NUMERICAL CALCULATION OF THE ELECTRON MOBILITY OF GaN ...

NUMERICAL CALCULATION OF THE ELECTRON MOBILITY OF GaN SEMICONDUCTOR COMPOUND S. AYDOĞU, M. AKARSU and Ö. ÖZBAª Osmangazi University, Art and Science Faculty, Department of Physics, Eskiºehir, TURKEY [email protected], [email protected], [email protected] Received December 21, 2004

In this work, the electron mobility of GaN semiconductor compound was calculated using iterative method in range of 30–600 K. We primarily considered polar optic phonon scattering, deformation-potential acoustic phonon scattering, piezoelectric scattering and impurity scattering mechanisms. Boltzmann transport equation was solved using iterative method. The band structure of GaN has taken to be non-parabolic. In addition, we took into account the mixing of wave functions and electron screening, and we investigated temperature dependence of mobility for the given compound. Key words: GaN, scattering mechanism, electron mobility, iterative method, III – nitrides.

1. INTRODUCTION

Recently, the previous studies about III–V semiconductor compounds are considered important. III–V semiconductor compounds, InN, GaN and AlN, respectively 1.89 eV, 3.4 eV, 6.2 eV, have the wide band gaps [1]. Because of these properties III – nitrides are used in the blue and UV light emitting diodes (LED’s), blue lasers, UV detectors and high power, high temperature field effect transistors [2, 3, 4, 5]. Since GaN semiconductor compounds possess the band gap ~ 3,4 eV, GaN’s have much larger breakdown electric field strength than GaAs semiconductor compounds [6]. Since GaN semiconductor compounds have large peak velocities, they are suitable for the high frequency applications [4]. GaN semiconductor compounds belong to both zinc blende structure and wurtzite structure. But their structures have thermally wurtzite structures [7]. Their lattice constants is a = 3.189 Å and c = 5.185 Å [8]. All III – nitrides have anisotropic band structures due to their wurtzite structures [9]. In the GaN’s which have nonparabolic band structures the minimum of conduction band takes part in Γ valley [10].

Paper presented at the 5th International Balkan Workshop on Applied Physics, 5–7 July 2004, Constanþa, Romania. Rom. Journ. Phys., Vol. 50, Nos. 9– 1 0 , P. 1047–1053, Bucharest, 2005

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S. Aydoğu, M. Akarsu, Ö. Özbaº

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In GaN semiconductor compounds the dominant scattering mechanism is primarily the polar optic phonon scattering mechanism. The others are ionized impurity scattering, acoustic phonon deformation potential scattering and acoustic phonon piezoelectric scattering mechanisms [11]. As a result of being elastic process of acoustic phonon deformation potential scattering, acoustic piezoelectric scattering and ionized impurity scattering, their mobilities can be calculated with the relaxation time approach. But if the electron energy can be compared with the phonon energy, this is important and such a scattering is inelastic scattering. The compensated changing in the energy is taken place after inelastic scattering. Since in inelastic scattering process the phonon energy (=ωA ) is the bigger than the electron energy (kBT); that is,

=ω A >> kBT. =, reduced Planck constant, kB is Boltzmann constant and ωA optic phonon frequency. Thus, relaxation time approach is not valid for inelastic scattering process [12]. Because of this the calculation of polar optic phonon scattering is used the other numerical method instead of relaxation time approach. In this work it was used an iterative method for the calculations. 2. NUMERICAL CALCULATION OF MOBILITY

We calculated the electron mobility solving Boltzmann transport equation with iterative method [13, 14, 15]. We have taken Boltzmann transport equation as follows,

Lc φ =

1 γ ′( E )

(1)

In this equation Lc φ = S0 ( E ) φ( E ) −

∑∑ S j ± (k )φ(E ± =ω j )

(2)

j + ,−

In Equation (2), S0 ( E ) represents all elastic scattering and out-scattering of inelastic scattering processes, and other term represent in-scattering of inelastic scattering process. φ is expressed as the perturbation by electric field or magnetic field [15]. Boltzmann transport equation is involved in scattering mechanisms may have occurred in the material. In this work we regarded that it was taken place acoustic phonon deformation potential scattering, acoustic piezoelectric scattering, ionized impurity scattering and polar optic phonon scattering for given materials. We took acoustic phonon deformation potential scattering, acoustic piezoelectric scattering, ionized impurity scattering as elastic process and also polar optic phonon scattering as inelastic process.

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Calculation of the electron mobility of GaN

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Acoustic phonon deformation potential scattering rate is defined as follows [16], +1

Sac ( E ) =

2 E12 kBTL m*3 /'2 1/ 2 γ ′ (1 − x )O( x )Sc ( x )dx γ 2 πρ = 4 υ2a

∫

(3)

−1

where E1 is acoustic deformation potential, TL is lattice temperature, m* is effective mass, ρ is mass density, υa is sound velocity, γ is band nonparabolicity. Acoustic phonon piezoelectric scattering rate can be taken as following equation [16], +1

2 k T m *1/ 2 e2 e14 B L −1/ 2 γ ′ S pz ( E ) = O( x )Sc ( x )dx γ 2 2 2 2 2ε 0 πρ = υ2a −1

∫

(4)

where e is electron charge, e14 is piezoelectric constant, ε0 is permittivity. We used Brooks-Herring approach [17] in the calculation of ionized impurity scattering. Ionized impurity scattering rate is [16], +1

N I e4 1 O ( x ) S ( E ) dx γ −3 / 2 γ ′ Sim ( E ) = c 1− x 16 2 πε20 m *1/ 2 −1

∫

(5)

Here NI is ionized impurity atoms concentration. Undoped GaN semiconductor compound is n-type semiconductor [11, 18]. N nitrogen vacancy is considered as donor in such a GaN semiconductor compound [11]. If Ga density is increased, GaN compound indicated the variation tendency of semi-insulator [19]. Polar optic phonon scattering which is inelastic scattering process is produced as a result of scattering the electrons with potential constituted with dipole moment which is resulted in the neighbour atoms place changing with opposite ionic charges [20]. Polar optic phonon scattering rate is [16],

Sop ( E ) =

e2 m*1/ 2 (χ s − χ∞ )kB θ0 × [ F+ ( E ) + B1h( E − kB θ0 ) F− ( E )] θ0 ⎡ ⎤ 2 4 2 πε 0 = χ s χ∞ ⎢ exp TL − 1⎥⎦ ⎣

( )

(6)

Here χ∞ and χs is respectively high frequency and low frequency lattice permittivity, θ0 is phonon Debye temperature, ε0 is permittivity. The other terms in Equation (4) can be found in Ref. 16. Relaxation time for each one scattering is given with the following equation [15]: τ( E ) =

1 S0 ( E )

(7)

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After we calculated total relaxation time, the perturbed values φ was obtained with a few of iteration steps. Thus mobility is calculated in the following equation [15]: μ = 2 e* 3m

∫ γ3 / 2 (E ) ∂ E f (E ) φ(E ) dE ∫ γ′(E ) γ1/ 2 (E ) f (E )

(8)

We took the structure of GaN compound as wurtzite structure. Further we supposed that GaN semiconductor compound has the isotropic nonparabolic band structure. We took into account electron screening and mixing of s and p wave functions. It was given the material parameters of GaN used in the calculations in Table 1. Table 1 The material parameters of GaN compounds used in calculations Parameter

Symbol

(unit)

GaN

High Frequency Dielectric Constant Low Frequency Dielectric Constant Polar Phonon Debye Temperature

ε∞ ε θA

[F/m] [F/m] [K]

5.47ε0a 10.4 ε0a 1044a

Mass Density Sound Velocity Piezoelectric Constant Acoustic Deformation Potential Effective Mass

ρ υs e14 Eds m*

[kg/m3] [m/s] [C/m3] [eV] [kg]

a

6.10 × 103b 6.59 × 103b 0.5c 9.2a 0.22m0a

Ref. [24], b Ref. [5], c Ref. [10], d Ref. [9].

3. RESULTS AND CONCLUSIONS

In this work, the mobility for GaN compound is calculated using iterative method. We investigated temperature dependence of electron mobility in range of 30–600 K. The electron concentration of GaN compound was taken as 1016 cm–3, 1017 cm–3 and 1018 cm–3. Further in this work we calculated the mobility for various electron concentrations. As a result of investigating temperature dependence of electron mobility, we found the electron mobility at 300 K about 1551 cm2/Vs, about 1039 cm2/Vs and about 830 cm2/Vs for electron concentrations of 1016 cm–3, 1017 cm–3 and 1018 cm–3, respectively. This has been shown in Fig. 1. These values are corresponding to the values in the literature, but these values were found as fewer excess than the values in the literature [6, 9].

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Since we were taken that carrier concentration was equal to impurity concentration, the electron mobility is increased up to 100 K. As shown in Fig. 1, it has shown that it is effective that ionized impurity scattering up to 100 K. Also

Fig. 1. – Temperature dependence of mobility for GaN.

Fig. 2. – Carrier concentration dependence of mobility for GaN.

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Fig. 3. – Carrier concentration dependence of mobility for GaN. It was used to 0.60 compensational ratio.

the electron mobility of this compound is decreased with being dominant of the polar optic phonon scattering above 200 K [9]. The electron mobility peak value for GaN compound is between 100 K and 200 K [9]. When we found the mobility peak value for 1016 cm–3 electron concentration at 130 K, we found the mobility peak value for 1017 cm–3 and 1018 cm–3 electron concentrations at 140 K. We regarded that the electron concentration equals to impurity concentration. Because of this we took into account screening by free carriers [23]. We didn’t use to compensation ratio in Fig. 2. But using 0.60 compensation ratio we calculate the electron mobility. In Fig. 3 it was given carrier concentration dependence of mobility using 0.60 compensation ratio. It was shown that these values are corresponding with the values of Ref. [9]. We calculated the mobility for various electron concentrations. We showed that degeneration was begun to occur in the electron concentrations above 2 × 1018 cm–3. It was shown that Found value is corresponding to literature [22]. REFERENCES 1. C. G. Van de Walle, M. D. McClustey, C. P. Master, L. T. Romano, N. M. Johnson, “Large and composition-dependent band gap bowing in InxGa1-xN alloy”, Materials Science and Engineering B., 59, 274–278 (1999).

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