of AlN PAiN(m), C/m2 -1.8088b(m) + 5.6248j;(m)
Piezo-electric polarization of GaN PGPN(m), C/m2 -0.9188b(m) + 9.5418b(m)
The basal strain 86(m) (a(0) - a(m))/a(m)
The lattice constant a(m), nm 0.031986 - 0.000891m
Consequently, the drain current from equation (4) will be expressed by:
dVdx)\
Ids(T m, x) = Zq^(x) ns(T, m)
dx I
(8)
We substitute the equations (5) and (7) in the drain current equation (8), and a new equation will appear which describes the drain current at each position x of the channel:
¡ds^ m x) = Zq^0x x I~K,(T) + im.T) + 4Um)(Vgs - Vth(m) - x
" \ 2^4(7» I X
dVdx)
* (9)
' + (^oEc -vSai)dVc{x)\ ECVSat dx I
Fig. 2 shows the equivalent electrical circuit diagram of the high-electron-mobility transistor HEMT (AlGaN-GaN), where Rs and Rd are the parasitic resistances of the source and drain, respectively. Rc represents the resistance of the 2-DEG channel.
From the schema in Fig. 2, the channel potential for x = 0 and x = L will be given by:
VC(x)|x=0 = ¡d^
Vc(x)|x=L = Vds - (¡dsRd).
With the above given boundary conditions, the various steps of integration of the equation (9) along the transistor channel will be written as follows:
LI (tipEc-vsat)dFc(x))t
¡ds) 1 +-----— dx =
0 \ Ecvsat dx I
L
= Zqns(T, m)^} j^^^Jdx, 0 \ dx '
Ids [x]L + (^C~V'aVc(x)]L = Zqns(T, m)^o[Vc(x)]L, \ Ecvsat )
-йН^к + Rd)) +
ECVSc
(10)
((HoEc - v.at)
L + -'~Vds + Zqns(T, m)^R + Rd))-
ECVsat
- Zqns(T, m)^o Vds = 0.
From equation (10), by replacing ns(T, m) with its value, the first term is multiplied by ¡ds into ctj, the second
0 L
Fig. 2. Equivalent electrical schema of the HEMT
term is multiplied by ¡ds into a2 and the last term that does not contain ¡ds by a3, we can obtain:
/(jioEc -vjai)
a> = -
~(Rs + Rd)
EcVsat /
a2 = L + Zq^R + Rd)x
-K2(T) + ШТ) + 4UT)(Vgs - Vth(T, m) - ВД) \2
2K4(T) (Цо^с - vsai)
Vd
ds
(11) (12)
EcVsat a3 = -Zq д0
-K2(T) + УВД + 4ЩТ, m)(Vgs - Vth(T, m) - ^(7))V2
2ВД m) J
x Vds-
The expression of the drain current Ids in linear regime will be given by the root of equation (10) and can be written in the following form:
Ids =
-a2 + Val - 4<xi<x3 2ai
The saturation region (Vds > Vdsat)
The application of an electric field in a semiconductor leads to free charge carriers (electrons and holes) to acquire a velocity of displacement u proportional to the applied field. The following relationship describes this velocity:
и = ^(x)E(x) =
Ho
dVdx)
1 | (ЦоEc-vsa^dVc{x)\ dx EcVsat dx I
At high electric field values, the velocity of the carriers is saturated at the value usat. This fact has allowed us to replace the velocity u by usat in expression (9) of the drain current lds\
Idsat= Zqusc
(13)
-K2(T) + -4ШТ) + 4ЩТ, m)(Vgs - Vth(T, m) -
2ВД m) J
In the same way, we replace the voltage Vds by the saturation voltage Vdsat in the equations (11) and (12).
In this case, the expression of the drain current in the saturation regime can be written in the form:
Idsat
-ß2 + Vß22-4a1ß3 2a!
With,
and
= . + (HOEç-Vsat)
ß2 = ö2 + „ Vdsab
Ecvsat
ß3 = 03^
(14)
(15)
(16)
+
X
x
x
x
2
x
where
where:
52 = L + Zqvo(Rs + Rd) x /-ВД + + 4ВД - Vth{T, m) -K,{T))V
\ 2FUT, m)
5з = -Zq^o x
ЩТ) + УВД + 4КЛ(Т, m)(Vgs - Vth(T, m) - К,(Т)У2 \ 2K4(T, m)
To determine the expression of the saturation voltage Vdsat, we equalize the equations (13) and (14) of the drain current Ids:
-P2 + Vp|-4aip3 „
2d!
" = ZqVsat :
x i-K2(T) + iKjjT) + 4K4(T, m)(Vgs - Vth{T, m) - K,(T)) X \ 2^(7»
We put:
S1 = 2a1Zqvsat X x I-K2(T) + V£I(7) + - Vth(T, m) - ^(7))\2
X \ 2^(7» /'
And equation (17) will be described as:
-P2 + Vp22 - 4aiP3 = Si. (18)
According to the expressions of (15) and (16), equation (18) is as follows:
„ L (HoEc-vsat)T Si = - S2 + "
EcVsc
-Vdsat) +
и , (ЦоEç-vmt)Tr Г
+ M 52 +---Vdsat - 4а153 Vdsat,
EcVsc
As a result, the expression of the saturation voltage Vdsat can be deduced and expressed by the formula:
_ -61(282 + 80 dSa' ' (VoEc-vsat) ^
25i-+ 4ai83
EcVsat
Small signal parameters
At this stage, for better modelling of the HEMT transistor, we need to determine and calculate its small signals parameters like the transconductance gm and the cut-off frequency fT, which may reflect the performance of this transistor. Transconductance gm
The transconductance is considered as one of the most important parameters in evaluating the performance of transistors for high-frequency applications. The transconductance gm of the AlmGa1-mN/GaN HEMT transistor is defined as [17].
gm
dV„
(19)
^ds =
-02 + V012 - 4aia3 2ai
(20)
From (19) and (20), we obtained the expression of the transconductance written by the equation:
1
2di
da2
2a,
da2
-4ai
8a3 8Vm
\ dVgs 2Val- 4aia3 I
Cut-off frequency fT
The characterization of the high-frequency performance of the HEMT transistor requires a detailed study of its cut-off frequency fT. This later determines the factor of the microwave operation of HEMT transistor components. The cut-off frequency can be calculated versus the transconductance gm as follows [10]:
fT = -
gmHo
Z\i0e(m) \dd + di + Ad)
(21)
2тiL~
where Ad presents the thickness of two-dimensional electron gas 2-DEG.
Results and discussion
To highlight the models of electrical and technological parameters (Ids, gm and fT) of the HEMT transistor, we made a comparison with experimental results [18]. The electrical, geometrical and technological parameters used in the proposed model are presented in Table 2.
The values of the constants K1, K2 and K3 used in the theoretical calculations for different values of the gate voltage Vgs with m = 0.15 are depicted in Table 3.
Fig. 3 shows the electrical characteristics of the HEMT transistor (Al0.15Ga0.85N/GaN) as compared to experimental measurements [18]. The length and the width gate of this transistor are equal to 1 ^m and 75 ^m, respectively. Fig. 3, a shows the variation of the drain current as a function of the drain voltage for different values of the gate voltage. For the calculation, we took a gate voltage sweep between -2 V and 1 V with a step of 1 V.
The results of the simulation indicate that there are two operating regimes. The first one is a linear regime, implying that the drain voltage increases, then the drain current Ids
Table 2. Different parameters used in the proposed model
Vds=cte
Parameter Description Value
Vth, V Threshold voltage -2.6
dd, nm Thickness of the doped layer 22
dj, nm Thickness of the undoped layer 3
Z, ^m Gate width 75
L, ^m Gate length 1
Vsab m/s Saturation velocity 1.19x105
Rs, n Parasitic source resistance 0.6
Rd, n Parasitic drain resistance 0.9
x
X
+
g
m
Table 3. Values of the parameters K1; K2 and K3 for different gate voltage values Vgs. They are calculated with the same method [19]
using the effective mass m* = 0.22m0 [7]
Gate voltage , Vgs, V K1, V K2 x 10-8, Vm K3 x 10-18, V-m2
1 -0.27750 1.68742 -4.23715
0 -0.27573 1.45776 -3.21042
-1 -0.27423 1.26515 -2.34936
-2 -0.27422 1.26274 -2.33856
also increases. In a saturated regime, the drain current keeps constant and independent to the drain voltage Vds. This can be explained by the pinching of the canal and the saturation of the electron velocity. These results coincide with the experimental measurements and prove the validity of the proposed model.
The curve in Fig. 3, b represents the dependence of the drain current vs. the gate voltage for a drain voltage set at the value 5 V. We concluded that the intersection of the asymptote of the curve with the axis of the voltages Vgs is equal to -2.6 V. This value is equivalent to the threshold voltage Vth of the HEMT transistor under study. In other words, the results of our model are in concordance with the experimental data and confirm the validity of our model.
Fig. 3, c displays the variation of the transconductance gm as a function of the gate voltage Vgs with the value of
the drain voltage Vds equal to 5 V. This transconductance gm increases with the voltage Vgs till it reaches a maximum gmMax (160 mS/mm) for the value of the gate voltage equal to -0.74 V. Consequently, the HEMT transistor achieves its optimum operating point. After this maximum, the transconductance decreases progressively, which can be attributed to the saturation of the carrier's velocity and the reduction of their mobility in the channel. This is caused by the increase in channel resistance. In addition, the thermal effects and presence of defects can be responsible for this drop transconductance.
Fig. 4, a shows the cut-off frequencies fT and the transcanductance gm as a function of the drain current ¡ds with gate width Z = 2 x 75 ^m and the fixed drain voltage Vds = 5 V. We observe that the maximum of the cut-off frequency is 9.66 GHz which corresponds to a drain current
0.52-
: 0.26 -
о
fl 'fi
П
0.00
« Experimental Data -Proposed Model V = r gs 1
V = Г gs 0
9 «
V = у gs -1
Vgs -2
Drain Voltage F&, V 0.20
Vth -2 0
Gate Voltage Vgs, V
&
0.10
■3
s о о
0.00
Gate Voltage Vgs, V
Fig. 3. Variation of the drain current lds as a function of: the drain voltage Vds for different gate voltage values Vgs (a) and gate voltage Vgs by fixing the drain voltage Vds at 5 V (b); variation of the transconductance gm as a function of gate voltage Vgs (c)
20
0 о
о
10
-Proposed Model of fT and gm
m Experimental Data of fT and gm
gm
У
9
Xе
fr
100 200 Ids, mA/mm with V^ = 5 V
300
Drain Current V
Fig. 4. Cut-off frequencies vs. drain current Ids at drain voltage Vds at 5 V (Z = 2x75 ^m) (a); dependence of the cut-off frequency fT
with polarization voltages Vgs and Vds (b)
of 250 mA/mm, where the maximum of transcanductance is located.
Fig. 4, b presents the dependence of the cut-off frequency fT on the gate and drain polarization voltages. It is obvious that the cut-off frequency rises with the gate voltage and decreases after reaching a maximum. Also, this cut-off frequency increases with the applied drain voltage. This is due to the impact of the gate voltage on the depth of the potential well and favors important diffusion of free electrons from the semiconductor donors.
Conclusion
In summary, we have studied a new analytical model of the drain current and small signal parameters of AlGaN-
GaN HEMTs devices and considered the nonlinear polarization effect using the 2-DEG channel current equation. With the help of the charge control model, a detailed investigation of different functioning regimes of the HEMT transistor has been conducted. In this work, we optimized the difference between the proposed model and the experimental data. This later extracted from the current-voltage characteristics evidence that this model is useful for the prediction of drain current. Basing on this model, we can calculate the values of the transconductance gm and the cut-off frequency fT. These results showed that the simulation model is satisfactory and consistent with the experimental measurements, evidencing the validity of our proposed model.
References
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Литература
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Authors
Azzeddine Farti — PhD, Associate Professor, Hassan II University of Casablanca, Casablanca, 20190, Morocco, https://orcid.org/0000-0002-8694-1471, azzeddine.farti-etu@etu.univh2c.ma
Abdelkader Touhami — D.Sc, Full Professor, Hassan II University of Casablanca, Casablanca, 20190, Morocco, gg 8158211000, https://orcid. org/0000-0001-8582-1884, abdelkadertouhami2016@gmail.com
Авторы
Фарти Аззеддин — PhD, доцент, Университет Хасана II, Факультет наук Касабланки, Касабланка, 20190, Марокко, https://orcid.org/0000-0002-8694-1471, azzeddine.farti-etu@etu.univh2c.ma Тухами Абделькадер — D.Sc, профессор, профессор, Университет Хасана II, Факультет наук Касабланки, Касабланка, 20190, Марокко, S3 8158211000, https://orcid.org/0000-0001-8582-1884, abdelkadertouhami2016@gmail.com
Received 08.09.2021
Approved after reviewing 03.12.2021
Accepted 28.01.2022
Статья поступила в редакцию 08.09.2021 Одобрена после рецензирования 03.12.2021 Принята к печати 28.01.2022
