# Determination of Temperature-Dependent Stress State in Thin AlGaN Layer of AlGaN/GaN HEMT Heterostructures by Near-Resonant Raman Scattering

Authors: Yanli Liu; Xifeng Yang; Dunjun Chen; Hai Lu; Rong Zhang; Youdou Zheng
Journal: Advances in Condensed Matter Physics
 (2015)

---

## Abstract

The temperature-dependent stress state in the AlGaN barrier layer of AlGaN/GaN heterostructure grown on sapphire substrate was investigated by ultraviolet (UV) near-resonant Raman scattering. Strong scattering peak resulting from the A1(LO) phonon mode of AlGaN is observed under near-resonance condition, which allows for the accurate measurement of Raman shifts with temperature. The temperature-dependent stress in the AlGaN layer determined by the resonance Raman spectra is consistent with the theoretical calculation result, taking lattice mismatch and thermal mismatch into account together. This good agreement indicates that the UV near-resonant Raman scattering can be a direct and effective method to characterize the stress state in thin AlGaN barrier layer of AlGaN/GaN HEMT heterostructures.


---

## Body

## 1. Introduction

Recently, AlGaN/GaN heterostructures have attracted considerable attention due to their potential use in high-power, high-temperature, and high-frequency electronic devices [1–5]. The high-temperature application is one important advantage of the AlGaN/GaN-based devices over GaAs-based and Si devices [6–8]. It is well known that the strain and stress in the AlGaN barrier layer due to lattice mismatch (LMM) and thermal mismatch between AlGaN and the underlying layers have important effect on the formation and transport properties of two-dimensional electron gas (2DEG) in AlGaN/GaN heterostructures [9–11]. Therefore, the investigation on the temperature dependence of stress or strain in AlGaN barrier layer is necessary for understanding the temperature-dependent electrical properties of AlGaN/GaN heterostructure and improving the reliability of the AlGaN/GaN based devices.In previous reports, the strain or stress in the AlGaN layer of AlGaN/GaN heterostructures was characterized typically by using X-ray diffraction [12, 13]. However, the reflection peaks of some asymmetric planes in AlGaN barrier layer are always invisible due to the thin thickness and poor interference of the plane [12]. So, the in-plane lattice constant and the biaxial strain of AlGaN layer cannot be measured directly using this method. Raman spectroscopy is an effective method for the residual stress measurement of crystal films. However, in the prior studies on Raman measurements of AlGaN/GaN heterostructures, the visible (532 nm, 488 nm) Raman spectroscopy is mainly used to detect the stress and 2DEG channel temperature by measuring the phonon frequency of GaN averaged over the whole buffer layer [14, 15], which cannot reflect directly the stress state in the AlGaN barrier layer.In this work, we investigated the temperature-dependent stress state in the thin AlGaN barrier layer of AlGaN/GaN heterostructure by means of UV near-resonant Raman scattering. The Raman measured results are in good agreement with those from theoretical calculation, taking LMM and thermal mismatch into account together.


## 2. Experiment

The AlGaN/GaN heterostructure used in this study was grown on sapphire substrate by metal-organic chemical vapor deposition. The sample consists of a 2μm thick unintentionally doped GaN layer, a 1 nm thick AlN spacer layer, and a 25 nm thick undoped Al0.27Ga0.73N barrier layer.The Raman scattering spectra were recorded by using an HR800 Jobin-Yvon spectrometer equipped with a liquid-nitrogen-cooled charge-coupled device in a backscattering geometry. A 325 nm He-Cd laser was used as an excitation source. A temperature stage with a quartz window was used to heat the sample from 80 to 600 K in flowing nitrogen.


## 3. Results and Discussion

### 3.1. The Stress in AlGaN Barrier Layer Determined by Near-Resonant Raman Scattering

The temperature-dependent UV Raman spectra of AlGaN/GaN heterostructure are shown in Figure1. Compared to the visible Raman spectrum of AlGaN/GaN heterostructure [19], a new peak near 785 cm−1 occurs in the UV Raman spectrum. This peak corresponds to the A1(LO) mode of the AlGaN layer according to the Al-composition dependent A1(LO) phonon frequency [20]. As shown in Figure 1, the A1(LO) phonon mode of the AlGaN layer shows enhancement effect in intensity and red shift in frequency with increasing temperature.Figure 1
(a) UV Raman spectra of AlGaN/GaN heterostructure with varying temperature and (b) the local amplified graph of the A1(LO) mode of AlGaN.
(a)
(b)The temperature dependence of the intensity of the A1(LO) phonon mode in AlGaN can be explained by studying the resonant Raman scattering in the structure with varying temperatures. By solving the Schrodinger and Poisson equations self-consistently using the Silvaco Atlas software, we can get the band diagram of the structure with varying temperature. The band gaps of AlGaN and GaN in the temperature range of 80–600 K are shown in Figure 2. The band gap of the AlGaN barrier layer is closer to the excitation energy than that of the GaN layer in the whole temperature range. The resonant Raman scattering arises from the AlGaN barrier layer. The band gap of the AlGaN barrier layer decreases and becomes closer and closer to the excitation energy with the increasing temperature. So, the intensity of the A1(LO) phonon mode of AlGaN increases with the increasing temperature.Figure 2
Temperature-dependent band gaps of the AlGaN/GaN heterostructure.There are several reasons for the frequency shift of phonon mode with varying temperature. The anharmonicity of the crystal lattice gives rise to the thermal expansion of lattice and phonon decay [21, 22]. The frequency shifts due to these two effects are denoted as Δ
ω
e
(
T
) and Δ
ω
d
(
T
), respectively. In an isotropic approximation, the term Δ
ω
e
(
T
) is given by [21, 22](1)
Δ
ω
e
T
=
-
ω
0
γ
∫
0
T
α
c
T
~
+
2
α
a
T
~
d
T
~
,where α
a and α
c are the temperature-dependent thermal expansion coefficients along a- and c-directions, ω
0 is the harmonic frequency of the optical phonon mode, and γ is the Grüneisen parameter. Here, the thermal expansion coefficient with variable temperature was described within multifrequency Einstein model [16]. Consider(2)
α
=
∑
i
=
1
n
X
i
θ
i
/
T
2
exp
⁡
θ
i
/
T
exp
⁡
θ
i
/
T
-
1
2
,where X
i and θ
i are model parameters listed in Table 1.Table 1
Model parameters for AlGaN, GaN, and substrate [16].
Material
θ
1 (K)
θ
2 (K)
θ
3 (K)
θ
4 (K)
X
1 (10−7/K)
X
2 (10−7/K)
X
3 (10−7/K)
X
4 (10−7/K)
GaN
a
75
581.25
1684.375
0.487
52.152
4.21
c
75
590.625
1675
0.621
47.312
1.125
AlN
a
125
600
1852.5
−4.348
44.074
35.056
c
100
528.75
1723.75
−5.174
29.857
39.565
Al0.27Ga0.73N
a
88.5
586.313
1729.77
−0.0818
4.997
1.254
c
81.75
573.919
1688.16
−0.0943
4.26
1.1504
Sapphire
a
135
565.625
1231.25
5468.75
1.2176
53.401
35.613
23.661
c
135
598.438
1468.75
5198.438
2.856
72.079
23.202
29.087Taking into account symmetric decays of the zone-center phonons into two phonons and three phonons with frequenciesω
0
/
2 and ω
0
/
3, respectively, the term Δ
ω
d
(
T
) can be described by [23](3)
Δ
ω
d
T
=
A
1
+
2
n
T
,
ω
0
2
+
B
1
+
3
n
T
,
ω
0
3
+
3
n
2
T
,
ω
0
3
,where A and B are constants and n
(
T
,
ω
)
=
[
exp
⁡
(
ħ
ω
/
k
B
T
)
-
1
]
-
1 is the Bose-Einstein distribution function which describes the thermal occupation number of phonon states. The parameters γ, ω
0, A, and B for AlGaN are 1.56, 793 cm−1, −4.646 cm−1, and −0.115 cm−1, respectively. The contributions of the thermal expansion of lattice and phonon decay effect to the frequency shift of A1(LO) mode in AlGaN are shown in Figure 3(a).Figure 3
(a) Raman shift of A1(LO) mode of AlGaN as a function of temperature together with the contribution of thermal expansion of lattice and phonon decay effects. (b) Temperature-dependent stress in AlGaN and its contribution to the Raman shift.
(a)
(b)Besides the phonon frequency shift due to the thermal expansion of lattice and the decay of optical phonon into phonon with lower energy, the temperature-dependent stress in crystalline also contributes to the frequency shift, which is denoted asΔ
ω
s
(
T
) [24]. Consider(4)
Δ
ω
s
T
=
2
a
~
λ
σ
x
x
+
b
~
λ
σ
z
z
.For A1(LO) mode in Al0.27Ga0.73N, the phonon deformation potentials a
~
λ, b
~
λ equal 1.001 and −1.576 cm−1/GPa, respectively [24]. The temperature-dependent phonon frequency should be written as(5)
ω
T
=
ω
0
+
Δ
ω
e
T
+
Δ
ω
d
T
+
Δ
ω
s
T
.According to the measured ω
(
T
) and the calculated Δ
ω
e
(
T
) and Δ
ω
d
(
T
) as shown in Figure 3(a), the temperature-dependent Δ
ω
s
(
T
) and the corresponding stress in AlGaN can be obtained. The results are shown in Figure 3(b). The measured stress in AlGaN increased from 1.88 GPa at 80 K to 2.28 GPa at 600 K.


### 3.2. Theoretical Calculation

In order to identify the accuracy of the stress determination in thin AlGaN barrier layer by analyzing near-resonant Raman spectroscopy, we calculate the temperature-dependent stress state of AlGaN layer theoretically by applying a stress model with multilayer structure. The total stress in the AlGaN barrier layer of AlGaN/GaN heterostructure grown on sapphire substrate consists of two parts: one is thermal stress due to thermal mismatch between AlGaN and the underlying GaN/substrate and the other is induced by LMM between AlGaN and GaN.Figure4 shows the analysis of thermal stress generated in multilayer structure [25]. An elastic multilayer structure at growth temperature T
g is shown schematically in Figure 4(a), where i denote layer number. When temperature decreases Δ
T, there are unconstrained strains in different layers. Hence, the free thermal strain, α
i
Δ
T, is generated in this layer i, as shown in Figure 4(b). Then, in order to achieve displacement compatibility, uniform tensile/compressive stresses are imposed on the individual layers (Figure 4(c)). Finally, the whole structure bends due to the asymmetric stresses in the multilayer structure (Figure 4(d)).Figure 4
Schematic of the generation of thermal stress in multilayer structure.
(a)
(b)
(c)
(d)Based on the logical analysis described in Figure4, the thermal stress in the AlGaN/GaN/sapphire structure can be calculated using the analytical model proposed by Hsueh and Evans [26] which decomposes thermal strain into a uniform component and a bending component. The thermal stress in AlGaN by taking a first-order approximation (i.e., ignoring terms with orders of t
i higher than one) is expressed as follows [25]:(6)
σ
thermal
=
∫
T
g
T
Y
2
α
a
,
s
-
α
a
,
2
+
4
Y
1
t
1
α
a
,
1
-
α
a
,
s
Y
s
t
s
+
4
Y
2
t
2
α
a
,
2
-
α
a
,
s
Y
s
t
s
d
T
~
,where the subscripts s, 1, and 2 denote the substrate, GaN, and AlGaN, respectively, Y is biaxial modulus given in terms of elastic constants C
i
j as Y
=
C
11
+
C
12
-
2
C
13
2
/
C
33, and t is layer thickness. Here, the elastic constants, biaxial modulus, and lattice constant of AlGaN are calculated from Vegard’s law. The above parameters are listed in Table 2.Table 2
Parameters used in theoretical calculation.
Material
C
11 (GPa)
C
12 (GPa)
C
13 (GPa)
C
33 (GPa)
Y (GPa)
t (μm)
a (Å)
GaNa
390
145
106
398
478.5
2
3.206
AlNb
410
149
99
389
508.6
3.131
Al0.27Ga0.73N
395.5
146
104
395.6
486.7
0.025
3.1858
sapphire
496
164
115
498
606.9
800
aPolian et al. [17].
bMcNeil et al. [18].Based on the above analysis, the temperature-dependent thermal stress in AlGaN layer can be calculated numerically. The calculated result as shown in the insert of Figure5 indicates that the biaxial compressive stress in AlGaN layer decreases with the increasing temperature in the temperature range of 80–600 K below growth temperature.Figure 5
Temperature-dependent stress in AlGaN barrier layer determined by Raman scattering and theoretical calculation. The inset shows the temperature dependence of thermal stress in AlGaN.Besides the thermal stress, the stress due to LMM between AlGaN and GaN also contributes to the total stress in AlGaN. This stress can be calculated using the following equations [10]:(7)
ε
x
x
=
a
1
-
a
2
a
2
=
a
1
a
2
-
1
,
σ
LMM
=
C
11
+
C
12
-
2
C
13
2
C
33
ε
x
x
,where a
1, a
2 are lattice constants of strain-free GaN and Al0.27Ga0.73N in c-plane, respectively. C
i
j is elastic constant of Al0.27Ga0.73N. These parameters are also listed in Table 2. The stress due to LMM between AlGaN and GaN is 3.272 GPa. The total stress in AlGaN which is the sum of σ
thermal and σ
LMM with varying temperature is also shown in Figure 5. The total stress increases from 1.89 GPa at 80 K to 2.27 GPa at 600 K, which is consistent with the result obtained from near-resonant Raman scattering.


## 3.1. The Stress in AlGaN Barrier Layer Determined by Near-Resonant Raman Scattering

The temperature-dependent UV Raman spectra of AlGaN/GaN heterostructure are shown in Figure1. Compared to the visible Raman spectrum of AlGaN/GaN heterostructure [19], a new peak near 785 cm−1 occurs in the UV Raman spectrum. This peak corresponds to the A1(LO) mode of the AlGaN layer according to the Al-composition dependent A1(LO) phonon frequency [20]. As shown in Figure 1, the A1(LO) phonon mode of the AlGaN layer shows enhancement effect in intensity and red shift in frequency with increasing temperature.Figure 1
(a) UV Raman spectra of AlGaN/GaN heterostructure with varying temperature and (b) the local amplified graph of the A1(LO) mode of AlGaN.
(a)
(b)The temperature dependence of the intensity of the A1(LO) phonon mode in AlGaN can be explained by studying the resonant Raman scattering in the structure with varying temperatures. By solving the Schrodinger and Poisson equations self-consistently using the Silvaco Atlas software, we can get the band diagram of the structure with varying temperature. The band gaps of AlGaN and GaN in the temperature range of 80–600 K are shown in Figure 2. The band gap of the AlGaN barrier layer is closer to the excitation energy than that of the GaN layer in the whole temperature range. The resonant Raman scattering arises from the AlGaN barrier layer. The band gap of the AlGaN barrier layer decreases and becomes closer and closer to the excitation energy with the increasing temperature. So, the intensity of the A1(LO) phonon mode of AlGaN increases with the increasing temperature.Figure 2
Temperature-dependent band gaps of the AlGaN/GaN heterostructure.There are several reasons for the frequency shift of phonon mode with varying temperature. The anharmonicity of the crystal lattice gives rise to the thermal expansion of lattice and phonon decay [21, 22]. The frequency shifts due to these two effects are denoted as Δ
ω
e
(
T
) and Δ
ω
d
(
T
), respectively. In an isotropic approximation, the term Δ
ω
e
(
T
) is given by [21, 22](1)
Δ
ω
e
T
=
-
ω
0
γ
∫
0
T
α
c
T
~
+
2
α
a
T
~
d
T
~
,where α
a and α
c are the temperature-dependent thermal expansion coefficients along a- and c-directions, ω
0 is the harmonic frequency of the optical phonon mode, and γ is the Grüneisen parameter. Here, the thermal expansion coefficient with variable temperature was described within multifrequency Einstein model [16]. Consider(2)
α
=
∑
i
=
1
n
X
i
θ
i
/
T
2
exp
⁡
θ
i
/
T
exp
⁡
θ
i
/
T
-
1
2
,where X
i and θ
i are model parameters listed in Table 1.Table 1
Model parameters for AlGaN, GaN, and substrate [16].
Material
θ
1 (K)
θ
2 (K)
θ
3 (K)
θ
4 (K)
X
1 (10−7/K)
X
2 (10−7/K)
X
3 (10−7/K)
X
4 (10−7/K)
GaN
a
75
581.25
1684.375
0.487
52.152
4.21
c
75
590.625
1675
0.621
47.312
1.125
AlN
a
125
600
1852.5
−4.348
44.074
35.056
c
100
528.75
1723.75
−5.174
29.857
39.565
Al0.27Ga0.73N
a
88.5
586.313
1729.77
−0.0818
4.997
1.254
c
81.75
573.919
1688.16
−0.0943
4.26
1.1504
Sapphire
a
135
565.625
1231.25
5468.75
1.2176
53.401
35.613
23.661
c
135
598.438
1468.75
5198.438
2.856
72.079
23.202
29.087Taking into account symmetric decays of the zone-center phonons into two phonons and three phonons with frequenciesω
0
/
2 and ω
0
/
3, respectively, the term Δ
ω
d
(
T
) can be described by [23](3)
Δ
ω
d
T
=
A
1
+
2
n
T
,
ω
0
2
+
B
1
+
3
n
T
,
ω
0
3
+
3
n
2
T
,
ω
0
3
,where A and B are constants and n
(
T
,
ω
)
=
[
exp
⁡
(
ħ
ω
/
k
B
T
)
-
1
]
-
1 is the Bose-Einstein distribution function which describes the thermal occupation number of phonon states. The parameters γ, ω
0, A, and B for AlGaN are 1.56, 793 cm−1, −4.646 cm−1, and −0.115 cm−1, respectively. The contributions of the thermal expansion of lattice and phonon decay effect to the frequency shift of A1(LO) mode in AlGaN are shown in Figure 3(a).Figure 3
(a) Raman shift of A1(LO) mode of AlGaN as a function of temperature together with the contribution of thermal expansion of lattice and phonon decay effects. (b) Temperature-dependent stress in AlGaN and its contribution to the Raman shift.
(a)
(b)Besides the phonon frequency shift due to the thermal expansion of lattice and the decay of optical phonon into phonon with lower energy, the temperature-dependent stress in crystalline also contributes to the frequency shift, which is denoted asΔ
ω
s
(
T
) [24]. Consider(4)
Δ
ω
s
T
=
2
a
~
λ
σ
x
x
+
b
~
λ
σ
z
z
.For A1(LO) mode in Al0.27Ga0.73N, the phonon deformation potentials a
~
λ, b
~
λ equal 1.001 and −1.576 cm−1/GPa, respectively [24]. The temperature-dependent phonon frequency should be written as(5)
ω
T
=
ω
0
+
Δ
ω
e
T
+
Δ
ω
d
T
+
Δ
ω
s
T
.According to the measured ω
(
T
) and the calculated Δ
ω
e
(
T
) and Δ
ω
d
(
T
) as shown in Figure 3(a), the temperature-dependent Δ
ω
s
(
T
) and the corresponding stress in AlGaN can be obtained. The results are shown in Figure 3(b). The measured stress in AlGaN increased from 1.88 GPa at 80 K to 2.28 GPa at 600 K.


## 3.2. Theoretical Calculation

In order to identify the accuracy of the stress determination in thin AlGaN barrier layer by analyzing near-resonant Raman spectroscopy, we calculate the temperature-dependent stress state of AlGaN layer theoretically by applying a stress model with multilayer structure. The total stress in the AlGaN barrier layer of AlGaN/GaN heterostructure grown on sapphire substrate consists of two parts: one is thermal stress due to thermal mismatch between AlGaN and the underlying GaN/substrate and the other is induced by LMM between AlGaN and GaN.Figure4 shows the analysis of thermal stress generated in multilayer structure [25]. An elastic multilayer structure at growth temperature T
g is shown schematically in Figure 4(a), where i denote layer number. When temperature decreases Δ
T, there are unconstrained strains in different layers. Hence, the free thermal strain, α
i
Δ
T, is generated in this layer i, as shown in Figure 4(b). Then, in order to achieve displacement compatibility, uniform tensile/compressive stresses are imposed on the individual layers (Figure 4(c)). Finally, the whole structure bends due to the asymmetric stresses in the multilayer structure (Figure 4(d)).Figure 4
Schematic of the generation of thermal stress in multilayer structure.
(a)
(b)
(c)
(d)Based on the logical analysis described in Figure4, the thermal stress in the AlGaN/GaN/sapphire structure can be calculated using the analytical model proposed by Hsueh and Evans [26] which decomposes thermal strain into a uniform component and a bending component. The thermal stress in AlGaN by taking a first-order approximation (i.e., ignoring terms with orders of t
i higher than one) is expressed as follows [25]:(6)
σ
thermal
=
∫
T
g
T
Y
2
α
a
,
s
-
α
a
,
2
+
4
Y
1
t
1
α
a
,
1
-
α
a
,
s
Y
s
t
s
+
4
Y
2
t
2
α
a
,
2
-
α
a
,
s
Y
s
t
s
d
T
~
,where the subscripts s, 1, and 2 denote the substrate, GaN, and AlGaN, respectively, Y is biaxial modulus given in terms of elastic constants C
i
j as Y
=
C
11
+
C
12
-
2
C
13
2
/
C
33, and t is layer thickness. Here, the elastic constants, biaxial modulus, and lattice constant of AlGaN are calculated from Vegard’s law. The above parameters are listed in Table 2.Table 2
Parameters used in theoretical calculation.
Material
C
11 (GPa)
C
12 (GPa)
C
13 (GPa)
C
33 (GPa)
Y (GPa)
t (μm)
a (Å)
GaNa
390
145
106
398
478.5
2
3.206
AlNb
410
149
99
389
508.6
3.131
Al0.27Ga0.73N
395.5
146
104
395.6
486.7
0.025
3.1858
sapphire
496
164
115
498
606.9
800
aPolian et al. [17].
bMcNeil et al. [18].Based on the above analysis, the temperature-dependent thermal stress in AlGaN layer can be calculated numerically. The calculated result as shown in the insert of Figure5 indicates that the biaxial compressive stress in AlGaN layer decreases with the increasing temperature in the temperature range of 80–600 K below growth temperature.Figure 5
Temperature-dependent stress in AlGaN barrier layer determined by Raman scattering and theoretical calculation. The inset shows the temperature dependence of thermal stress in AlGaN.Besides the thermal stress, the stress due to LMM between AlGaN and GaN also contributes to the total stress in AlGaN. This stress can be calculated using the following equations [10]:(7)
ε
x
x
=
a
1
-
a
2
a
2
=
a
1
a
2
-
1
,
σ
LMM
=
C
11
+
C
12
-
2
C
13
2
C
33
ε
x
x
,where a
1, a
2 are lattice constants of strain-free GaN and Al0.27Ga0.73N in c-plane, respectively. C
i
j is elastic constant of Al0.27Ga0.73N. These parameters are also listed in Table 2. The stress due to LMM between AlGaN and GaN is 3.272 GPa. The total stress in AlGaN which is the sum of σ
thermal and σ
LMM with varying temperature is also shown in Figure 5. The total stress increases from 1.89 GPa at 80 K to 2.27 GPa at 600 K, which is consistent with the result obtained from near-resonant Raman scattering.


## 4. Conclusions

The temperature-dependent stress state in the AlGaN barrier layer of AlGaN/GaN heterostructure was investigated by UV near-resonant Raman scattering. Strong scattering peak resulting from the A1(LO) phonon mode of AlGaN is observed under near-resonance condition. The temperature-dependent stress in the AlGaN layer determined by the resonance Raman spectra is consistent with the theoretical calculation result. This good agreement indicates that the UV near-resonant Raman scattering can be a direct and effective method to characterize the stress state in thin AlGaN barrier layer of AlGaN/GaN HEMT heterostructures.



---

*Source: 918428-2015-03-22.xml*