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The effect of hydrostatic pressure on material parameters and electrical transport properties in bulk GaN
Physics Letters A 373 (2009) 1773–1776
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The effect of hydrostatic pressure on material parameters and electrical transport properties in bulk GaN Hosein Eshghi ∗ Department of Physics, Shahrood University of Technology, Shahrood 316-36155, Iran
a r t i c l e
i n f o
Article history: Received 18 January 2009 Received in revised form 2 March 2009 Accepted 9 March 2009 Available online 14 March 2009 Communicated by A.R. Bishop PACS: 72.10.-d 73.20.Hb 73.50.Dn
a b s t r a c t Experimental data for temperature dependence of electron transport properties in a bulk, low dislocation density, GaN sample at atmospheric pressure and 7.1 kbar have been presented. The data are representing a weak hydrostatic pressure dependence. Our quantitative analysis on its material parameters including: high and low dielectric constants (ε∞ , εs ), longitudinal and transverse optical phonons (ωLO , ωTO ), and electronic effective mass (me∗ ) show a small fractional change of −0.12, −0.14, 0.05, 0.058 and 0.089 (percent/kbar), respectively. These results are confirmed by the Hall-effect data analysis on the basis of charge neutrality condition and various scattering mechanisms. © 2009 Elsevier B.V. All rights reserved.
Keywords: Semiconductor GaN Material parameters Hydrostatic pressure Transport properties
1. Introduction One method often used to characterize semiconductors is to measure the free-carrier mobility and concentration, and to analyze the results obtained to determine the impurity content of the material. However, a relatively new approach has been to investigate electron transport as a function of hydrostatic pressure [1]. This enables parameters varied and thus provides a further test of the transport theories for electrons in a number of semiconductors. In this Letter we report the electron mobility and carrier concentration both as a function of temperature and pressure of a bulk GaN layer, and analyze the results.
with pressure. According to the k . p theory, the pressure dependence of the electronic effective mass is given by [2]: m0 m∗ ( P )
=1+
C E g(P )
(1)
where C is constants for different materials, and E g is the band gap with a pressure coefficient α = dE g /d P . For the pressure dependence of high frequency dielectric constant, we use the equation given by Goni et al. [3]:
∂ ln ε∞ 5(ε∞ − 1) ≈ (0.9 − f i ), ∂ ln V 3ε∞
(2)
2. Theory
where f i is the ionicity of the material under pressure and V is the volume of the material. It satisfies approximately the following linear relation in the low pressure range [4]:
2.1. Pressure coefficients
B ( P ) = −V
As applying hydrostatic pressure on the material, it changes the structure of energy band and the electronic effective mass change
*
Tel.: +98 9122733496; fax: +98 2733335270. E-mail address: [email protected].
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dP dV
= B 0 + B 0 . P .
(3)
Here B is the bulk modulus and is assumed being linear with the pressure for not very high pressures, B 0 and B 0 are the bulk modulus at zero and its first order pressure deviation at zero pressure, respectively. Thus we can write the pressure coefficient for the high frequency dielectric constant as:
∂ ε∞ −5(ε∞ − 1) = (0.9 − f i ). ∂P 3B
(4)
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H. Eshghi / Physics Letters A 373 (2009) 1773–1776
The static dielectric constant is then determined by the famous Lyddane–Sachs–Teller relation:
εs = ε∞
h¯ ωLO
2 (5)
,
h¯ ωTO
2.2.2.3. Polar optical phonon scattering Ehrenreich has derived an analytical expression for the mobility limit due to polar optical scattering as follows [12]:
μPOP = 0.199
γ = B0
(6)
ω0 ∂ P
we can obtain the optical phonon energy dependence on pressure. In this equation ω and ω0 are the optical phonon frequencies under hydrostatic pressure P = 0 and P = 0, respectively. The values of the Grüneisen parameter γ and the other parameters used in our pressure calculations are given in Table 1. 2.2. Analysis of the Hall-effect measurements 2.2.1. The charge neutrality condition The temperature dependence of the electron concentration can be employed to derive parameters for characterizing shallow donors. For n-type semiconductors, the electron concentration, n, is given by charge neutrality condition [5]: n=
Nd 1+
ngn Nc
Ed BT
exp( k
)
− Na
2.2.2.1. Deformation potential scattering The mobility limited by acoustic-mode deformation potential scattering is [6]:
μac =
π
ρυ
(8)
where ρ is the crystal mass density, mn∗ is the electron effective mass, υs is the average sound speed in the crystal, and E dp is the deformation potential. 2.2.2.2. Piezoelectric scattering The mobility limited by piezoelectric scattering is given by [6]:
μPZ =
16(2π )1/2 ρυs2 h¯ 2 q
3(qh pz /ε0 εs )2 (mn∗ )3/2 (k B T )1/2
where h pz is the piezoelectric constant.
3/ 2
mn∗
q∗
300
m0
h¯ ω
kB T (10)
where q∗
= M ω a ε0 (1/ε∞ − 1/εs ) and G (¯hω/k B T ) is a slow varying function of order unity; we have taken it as constant and equal to mass (1.936 × 10−26 kg in GaN). √0.5.2 M is the reduced−ion 29 V a [= ( 3a c )/4 = 2.283 × 10 m3 ] is the volume for a Ga and N ion pair. 2V
2.2.2.4. Ionized impurity scattering The mobility limited by ionized impurity has been formulated by Brooks and Herring after taking into account the screening effect of both the free carriers and the impurity ions, and has the following expression [13]:
μi−im =
128(2π )1/2 ε02 εs2 (k B T )1/2 q3 (m∗ )1/2 (n + 2N n
a)
ln(1 + b) −
b
−1 (11)
1+b
where b=
24mn∗ ε0 εs (k B T ) q2 h¯ 2 n
and n = n +
( N d − N a − n)(n + N a ) Nd
.
2.2.2.5. Neutral impurity scattering The neutral impurity mobility has been modeled by Erginsoy [14] as:
2.2.2. Scattering processes in n-type GaN Scattering mechanisms including ionized and neutral impurities, dislocations, deformation potential and piezoelectric potential of acoustic phonons, and polar optical phonons are the main mechanisms that limit the electron mobility in GaN in the temperature range of interest here (60–300 K). An accurate calculation of the mobility requires numerical iterative solution of the Boltzmann transport equation. In this work, a simpler analytical approach is used. The mobility limit due to each individual scattering process is calculated independently with the corresponding analytical expressions. To a first approximation, the total mobility with different scattering events is obtained according to Matthiesens’s rule, μtot ∼ = 1/ i 1/μi .
2 4 2(2 ) ¯ q sh 2 ∗ 5 / 2 3E dp (mn ) (k B T )3/2
2
(7)
where N d is the donor density, E d its activation energy, gn the degeneracy of the donor states, N a the acceptor concentration, and N c the conduction band effective density of states.
1/ 2
q
× 1022 M . 1023 V a 10−13 ω p eh¯ ω/k B T − 1 . G
where h¯ ωTO and h¯ ωLO are the transverse optical (TO) and longitudinal optical (LO) phonon energies, respectively. According to the Grüneisen parameter: 1 ∂ω
1/2
T
(9)
μn−im =
q
me∗ /m0
20a B h¯
εs Nn−im
(12)
where N n−im and a B are the neutral impurity concentration and Bohr radius in a hydrogen atom, respectively. 2.2.2.6. Dislocation scattering Podor [15] used a screened potential and found the dislocation scattering limited mobility as:
√
μdis =
30 2π εs2 ε02 d2 (k B T )3/2 ∗ 1/ 2
q3 N dis f 2 L D mn
(13)
where the quantities d, N dis , f , and L D are the distance between the defect centers along the dislocation line (∼ 5 Å in GaN), the dislocation density, the occupation rate of these sites (assuming a value of unity), and the Debye screening length ( εs ε0 k B T /q2 n), respectively. 3. Results and discussion 3.1. Electron concentration analysis Our interested sample is a non-intentional doped layer of 1.34 ± 0.01 μm thick which is grown by MOCVD method on sapphire substrate. Fig. 1 shows Hall-effect results, including the temperature dependence of the electron density and the electron mobility (inset), at the atmospheric pressure (open circles) and 7.1 kbar (filled squares). As is obvious with increasing the pressure there is a small deviation between two sets of data. Here we aim to investigate this phenomenon to find out the influence of pressure on material parameters which can be the reason for the lower electron mobility and faster change of electron density in the conduction band. The fitted curves show the theoretical predictions for the Hall electron density and mobility in this material. As is clear, except at low temperatures (below 80 K) the theory agrees quite well with
H. Eshghi / Physics Letters A 373 (2009) 1773–1776
1775
Table 1 GaN parameters used in the computation. Phonon energies are measured in units of meV, electronic effective mass in units of the bare electron mass m0 , bulk modulus B 0 in units of kbar, energy band gap in units of eV, pressure coefficient of band gap α = dE g /d P in units of meV/kbar. h¯ ωLO
ε∞ a
5.41 a
mn∗
h¯ ωTO b
b
92.99
Karch et al., Ref. [7].
69.43 b
fi c
0.19
Wagner et al., Ref. [8].
c
B0 d
e
0.5
Wang et al., Ref. [9].
2054 d
Phillips, Ref. [10].
Fig. 1. Hall electron concentration and mobility (the inset) vs. temperature. ◦ and experimental data, also dashed and solid fitted curves in both figures corresponds to our calculations at atmospheric and 7.1 kbar pressures, respectively. Table 2 Other material parameters of GaN (atmospheric pressure). Parameter
Value
Ref.
Sound speed (υs ), m/s Piezoelectric constant (h pz ), C/m2 Density (ρ ), kg/m3 Deformation potential (E dp ), eV Lattice constant (a), Å
6.59 × 103 0.5 6.1 × 103 12 5.125
[5] [5] [16] [17] [18]
the experimental data at higher temperatures while the conduction band electrons controls the conductivity of the material. The deviation between the conduction band transport and the experimental data at low T s has been related to the parallel conduction path in the impurity band and/or the interfacial layer [19–22], which is not under our investigation in this Letter. To describe the variation of electron density vs. temperature, Fig. 1, we used the usual activated donor process, Eq. (7), first for atmospheric and then for 7.1 kbar pressure data. Using the least square fit method we found at ambient pressure the best fitting parameters are: N d = 1 × 1018 cm−3 , E d = 7 meV and N a = 1.5 × 1017 cm−3 . For the 7.1 kbar pressure data, considering the same concentrations for N d and N a as the above values, we found the activation energy should be 7.2 ± 0.1 meV. 3.2. Electron mobility analysis To complete the modeling of electron transport properties first we have used the atmospheric pressure of μ vs. T data, Fig. 2. Here we have shown the most important limited mobility components, Eqs. (8)–(13), and the total conduction band mobility μc (tot). The material parameters of GaN used for the mobility calculations, in addition to those in Table 1 are listed in Table 2. The dislocation density, N dis = 1 × 109 cm−2 , as the fitting parameter, seems is not significant in this sample in our interested temperature range. To describe the temperature dependence of electron mobility at the higher hydrostatic pressure we need to find the pressure
e
B 0
Eg
α
γLO
γTO
4.6e
3.3c
3.6e
1.02b
1.19b
Wei et al., Ref. [11].
Fig. 2. Temperature dependence of electron mobility at atmospheric pressure. The curves show various scattering limited mechanisms and the total theoretical electron mobility (see the text). Table 3 Results of calculations for relative variations of some GaN material parameters with pressure. 1 dε∞ ε∞ d P
1 dεs εs d P
1 dωLO ωLO d P
1 dωTO ωTO d P
∗ 1 dme me∗ d P −4
(10−3 /kbar)
(10−3 /kbar)
(10−4 /kbar)
(10−4 /kbar)
(10
−1.2
−1.38
4.97
5.8
8.9
/kbar)
variations of the intrinsic material (GaN) parameters. Results of our calculations, based on Eqs. (1)–(6), for the rate of variations for low and high frequency dielectric constants, also LO and TO phonon energies and electron effective mass as a function of hydrostatic pressure are summarized in Table 3 and are shown in Figs. 3 and 4. Considering these variations in the material parameters involving in the analytical expressions for various scattering mechanisms at atmospheric pressure, we found the total conduction band mobility at 7.1 kbar, shown by dashed line in the inset of Fig. 1, is well fitted to experimental data. The accuracy of our analysis can also be examined using the simple hydrogenic model in calculating the activation energy of donor impurities, i.e. E d = 13.6(m∗ /εs2 m0 ) eV. According to this relation the pressure variation of εs from 9.705 at atmospheric pressure to 9.61 at 7.1 kbar tend to a relative variation in this quantity, i.e. {[ E d ( P ) − E d (0)]/ E d (0)} of about 3%. This is close to that we found from n vs. T data analysis in Section 3.1. 4. Conclusion Data have been presented for the temperature dependence of the electron concentration and mobility in a bulk GaN layer at ambient pressure and 7.1 kbar hydrostatic pressure. The Hall data represent a poor pressure variation. Analyses show that the main reason for this effect can be attributed mainly to its big bulk modulus, which in turn affects on its material parameters pressure dependence (Table 3).
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H. Eshghi / Physics Letters A 373 (2009) 1773–1776
Fig. 3. Our analytical calculations for pressure dependence of material parameters in GaN for (a) dielectric constants and (b) longitudinal and transverse optical phonon energies up to 10 kbar, based on Eqs. (2)–(6).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Fig. 4. The relative effective mass vs. hydrostatic pressure, based on Eq. (1), in the bulk GaN.
Acknowledgements The author would thank Dr. D. Lancefield, at the University of Surrey, UK, for his helps with the Hall measurements.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
J.R. Hayes, D. Patel, A.R. Adams, P.D. Greene, J. Electron. Mater. 11 (1982) 155. D.Z.Y. Ting, Y.C. Chang, Phys. Rev. B 36 (1987) 4359. A.R. Goni, K. Syassen, M. Cardona, Phys. Rev. B 41 (1990) 10104. O.L. Anderson, J. Phys. Chem. Solids 27 (1966) 547. H. Morkoc, Nitride Semiconductors and Devices, Springer-Verlag, Heidelberg/Berlin/New York, 1999. D.A. Anderson, N. Aspley, Semicond. Sci. Technol. 1 (1986) 187. K. Karch, F. Bechstedt, T. Pletl, Phys. Rev. B 56 (1997) 3560. J.M. Wagner, J. Bechstedt, Phys. Rev. B 62 (2000) 4526. H. Wang, G.A. Farias, V.N. Freire, Phys. Rev. B 60 (1997) 5705. J.C. Phillips, Bonds and Bands in Semiconductors, Academic Press, San Diego, 1973. S.H. Wei, A. Zunger, Phys. Rev. B 60 (1999) 5404. H. Ehrenreich, J. Phys. Chem. Solids 8 (1959) 130. H. Brooks, Phys. Rev. 83 (1951) 879. C. Erginsoy, Phys. Rev. 79 (1950) 1013. B. Podor, Phys. Status Solidi 16 (1966) K167. M. Fonciulli, T. Lei, T.D. Moustakas, Phys. Rev. B 48 (1993) 48. A.D. Bykhovski, V.V. Kaminski, M.S. Shur, Q.C. Chen, M.A. Khan, Appl. Phys. Lett. 68 (1996) 818. O. Lagerstedt, B. Monemar, Phys. Rev. B 19 (1979) 3064. H. Eshghi, D. Lancefield, Phys. Status Solidi B 216 (1999) 733. J.J. Harris, K.J. Lee, Phys. Status Solidi A 176 (1999) 363. Z. Bougrioua, J.L. Farvacque, Phys. Status Solidi B 216 (1999) 571. D.K. Gaskill, A.E. Wickenden, J. Electron. Mater. 24 (1995) 1525.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The effect of hydrostatic pressure on material parameters and electrical transport properties in bulk GaN Hosein Eshghi ∗ Department of Physics, Shahrood University of Technology, Shahrood 316-36155, Iran
a r t i c l e
i n f o
Article history: Received 18 January 2009 Received in revised form 2 March 2009 Accepted 9 March 2009 Available online 14 March 2009 Communicated by A.R. Bishop PACS: 72.10.-d 73.20.Hb 73.50.Dn
a b s t r a c t Experimental data for temperature dependence of electron transport properties in a bulk, low dislocation density, GaN sample at atmospheric pressure and 7.1 kbar have been presented. The data are representing a weak hydrostatic pressure dependence. Our quantitative analysis on its material parameters including: high and low dielectric constants (ε∞ , εs ), longitudinal and transverse optical phonons (ωLO , ωTO ), and electronic effective mass (me∗ ) show a small fractional change of −0.12, −0.14, 0.05, 0.058 and 0.089 (percent/kbar), respectively. These results are confirmed by the Hall-effect data analysis on the basis of charge neutrality condition and various scattering mechanisms. © 2009 Elsevier B.V. All rights reserved.
Keywords: Semiconductor GaN Material parameters Hydrostatic pressure Transport properties
1. Introduction One method often used to characterize semiconductors is to measure the free-carrier mobility and concentration, and to analyze the results obtained to determine the impurity content of the material. However, a relatively new approach has been to investigate electron transport as a function of hydrostatic pressure [1]. This enables parameters varied and thus provides a further test of the transport theories for electrons in a number of semiconductors. In this Letter we report the electron mobility and carrier concentration both as a function of temperature and pressure of a bulk GaN layer, and analyze the results.
with pressure. According to the k . p theory, the pressure dependence of the electronic effective mass is given by [2]: m0 m∗ ( P )
=1+
C E g(P )
(1)
where C is constants for different materials, and E g is the band gap with a pressure coefficient α = dE g /d P . For the pressure dependence of high frequency dielectric constant, we use the equation given by Goni et al. [3]:
∂ ln ε∞ 5(ε∞ − 1) ≈ (0.9 − f i ), ∂ ln V 3ε∞
(2)
2. Theory
where f i is the ionicity of the material under pressure and V is the volume of the material. It satisfies approximately the following linear relation in the low pressure range [4]:
2.1. Pressure coefficients
B ( P ) = −V
As applying hydrostatic pressure on the material, it changes the structure of energy band and the electronic effective mass change
*
Tel.: +98 9122733496; fax: +98 2733335270. E-mail address: [email protected].
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.03.013
dP dV
= B 0 + B 0 . P .
(3)
Here B is the bulk modulus and is assumed being linear with the pressure for not very high pressures, B 0 and B 0 are the bulk modulus at zero and its first order pressure deviation at zero pressure, respectively. Thus we can write the pressure coefficient for the high frequency dielectric constant as:
∂ ε∞ −5(ε∞ − 1) = (0.9 − f i ). ∂P 3B
(4)
1774
H. Eshghi / Physics Letters A 373 (2009) 1773–1776
The static dielectric constant is then determined by the famous Lyddane–Sachs–Teller relation:
εs = ε∞
h¯ ωLO
2 (5)
,
h¯ ωTO
2.2.2.3. Polar optical phonon scattering Ehrenreich has derived an analytical expression for the mobility limit due to polar optical scattering as follows [12]:
μPOP = 0.199
γ = B0
(6)
ω0 ∂ P
we can obtain the optical phonon energy dependence on pressure. In this equation ω and ω0 are the optical phonon frequencies under hydrostatic pressure P = 0 and P = 0, respectively. The values of the Grüneisen parameter γ and the other parameters used in our pressure calculations are given in Table 1. 2.2. Analysis of the Hall-effect measurements 2.2.1. The charge neutrality condition The temperature dependence of the electron concentration can be employed to derive parameters for characterizing shallow donors. For n-type semiconductors, the electron concentration, n, is given by charge neutrality condition [5]: n=
Nd 1+
ngn Nc
Ed BT
exp( k
)
− Na
2.2.2.1. Deformation potential scattering The mobility limited by acoustic-mode deformation potential scattering is [6]:
μac =
π
ρυ
(8)
where ρ is the crystal mass density, mn∗ is the electron effective mass, υs is the average sound speed in the crystal, and E dp is the deformation potential. 2.2.2.2. Piezoelectric scattering The mobility limited by piezoelectric scattering is given by [6]:
μPZ =
16(2π )1/2 ρυs2 h¯ 2 q
3(qh pz /ε0 εs )2 (mn∗ )3/2 (k B T )1/2
where h pz is the piezoelectric constant.
3/ 2
mn∗
q∗
300
m0
h¯ ω
kB T (10)
where q∗
= M ω a ε0 (1/ε∞ − 1/εs ) and G (¯hω/k B T ) is a slow varying function of order unity; we have taken it as constant and equal to mass (1.936 × 10−26 kg in GaN). √0.5.2 M is the reduced−ion 29 V a [= ( 3a c )/4 = 2.283 × 10 m3 ] is the volume for a Ga and N ion pair. 2V
2.2.2.4. Ionized impurity scattering The mobility limited by ionized impurity has been formulated by Brooks and Herring after taking into account the screening effect of both the free carriers and the impurity ions, and has the following expression [13]:
μi−im =
128(2π )1/2 ε02 εs2 (k B T )1/2 q3 (m∗ )1/2 (n + 2N n
a)
ln(1 + b) −
b
−1 (11)
1+b
where b=
24mn∗ ε0 εs (k B T ) q2 h¯ 2 n
and n = n +
( N d − N a − n)(n + N a ) Nd
.
2.2.2.5. Neutral impurity scattering The neutral impurity mobility has been modeled by Erginsoy [14] as:
2.2.2. Scattering processes in n-type GaN Scattering mechanisms including ionized and neutral impurities, dislocations, deformation potential and piezoelectric potential of acoustic phonons, and polar optical phonons are the main mechanisms that limit the electron mobility in GaN in the temperature range of interest here (60–300 K). An accurate calculation of the mobility requires numerical iterative solution of the Boltzmann transport equation. In this work, a simpler analytical approach is used. The mobility limit due to each individual scattering process is calculated independently with the corresponding analytical expressions. To a first approximation, the total mobility with different scattering events is obtained according to Matthiesens’s rule, μtot ∼ = 1/ i 1/μi .
2 4 2(2 ) ¯ q sh 2 ∗ 5 / 2 3E dp (mn ) (k B T )3/2
2
(7)
where N d is the donor density, E d its activation energy, gn the degeneracy of the donor states, N a the acceptor concentration, and N c the conduction band effective density of states.
1/ 2
q
× 1022 M . 1023 V a 10−13 ω p eh¯ ω/k B T − 1 . G
where h¯ ωTO and h¯ ωLO are the transverse optical (TO) and longitudinal optical (LO) phonon energies, respectively. According to the Grüneisen parameter: 1 ∂ω
1/2
T
(9)
μn−im =
q
me∗ /m0
20a B h¯
εs Nn−im
(12)
where N n−im and a B are the neutral impurity concentration and Bohr radius in a hydrogen atom, respectively. 2.2.2.6. Dislocation scattering Podor [15] used a screened potential and found the dislocation scattering limited mobility as:
√
μdis =
30 2π εs2 ε02 d2 (k B T )3/2 ∗ 1/ 2
q3 N dis f 2 L D mn
(13)
where the quantities d, N dis , f , and L D are the distance between the defect centers along the dislocation line (∼ 5 Å in GaN), the dislocation density, the occupation rate of these sites (assuming a value of unity), and the Debye screening length ( εs ε0 k B T /q2 n), respectively. 3. Results and discussion 3.1. Electron concentration analysis Our interested sample is a non-intentional doped layer of 1.34 ± 0.01 μm thick which is grown by MOCVD method on sapphire substrate. Fig. 1 shows Hall-effect results, including the temperature dependence of the electron density and the electron mobility (inset), at the atmospheric pressure (open circles) and 7.1 kbar (filled squares). As is obvious with increasing the pressure there is a small deviation between two sets of data. Here we aim to investigate this phenomenon to find out the influence of pressure on material parameters which can be the reason for the lower electron mobility and faster change of electron density in the conduction band. The fitted curves show the theoretical predictions for the Hall electron density and mobility in this material. As is clear, except at low temperatures (below 80 K) the theory agrees quite well with
H. Eshghi / Physics Letters A 373 (2009) 1773–1776
1775
Table 1 GaN parameters used in the computation. Phonon energies are measured in units of meV, electronic effective mass in units of the bare electron mass m0 , bulk modulus B 0 in units of kbar, energy band gap in units of eV, pressure coefficient of band gap α = dE g /d P in units of meV/kbar. h¯ ωLO
ε∞ a
5.41 a
mn∗
h¯ ωTO b
b
92.99
Karch et al., Ref. [7].
69.43 b
fi c
0.19
Wagner et al., Ref. [8].
c
B0 d
e
0.5
Wang et al., Ref. [9].
2054 d
Phillips, Ref. [10].
Fig. 1. Hall electron concentration and mobility (the inset) vs. temperature. ◦ and experimental data, also dashed and solid fitted curves in both figures corresponds to our calculations at atmospheric and 7.1 kbar pressures, respectively. Table 2 Other material parameters of GaN (atmospheric pressure). Parameter
Value
Ref.
Sound speed (υs ), m/s Piezoelectric constant (h pz ), C/m2 Density (ρ ), kg/m3 Deformation potential (E dp ), eV Lattice constant (a), Å
6.59 × 103 0.5 6.1 × 103 12 5.125
[5] [5] [16] [17] [18]
the experimental data at higher temperatures while the conduction band electrons controls the conductivity of the material. The deviation between the conduction band transport and the experimental data at low T s has been related to the parallel conduction path in the impurity band and/or the interfacial layer [19–22], which is not under our investigation in this Letter. To describe the variation of electron density vs. temperature, Fig. 1, we used the usual activated donor process, Eq. (7), first for atmospheric and then for 7.1 kbar pressure data. Using the least square fit method we found at ambient pressure the best fitting parameters are: N d = 1 × 1018 cm−3 , E d = 7 meV and N a = 1.5 × 1017 cm−3 . For the 7.1 kbar pressure data, considering the same concentrations for N d and N a as the above values, we found the activation energy should be 7.2 ± 0.1 meV. 3.2. Electron mobility analysis To complete the modeling of electron transport properties first we have used the atmospheric pressure of μ vs. T data, Fig. 2. Here we have shown the most important limited mobility components, Eqs. (8)–(13), and the total conduction band mobility μc (tot). The material parameters of GaN used for the mobility calculations, in addition to those in Table 1 are listed in Table 2. The dislocation density, N dis = 1 × 109 cm−2 , as the fitting parameter, seems is not significant in this sample in our interested temperature range. To describe the temperature dependence of electron mobility at the higher hydrostatic pressure we need to find the pressure
e
B 0
Eg
α
γLO
γTO
4.6e
3.3c
3.6e
1.02b
1.19b
Wei et al., Ref. [11].
Fig. 2. Temperature dependence of electron mobility at atmospheric pressure. The curves show various scattering limited mechanisms and the total theoretical electron mobility (see the text). Table 3 Results of calculations for relative variations of some GaN material parameters with pressure. 1 dε∞ ε∞ d P
1 dεs εs d P
1 dωLO ωLO d P
1 dωTO ωTO d P
∗ 1 dme me∗ d P −4
(10−3 /kbar)
(10−3 /kbar)
(10−4 /kbar)
(10−4 /kbar)
(10
−1.2
−1.38
4.97
5.8
8.9
/kbar)
variations of the intrinsic material (GaN) parameters. Results of our calculations, based on Eqs. (1)–(6), for the rate of variations for low and high frequency dielectric constants, also LO and TO phonon energies and electron effective mass as a function of hydrostatic pressure are summarized in Table 3 and are shown in Figs. 3 and 4. Considering these variations in the material parameters involving in the analytical expressions for various scattering mechanisms at atmospheric pressure, we found the total conduction band mobility at 7.1 kbar, shown by dashed line in the inset of Fig. 1, is well fitted to experimental data. The accuracy of our analysis can also be examined using the simple hydrogenic model in calculating the activation energy of donor impurities, i.e. E d = 13.6(m∗ /εs2 m0 ) eV. According to this relation the pressure variation of εs from 9.705 at atmospheric pressure to 9.61 at 7.1 kbar tend to a relative variation in this quantity, i.e. {[ E d ( P ) − E d (0)]/ E d (0)} of about 3%. This is close to that we found from n vs. T data analysis in Section 3.1. 4. Conclusion Data have been presented for the temperature dependence of the electron concentration and mobility in a bulk GaN layer at ambient pressure and 7.1 kbar hydrostatic pressure. The Hall data represent a poor pressure variation. Analyses show that the main reason for this effect can be attributed mainly to its big bulk modulus, which in turn affects on its material parameters pressure dependence (Table 3).
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Fig. 3. Our analytical calculations for pressure dependence of material parameters in GaN for (a) dielectric constants and (b) longitudinal and transverse optical phonon energies up to 10 kbar, based on Eqs. (2)–(6).
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Fig. 4. The relative effective mass vs. hydrostatic pressure, based on Eq. (1), in the bulk GaN.
Acknowledgements The author would thank Dr. D. Lancefield, at the University of Surrey, UK, for his helps with the Hall measurements.
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