Hall measurements were carried out in darkness in the temperature range between T = 9 and 300 K. The current flowing through the sample was kept relatively low (I < 50 μA) to ensure ohmic conditions. Both the mobility and carrier concentration were found to be independent of current and magnetic field at all temperatures.

The temperature dependence of the dark conductivity (DC) for the undoped (GS 1509) and Mg-doped (GS 1989) samples are shown in Figure 1. It is clear that the room temperature conductivity in the Mg-doped sample is about an order of magnitude smaller than that in the undoped material (at 2 K it is two orders of magnitude smaller).

The weak temperature dependence of both Hall density and mobility in the undoped material GS1509 that is shown in Figure 2a,b is a typical feature in undoped GaInN (and InN). The Hall carrier density at T = 2 K is n = 1.7 10¹⁴ cm⁻² and it changes only by about 5% in the whole temperature range between 2 and 300 K. The parallel conductivity from the degenerate surface layer results in very small temperature sensitivity 1 2 3 . The electron mobility, as plotted in Figure 2b which is μ = 63 cm²/Vs at T = 2 K and μ = 72 cm²/Vs at T = 300 K, i.e., has a very weak temperature dependence. This temperature dependence has also been observed by other research groups and explained as either being due to a reduced electron–phonon interaction or high degeneracy 24 . Since InN which is similar to the other group III-N semiconductors is a highly polar material, scattering due to polar optical phonons is expected to play an important role. Therefore, the observed weak temperature dependence is more likely to be associated with the latter, i.e. the screening of polar interaction with high carrier density 25 26 .

The temperature behaviour of Hall data which is similar to ours is commonly interpreted in the literature in terms of two parallel conducting channels as described by Jones et al. 20 and Mahboob et al. 16 17 as an electron accumulation layer on the surface and a 3-D bulk layer. However, variable magnetic field Hall measurements and the quantitative mobility spectrum analysis 27 of the Hall data must be considered in order to distinguish the contribution of each channel though the analysis which may be further complicated as a result of the small discrepancy between the electron densities and the mobilities in the two channels 28 .

The temperature dependence of the Hall density and mobility for the Mg-doped material (GS1989) is shown in Figure 2c,d. Electron mobility decreases with increasing temperature from 0.16 m²V⁻¹ s⁻¹ at T = 2 K to 0.003 m²V⁻¹ s⁻¹ at T = 300 K, while the carrier density changes considerably more than that in the undoped material from around 2 × 10¹¹ to 9.0 × 10¹³ m⁻² in the same temperature range. We have plotted the logarithm of the Hall carrier density as a function of inverse temperature in Figure 3. There is a clear tendency to an activated behaviour at high temperatures. This observation may be explained speculatively, in terms of a mechanism involving two-channel conductivity with both types of type carriers, namely: (1) the 2DEG accumulation at the surface layer where the density for these carriers will be temperature independent as it is the case in undoped material as shown in Figure 2a and (2) the bulk layer. For the bulk conductivity, we may assume either of the equally plausible three scenarios: (1) The whole bulk layer is p-type or (2) p-type conductivity occurs in the bulk layer only below a certain depth from the surface and adjacent to the bottom buffer layer as suggested by Jones et al. and Ager et al. 20 29 . Alternatively, (3) p-type conductivity is not within a uniform layer but occurs through p-type percolation channels around slightly compensated n-type islands, the sizes of which may vary, depending on the local electron and hole densities, from nano to microscale islands or chain (similar to the model that was proposed by us and Ridley 30 ) to explain successfully hot electron percolation in 2-D GaAs and superconductivity in InN 31 . These three mechanisms are depicted schematically in Figure 4. In either of the three models, at high temperatures as the Mg acceptors 1 32 are ionized, the contribution of thermally activated holes to the conductivity increases as observed. From Figure 3 at the highest measurement temperatures, we obtain an activation energy close to ΔE = 108 ± 20 meV. This value is within the range of the reported binding energy of Mg in indium-rich GaInN and InN 22 . It should be noted that the accuracy of the activation energy obtained from the temperature dependence of the Hall density in our limited range of high temperatures is within about ±20%.

Following the analysis for parallel conduction by Kane et al. 33 and considering the above model for the bulk and surface conductivity channels in Mg-doped GaInN 1 32 , the expressions for the effective Hall mobility and carrier density due to the two n type and p-type conducting channels may be written as

n H = n e μ e + p μ p 2 n e μ e 2 + p μ p 2

μ H = n e μ e 2 + p μ p 2 n e μ e + p μ p ,

where n _e and μ _e , are the electron density and the mobility in the GaInN accumulation layer, and p and μ _p are the hole density and mobility in the p-channels of the bulk layer. Here it is assumed that the electron density in the n-type regions in the bulk (n _B) is negligible compared to the surface accumulation density (n _B << n _e).

The temperature dependences of n _e and μ _e in the undoped material are expected to be weak as it is observed by us in Figure 2a,b and by others 24 . Therefore, we can assume that the temperature dependence of the Hall carrier density and mobility in the Mg-doped sample is fundamentally due to the temperature dependence of both p, and μ _p in the p-channels in the bulk layer. Furthermore, at low temperatures, because of the freeze-out of the free hole density, n _e > > p and the conductivity are fundamentally due to the conduction in the accumulation layer, i.e.

T → 0

μ H = μ e a n d n H = n e .

At higher temperatures according to Equations 1 and 2, the measured Hall carrier density and mobility will be the complicated functions of temperature as determined by the temperature dependence of the effective scattering mechanisms in the accumulation layer and the bulk. However, the temperature dependence of the Hall density will be primarily due to the thermally activated behaviour of the holes as we have observed in Figure 3. As a result the activation energy obtained from Figure 3 should correspond to the Mg binding energy.

Since the conductivity in the p-type material is about a factor of 100 smaller than the corresponding undoped material, it is possible to have a detectable spectral PC in the Mg-doped material. We carried out the PC spectrum measurements on both the undoped and the Mg-doped samples at temperatures between T = 33 and 300 K. In the measurements a focused beam of light from a halogen lamp was dispersed using a 0.5-m monochromator. A small fraction of the dispersed light from the monochromator was then split and directed onto a pyroelectric detector to monitor the excitation intensity. The rest of the light was directed onto the sample. A dc voltage was applied to the sample with a series resistor, and the PC signal was detected using a lock-in amplifier. Spectral photoconductivity was measured as a function of temperature under constant illumination intensity. Therefore, to the first approximation, we assume that the photo-generated carrier density is independent of temperature. Therefore, the change in the photoconductivity with temperature is attributed to temperature dependence of the photo-generated carrier mobility.

There was no detectable PC signal in the undoped material in the range of measurement temperatures (30 > T > 300 K). Mg-doped sample gave strong PC where the results are plotted in Figure 5a,b. The measurements were taken using two experimental sets one with a grating for better resolution at high energies (a) and the other with a better resolution at low energies (b). It is clear that the data agree well in the overlapping energy region.

The main features of the PC spectra are the presence of three broad peaks at E approximately 0.65 eV (λ approximately 1.90 μm), E approximately 1.00 eV (λ approximately 1.24 μm) and E approximately 1.40 eV (λ approximately 0.89 μm). The peak at 1.00 eV corresponds to band-to-band transitions with the nominal indium concentration of 80% 31 34 . This is commonly accepted value for the band gap of approximately 1.0 eV for Ga_0.2In_0.8 N. The other two peaks may be explained tentatively in terms of a non uniform composition probably associated with indium segregation. The 1.90-μm peak may be due to the Ga-deficient regions with band-to-band transition energy of 0.65 eV corresponding to the InN band gap 34 35 . The higher energy peak at 1.4 eV may be the regions with reduced average indium concentration of 50% 31 34 . The broad nature of the PC peaks does not allow the identification or the existence of the Mg-related transitions.

It is clear from Figure 5 that the peak wavelengths of the PC peaks have no notable temperature dependence. This observation is similar to the behaviour of PL spectra as reported in literature, where there is an extremely weak dependence of the PL peak on temperature 7 . The intensities of PC peaks, however, depend strongly on the temperature of the measurements particularly at temperatures T > 90 K where PC drops rapidly with increasing temperature.

In Figure 6 we plotted the logarithm of the peak photoconductivity against the inverse temperature for the two prominent peaks in Figure 5a, i.e. E = 1.0 and E = 1.4 eV for temperatures T > 150 K. There is a clear thermally activated behaviour in the form

σ ∝ exp Φ k T ,

with an activation energy of Φ = 175 ± 15 meV. In order to attach any significance to this activation energy, we consider the dynamics of photoconductivity. Here we assume that most of the absorption occurs in the bulk layer so that the PC signal is fundamentally associated with the bulk.

Following the theoretical model developed by us for photo-voltage 36 , the photoconductivity (Δσ) due to the excess electron (Δn) and hole (Δp) densities is as follows:

Δ σ = e Δ n μ e + e Δ p μ n ,

where e, μ _e and μ _h are the electronic charge, electron and hole mobilities respectively.

In the absence of reflection and surface recombination and when the radiation is uniformly absorbed by the sample the rate of generation of electron hole pairs 36 37

ℜ = q α I h v

where q, α, I and hν are the quantum efficiency, absorption coefficient, intensity of the light absorbed and the photon energy respectively.

In the absence of trapping and in n-type material, the time dependence of the excess hole concentration, i.e. the rate equation is

d Δ p d t = ℜ − Δ p τ p ,

where τ _p is the minority carrier radiative lifetime for holes.

In the steady state

d Δ p d t = 0

thus,

ℜ = Δ p 0 τ p = q α I h ν p Δ p 0 = q α I h ν τ p ,

Where Δp ₀ is the excess hole density in the steady state.

When the illumination intensity is low or the thermal equilibrium hole density (p ₀) is high, i.e.

Δσ ₀ < σ and Δp ₀ < p ₀

If there is no trapping, n ₀ = Δp ₀, the photoconductivity in the steady state becomes

Δ σ 0 = e Δ n 0 μ e + e Δ p 0 μ h = e Δ p 0 1 + β μ h = e ℜ τ p 1 + β μ h = e q α I h ν τ p 1 + β μ h ,

where β = μ e μ h .

Photoconductivity will increase linearly with intensity. However, the experiments were performed at constant illumination intensity. Therefore, at high temperatures (in the optical phonon scattering regime) when the quantum efficiency, absorption coefficient and the recombination lifetime have weak temperature dependences, the temperature dependence of the PC will be determined primarily by the minority carrier mobility.

In the presence of differential trapping (Δn ≠ Δp), the rate equation for excess holes becomes

d Δ p d t = ℜ − Δ p τ p − Δ p τ 1 + Δ N s τ 2

Here τ ₁ , τ ₂ and ΔN _S are the average lifetime before the hole is caught in a trap, average lifetime that a hole stays in the trap (de-trapping time) and the excess hole concentration in the trap.

The time dependence of excess holes in the trap will be

d Δ N s d t = Δ p τ 1 − Δ N s τ 2

In the steady state,

d Δ N s d t = 0 a n d Δ N s 0 = Δ p 0 τ 1 τ 2

Space charge neutrality requires Δn ₀ = Δp ₀ + ΔN _s0.

By substituting ΔN _s0 from Equation 10, the excess electron density in thermal equilibrium, Δn ₀ , can be obtained as follows:

Δ n 0 = Δ p 0 1 + τ 2 τ 1

Thus, the photoconductivity in the steady state:

Δ σ 0 = e Δ p 0 1 + τ 2 τ 1 μ e + μ h .

By substituting Δp ₀ from Equation 6, we obtain

Δ σ 0 = e ℜ τ p 1 + β + β τ 2 τ 1 μ h .

At high temperatures the temperature dependence of the radiative lifetime and the trapping lifetimes will have weak temperature dependences. Therefore, at high temperatures in the longitudinal-optical (LO) phonon, scattering regime conductivity will be dominated by the product of the two exponential terms, i.e. the de-trapping time 35 38 is

τ 2 = A exp E th k T L

and in the LO scattering limited hole mobility 2 3 25 26

μ h = e τ m m * exp ℏ ω LO k T L ,

where E _th is the energy difference between the hole trap and the valence band (trap emission energy). m*, ℏω _LO , k _B , T _L and τ _m are respectively, the effective mass, optical phonon energy Boltzmann's constant, lattice temperature and electron-LO phonon scattering (momentum relaxation) time which is given by 2 25 26

τ m = e 2 ω LO 2 π ℏ m * 2 ℏ ω LO 1 2 1 ε ∞ − 1 ε s − 1

Here ε _∞  = 8.4 ε _∞ and = ε _s  = 15.3 ε ₀ are the high frequency and static permittivity, respectively.

Therefore, Δσ ₀ will have an exponential temperature dependence, i.e.

Δ σ 0 ∝ exp E th + ℏ ω LO k T

At high temperatures a plot of the logarithm of the peak photoconductivity versus inverse temperature should give a straight line with a slope

E th + ℏ ω LO k .

Therefore, the thermal activation energy should correspond to

Φ = E th + ℏ ω LO .

We obtain from Figure 6 that

Φ = 175 ± 15 m e V .

Since the optical phonon energy in indium-rich GaInN is ℏω _LO  = 72 meV 2 3 the hole traps will have an average depth from the valence band edge of E _th = 103 ± 15 meV. This value is very close to the Mg binding energy in indium-rich GaInN obtained from the C-V and variable magnetic field Hall measurements, in 23 39 and from Figure 3.

In order to confirm the presence of Mg acceptors, we measured the PL spectrum of both doped and undoped material at T = 10 K. The results are shown in Figure 7. In the undoped material, the PL due to band-to-band recombination has a peak energy of E _B−B = 0.98 eV ± 10%. In the Mg-doped sample however, the emission is due to the e-A recombination with the peak energy of E = 0.88 eV ± 10%. The difference of ΔE approximately 100 meV between the two PL peaks is in the right range for the Mg binding energy and agrees well with the values for the acceptor level obtained from the PC and DC Hall data.