Intrinsic broadening of the mobility spectrum of bulk n-type GaAs

Historically, the mobility of semiconductor charge carriers is treated as a single-valued function of temperature and other parameters such as the doping concentration. Such a description has been adequate for electronic devices with behaviors that are dominated by bulk carrier transport. Treating carrier mobility as a single valued function is consistent with conventional Hall measurements which are performed at a single value of magnetic field strength and provide only a weighted average of the electron mobility and carrier concentration of a semiconductor structure. Modern electronic devices consisting of numerous semiconductor layers often result in the population of numerous distinct carrier species, therefore, treating carrier mobility as a single valued function of temperature and other semiconductor device parameters such as carrier concentration can be a completely inadequate description. Mobility spectrum analysis is an experimental technique based on magnetic field-dependent conductivity-tensor measurements for the extraction of a mobility distribution. Mobility spectrum analysis has been applied to the study of the carrier conductivity mechanisms of numerous semiconductor structures and devices. However, there is a severe lack of reported studies on theoretical calculations of the mobility distribution of semiconductor structures or devices. In addition, the majority of reports on experimental mobility spectrum analysis are of complex multi-layered structures such as type-II superlattices and the interpretation of the mobility spectra has been difficult. Therefore, a good understanding of mobility spectra has yet to be developed. We present calculations of the electron mobility distribution of bulk GaAs which predicts the existence of multiple mobility distribution peaks resulting from electron conductivity in the Γ conduction band. This report serves as an important and simple test case upon which experimentally measured mobility spectra can be compared and presents insight into the general nature of electron mobility distributions.


Introduction
Historically, the mobility of a semiconductor charge carrier is treated as a single valued function of temperature and other semiconductor parameters such as the doping density. This, in part, is due to the fact that common techniques to measure mobility are acquired at a single magnetic field strength, which results in a single value of mobility and carrier concentration. Modern semiconductor devices and structures often consist of multiple semiconductor layers, and thus contain populations of distinct carrier species caused by n-and p-doped regions, twodimensional conduction of surface and interface layers, and thermally generated minority carriers, among other possibilities. Standard, single-magnetic-field measurements of the resistivity and Hall coefficient provide only an averaged value of the mobility and carrier concentration. Such averaged values have little physical meaning for devices where multiple carrier species, in addition to bulk majority carriers, have a strong influence on device operation. Knowledge of the mobility and carrier concentration of the individual carrier species provides valuable insight into carrier interactions and device operation, which has motivated the development of mobility spectrum analysis techniques.
MSA is based upon an analysis of magnetic field-dependent conduction, which is assumed to be of the form x y x y 2 2 2 2 where n, e, μ, and B are the carrier density, the magnitude of the charge on an electron, the carrier mobility, and the magnetic field strength, respectively. S is −1 for electrons and +1 for holes. The magnetic field is applied in the z-direction and the current is constrained to flow in the x-and y-directions.
Within the MSA framework, the conductivity relationship is generalized to allow the conductivity-tensor components to be continuous functions of mobility, . (2) x y x y 2 2 2 2 MSA algorithms are analogous to a Fourier transform in the sense that conductivity is transformed from the magnetic field-strength domain to the mobility domain. To provide an understanding of mobility spectra, we present the calculation results of the mobility distribution of bulk GaAs. A complicated mobility distribution with multiple peaks resulting from electron conductivity in only the Γ conduction band is predicted. The mobility distribution is dependent on the nature of the scattering interactions, and therefore is influenced by temperature and the doping concentration. The electron kinetic energy distribution directly leads to a broadening of the mobility distribution, which is persistent over a wide temperature range for nondegenerately doped GaAs. The calculated results provide a means to evaluate and interpret experimentally derived mobility spectra.

Model details
To calculate the mobility distribution of n-type GaAs, the Boltzmann transport equation (BTE) is solved numerically only for conduction band electrons under the influence of an applied electric field. Consider the distribution function, f t k ( , ), which represents the probability that the electron momentum state, 〉 k | , is occupied, and t is a time parameter. Since we are dealing with homogeneous bulk material under steady-state conditions, the distribution function has no spatial dependence.
The Boltzmann transport equation expresses the fact that under steady-state conditions, the occupation probability of any given momentum state, 〉 k | , is time independent, Considering the actions of the applied electric field and electron-scattering mechanisms, the steady-state condition gives where e is the magnitude of the electron charge, ℏ is Plankʼs constant, and E is the applied electric field. The right-hand-side term of the above equation represents the rate of change of the electron momentum state occupation due to scattering events. Three scattering interactions are taken into consideration: polar optical phonons, acoustic phonons, and ionized impurities. Solving the Boltzmann transport equation provides information about how the electrons are distributed among the momentum states. From the momentum distribution, we obtained steadystate distributions such the conductivity, σ E ( ), and mobility, μ E ( ), as a function of carrier energy, and the conductivity as a function of mobility, σ μ ( ). The longitudinal polar optical (LO) phonon interaction takes the usual form. The rate at which an electron in an initial momentum state, 〉 k | i , scatters to a final state, 〉 k | f , is given by the following equation, f is the magnitude of the phonon wavevector, ω LO is the LO phonon frequency (assumed to be dispersionless), ε s is the static permitivity, and ε e is the high frequency permitivity. N q is the number of phonons occupying the mode, where T is the temperature and k b is the Boltzmann constant. The upper (lower) sign of ± corresponds to emission (absorption) of phonons. The total rate at which electrons scatter from the state k i is an integration over all final states, The total rate at which electrons scatter into the state k i is given by The − f k (1 ( )) i terms are important for degenerately doped semiconductors, but are very close to unity for the nondegenerate GaAs. Nevertheless, as considered in this work, they were included in the computational routines.
The longitudinal acoustic phonon scattering rates are evaluated by a Bardeen-Shockley deformation-potential approximation method [28], the details of which can be found in the appendix. Ionized impurity scattering is calculated within the usual Brooks-Herring approach for nondegenerate semiconductors. The scattering rate of an electron in the state k i is given by i s 0 ε s is the static permitivity and q 0 is the screening parameter, which is dependent on the doping density, N D , The band structure of the GaAs is provided as input to the scattering routines, which is calculated from an eight-band k p . model. After the total scattering rates are calculated, including the contribution of the applied electric field, a correction is made to the electron distribution function, f k ( ), based on the imbalance between the rate at which electrons leave and enter a given momentum state. The scattering rates are re-evaluated, and this process is repeated until the net scattering rate converges to zero. This fully numerical method is similar to that of Rode [29], and is commonly used for calculations involving anisotropic and inelastic scattering processes [30,31].
For all calculations, the standard material parameters for GaAs have been provided as input, as taken from [32]. The relative static and high-frequency permitivity values are taken as 12.9 [33] and 10.89 [34], respectively.
All distributions are calculated from f k ( ). The conductivity density (in the direction of the applied electric field) as a function of the electron energy is calculated by numerically evaluating the following integral where θ is the angle with respect to the direction of the applied electric field, is the angular frequency of the electronic state and the electron group velocity, v g , is in the direction of k.
The mobility as a function of electron energy is obtained from the conductivity density and the differential carrier density.
2 μ E ( ) and σ E ( ) allow for the construction of the mobility distribution function, σ μ ( ), which indicates the contribution to the conductivity of carriers with a particular mobility. A mobility slice of finite width is chosen, (μ μ Δμ → + ), and the energy ranges, for which the mobility distribution (μ E ( )) falls within the mobility of the chosen slice are determined. The conductivity is then determined by, computational results for a temperature of 300 K and an electron concentration of × 1 10 16 cm −3 . Figure 1 shows the electron mobility and conductivity density as a function of electron kinetic energy, and figure 2 is a plot of the resultant mobility distribution. The calculated average mobility is 6820 cm 2 (Vs) −1 , which is in agreement with experimentally determined values of the drift mobility for room-temperature bulk GaAs with a carrier concentration of × 1 10 16 cm −3 . An interesting feature of the μ E ( ) and σ E ( ) plots is their sudden reductions in magnitude, which occur at integer multiples of the LO phonon energy (36 meV). A reduction of mobility and conductivity is expected at the LO phonon energy point, since LO phonon scattering is the dominant scattering mechanism for GaAs at room temperature and E = 36 meV is the threshold for LO phonon emission, which results in a dramatic increase in the electron scattering rate. The subsequent reductions at larger energies are a result of the tendency for LO phonon scattering to occur for smaller phonon wavevectors. Consider the change in the distribution function due to the applied electric field,  Plots of σ μ ( ) for a range of temperatures are shown in figure 3. Decreasing temperature leads to a reduction in magnitude of the σ μ ( ) peaks at higher mobility. This is a consequence of the fact that fewer electrons have enough thermal energy to populate states with an energy greater than the LO phonon energy. For a temperature of 100 K, the σ μ ( ) distribution is very broad and only a single peak exists. Some general observations can be made regarding the nature of the calculated mobility distribution. First, the mobility distributions are rather broad, and second, the subsequent peaks of the mobility distribution occur at larger mobility. The broad, calculated mobility distribution is a result of the approximate square root dependence of the GaAs, electron group velocity on the electron energy. The group velocity term in  equation (12), together with equation (13), results in a large mobility spread with electron kinetic energy. This directly leads to the broad mobility distribution, particularly at low temperatures, such that there are no polar optical phonon-related peaks. For semiconductors that are heavily doped, the mobility distribution may become narrow at low temperatures, since carrier interactions and transport predominantly occur at the Fermi level. However, for the results presented here, at a temperature of 100 K and an electron concentration of × 1 10 16 cm −3 , the Fermi level is 18 meV below the conduction band edge, and therefore conduction occurs over a large relative electron energy range. As shown in figure 1, the mobility increases with each polar optical phonon emission-related dip. Therefore, each subsequent peak of the mobility distributions occurs at larger mobility.
The nature of the scattering interactions, and therefore of the electron concentration, has an impact on the mobility distribution. Figures 4 and 5 show results for a temperature of 300 K and a carrier concentration of × 1 10 17 cm −3 . An increase of the carrier concentration results in a relatively larger ionized impurity scattering rate, which tends to mask the effects of the LO phonon scattering. This results in shallower dips in the mobility function, μ E ( ), and the second peak of the σ μ ( ) distribution has a reduced magnitude relative to the first peak. A mobility distribution, σ μ ( ), that features more that a single peak is expected to occur for a wide variety of semiconductor materials and structures, provided that the LO phonon scattering dominates the total carrier scattering rate. These conditions are likely to hold for polar semiconductors with a low to moderate concentration of impurity atoms at a temperature greater than about 200 K. For structures such as superlattices with mobilities that are limited by interface roughness scattering, LO phonon interactions are less likely to have a significant influence on the qualitative features of the mobility distributions.
The development of mobility spectrum algorithms initially considered populations of distinct carrier species, and assumptions had to be made regarding the number of species present. Modern algorithms consider a continuum of carrier mobility, and therefore no prior assumptions need to be made regarding the number of carrier species that are present in a semiconductor sample under consideration. However, there is a tendency to assume that the individual peaks of a mobility spectrum originate from distinctly different carrier conduction mechanisms such as the surface and bulk conduction of narrow-band-gap semiconductors. Designating carriers into distinct species can be a completely inadequate description, since carriers generally occupy a continuum of states with a coupling between states that can vary greatly depending on the available phonon modes and the overlap of wavefunctions. For example, the bulk and surface carrier conduction of a narrow-band-gap semiconductor occur in spatially separate regions, and therefore the coupling of carriers between these two conduction mechanisms is minimal, whereas the carriers occupying the ground and excited state subbands of a quantum well superlattice [35] are tightly coupled by phonon transitions. Given the thermal energy distribution of carriers, it is evident that within a mobility spectrum approach, carriers should be considered to occupy a continuum of states, which can have important consequences on the mobility spectrum.
An exact solution of the BTE can be computationally expensive, and other methods of determining carrier mobility are common in the literature. A particularly popular method for calculating mobility is a momentum relaxation approach of the following form,  where E is the electron kinetic energy and τ is the carrier momentum relaxation lifetime derived from scattering rates of the following form, ⎛ i f is the scattering rate from state k i to k f and θ if is the scattering angle between k i and k f .
An energy dependence can be introduced into the mobility, An energy-dependent conductivity density can also be obtained by the following description, c where n c (E) is the carrier density per unit of carrier energy. To demonstrate the influence that the common momentum relaxation approach described above has on the calculated mobility distribution, calculations have been performed for various temperatures and an electron concentration of × 1 10 16 cm −3 . The momentum relaxation approach described above cannot obtain mobility or conductivity functions with the level of detail of more complete solutions of the BTE. This can be seen by the results shown in figure 6. The corresponding mobility distributions shown in figure 7 contain single mobility peaks that result from the plateau of the mobility as a function of energy, for energies just beyond the LO phonon energy.
Reports on MSA measurements cover a wide range of semiconductor materials and structures, the majority of which are complex multilayered structures, so there are few reports that the work presented in this paper can be directly compared to. However, it can be concluded that MSA measurements tend to return mobility spectra with conductivity that occurs over a considerably narrower mobility range. See, for example, the MSA reports by Brown et al of single-layer n-type HgCdTe and InP epi layers [27], Hudait et al on InAs 1 P −x 1 [36], and Umana-Membreno et al on LPE-grown Hg x Cd x Te layers [26].
There is no compelling evidence of multiple MSA peaks due to a polar optical phonon interaction in the literature. Hudait et al [36] performed an analysis on 1.5 μm thick epilayers of InAs x P −x 1 , Si-doped to a concentration of × 2 10 16 cm −2 , using the quantitative MSA algorithm, which indicates the presence of multiple peaks in the electron mobility spectrum due to conduction within the epilayers. An earlier work by Achard et al [10], which studied the mobility spectrum of InP epilayers using an older mobility spectrum algorithm, reported only a single mobility peak at higher temperatures. Acar et al [25] reported on the measured mobility spectrum of bulk GaSb, indicating the possibility that LO phonon scattering resulted in two peaks in the mobility spectrum. For higher temperatures, the mobility spectrum splits into two peaks that are closely spaced in mobility. The occurrence of the second peak is attributed by Acar et al to the population of the L conduction band with electrons, due to its small energy separation (≈ 75 meV) from the Γ band. Considering that the ratio of electron mobilities in the L and Γ bands, μ μ Γ , L is about 0.1 [37], it is possible that the L band conduction occurs at a lower mobility range that is outside of the plots presented in [25], and that both peaks result from conduction in only the Γ band. A particularly interesting report published recently by Brown et al [27] studied the mobility spectrum of nondegenerate molecular beam epitaxy-  grown n-type Hg −x 1 Cd x Te epilayers. The mobility spectrum of these epilayers exhibited a double peak for temperatures greater than about 150-200 K. These two peaks are attributed to conduction in regions of different Hg content, and hence different mobility. This view is supported by measurements of the composition profiles of the Hg −x 1 Cd x Te layers, which show a dramatic increase of the cadmium fraction for regions towards the epilayer/substrate interface. However, the evolution of the lower mobility peak, which becomes evident for temperatures greater than 150 K, is closely separated in mobility from the higher mobility peak (within a factor of two), which is qualitatively very similar to the GaA results calculated in this work. Therefore, it seems possible that the double peak is due to the inherent mobility spectrum of − Hg x 1 Cd x Te. This is supported by the fact that the mobility of mercury cadmium telluride is strongly dependent on the cadmium content, and all three Hg −x 1 Cd x Te epilayer samples grown on different substrates display similar mobility spectra. Similarly, Umana-Membreno et al [26] studied liquid-phase epitaxy-grown HgCdTe epilayers of 20 μm thickness. The mobility spectra displayed closely spaced double peaks, which are attributed to electron conduction in regions of different cadmium content. Given the computational results presented in this work, predicting that multiple peaks of a mobility spectrum can naturally arise from carrier transport in bulk material due to electron-polar optical phonon scattering, the assumption that different peaks of a mobility spectrum are due to different carrier transport mechanisms may not be valid.
It is not a forgone conclusion that the results presented here would match those of the experiment. A central question is whether the various nonideal aspects of experimental Hall measurements should be expected to substantially influence the derived spectra. Particular concerns include effects such as a nonideal Hall bar or van der Pauw geometry, lateral inhomogeneities of the material, contact resistance, finite magnetic field range, possible effects of the Hall factor, and the range of validity of the conductivity-tensor components given by equation (2). Further work must be done on understanding the limitations of MSA and the effects of nonideal experimental influences on the derived mobility spectrum.

Conclusions
We have presented calculations of the electron mobility distribution of bulk GaAs that reveal some interesting features of the mobility distribution due to conduction within the Γ band. We found that the mobility distribution of GaAs is considerably complex, with a shape that depends on the nature of the scattering interactions. We predicted that under conditions where the optical phonon interaction is the dominant scattering mechanism, multiple peaks are present in the σ μ ( ) spectrum. This suggests that the interpretation of experimental quantitative mobility spectrum analysis requires careful consideration and, in particular, the assignment of individual peaks of the mobility spectrum to different conduction mechanisms may not be valid in some situations. In addition, the calculated conductivity occurred over a broad mobility range, which is a consequence of the thermal energy distribution of electrons.
The classical continuum elastic acoustic modes are calculated, from which the hydrostatic strain is determined for an acoustic mode equal to the phonon energy, ω =  E . Acoustic mode solutions are assumed to take the following form, where t p r ( , ) is the atomic displacement function of the acoustic mode, a is an atomic displacement vector, q is the acoustic wavevector, and ω is the frequency of the acoustic mode.
a C a q C C a q q a q q C a q q a C a q C C a q q a q q C a q q a C a q C C a q q a q q C a q q (A. where a c and ε hyd are the conduction band deformation parameter and the hydrostatic strain, respectively. The commonly used GaAs, conduction band deformation parameter of 7.17 eV [32] is used in this work. Using a similar method to that described above for LO phonons, the total scattering rates are calculated for each k i momentum state.