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GaN hetero-interface under pressure and impurity effects
Journal Pre-proof Two-dimensional electron gas modeling in strained InN/GaN hetero-interface under pressure and impurity effects H. El Ghazi, R. En-nadir, H. Abboudi, F. Jabouti, A. Jorio, I. Zorkani PII:
S0921-4526(19)30831-2
DOI:
https://doi.org/10.1016/j.physb.2019.411951
Reference:
PHYSB 411951
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 26 October 2019 Revised Date:
19 November 2019
Accepted Date: 13 December 2019
Please cite this article as: H. El Ghazi, R. En-nadir, H. Abboudi, F. Jabouti, A. Jorio, I. Zorkani, Two-dimensional electron gas modeling in strained InN/GaN hetero-interface under pressure and impurity effects, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/ j.physb.2019.411951. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Two-dimensional electron gas modeling in strained InN/GaN heterointerface under pressure and impurity effects
H. El Ghazi (1,2,*), R. En-nadir (2), H. Abboudi (2), F. Jabouti (2), A. Jorio (2), I. Zorkani(2)
(1)
(2)
ENSAM Laboratory, ENSAM, Hassan II University, Casablanca, Morocco
Faculty of sciences Dhar El Mehrez, Mohammed Ben Abdellah University, Fes, Morocco
Abstract The two-dimensional electron gas (2DEG) density modeling in WZ strained InN/GaN hetero-interface with Ga-face arrangement is considered in this paper. We theoretically investigate the role of an externally applied hydrostatic pressure and impurity’s position by analyzing their impact in the presence of built electric field due to the dielectric mismatch between the system and its surrounding matrix. To solve the coupling Schrödinger-Poisson equations, as well as the polarization induced charges, finite element method code is used. Pressure-dependence of dielectric constant, band-gap energy, lattices and internal electric field is considered. The 2DEG appears to be strongly-dependent internal and external perturbations exhibiting a non-monotonic behavior. Our results prove that: (1) the InN channel is more populated for on-center impurity compared to off-center one, (2) the electron density in the triangular shape quantum well increases with the pressure; (3) the 2DEG density is enhanced versus the positive impurity’s position and dropped versus pressure.
Keywords: 2DEG, Hydrostatic pressure, Impurity, Electron density, Hetero-interface.
*corresponding author: [email protected]
1
1. Introduction: For decades, nitrides binary (InN, GaN and AlN) and ternary (InGaN and AlGaN) based hetero-structures are systems of paramount interest in nano-science and nanotechnology. They have been a subject of intense recent investigations and have emerged as attractive candidates for nano-devices such as solar cell (SC), light emitting diode (LED) and HEMT transistors. Due to their high breakdown voltage, high peak electron velocity, robust thermal stability, chemical stability, strong intrinsic spontaneous and piezoelectric polarization effects and large saturation velocity, semiconductor hetero-structures based on III-nitride are presented as adequate materials for generating two dimensional (2D) electron gas (2DEG) systems with high mobility. In the midst of III-nitrides, InN has emerged as a good candidate to be the best conductive-channel material because the electrons effective mass is the smallest and the conduction band offsets is the largest at the hetero-interface with GaN and AlN. Many papers have been reported studying the characteristics of the two dimensional electron gas (2DEG) due to the spontaneous (SP) and piezoelectric (PZ) polarizations [1-29]. In the literature, herero-structures such SiO2/Si [5], (Mg,Zn)O/ZnO [6], (Al,Ga)N/GaN [7,8], (Al,Ga)As/GaAs [9,10] and LaAlO3/SrTiO3 [11] have been studied intensely. It has been shown that the GaN-based 2DEG at the (Al,Ga)N/GaN [13], AlN/GaN [15], (In,Ga)N/GaN [16]
and (In,Al)N/GaN [14] hetero-interfaces leads to successful high
electron-mobility transistors (HEMTs) which open the door for others communication applications such 5G high-speed wireless generation [17,18]. Important experimental discoveries and conceptual developments for high power and high speed electronics are also reported [20-25]. Hasan et al. [26] have examined the In mole composition effects on 2DEG density at (In,Ga)N/GaN interface. It found that the sheet carrier density maximum is about
2.12 10 and 7.08 10 at InN/AlN and InN/GaN hetero-interfaces respectively. In the same trend, Atmaca et al [27] have shown that the 2DEG density is enhanced as the In fraction increases in asymmetric GaN/(In,Ga)N/ZnO hetero-structure. The effect of (In,Ga)N interfacial layer on the 2DEG density in GaN-based HEMT is investigated by Muhtadi et al. [28]. They have shown that the insertion of (In,Ga)N layer induces a very strong enhancement
of 2DEG density compared to the usual GaN-based one. Recently, experimental study of 2DEG at InN/(In,Ga)N hetero-interface grown by molecular beam epitaxy is reported by Wang et al. [30]. Based on the Hall-effect measurements at room temperature, the directly probed sheet electron density is obtained
2
around of 2.14 10 . At 3, the authors have deduced an electron density about
3.30 10 based on the measured multi-frequency Shubnikov-de Hass oscillations of 2DEG. In the same tendency, Gazzah et al. [16] have recently examined the joint effects of
heat transfer and electronic properties in (In,Ga)N/GaN based QW. They have found that the 2DEG density increases with increasing the indium fraction in the (In,Ga)N/GaN based QW. It is well known that the pressure is one of the paramount tools for understanding the electronic and electric characteristics in nano-devices. To the best of our knowledge, no paper has been done to treat theoretically the influence of the impurity on 2DEG density. In the following, we will concentrate on the theoretical investigation of the 2DEG at i-InN/n-GaN hetero-interface under the combined effects of an externally applied hydrostatic pressure and impurity’s position considering the pressure-dependent internal built-electric field.
2. Theoretical background: In our model, electronic characteristics of − / − / − hetero-structure
with [0001] orientation are investigated. A cross section schematic drawing of this hetero-
structure with Ga-face arrangement is presented in Fig. 1. The device geometry is similar to that of a conventional GaN-based, solar cell, HEMTs and LEDs with a planar cross-section.
Without pressure effect, the thickness of the well and the barrier are respectively 10∗ and 8∗ where ∗ = 2.8 is the GaN effective Bohr radius.
[0001]
p-GaN
2DEG 0
i-InN n-GaN
Ga-face
!
- -- - - -- -- - - - - + + + + + + + +++
& + '
!
!"
- - - - - -- - - - - - - - ++++++++ !
- - - - - - - - --- - - + + + + + + + + ++ ++
#%
#$ #%
Substrate (Bulk GaN)
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges. Based on the single band parabolic and the effective mass approximations, the electron Hamiltonian is described as follow:
3
−
ℏ) *
*+
,
*
-∗ .,+ *+
Ψ1,23 4 5 −
6)
789 8 ∗ .,+ |;< ;<9 |
Ψ1,2= 4 + ?@ A, 4 Ψ1,2= 4 = E1 Ψ1,2= 4 (1)
Where 4, C< and C
constant respectively while GD is the vacuum permittivity. k + and n are respectively the
momentum vector parallel to the x-growth direction (c-axis structure) and the subband index while E1 is the electron energy. ?@ A, 4 is the pressure and position-dependent total potential energy expressed as follow:
?@ A, 4 = ?J A, 4 + ?K A, 4 + ?6+ A, 4 + ?L A, 4
(2)
The first term describes the conduction band offset between GaN and InN materials which is expressed as: ?J A, 4 = M
0 |4| ≤
OP
QRSTJ A − STU A V WXYWZℎWCW
(3)
Q = 0.7 is the conduction band off-set parameter. The band gap pressure dependence is
expressed as [31]:
ST A = ST 0 + \A + ]A
(4)
ST 0 is the band-gap energy without pressure effect.
Taking into account of the lattice mismatch between the well and the barrier, the internal electric field effect on the polarization charges is considered and included via the second term. It is given as: ?K A, 4 = −W^ A, 4 4
(5)
We have used for internal electric field the same expression adopted in our previous paper [12] and it is given according to Bigenwald as [32]: ^ A, 4 =
de ei de b p |4| ≤ ∗ ∗ qr ` 89 l8P . ' 8m . nooP p
a2 ` _
c. fgh '.fgh .jkh c
m fgh '. fgh .jkh c c.de ei de ∗ . s om p t' 8 ∗ . 5 89 ,8P m oP p
OP
WXYWZℎWCW
(6)
4
uU , uJ and Gv∗ are respectively the thickness layer of the structure and the relative dielectric constant.
According to Ambacher et al. [7], InN and GaN spontaneous and piezoelectric polarizations are given as: yz{ Awx = −0.042
(7)
yz{ Ax| = −1.373G + 7.559G { Awx = −0.029
(8) (9)
The strain parameter, G, is expressed from the and the lattices as follow: G A =
jkh . fgh . fgh .
(10)
According to Ref. [33], pressure-dependence of lattice, well and barrier widths is given as follow: uv A = uDv 1 − + 2 A
(11)
uDv is lattice, width of well and the barrier without pressure effect while and are the compliance constants expressed via the elastic constants, v , as follow: = )' ) ) ) = ' '
)
(12)
)
According to Ben Jazia et al. [34], the exchange-correlation potential is given as: ?6+ = −0.961 78
6) z .,+ , 5 ∗ 7 9 8 .,|
(13)
The density of confined electrons, A, 4 , is given as: A, 4 =
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5 cΨ1,2= A, 4 c
(14)
Where and are the Boltzmann constant and temperature respectively. The Fermi energy, SK , is obtained by the neutrality equation [16]:
5
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5 =
{)
' 6+xn q
+
d 6
(15)
S = 26 W? is the shallow donor binding energy while = 2. 10 is the
doping density per unit of area [35].
The 2DEG density is expressed following the recent paper [16] as: =
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5
(16)
The fourth term in Eq. (2) represents the Hartree potential implied in Poisson equation [36]: GD *+ ,G ∗ A, 4 *
5 = W * 4 − A, 4
* ¡ .,+ *+
(17)
* 4 is the x-dependent density of ionized donors. The doping concentration in n-GaN and p-GaN regions are respectively = 2. 10¢ and £ = 2. 10¤ .
∗ A, 4 and G ∗ A, 4 are respectively the effective mass and relative dielectric constant. According to Ref. [37], they are given as:
∗ A,
G
4 =
∗ A,
b `
-9
¥P ¦ ' fgh § p
|4| ≤
OP
9 WXYWZℎWCW a ¥m `' jkh¦ _ § p
-
∗ A |4| Gyz{ ≤
(18)
OP
4 = ¨ ∗ A WXYWZℎWCW G{
(19)
Where, D is the free electron mass and 6v is the energy related momentum matrix element
obtained by the previous equation without pressure effect.
The hydrostatic pressure effect on the dielectric constant is expressed according to Xia et al. [38] as: G ∗ A = ,1 + G© 0 − 1 W4 s−
.
t5 ,®o¯ . 5
ª D.« ¬.
®
¯
(20)
For transversal and longitudinal compounds, the angular frequency, ≡ ±, u± , is given as: ²v A = ²v 0 W4 ,
£³ .
5
(21)
6
3. Results and discussion In the present paper, the Finite element Method (FEM) is used for the space discretization of the coupling Schrödinger and Poisson equations. Our numerical results are obtained for p-GaN/i-InN/n-GaN hetero-structure. All physical parameters concerned the numerical calculations are listed in table 1. To avoid the deformation of the material structure,
we restrict ourselves to the case corresponding to hydrostatic pressure values in 0 − 40GPa
range.
Table 1: Parameters used in our numerical calculation [39-42] Parameter
ST P = 0 W?
GaN
InN
\ (meV/GPa)
3.42
0.78
40
16
GPa
-0.38
-0.02
293
187
159
125
² · P = 0 1/cm
731.51
621.53
ºO·
525.56
487.67
1.08
1.27
1.47
1.52
»
4.942
6.723
0.365
0.406
190
136
5.185
5.703
] (meV/GPa2) GPa
²O· P = 0 1/cm º·
G© P = 0 ¼ GPa
C P = 0 10 D m
Fig. 1 displays the structure under study. Taking into account that the GaN lattice parameter is lower than that of InN, the GaN layer is consequently under tensile strain. Positive charges are created in lower InN layer compensated by negative charges at the upper interface of InN layer forming a gas confined at i-InN/n-GaN interface called as the two dimensional electron gas (2DEG). This density can be attributed to different sources such as extrinsic and intrinsic defects, surface capture centers, InN barrier and high dislocation densities. In this trend, according to the study of the full width at half maximum of the rocking curves on (002) and (102) InN planes, Wang et al. [30] have shown that the growth of GaN layer on InN one is accompanied by high dislocation densities. They have obtained 7
dislocation densities of InN in InN/GaN hetero-interface around 5.88 10¢ and 7.23 10D respectively (Fig. S1b: Supporting Information). In order to investigate our model, we start showing the effects of an externally applied pressure on dielectric constant, effective mass, band gap energy, conduction band offset and internal electric field. Figure 2 reveals that the effective mass, the band gap energy and then the band offset increase linearly versus the pressure at least in the studied pressure domain. It is also shown that the relative dielectric constant decreases as a function of the pressure contrary to one. For the internal electric field, two behaviors are obtained limited by pressure value crossover A@ ~30 A . For A ¿ A@ , the barrier electric field is greater than the well one and they decrease via the pressure. For higher values of pressure A À A@ , the well one is greater and they increase according to hydrostatic pressure. These results are in good conformity with those obtained by Baskoutas et al. for ZnO quantum dots and nanorods subjected to externally applied hydrostatic pressure and CdS quantum dots capped by various dielectric matrix materials [43-46].
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
8
In Figure 3, we present the conduction band energy diagram computed using the selfconsistent code for two different values of hydrostatic pressure under the effects of different terms of potential barrier. It is clear that the conduction band has a triangular shape nearest the O
higher interface of the structure located at s P t. As the pressure increases the curve becomes
less deep compared to lower one. Principally, this is due to the internal quantum-confined Stark effect (QCSE) which diminishes with increasing the pressure. This result is in good agreement with that reported by Ben Jaziad et al. [34]. Consequently, a 2DEG is cumulated at the upper InN/GaN hetero-interface due to difference in spontaneous and piezoelectric polarizations between the hetero-structure layers in the one hand and the conduction band offset in the other hand. Additionally, it is revealed from the same figure that the band offset increases as a function of the pressure. For instance, it is equal to 1.97 W? and 2.17 W? for pressure values equal to 0 and 20 GPa respectively.
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure.
9
The results regarding the Fermi energy and electron density as a function of the xthickness along the growth direction of the studied structure are exhibited on Fig. 4. The effects of pressure and the positive donor’s position are also examined. It is clearly shown that the electron density is not monotonic but presents a maximum value located in the vicinity of the higher interface. It is also clear that its maximum is enhanced versus pressure for two different position of the impurity. Moreover, the electron density is found to be more spread in GaN layer for higher value of pressure as expected. For the impurity located at
OP
, the
electron density is found to be displaced to the upper interface and be more broaden in the barrier. This result can be explained by the fact that the electron density located in the triangular shape QW is more attracted by the impurity placed at the higher interface compared with the on-center one. Furthermore, the InN layer is more populated for on-center impurity than for off-center one. This result can be due to the fraction of the electron density extended in the barrier which is removed toward the well under electrostatic interaction. It can be seen also that as the impurity moves far away the structure center in the positive sense, the electron density decreases for all pressure values. This is due to the fact that a fraction of the electron’s population is captured by the defects located in the interface. For example, the density is equal
to 2.105 10¢ and 1.961 10¢ for on-center impurity and off-center case
respectively. From the earlier results, one can speculate that the 2DEG density can only enhance as a function of the donor’s position. In the same trend, the electron density augments via the pressure for all position of the impurity which can be assigned to the conduction band offset increasing. As the pressure
changes from 0 A to 20 A, the electron density varies respectively from
2.105 10¢ to 2.291 10¢ showing 8.8% enhancement. So, the increasing of the pressure forces the density to be less spread in the barrier and to move into the well which
induces a reduction of the 2DEG density. For instance, this latter is equal to 4.235 10 ,
2.753 10 and 1.426 10 for pressure values equal to 0 A, 20 A and 40 A respectively. The drop in the 2DEG density is about 35% and 66% for 0 − 20A
and 0 − 40A pressure ranges respectively which is not suitable for HEMT applications. To better show and clarify that, we present in table 2 the obtained results for three different values of pressure 0, 20, 40 A and position of the impurity 0, uU ⁄2 , uU + uJ ⁄2 .
10
Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included. The obtained results are in quite agreement with those of several authors concerning different semiconductor materials but some discrepancies remain. For example, Gazzah et al. [16] have reported that as the In fraction in (In,Ga)N well channel increases the depth of the triangular QW increases and then the densities of confined electrons and 2DEG are found to be enhanced. The In-dependent 2DEG behavior is completely different of our results even if the In mole and pressure induce both band offset increasing. Incidentally, it should be noted that these discrepancies are due mainly to the electric field neglected in the barrier on the one hand and to the presence of the impurity neglected in all available papers. In the same tendency, Hasan et al. [26] have proved that the sheet carrier density decreases versus In mole composition at (In,Ga)N hetero-interface (Fig. 5 [26]: For more details). This is completely in disagreement with the results reported in Ref. [16]. It found that it is equal respectively to 7.08 10 and 3.23 10 at InN/GaN and .ª .ª hetero-interfaces.
11
Also, it can be attributed to It is important to notice that as the impurity moves far away the structure center in the negative sense of the x-axis, it is expected that the 2DEG density shows a drop while the electron density shows a significant enhancement. Table 2: Electronic properties of InN/GaN hetero-structure for three different values of pressure 0, 20, 40 A and position of the impurity 0, uU ⁄2 , uU + uJ ⁄2 . A A
0
ÃD ∗
0 5
0
9
0 20 20
0
40
−0.728
−0.084
0.391 0.194
0.155
10
1.961 1.895
0.017
2.094
0.234
0.187
2.105
2.291
0
−0.187
-+ 10¢
0.121
0.072
3.348
40
−0.531
0.625
−0.132
0.851
40
−0.297
SK − S-vz W?
2.926 6.174
20
SK W?
4.235
4.873 6.385
2.753
2.161
3.291
0.086
2.341
1.426
0.007
2.024
0.039
2.185
4.602
2.142 3.853
4. conclusion In summary, the impact of an externally applied hydrostatic pressure and the impurity’s position on the 2DEG properties in p-GaN/i-InN/n-GaN hetero-structure with Ga-face is investigated numerically by solving the coupling one-dimensional non linear Schrödinger and Poisson equations self-consistently within the effective mass and one parabolic band approximations. The pressure-dependent internal built-electric field due to spontaneous and piezoelectric polarizations is considered. Two-dimensional electron gas (2DEG) density, created at the InN/GaN upper hetero-interface under the joint effects of polarization charge and conduction band-offset, appears to be highly dependent pressure, size and impurity. Our results reveal that: •
The InN channel is more populated for on-center impurity,
•
The triangular shape QW becomes less deeper versus the pressure,
•
The electron density in triangular shape QW increases with the pressure,
12
•
2DEG density enhances versus the positive impurity’s position and drops versus hydrostatic pressure. Due to the importance of the III-Nitrides materials, we think that present work provides
a new model for significant potential applications such as high efficiency power transistor, laser, light emitting diode, photovoltaic cells, infra-red photo-detectors and would guide further theoretical and experimental research works in group III-nitrides.
13
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List of Figures
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges.
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure. Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included.
17
Ga-face
- -- - - -- -- - - - -
n-GaN
[0001]
i-InN p-GaN
#%
+ + + + + + + +++
2DEG 0
!
- - - - - -- - - - - - - -
&'
!
!"
#$
+ ++++++++
- - - - - - - - --- - - ! -
#%
+ + + + + + + + ++ ++
Substrate (Bulk GaN)
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges.
18
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
19
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure.
20
Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included.
21
In order to investigate our model, we start showing the effects of an externally applied pressure on dielectric constant, effective mass, band gap energy, conduction band offset and internal electric field. Figure 2 reveals that the effective mass, the band gap energy and then the band offset increase linearly versus the pressure at least in the studied pressure domain. It is also shown that the relative dielectric constant decreases as a function of the pressure contrary to one. For the internal electric field, two behaviors are obtained limited by pressure value crossover ~30 . For < , the barrier electric field is greater than the well one and they decrease via the pressure. For higher values of pressure > , the well one is greater and they increase according to hydrostatic pressure. These results are in good conformity with those obtained by Baskoutas et al. for ZnO quantum dots and nanorods subjected to externally applied hydrostatic pressure and CdS quantum dots capped by various dielectric matrix materials [43-46].
Conflict of interest statment
We bring to your knowledge that the manuscript has not been previously published and is not currently in press under consideration, or being considered for publication elsewhere. All authors have read and approved the manuscript being submitted and agree to its submission to Physica B: Condensed Matter Journal. We have no conflicts of interest to disclose.
Prof: Haddou El Ghazi ENSAM, H2 University, Casablanca, Morocco
S0921-4526(19)30831-2
DOI:
https://doi.org/10.1016/j.physb.2019.411951
Reference:
PHYSB 411951
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 26 October 2019 Revised Date:
19 November 2019
Accepted Date: 13 December 2019
Please cite this article as: H. El Ghazi, R. En-nadir, H. Abboudi, F. Jabouti, A. Jorio, I. Zorkani, Two-dimensional electron gas modeling in strained InN/GaN hetero-interface under pressure and impurity effects, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/ j.physb.2019.411951. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Two-dimensional electron gas modeling in strained InN/GaN heterointerface under pressure and impurity effects
H. El Ghazi (1,2,*), R. En-nadir (2), H. Abboudi (2), F. Jabouti (2), A. Jorio (2), I. Zorkani(2)
(1)
(2)
ENSAM Laboratory, ENSAM, Hassan II University, Casablanca, Morocco
Faculty of sciences Dhar El Mehrez, Mohammed Ben Abdellah University, Fes, Morocco
Abstract The two-dimensional electron gas (2DEG) density modeling in WZ strained InN/GaN hetero-interface with Ga-face arrangement is considered in this paper. We theoretically investigate the role of an externally applied hydrostatic pressure and impurity’s position by analyzing their impact in the presence of built electric field due to the dielectric mismatch between the system and its surrounding matrix. To solve the coupling Schrödinger-Poisson equations, as well as the polarization induced charges, finite element method code is used. Pressure-dependence of dielectric constant, band-gap energy, lattices and internal electric field is considered. The 2DEG appears to be strongly-dependent internal and external perturbations exhibiting a non-monotonic behavior. Our results prove that: (1) the InN channel is more populated for on-center impurity compared to off-center one, (2) the electron density in the triangular shape quantum well increases with the pressure; (3) the 2DEG density is enhanced versus the positive impurity’s position and dropped versus pressure.
Keywords: 2DEG, Hydrostatic pressure, Impurity, Electron density, Hetero-interface.
*corresponding author: [email protected]
1
1. Introduction: For decades, nitrides binary (InN, GaN and AlN) and ternary (InGaN and AlGaN) based hetero-structures are systems of paramount interest in nano-science and nanotechnology. They have been a subject of intense recent investigations and have emerged as attractive candidates for nano-devices such as solar cell (SC), light emitting diode (LED) and HEMT transistors. Due to their high breakdown voltage, high peak electron velocity, robust thermal stability, chemical stability, strong intrinsic spontaneous and piezoelectric polarization effects and large saturation velocity, semiconductor hetero-structures based on III-nitride are presented as adequate materials for generating two dimensional (2D) electron gas (2DEG) systems with high mobility. In the midst of III-nitrides, InN has emerged as a good candidate to be the best conductive-channel material because the electrons effective mass is the smallest and the conduction band offsets is the largest at the hetero-interface with GaN and AlN. Many papers have been reported studying the characteristics of the two dimensional electron gas (2DEG) due to the spontaneous (SP) and piezoelectric (PZ) polarizations [1-29]. In the literature, herero-structures such SiO2/Si [5], (Mg,Zn)O/ZnO [6], (Al,Ga)N/GaN [7,8], (Al,Ga)As/GaAs [9,10] and LaAlO3/SrTiO3 [11] have been studied intensely. It has been shown that the GaN-based 2DEG at the (Al,Ga)N/GaN [13], AlN/GaN [15], (In,Ga)N/GaN [16]
and (In,Al)N/GaN [14] hetero-interfaces leads to successful high
electron-mobility transistors (HEMTs) which open the door for others communication applications such 5G high-speed wireless generation [17,18]. Important experimental discoveries and conceptual developments for high power and high speed electronics are also reported [20-25]. Hasan et al. [26] have examined the In mole composition effects on 2DEG density at (In,Ga)N/GaN interface. It found that the sheet carrier density maximum is about
2.12 10 and 7.08 10 at InN/AlN and InN/GaN hetero-interfaces respectively. In the same trend, Atmaca et al [27] have shown that the 2DEG density is enhanced as the In fraction increases in asymmetric GaN/(In,Ga)N/ZnO hetero-structure. The effect of (In,Ga)N interfacial layer on the 2DEG density in GaN-based HEMT is investigated by Muhtadi et al. [28]. They have shown that the insertion of (In,Ga)N layer induces a very strong enhancement
of 2DEG density compared to the usual GaN-based one. Recently, experimental study of 2DEG at InN/(In,Ga)N hetero-interface grown by molecular beam epitaxy is reported by Wang et al. [30]. Based on the Hall-effect measurements at room temperature, the directly probed sheet electron density is obtained
2
around of 2.14 10 . At 3, the authors have deduced an electron density about
3.30 10 based on the measured multi-frequency Shubnikov-de Hass oscillations of 2DEG. In the same tendency, Gazzah et al. [16] have recently examined the joint effects of
heat transfer and electronic properties in (In,Ga)N/GaN based QW. They have found that the 2DEG density increases with increasing the indium fraction in the (In,Ga)N/GaN based QW. It is well known that the pressure is one of the paramount tools for understanding the electronic and electric characteristics in nano-devices. To the best of our knowledge, no paper has been done to treat theoretically the influence of the impurity on 2DEG density. In the following, we will concentrate on the theoretical investigation of the 2DEG at i-InN/n-GaN hetero-interface under the combined effects of an externally applied hydrostatic pressure and impurity’s position considering the pressure-dependent internal built-electric field.
2. Theoretical background: In our model, electronic characteristics of − / − / − hetero-structure
with [0001] orientation are investigated. A cross section schematic drawing of this hetero-
structure with Ga-face arrangement is presented in Fig. 1. The device geometry is similar to that of a conventional GaN-based, solar cell, HEMTs and LEDs with a planar cross-section.
Without pressure effect, the thickness of the well and the barrier are respectively 10∗ and 8∗ where ∗ = 2.8 is the GaN effective Bohr radius.
[0001]
p-GaN
2DEG 0
i-InN n-GaN
Ga-face
!
- -- - - -- -- - - - - + + + + + + + +++
& + '
!
!"
- - - - - -- - - - - - - - ++++++++ !
- - - - - - - - --- - - + + + + + + + + ++ ++
#%
#$ #%
Substrate (Bulk GaN)
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges. Based on the single band parabolic and the effective mass approximations, the electron Hamiltonian is described as follow:
3
−
ℏ) *
*+
,
*
-∗ .,+ *+
Ψ1,23 4 5 −
6)
789 8 ∗ .,+ |;< ;<9 |
Ψ1,2= 4 + ?@ A, 4 Ψ1,2= 4 = E1 Ψ1,2= 4 (1)
Where 4, C< and C
constant respectively while GD is the vacuum permittivity. k + and n are respectively the
momentum vector parallel to the x-growth direction (c-axis structure) and the subband index while E1 is the electron energy. ?@ A, 4 is the pressure and position-dependent total potential energy expressed as follow:
?@ A, 4 = ?J A, 4 + ?K A, 4 + ?6+ A, 4 + ?L A, 4
(2)
The first term describes the conduction band offset between GaN and InN materials which is expressed as: ?J A, 4 = M
0 |4| ≤
OP
QRSTJ A − STU A V WXYWZℎWCW
(3)
Q = 0.7 is the conduction band off-set parameter. The band gap pressure dependence is
expressed as [31]:
ST A = ST 0 + \A + ]A
(4)
ST 0 is the band-gap energy without pressure effect.
Taking into account of the lattice mismatch between the well and the barrier, the internal electric field effect on the polarization charges is considered and included via the second term. It is given as: ?K A, 4 = −W^ A, 4 4
(5)
We have used for internal electric field the same expression adopted in our previous paper [12] and it is given according to Bigenwald as [32]: ^ A, 4 =
de ei de b p |4| ≤ ∗ ∗ qr ` 89 l8P . ' 8m . nooP p
a2 ` _
c. fgh '.fgh .jkh c
m fgh '. fgh .jkh c c.de ei de ∗ . s om p t' 8 ∗ . 5 89 ,8P m oP p
OP
WXYWZℎWCW
(6)
4
uU , uJ and Gv∗ are respectively the thickness layer of the structure and the relative dielectric constant.
According to Ambacher et al. [7], InN and GaN spontaneous and piezoelectric polarizations are given as: yz{ Awx = −0.042
(7)
yz{ Ax| = −1.373G + 7.559G { Awx = −0.029
(8) (9)
The strain parameter, G, is expressed from the and the lattices as follow: G A =
jkh . fgh . fgh .
(10)
According to Ref. [33], pressure-dependence of lattice, well and barrier widths is given as follow: uv A = uDv 1 − + 2 A
(11)
uDv is lattice, width of well and the barrier without pressure effect while and are the compliance constants expressed via the elastic constants, v , as follow: = )' ) ) ) = ' '
)
(12)
)
According to Ben Jazia et al. [34], the exchange-correlation potential is given as: ?6+ = −0.961 78
6) z .,+ , 5 ∗ 7 9 8 .,|
(13)
The density of confined electrons, A, 4 , is given as: A, 4 =
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5 cΨ1,2= A, 4 c
(14)
Where and are the Boltzmann constant and temperature respectively. The Fermi energy, SK , is obtained by the neutrality equation [16]:
5
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5 =
{)
' 6+xn q
+
d 6
(15)
S = 26 W? is the shallow donor binding energy while = 2. 10 is the
doping density per unit of area [35].
The 2DEG density is expressed following the recent paper [16] as: =
-∗ .,+ 7ℏ)
∑z u ,1 + W4 s
g
t5
(16)
The fourth term in Eq. (2) represents the Hartree potential implied in Poisson equation [36]: GD *+ ,G ∗ A, 4 *
5 = W * 4 − A, 4
* ¡ .,+ *+
(17)
* 4 is the x-dependent density of ionized donors. The doping concentration in n-GaN and p-GaN regions are respectively = 2. 10¢ and £ = 2. 10¤ .
∗ A, 4 and G ∗ A, 4 are respectively the effective mass and relative dielectric constant. According to Ref. [37], they are given as:
∗ A,
G
4 =
∗ A,
b `
-9
¥P ¦ ' fgh § p
|4| ≤
OP
9 WXYWZℎWCW a ¥m `' jkh¦ _ § p
-
∗ A |4| Gyz{ ≤
(18)
OP
4 = ¨ ∗ A WXYWZℎWCW G{
(19)
Where, D is the free electron mass and 6v is the energy related momentum matrix element
obtained by the previous equation without pressure effect.
The hydrostatic pressure effect on the dielectric constant is expressed according to Xia et al. [38] as: G ∗ A = ,1 + G© 0 − 1 W4 s−
.
t5 ,®o¯ . 5
ª D.« ¬.
®
¯
(20)
For transversal and longitudinal compounds, the angular frequency, ≡ ±, u± , is given as: ²v A = ²v 0 W4 ,
£³ .
5
(21)
6
3. Results and discussion In the present paper, the Finite element Method (FEM) is used for the space discretization of the coupling Schrödinger and Poisson equations. Our numerical results are obtained for p-GaN/i-InN/n-GaN hetero-structure. All physical parameters concerned the numerical calculations are listed in table 1. To avoid the deformation of the material structure,
we restrict ourselves to the case corresponding to hydrostatic pressure values in 0 − 40GPa
range.
Table 1: Parameters used in our numerical calculation [39-42] Parameter
ST P = 0 W?
GaN
InN
\ (meV/GPa)
3.42
0.78
40
16
GPa
-0.38
-0.02
293
187
159
125
² · P = 0 1/cm
731.51
621.53
ºO·
525.56
487.67
1.08
1.27
1.47
1.52
»
4.942
6.723
0.365
0.406
190
136
5.185
5.703
] (meV/GPa2) GPa
²O· P = 0 1/cm º·
G© P = 0 ¼ GPa
C P = 0 10 D m
Fig. 1 displays the structure under study. Taking into account that the GaN lattice parameter is lower than that of InN, the GaN layer is consequently under tensile strain. Positive charges are created in lower InN layer compensated by negative charges at the upper interface of InN layer forming a gas confined at i-InN/n-GaN interface called as the two dimensional electron gas (2DEG). This density can be attributed to different sources such as extrinsic and intrinsic defects, surface capture centers, InN barrier and high dislocation densities. In this trend, according to the study of the full width at half maximum of the rocking curves on (002) and (102) InN planes, Wang et al. [30] have shown that the growth of GaN layer on InN one is accompanied by high dislocation densities. They have obtained 7
dislocation densities of InN in InN/GaN hetero-interface around 5.88 10¢ and 7.23 10D respectively (Fig. S1b: Supporting Information). In order to investigate our model, we start showing the effects of an externally applied pressure on dielectric constant, effective mass, band gap energy, conduction band offset and internal electric field. Figure 2 reveals that the effective mass, the band gap energy and then the band offset increase linearly versus the pressure at least in the studied pressure domain. It is also shown that the relative dielectric constant decreases as a function of the pressure contrary to one. For the internal electric field, two behaviors are obtained limited by pressure value crossover A@ ~30 A . For A ¿ A@ , the barrier electric field is greater than the well one and they decrease via the pressure. For higher values of pressure A À A@ , the well one is greater and they increase according to hydrostatic pressure. These results are in good conformity with those obtained by Baskoutas et al. for ZnO quantum dots and nanorods subjected to externally applied hydrostatic pressure and CdS quantum dots capped by various dielectric matrix materials [43-46].
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
8
In Figure 3, we present the conduction band energy diagram computed using the selfconsistent code for two different values of hydrostatic pressure under the effects of different terms of potential barrier. It is clear that the conduction band has a triangular shape nearest the O
higher interface of the structure located at s P t. As the pressure increases the curve becomes
less deep compared to lower one. Principally, this is due to the internal quantum-confined Stark effect (QCSE) which diminishes with increasing the pressure. This result is in good agreement with that reported by Ben Jaziad et al. [34]. Consequently, a 2DEG is cumulated at the upper InN/GaN hetero-interface due to difference in spontaneous and piezoelectric polarizations between the hetero-structure layers in the one hand and the conduction band offset in the other hand. Additionally, it is revealed from the same figure that the band offset increases as a function of the pressure. For instance, it is equal to 1.97 W? and 2.17 W? for pressure values equal to 0 and 20 GPa respectively.
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure.
9
The results regarding the Fermi energy and electron density as a function of the xthickness along the growth direction of the studied structure are exhibited on Fig. 4. The effects of pressure and the positive donor’s position are also examined. It is clearly shown that the electron density is not monotonic but presents a maximum value located in the vicinity of the higher interface. It is also clear that its maximum is enhanced versus pressure for two different position of the impurity. Moreover, the electron density is found to be more spread in GaN layer for higher value of pressure as expected. For the impurity located at
OP
, the
electron density is found to be displaced to the upper interface and be more broaden in the barrier. This result can be explained by the fact that the electron density located in the triangular shape QW is more attracted by the impurity placed at the higher interface compared with the on-center one. Furthermore, the InN layer is more populated for on-center impurity than for off-center one. This result can be due to the fraction of the electron density extended in the barrier which is removed toward the well under electrostatic interaction. It can be seen also that as the impurity moves far away the structure center in the positive sense, the electron density decreases for all pressure values. This is due to the fact that a fraction of the electron’s population is captured by the defects located in the interface. For example, the density is equal
to 2.105 10¢ and 1.961 10¢ for on-center impurity and off-center case
respectively. From the earlier results, one can speculate that the 2DEG density can only enhance as a function of the donor’s position. In the same trend, the electron density augments via the pressure for all position of the impurity which can be assigned to the conduction band offset increasing. As the pressure
changes from 0 A to 20 A, the electron density varies respectively from
2.105 10¢ to 2.291 10¢ showing 8.8% enhancement. So, the increasing of the pressure forces the density to be less spread in the barrier and to move into the well which
induces a reduction of the 2DEG density. For instance, this latter is equal to 4.235 10 ,
2.753 10 and 1.426 10 for pressure values equal to 0 A, 20 A and 40 A respectively. The drop in the 2DEG density is about 35% and 66% for 0 − 20A
and 0 − 40A pressure ranges respectively which is not suitable for HEMT applications. To better show and clarify that, we present in table 2 the obtained results for three different values of pressure 0, 20, 40 A and position of the impurity 0, uU ⁄2 , uU + uJ ⁄2 .
10
Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included. The obtained results are in quite agreement with those of several authors concerning different semiconductor materials but some discrepancies remain. For example, Gazzah et al. [16] have reported that as the In fraction in (In,Ga)N well channel increases the depth of the triangular QW increases and then the densities of confined electrons and 2DEG are found to be enhanced. The In-dependent 2DEG behavior is completely different of our results even if the In mole and pressure induce both band offset increasing. Incidentally, it should be noted that these discrepancies are due mainly to the electric field neglected in the barrier on the one hand and to the presence of the impurity neglected in all available papers. In the same tendency, Hasan et al. [26] have proved that the sheet carrier density decreases versus In mole composition at (In,Ga)N hetero-interface (Fig. 5 [26]: For more details). This is completely in disagreement with the results reported in Ref. [16]. It found that it is equal respectively to 7.08 10 and 3.23 10 at InN/GaN and .ª .ª hetero-interfaces.
11
Also, it can be attributed to It is important to notice that as the impurity moves far away the structure center in the negative sense of the x-axis, it is expected that the 2DEG density shows a drop while the electron density shows a significant enhancement. Table 2: Electronic properties of InN/GaN hetero-structure for three different values of pressure 0, 20, 40 A and position of the impurity 0, uU ⁄2 , uU + uJ ⁄2 . A A
0
ÃD ∗
0 5
0
9
0 20 20
0
40
−0.728
−0.084
0.391 0.194
0.155
10
1.961 1.895
0.017
2.094
0.234
0.187
2.105
2.291
0
−0.187
-+ 10¢
0.121
0.072
3.348
40
−0.531
0.625
−0.132
0.851
40
−0.297
SK − S-vz W?
2.926 6.174
20
SK W?
4.235
4.873 6.385
2.753
2.161
3.291
0.086
2.341
1.426
0.007
2.024
0.039
2.185
4.602
2.142 3.853
4. conclusion In summary, the impact of an externally applied hydrostatic pressure and the impurity’s position on the 2DEG properties in p-GaN/i-InN/n-GaN hetero-structure with Ga-face is investigated numerically by solving the coupling one-dimensional non linear Schrödinger and Poisson equations self-consistently within the effective mass and one parabolic band approximations. The pressure-dependent internal built-electric field due to spontaneous and piezoelectric polarizations is considered. Two-dimensional electron gas (2DEG) density, created at the InN/GaN upper hetero-interface under the joint effects of polarization charge and conduction band-offset, appears to be highly dependent pressure, size and impurity. Our results reveal that: •
The InN channel is more populated for on-center impurity,
•
The triangular shape QW becomes less deeper versus the pressure,
•
The electron density in triangular shape QW increases with the pressure,
12
•
2DEG density enhances versus the positive impurity’s position and drops versus hydrostatic pressure. Due to the importance of the III-Nitrides materials, we think that present work provides
a new model for significant potential applications such as high efficiency power transistor, laser, light emitting diode, photovoltaic cells, infra-red photo-detectors and would guide further theoretical and experimental research works in group III-nitrides.
13
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List of Figures
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges.
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure. Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included.
17
Ga-face
- -- - - -- -- - - - -
n-GaN
[0001]
i-InN p-GaN
#%
+ + + + + + + +++
2DEG 0
!
- - - - - -- - - - - - - -
&'
!
!"
#$
+ ++++++++
- - - - - - - - --- - - ! -
#%
+ + + + + + + + ++ ++
Substrate (Bulk GaN)
Figure 1: (On-line color) Schematic cross section of p-GaN/i-InN/n-GaN heterostructure showing the polarization charges.
18
Figure 2: (On-line color)The relative dielectric constant, effective masse, band gap energy, conduction band off-set (rectangular form) and internal electric field as a function of the hydrostatic pressure.
19
Figure 3: (On-line color) The conduction band energy diagram versus the thickness, X, along the growth direction for three different potential barriers (Rectangular potential ?J , the electric field effect ?J > ?K , and total potential
? ) under the effect of the hydrostatic
pressure.
20
Figure 4: (On-line color) The electron density according to the thickness, X, along the growth direction for two different hydrostatic pressure values. The effect of shallow donor impurity’s position is included.
21
In order to investigate our model, we start showing the effects of an externally applied pressure on dielectric constant, effective mass, band gap energy, conduction band offset and internal electric field. Figure 2 reveals that the effective mass, the band gap energy and then the band offset increase linearly versus the pressure at least in the studied pressure domain. It is also shown that the relative dielectric constant decreases as a function of the pressure contrary to one. For the internal electric field, two behaviors are obtained limited by pressure value crossover ~30 . For < , the barrier electric field is greater than the well one and they decrease via the pressure. For higher values of pressure > , the well one is greater and they increase according to hydrostatic pressure. These results are in good conformity with those obtained by Baskoutas et al. for ZnO quantum dots and nanorods subjected to externally applied hydrostatic pressure and CdS quantum dots capped by various dielectric matrix materials [43-46].
Conflict of interest statment
We bring to your knowledge that the manuscript has not been previously published and is not currently in press under consideration, or being considered for publication elsewhere. All authors have read and approved the manuscript being submitted and agree to its submission to Physica B: Condensed Matter Journal. We have no conflicts of interest to disclose.
Prof: Haddou El Ghazi ENSAM, H2 University, Casablanca, Morocco