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Titre: Compact mole fraction-dependent modeling of I-V and C-V characteristics in Al Ga N/GaN HEMTs
Auteur: Nawel Kermas

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Compact mole fraction-dependent
modeling of I-V and C-V characteristics in
Al $$_{x}$$ x Ga $$_{1-x}$$ 1 - x N/GaN
HEMTs
Nawel Kermas, Bouaza Djellouli, Driss
Bouguenna, Wondwosen Eshetu, Oana
Moldovan, et al.
Journal of Computational Electronics
ISSN 1569-8025
J Comput Electron
DOI 10.1007/s10825-017-1067-7

1 23

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1 23

Author's personal copy
J Comput Electron
DOI 10.1007/s10825-017-1067-7

Compact mole fraction-dependent modeling of I-V and C-V
characteristics in Al x Ga1−x N/GaN HEMTs
Nawel Kermas1 · Bouaza Djellouli1 · Driss Bouguenna2 · Wondwosen Eshetu3 ·
Oana Moldovan3 · Benjamin Iñiguez3

© Springer Science+Business Media, LLC 2017

Abstract In this paper we present a compact mole fractiondependent modeling of the I-V and C-V characteristics in
Alx Ga1−x N/GaN HEMTs using Atlas TCAD. The C-V characteristics of the Alx Ga1−x N/GaN HEMTs are obtained
using a charge conserving model. Based on this modeling,
we have developed an analytical model for the threshold
voltage, the carrier sheet density, the drain current, and the
capacitance. The model covers all the different operating
regimes of the device. The model includes the Al mole fraction of the barrier layer effects and was incorporated into our
previously developed charge based on I-V and C-V characteristics, and we also simulated the device transconductance
of Alx Ga1−x N/GaN HEMTs. The results of the modeled I-V
and C-V characteristics are in excellent agreement with the
simulated data obtained by the Atlas TCAD simulator, which
demonstrates the validity of the proposed model.
Keywords I-V and C-V characteristics · Transconductance ·
Alx Ga1−x N/GaN · HEMTs · Mole fraction · Threshold
voltage · Numerical simulation · Atlas TCAD simulator

B

Driss Bouguenna
dbouguenna.um@gmail.com

1

Laboratory of Modelization and Calculation Methods,
Department of Electronics, University of Saida, 20000 Saida,
Algeria

2

Laboratory of Materials, Applications and Environment,
Department of Technical Science ST, Faculty of Science and
Technology, University Mustapha Stambouli of Mascara,
29000 Mascara, Algeria

3

ETSE DEEEA, University Rovira i Virgili (URV), Avda.
Països Catalans 26, 43007 Tarragona, Spain

1 Introduction
Recently, high performance Alx Ga1−x N/GaN HEMT devices
have been successfully used for various applications [1,2].
This is motivated by their potential in commercial and military applications, e.g., in the area of telecommunication
systems, base stations market, as well as radar, W-CDMA
mobile-phone applications [3,4], high temperature electronics, high power solid state switching, and hard radiation
in space electronics. In fact, Alx Ga1−x N/GaN HEMTs
exploit the advantages of the wide band gap GaN materials, namely high mobility and high carriers density reached
by the two dimensional electron gas (2-DEG) formed at the
hetero-interface, which is the main principle of the HEMTs
operation [5–7].
An analysis of the unified 2-DEG charge density n s for all
regimes of the device operation is a primary requirement for
the development of a physics based compact model for these
devices [8]. The models proposed in [9] and [10] presented
the I-V and the C-V characteristics of Alx Ga1−x N/GaN
HEMTs based on the device physics. However, the effect
of the mole fraction was not taken into consideration.
The maximum sheet charge caused by piezoelectric polarization can be reached by increasing the Al concentration of
the barrier in HEMTs, but increasing it into a critical limit
causes a strain relaxation and, therefore, reduction in piezoelectric polarization. For example, for a barrier thickness of
300 Å macroscopic strain relaxation was first observed for
mole fraction of x = 38% [11]. The degree of relaxation
increased linearly with increasing mole fraction for higher
values of x.
In this work, we describe the target Alx Ga1−x N/GaN
HEMTs structure in Sect. 2. In Sect. 3 we derive the mole
fraction dependent threshold voltage model and incorporate
it in the I-V and C-V model presented in [10]. In Sect. 4, the

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J Comput Electron
Table 1 List of symbols

Fig. 1 Cross sectional view of Alx Ga1−x N/GaN HEMTs

results of our model are compared with the simulation data
obtained by Atlas TCAD simulation and discussed. Finally,
we draw the conclusions in Sect. 5.

Symbol

Physical meaning

ε

Permittivity of AlGaN

q

Elementary charge

dd

AlGaN barrier layer thickness

di

AlGaN spacer layer thickness

d

AlGaN layer thickness

W

Channel width

L

Channel length

Ef

Position of Fermi level

E0

Position of first sub-band

Ppz

Piezoelectric polarization

Psp

Spontaneous polarization

Nd

Ionized donor density

μ

Carrier mobility

Vth

Threshold voltage

E ceff (x)

Conduction band offset

ns

2-DEG charge density

m∗

Effective mass of electron

2 HEMT device structure
Figure 1 shows the Alx Ga1−x N/GaN HEMT devices consisting of a 200 nm GaN buffer layer [12,13], deposited on SiC
substrate in the (0001) direction. SiC substrate is insulating.
On the top there is a 3 nm Alx Ga1−x N spacer layer, followed
by the growth of a 9 nm Alx Ga1−x N barrier. The source and
drain contacts are ohmic, while the gate is a Schottky barrier
of (Ni and Ti) with Si3 N4 oxide layer deployed on both sides
of the gate.

3 Analytical models
3.1 Sheet carrier density model
The 2-DEG charge density n s , can be calculated using
a relatively simple equation derived from the solution of
Schrodinger’s and Poisson’s equations in the quantum well
assuming a triangular potential profile, and considering only
the first subband. The latter approximation is accurate enough
in Alx Ga1−x N/GaN heterostructure as explained by [9,10].




n s = DVth ln exp

E f − E0
Vth



(1)



where D = 4πhm2 is the conduction band density of state of
2-DEG system.
This first subband is given by
.
E 0 = γ0 n 2/3
s

123

ns =


ε 
Vg0 − E f ,
qd

(3)

where γ0 is a constant estimated by Shubnikov De Hass or
from cyclotron resonance experiments [14], and Vg0 = Vg −
Voff , Voff is the cut-off or threshold voltage (all the other
symbols have standard definitions, given in Table 1).
An explicit expression of n s valid from sub-threshold to
above threshold was proposed by Khandelwal et al. [9].
From Eqs. (2) and (3), an important relation between the
applied gate voltage Vg0 , the channel potential V , and the
carrier density n s can be developed in [10]
qdn s
+ γ0 n 2/3
Vg0 − V =
s + Vth ln
ε



ns
DVth


.

(4)

3.2 Threshold voltage model


+1 ,

The carrier density depends on the applied bias as

(2)

The amount of Al mole fraction available in the barrier layer
of Alx Ga1−x N/GaN HEMTs greatly influences the device
behavior. The Schottky barrier height φeff , the dielectric constant of the barrier layer ε, the conduction band offset, and the
total polarization σtotal are among the important parameters
that are affected by the Al mole fraction. Based on [15,16],
these device parameters can be expressed as a function of the

Author's personal copy
J Comput Electron

mole fraction x as

The spontaneous polarization Psp can be expressed as [17]

φeff (x) = 0.84 + 1.3x,

(5)

ε (x) = 0.3x + 10.4,


Al Ga
N
E ceff (x) = 0.7 E g x 1−x − E gGaN ,

(6)

Al Ga

(7)

N

where E g x 1−x is the energy band gap of Alx Ga1−x N
layer as a function of Al mole fraction, and is given by [15]
Al x Ga1−x N

= 6.13x + (1 − x) 3.42 − x (1 − x) ,

Eg

(8)

and E gGaN = 3.42 eV is the energy gap of GaN.
The total induced polarization at the interface is given by
[17]
σtotal = Ppz + Psp ,

Al Ga1−x N

Al Ga

GaN
− Ppz
,

(10)

N

GaN are the piezoelectric polarizations
with Ppz x 1−x and Ppz
of Alx Ga1−x N and GaN, respectively.
Al Ga
N
Ppz x 1−x is the piezoelectric polarization of Alx Ga1−x N,
and can be expressed as [11]
Al Ga
N
Ppz x 1−x




aGaN − aAlx Ga1−x N
=2
aAlx Ga1−x N

Al Ga
N
e31 x 1−x

Al x Ga1−x N
Al Ga
Nc
− e33 x 1−x 13
Al Ga
N
c33 x 1−x

aAlx Ga1−x N = xaAlN + (1 − x) aGaN ,
N


,

(11)

Al Ga

(12)

N

Al
GaN
= xe31
+ (1 − x) e31
,

=
=
=

GaN and P AlN are the spontaneous polarizations of
where Psp
sp
GaN and AlN, respectively.
Al Ga
N
Psp x 1−x is the spontaneous polarization of Alx Ga1−x N
can be expressed as a Vegard’s interpolation is given by
Eq. (18)
Al Ga1−x N

Psp x

AlN
GaN
= x Psp
+ (1 − x) Psp
.

(18)

By solving Poisson’s equation in the barrier layer, the expression of threshold voltage is derived that depends on the
parameters influenced by the mole fraction, and is given by
[16]

AlN
xe33
AlN
xc13
AlN
xc33

+ (1 −
+ (1 −
+ (1 −

GaN
x) e33
,
GaN
x) c13
,
GaN
x) c33 .

Voff = φeff (x) − E ceff (x) −

q Nd dd2
σtotal

(dd + di ) ,
2ε (x)
ε (x)
(19)

where φeff (x) is the Schottky barrier height, E ceff (x) is
the conduction band offset between Alx Ga1−x N and GaN
and Nd is the doping density of Alx Ga1−x N layer, σtotal is
the total polarization induced charge.
In our simulations the doping density of Alx Ga1−x N layer
was assumed to be Nd = 1024 cm−3 .

An analytical drain current model is derived from Eq. (4)
based on the charge control model developed in [10]
Ids =

e31 x 1−x and e33 x 1−x are the piezoelectric coefficients
Al Ga
N
Al Ga
N
of Alx Ga1−x N, c13 x 1−x and c33 x 1−x are the elastic
constants of Alx Ga1−x N.They can be expressed by Vegard’s
interpolation are given by Eqs. (13), (14), (15) and (16).
Al Ga
N
e31 x 1−x
Al Ga
N
e33 x 1−x
Al Ga
N
c13 x 1−x
Al Ga
N
c33 x 1−x

(17)

3.3 Core drain current, charge and capacitance models

where aGaN and aAlx Ga1−x N are the lattice parameters of GaN
and of Alx Ga1−x N, respectively.
aAlx Ga1−x N it can be expressed by Vegard’s interpolation
is given by Eq. (12)

Al Ga

GaN
− Psp
,

(9)

where Ppz is the piezoelectric polarization can be weighed
using the Vigard’s interpolation formula [17]
Ppz = Ppz x

Al Ga1−x N

Psp = Psp x

(13)


2

qμW qd 2
5/3
5/3
n D − n 2S + γ0 n D − n S
L

5
+Vth (n S − n D ) .

where n S and n D are the charge carrier concentrations at the
source and drain, respectively.
The total gate charge can be written as [10]
QG


⎛ qd 
 1 8/3

⎞
8/3
3
3
+ 21 Vth n 2D − n 2S
3ε n D − n S + 4 γ0 n D − n S

⎠.


= W Lq

5/3
5/3
qd  2
2 + 2γ
n
+
V
n

n

n
(n

n
)
0
th
D
S
D
S
D
S

5

(21)
The gate-to-source and gate-to-drain capacitances can be calculated using the partial differentiations of the gate charge as
explained in [10]

(14)
(15)
(16)

(20)


d fmain (n s )
C Gx = W Lq

dVx

(n s )
g (n s ) − f (n s ) dgmain
dVx

(g (ns))2


.

(22)

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J Comput Electron
Table 2 Parameters used to plot the I-V and C-V characteristics
Parameter

Description

Sample 1 Al25 Ga75 N

Sample 2 Al30 Ga70 N

Sample 3 Al35 Ga65 N

Voff (V)

Cut-off voltage

−1.16

−1.61

−2.10

dd (nm)

Thickness of barrier

9

9

9

di (nm)

Thickness of spacer

3

3

3

Wg (μm)

Gate width

100

100

100

L g (μm)

Gate length

0.4

0.4

0.4

Table 3 Relevant material
parameters used in calculation

Parameter

Symbol

Value

Unit

Refs.

GaN spontaneous polarization

GaN
Psp

0.029

C/m2

[11]

AlN spontaneous polarization

AlN
Psp

0.081

C/m2

[11]

GaN lattice parameter

aGaN

0.316

nm

[18]

AlN lattice parameter

aAlN

0.311

nm

[18]

GaN elastic constants

GaN
c13

106

GPa

[18]

GaN
c33

398

GPa

[18]

Al N
c13
AlN
c33
GaN
e31
GaN
e33
AlN
e31
AlN
e33

99

GPa

[18]

389

GPa

[18]

−0.33

C/m2

[18]

0.65

C/m2

[18]

−0.58

C/m2

[18]

1.58

C/m2

[18]

AlN elastic constants
GaN piezoelectric coefficients
AlN piezoelectric coefficients

4 Results and discussion
The current-voltage and capacitance-voltage characteristics
of the device for different Al mole fractions (see Table 2)
have been simulated using our model and compared with the
simulation data obtained by the Atlas TCAD simulator. The
material parameters used in calculation are listed in Table 3.
Figure 2 shows the I-V characteristics of the Alx Ga1−x
N/GaN HEMT devices at Vgs = −1 V. The results show that,
for x = 35% the HEMT reaches the maximum drain source
saturation current compared with those obtained with x = (25
and 30) %.
On the other hand, Figs. 3 and 4 show the electrical
transfer characteristics (in linear a, logarithmic scale b) and
the transconductance, respectively, at Vdg = 7 V, where the
HEMT device reaches a low value of threshold voltage at
x = 35%. Thus, the current flowing in the device depends
strongly on the Al mole fraction of the top layers. This is
essentially due to an increase in the saturation of the drain
current. Our results indicate that an improvement of the performance of HEMT devices can be reached by varying the
mole fraction in the Alx Ga1−x N layer.
The simulations of the C-V characteristics have been carried out and the results were compared with data obtained
from the Atlas TCAD simulator.

123

Fig. 2 Comparison of modeled Ids −Vds characteristics (solid line)
with Atlas TCAD data (symbols) for different mole fractions x in
Alx Ga1−x N/GaN HEMT with Vgs = −1 V

The variation of the gate to source capacitance Cgs in terms
of the gate voltage Vgs at drain voltage Vds = 7 V is shown
in Fig. 5. Our results show the dependence of threshold voltage on mole fraction. Cgs is small when Vgs is below or
close to Voff and it is equal to the fringing capacitance. This
means that the effective distance between the gate and the
heterointerface is not affected appreciably, thus keeping the
Alx Ga1−x N layer fully depleted [19]. The value of the fring-

Author's personal copy
J Comput Electron

Fig. 5 Comparison of modeled (solid line) with Atlas TCAD data
(symbols) Cgs −Vg , for different mole fractions in Alx Ga1−x N/GaN
HEMT with Vds = 7 V

Fig. 3 Comparison of modeled (solid line) Ids −Vgs characteristics with
Atlas TCAD data (symbols), in linear a and logarithmic b scale for
different mole fractions in Alx Ga1−x N/GaN HEMT, Vds = 7 V
Fig. 6 Comparison of modeled (solid line) with Atlas TCAD data
(symbols) Cgd −Vd , for different mole fractions in Alx Ga1−x N/GaN
HEMT with Vg = −1 V

Fig. 4 Comparison of modeled (solid line) gm −Vgs characteristics
with Atlas TCAD data (symbols) for different mole fractions in
Alx Ga1−x N/GaN HEMT with Vds = 7 V

ing capacitance is found to be 0.78 pF/mm. As Vgs increases
Cgs increases due to the increase of the depth of the electron channel and the sheet carrier density increases, thereby
increasing the channel charge and thus affects the gate-tosource capacitance. It is clear that the simulated model and
the Cgs values obtained by Atlas TCAD software agree quite
well.
The dependence of gate-to-drain capacitance Cgd on the
gate voltage is plotted in Fig. 6 for varying values of the Al
mole fraction. The gate-to-drain capacitance increases with
increasing mole fraction. Thus, for higher Al mole fraction,
the conduction band offset E ceff increases, which leads
to a higher 2-DEG density and the drain current increases
owing to increase in the 2-DEG density, consistent with high
mobility. This indicates a steady decrease in Cgd as the drain
bias increases from 0 to 10 V, while gate bias is fixed to

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J Comput Electron

−1 V due to losing control over the channel charge, then
Cgd stays almost constant. Moreover, we observe that as the
capacitance decreases, the concentration of the mole fraction
of the barrier layer increases. The results of our model are
in excellent agreement with the simulation data obtained by
the Atlas TCAD simulator.

5 Conclusions
In this paper, we have incorporated the effect of the mole fraction in a compact model for Alx Ga1−x N/GaN HEMTs, by
means of the threshold voltage expression. The charge sheet
density expression depends on Voff , where the mole fraction
effect is accounted for. This leads to mole fraction dependent
expressions of the drain current and the capacitances. This
allows a better understanding of the ways to achieve higher
performance levels for GaN based high speed devices. The
results using Atlas TCAD simulation data and our model
show that the HEMT reaches a low value of threshold voltage
at x = 35 % with Vdg = 7 V and also the gate to source capacitance Cgs is small when Vgs is below or close to Voff and it
is equal to the fringing capacitance. The value of the fringing
capacitance is found to be 0.78 pF/mm. Also, the proposed
analytical model for HEMTs I-V and C-V characteristics for
different mole fractions shows an excellent agreement with
Atlas TCAD data.
Acknowledgements This work was supported by the Spanish Ministry of Economy and Competitiveness through project GREENSENSE
(TEC2015-67883- R) and by the ICREA Academia 2013 Award.

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