1. Local vibrational modes in doped GaN

Figure 1 shows Raman spectra of samples with the respective magnesium concentrations of 8·10¹⁹ cm⁻³ (A), 1.2·10¹⁹ cm⁻³ (B), 9·10¹⁸ cm⁻³ (C) and 6·10¹⁸ cm⁻³ (D) in the high-energy region. In the spectrum of sample A with the highest magnesium concentration a new mode appears at 2129 cm⁻¹ in addition to the four LVM described in Ref. 5. Apparently, the intensity of the modes correlates with the magnesium content. A Mg-concentration of around 1·10¹⁹ cm⁻³ is necessary for some of the high-energy modes to appear well-resolved in the spectra. The hydrogen concentration for all the samples investigated is about 1·10¹⁹ cm⁻³ as determined by secondary ion mass spectroscopy. Since hydrogen was not intentionally supplied during growth its incorporation possibly arises from the residual water vapor pressure in the growth chamber.

Figure 1: Room-temperature Raman spectra of GaN with different magnesium content in the high-energy range.

Yi et al. [Reference Yi and Wessels8] observed a similar structure of five modes around 2800 cm⁻¹ in Mg-doped GaN grown by metalorganic vapor phase epitaxy (MOVPE). These modes were attributed to symmetric and asymmetric C-H_n (n=1, 2, 3) vibrations due to carbon incorporation caused by the decomposition of the magnesium precursor during growth. Remarkably, the frequency ratio for all the five modes is between 1.33 and 1.34. This is close to

, a value expected for similar vibrations with carbon (m_C=12.01 u) replaced by magnesium (m_Mg=24.31 u). This implies that magnesium is built into the lattice on a nitrogen site with force constants similar to those of carbon, an interpretation which seems doubtful. Furthermore, there are experimental and theoretical articles that determined the frequency of the Mg-H vibration above 3000 cm⁻¹ [Reference Brandt, Ager, Götz, Johnson, Harris, Molnar and Moustakas5, Reference Neugebauer and van de Walle9].

Alternatively, hydrogen-decorated native defects or hydrogen at extended defects such as dislocations can give rise to high-energetic vibrations. When the Fermi level reaches a value below 2.1 eV the formation of V_Ga-H_n complexes with n=1, 2, 3 is likely to occur with a predicted frequency of 3100 cm⁻¹ for a V_Ga-H complex [Reference van de Walle10]. This is much higher than the frequencies we observe, even if anharmonic terms may lower the frequency considerably [Reference van de Walle10].

However, we rather believe that the magnesium incorporation causes the high-energy modes indirectly by creating defects or by lowering the Fermi level and therefore making the the hydrogen complex formation likely [Reference Kaschner, Siegle, Straßburg, Hoffmann, Thomsen, Birkle, Einfeldt and Hommel11].

Apart from the five high-energy modes the highly Mg-doped samples exhibit new vibrational modes in the region of the acoustic and optical GaN phonons. We found structures at 136 cm⁻¹, 262 cm⁻¹, 320 cm⁻¹, 595 cm⁻¹ and 656 cm⁻¹. Figure 2 displays the low-energy part of the Raman spectra of the four samples in the same sequence as in Figure 1. The broad structures centered around 320 cm⁻¹ and 595 cm⁻¹ may result from disorder-activated scattering, in which built-in defects yield a relaxation of the q=0 selection rule for first-order Raman scattering. This interpretation is supported by the fact that from sample D to A the forbidden A₁(TO) mode at 533 cm⁻¹ increases considerably in intensity. Limmer et al. [Reference Limmer, Ritter, Sauer, Mensching, Liu and Rauschenbach12] reported on a disorder-activated Raman-mode around 300 cm⁻¹ in ion-implanted GaN which is close to our 320 cm⁻¹ mode. Furthermore, the cut-off at 340 cm⁻¹ fits well with calculated phonon dispersion curves [Reference Siegle, Kaczmarczyk, Filippidis, Litvinchuk, Hoffmann and Thomsen13, Reference Nipko, Loong, Balkas and Davis14]. On the other hand, the modes at 136 cm⁻¹, 262 cm⁻¹ and 656 cm⁻¹ probably do not originate from disorder activated scattering since the phonon density of states (PDOS) does not exhibit marked structures in this frequency range [Reference Nipko, Loong, Balkas and Davis14]. Recently, a mode at 656 cm⁻¹ was observed in GaN after annealing at 1000°C and related to a damaged-sapphire substrate [Reference Kuball, Demangeot, Frandon, Renucci, Massies, Grandjean, Aulombard and Briot15]. The mode we observed at 656 cm⁻¹ is of different origin since it does not scale with the intensity of the sapphire mode at 418 cm⁻¹ nor with other modes of the damaged-sapphire at e.g. 770 cm⁻¹ but rather with the Mg-concentration.

We thus believe these structures in the low-energy range to be correlated with magnesium. Equation (1) gives us a rough estimate of the magnesium local vibrational mode frequency from the effective masses of the GaN and LVM vibrations.

Assuming that magnesium occupies a gallium site we obtain a value of about 640 cm⁻¹ for N-Mg vibrations using ω_GaN = ω(E₁ (TO)) = 560 cm^-1

Figure 2: Room-temperature Raman spectra (normalized to E₂ intensity) of GaN with different magnesium content in the low energy range. Apart from the host lattice phonons five additional modes are observed. The peak marked by an asterisk is a phonon from the sapphire substrate.

Though the 656 cm⁻¹ mode lies in the range of the optical phonons its observation may be possible because the PDOS is relatively low in the region from 640 cm⁻¹ to 675 cm⁻¹ as confirmed by second-order Raman-scattering [Reference Siegle, Kaczmarczyk, Filippidis, Litvinchuk, Hoffmann and Thomsen13] and time-of-flight neutron spectroscopy [Reference Nipko, Loong, Balkas and Davis14]. We hence assign the 656 cm⁻¹ mode to a local vibrational mode of magnesium in GaN. The Mg-concentration in the 10¹⁹ cm⁻³ range is apparently sufficient to observe the LVM without resonant excitation. The nature of the 136 cm⁻¹ and 262 cm⁻¹ mode remains unclear. Since the density of states is also relatively low in the energy range of the two modes in question it is conceivable that they are also LVM.

We did not observe any LVM for the Si- and C-doped GaN samples in low-energy range nor in the high-energy range.

2. Biaxial Stress due to incorporation of dopants

Due to its nonpolar character the E₂ phonon frequency is a good measure for biaxial stress in GaN layers. Biaxial compressive stress increases the phonon frequency. Values of 4.2 cm⁻¹/GPa [Reference Kisielowski, Krüger, Ruvimov, Suski, Ager, Jones, Lilienthal- Weber, Rubin, Weber, Bremser and Davis2], 6.2 cm⁻¹/GPa [Reference Kozawa, Kachi, Kano, Nagase, Koide and Manabe16] and 7.7 cm⁻¹/GPa [Reference Lee, Choi, Lee, Shin, Kim, Noh, Son, Lim and Lee17] have been reported. Figure 3 shows the dependence of the E₂ frequency on the dopant concentration. While the E₂ values remain nearly constant between 567 cm⁻¹ and 568 cm⁻¹ over the whole doping range in case of the silicon doping series, we see a strong hardening in the magnesium series for Mg-concentrations exceeding 1.10¹⁹ cm⁻³. This behavior can be explained by the different size of the dopant atom (r_Mg=0.14 nm) and the replaced host atom (r_Ga=0.126 nm). Using equation (2) one can calculate the influence of this effect on the strain [Reference Kisielowski, Krüger, Ruvimov, Suski, Ager, Jones, Lilienthal- Weber, Rubin, Weber, Bremser and Davis2].

Figure 3: Position of the E₂ mode for different doped GaN samples.

Here, N is the number of lattice sites of the host matrix, C the dopant concentration, c and c₀ the strained and unstrained lattice constant, respectively. Equation (3), where B is the bulk modulus and ν is the Poisson ratio, correlates the strain with the biaxial stress.

Figure 4 shows the experimental data of the Mg-doped samples and calculated curves with different values for the strain-free E₂ frequency and Δω(E₂)/Δσ. It is obvious that the observed behavior can be explained by a size effect qualitatively, but the uncertainty for the basic GaN material parameters renders exact calculations difficult.

Figure 4: E₂ position vs. Mg-concentration. The lines show the predicted behavior using eq. (2) and (3) with different values for the bulk E₂ frequency and the E₂ shift. The values for B and ν are taken from Ref. 2.

When substituting gallium by silicon one would expect the same effect but with opposite sign since the atomic radius of silicon is smaller than that of gallium. Surprisingly, we did not observe any significant shift for the E₂ frequency up to a silicon concentration of 1·10²⁰ cm⁻³. The highly Si-doped samples do also exhibit a large free-carrier concentration (see next chapter). Therefore we believe, that the size effect is compensated by the change of the lattice constant due to free electrons [Reference Lee, Choi, Lee, Shin, Kim, Noh, Son, Lim and Lee17]. Following Ref. 18 a free-carrier concentration of 5·10¹⁹ cm⁻³ should increase the lattice constant by 0.01% which equals the absolute value of the size effect caused by the same silicon concentration. The carbon concentration of the samples in our study is too low to observe a significant E₂ shift.

3. Determination of the free-carrier concentration

The interaction of free charge carriers with longitudinal optical phonons leads to the observation of longitudinal phonon plasmon modes (LPP) [Reference Perlin, Camassel, Knap, Taliercio, Chervin, Suski, Grzegory and Porowski3]. If damping can be neglected the frequency of the LPP modes is given by equation (4).

Here, ω_L and ω_T are the frequencies of the LO and TO phonons, respectively. The plasmon frequency is correlated with the free-carrier concentration.

In equation (5) n and m* are the concentration and effective mass of the charge carriers, ε_∞ the high-frequency dielectric constant.

In Figure 5 the spectra of five samples with silicon concentrations between 2.6·10¹⁵ cm⁻³ (top) and 1.0·10²⁰ cm⁻³ (bottom) are plotted. The two highest doped samples exhibit LPP⁻ modes at 450 cm⁻¹ and 520 cm⁻¹. The A₁(LO) mode already vanished and therefore the LPP⁺ mode appears at 2580 cm⁻¹ and 1070 cm⁻¹ (not shown). From equations (4) and (5) we calculated the corresponding carrier concentration to be 7.4·10¹⁹ cm⁻³ and 1.0·10¹⁹ cm⁻³, respectively. Notice that in both spectra a structure at around 650 cm⁻¹ (**) appears which is only known from highly doped GaN [Reference Demangeot, Frandon, Renucci, Meny, Briot and Aloumbard19]. The A₁(LO) mode of the sample with a silicon concentration of about 1.1·10¹⁸ cm⁻³ appears slightly asymmetric typical for carrier concentrations between 5·10¹⁷ cm⁻³ and 1·10¹⁸ cm⁻³. For the remaining two samples the free-carrier concentration is too low to cause any shift or broadening of the A₁(LO) mode and cannot be determined by this method. The inset of Figure 5 compares the free-carrier concentration determined by Hall measurements and by Raman spectroscopy as a function of the silicon concentration. Both data sets agree well. Raman spectroscopy offers the advantage that the free-carrier concentration can be determined with a spatial resolution better than 1 µm without contacting the samples.

Figure 5: Room-temperature Raman spectra of different Si-doped samples. The inset shows the carrier concentration determined by Hall and Raman measurements vs. Si-content.