Experimental studies of transport in GaPN dilute nitrides have evidenced low hole mobilities, which limit their applications in optoelectronics. Theoretical work to date has not explained the origin of such low hole mobilities. Here, we use full band cellular Monte Carlo methods to investigate hole transport in C-doped GaPN dilute nitrides as a function of hole concentration. Good agreement between simulation and experiment is obtained by introducing a doping-dependent self-compensation. The results suggest that the reduction in the hole mobility is caused by the compensation for the p-type C doping, which is likely due to the formation of C–N complexes that act as donor scattering centers. This agrees well with the low C activation ratio reported by Liu et al. [Appl. Phys. Lett. 96, 032106 (2010)] and other studies on C–N complexes in GaP.
The incorporation of a small fraction of nitrogen atoms into GaP(As) (so-called dilute nitrides) results in both large bandgap reduction1,2 and lattice constant reduction,3–5 providing the ability to both tune the bandgap and provide lattice matching during epitaxial growth. In particular, the possibility of III–V materials lattice-matched to Si has many potential applications, including advanced photovoltaics,6,7 lasers,8 and optoelectronic integrated circuits.9 However, short diffusion lengths in dilute nitrides, in general, have been a limiting factor for applications in photovoltaics.10,11 There has been a theory reported for electron mobility in dilute nitrides,12 which predicts that the incorporation of N impacts the electron mobility inversely as , where Ec is the conduction band edge and x is the N concentration. This model cannot be extended to the hole mobility in dilute nitrides, since N incorporation introduces a narrow band of states near the conduction band minimum while negligibly affecting the valence band maximum.13 Hence, the model would predict only an insignificant change in the hole mobility, which contradicts the experimental fact that N incorporation also greatly reduces hole mobility in different dilute nitrides, including GaInAsN14,15 and GaP(As)N.16,17 Although this is a general problem for dilute nitrides, we focus here on GaP(As)N alloys due to their importance for integrated III–V/Si applications as mentioned above. There have been relatively few experimental reports on hole transport in GaP(As)N.16–18 Reference 16 is the only one that systematically reports on GaPN with different N mole fractions, including pure GaP for comparison, over a wide doping concentration range. In the present work, we compare to the mobility data in Ref. 16 using the full band cellular Monte Carlo (CMC)19,20 simulation of the low field mobility to better understand the origin of the lower hole mobilities of p-type GaPN dilute nitrides compared to those of GaP.
The CMC method is a stochastic solution of the Boltzmann transport equation, which uses a full band description of the carrier dynamics as well as the random scattering processes responsible for the mobility. Here, we use the sp3d5s* tight-binding (TB) method21 to calculate the band structure throughout the entire Brillouin zone. Due to its semi-empirical nature, the tight-binding parameters (e.g., overlap integrals) are adjusted to provide a best fit to experimental data. We have previously optimized the sp3d5s* parameters for GaP and other III–V materials based on the experimental optical transition energies and the dielectric function, which provides a better description of the optical properties compared to other available sets.22 In the present work, an accurate description of the energy dispersion and effective mass near the top of the valence band is the main concern. Hence, we chose the sp3d5s* parameter set in NEMO523 for GaP, which gives good agreement with the measured effective mass data.
The scattering mechanisms used in the simulation include polar optical phonon scattering, deformation potential scattering, piezoelectric scattering, ionized impurity scattering, alloy scattering, and neutral impurity scattering, of which the last two may be important for dilute nitrides in general. However, the results of the present work suggest that, as discussed later, ionized impurity scattering is critical in explaining the low hole mobility in C-doped GaPN dilute nitrides. Here, we use the Ridley multi-ion scattering formalism,24 where the ionized impurity scattering rate from wavevector k in band v to a small region Ωk′ centered at wavevector k′ can be written as
where NB is the total ionized impurity density, vg is the carrier group velocity, and ΓBH is the Brooks–Herring screened ionized impurity scattering rate, written as
where Z is the number of charges of the ion species, e is the elemental charge, ℏ is Planck’s constant, ϵ0 is the vacuum permittivity, J(k, k′) is the overlap integral of the cell periodic part of the Bloch function evaluated using the tight-binding wave functions, Ev(k) is the carrier energy at wavevector k in band v, and βs is the Debye screening length, which depends on the free hole density.
We first calibrate the model by comparing with the experimental temperature-dependent hole mobility in GaP reported in Ref. 25. As shown in Fig. 1, the simulated hole mobility for GaP is in excellent agreement with the experiment from 50 K to 400 K, particularly at 300 K where we are particularly interested. From high temperature to low temperature, the mobility first increases, limited by phonon scattering. At around 90 K, ionized impurity scattering starts to dominate and bends down the curve. For temperatures lower than 40 K, hole transport becomes dominated by hopping from occupied to unoccupied acceptors,25 which is outside the description given by the CMC approach. As noted earlier, the NEMO tight-binding parameters gave a better fit to the mobility data as shown in Fig. 1 than those optimized for optical spectra, and hence, this parameter set is used as the basis for dilute nitrides as discussed in the following.
We then performed the CMC simulation of the GaP hole mobility at room temperature for different acceptor concentrations assuming full ionization and no (donor-like) compensation, i.e., p = NA. The resulting hole mobilities agree well with a series of reported experiments for GaP, as shown in Fig. 2, further validating the accuracy of the simulation. Ionized impurity scattering increasingly dominates and reduces the mobility with the increase in acceptor concentration as expected.
To simulate for GaPN, we use the sp3d5s*sN tight-binding model developed specifically for dilute nitrides by Shtinkov et al.26 In the present work, it is based on the NEMO GaP sp3d5s* parameters, which give a better fit to the GaP mobility data, using the same procedure as in our previous work.22 As mentioned earlier, the incorporation of N in III–Vs is generally considered to impact mainly the conduction band with negligible effect on the valence band.13 Consistent with this, the sp3d5s*sN model results in very little modification of the valence band dispersion for GaPN compared to GaP. Correspondingly, the full band Monte Carlo hole mobility results for GaPN are essentially the same as those for GaP within the numerical accuracy of the method, if no consideration of compensation or other additional effects is taken.
The experimental mobility data for GaPN are shown in Fig. 2 by the lower mobility set of colored data points. As seen, the experiments in Ref. 16 show much lower hole mobilities for GaPN than GaP, with no distinguishable difference in the GaPN hole mobilities with different N mole fractions ranging from x = 0.5% to 3%. This fact excludes alloy scattering as being a dominant mechanism, since it has an x(1 − x) prefactor in the scattering rate. As a check, we have included this mechanism in the simulations and found that it has a negligible effect. We have also found that neutral impurity scattering cannot account for the low mobilities observed for hole concentrations from 1 × 1017 cm−3 to 5 × 1019 cm−3, where the density of neutral impurities required to reach such low mobilities needs to vary continually from 2 × 1020 cm−3 to 1 × 1021 cm−3 for the same concentration range, which is physically difficult to justify in terms of a doping-dependent source with such high concentrations. What is needed to explain the reduction in mobility is a scattering mechanism that is independent of the N concentration but has an acceptor concentration dependence.
To explain the experimental mobility, we suggest a doping-dependent compensation mechanism leading to higher ionized impurity scattering rates. The basis for this hypothesis is that in Ref. 16, they reported that the activation ratio of C in pure GaP is almost 100%, whereas in GaPN, it varied from 5% to 40% with increase in the free hole concentration in Fig. 2 from 2.4 × 1017 cm−3 to 3.3 × 1019 cm−3, as determined by secondary ion mass spectroscopy (SIMS). We assume that this change in activation is entirely due to compensation and include this in the ionized impurity scattering rate to compare with experiment. The ionized acceptors are treated as singly charged negative ions, , and compensation is represented by singly charged positive ions of concentration . The free hole concentration is , and the total ionized impurity concentration is . Figure 3 shows that the CMC simulation results including compensation match well with the experimental data. To achieve this agreement, an increasing amount of compensation is needed as the free hole concentration increases. As shown in Fig. 4, the required compensation concentration needed in the simulation for a good fit increases from 5 × 1017 cm−3 to 3 × 1019 cm−3 for hole concentration from 1 × 1017 cm−3 to 3 × 1019 cm−3. Combined with the fact that the reduction in the hole mobility is related to the presence of N contents in GaPN, this could imply that some of the C atoms contribute to the compensation through some interaction with N species. The simulated hole to total ion concentration ratios, also shown in Fig. 4, increase from 9.6% to 35.7% for hole concentration from 1 × 1017 cm−3 to 1 × 1019 cm−3, which agrees roughly with the C activation ratio in GaP0.99N0.01 measured in Ref. 16, which is also shown.
Experimentally, it has been reported that triple bonded C–N complexes can form in GaP, and these complexes do not seem to contribute to p-type doping.28 Later, first-principles density functional calculations support the formation of triple-bond C–N complexes in GaP and find that these complexes act as donors (i.e., compensation) under p-type doping conditions.29 One major disagreement between Ref. 16 and the above two studies is that they did not detect C–N infrared absorption peaks in their samples. However, our simulations seem to support donor type of C–N complex formation in p-type GaP and that as more C–N complexes form, a smaller percentage of C contribute to the compensation, with the percentage being as high as 45% at low acceptor concentrations.
In summary, we have used full band Cellular Monte Carlo simulations including all relevant scattering mechanisms to investigate the low field hole mobility in C-doped GaPN dilute nitrides in comparison to experiment. Good agreement between simulation and the experiment was achieved by including a doping-dependent compensation, in agreement with the experimentally observed C activation. Our results suggest that some of the C and N atoms form donor complexes, which compensate the acceptors, leading to increased ionized impurity scattering for a given free hole concentration. More experimental work investigating compensation effects in dilute nitride materials would help in substantiating this conclusion.
This material is based upon the work primarily supported by the National Science Foundation (NSF) and the Department of Energy (DOE) under NSF CA Grant No. EEC-1041895. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the NSF or the DOE.
The data that support the findings of this study are available from the corresponding author upon reasonable request.