Electron mobility in semi-metal HgCdTe quantum wells: dependence on the well width

Energy spectra, carrier concentration and electron mobility are numerically modeled in intrinsic and n-type semi-metal HgCdTe quantum wells at T = 77 K. We present results for the electron mobility calculated in a model incorporating electron scattering on longitudinal optical phonons, charged impurities, and holes, and including the 2D electron gas screening for all mentioned scattering mechanisms. Inelasticity of electron–phonon scattering is treated by means of a direct iterative solution of Boltzmann transport equation. Comparison with the experimental data at liquid helium temperature is provided.

corresponds to a semi-metallic state) for well widths larger than the critical thickness of about 7 nm (Melezhik et al. 2014).
Three efficient electron scattering mechanisms are dominant in bulk Hg 1−x Cd x Te at nitrogen temperature: inelastic scattering on longitudinal optical (LO) phonons, residual charged impurities (CI) scattering, and electron-hole (EH) scattering (the latter two are elastic) (Dubowski et al. 1981). To calculate the impact of these scattering mechanisms on the electron mobility in quantum well, the linearized Boltzmann transport equation (lBTE) was iteratively solved. Direct solution of lBTE allows one to accurately include the inelasticity of electron scattering, and recovers how the carrier distribution function is perturbed by the applied electric field in the channel. The obtained approximation for the perturbed distribution function allows us to calculate the electron mobility.

Methods
Calculations of energy spectra of Hg 0.32 Cd 0.68 Te/Hg 1−x Cd x Te/Hg 0.32 Cd 0.68 Te QW with the (100) growth axis were performed in the framework of the 8-band k-p Hamiltonian (Novik et al. 2005). This method and the results obtained are described in detail in Melezhik et al. (2014). Here, we only briefly outline the calculated dependencies of carrier energy spectra and intrinsic concentrations.
The dependence of the 2D-confined carrier energy spectrum on the in-plane wavevector k is plotted in Fig. 1. We have neglected the misfit strain in the spectrum calculations since it can be compensated in a multi-layered heterostructure. Therefore there is no band overlap between heavy-hole and electron states in our case.
We have also calculated the hybridized electron wave functions and incorporated them into the calculation of matrix elements for electron scattering. For the sake of simplicity, in such calculations we used the electron wave functions taken at the Fermi level.
The perturbed distribution function is obtained from the lBTE. We followed the methodology of Kawamura and Das Sarma (1992) (100) axis on the in-plane wave vector k. Energy levels marked HH0 to HH3 denote the ground state and the first three heavy hole levels, respectively, and E1 marks the first electron level. HH0 and E1 levels belong to the conduction band, while HH1-HH3 are levels from the valence band electron level is populated, and all scattering processes take place within this level. The energy of the bottom of the ground level is denoted E 0 . We consider the distribution function f(r,k t) which gives the occupation probability of the state |k〉 by an electron in a volume element dr at the position r at time t. The rate of change of f(r,k,t) with respect to time is given by the familiar Boltzmann equation: where E is the electron energy, F is the force due to the external electric field (this force for electrons is F = −eE ext , where E ext is the external field, and e is the electron charge). The last term is the collision integral, which arises from the electron scattering and is given by: where S(k, k′) is the differential scattering rate from state k to state k ′ , bold characters are the vectors while non-bold characters are scalars. For a uniform electric field in a homogeneous system, the Boltzmann equation in the steady state becomes: In equilibrium, the carrier distribution is simply given by the Fermi-Dirac occupation factor f 0 (k). The Fermi level for an intrinsic system is found numerically by setting the concentrations of electrons and holes in the well equal to each other. For a doped system with the given electron concentration, the Fermi level is found by fitting the electron concentration in the well to the needed value. For weak electric fields, we suppose fieldinduced changes of the Fermi level to be negligible.
In the presence of a weak electric field, the distribution function f undergoes an axially symmetric perturbation and is shifted towards the field direction. In this case f may be presented as where α is the angle between k and F, and φ(E) is the perturbation distribution, which has the physical dimension of time. One should note that the coefficients for the perturbed part of f (k) can be written in an arbitrary form since it amounts to a rescaling of the unknown function φ(E), and the form (4) was chosen merely to simplify the further algebra. To obtain the linearized form of collision integral (2), f (k) in the form of (4) is substituted into the expression (2), which yields the following simplified form for the collision integral: (3) and neglecting the term proportional to F 2 in the lefthand side of (3), one arrives at the linearized Boltzmann transport equation (lBTE): where ϑ is the angle between k and k′, while S(k, k′) is the differential scattering rate.
Consider the scattering mechanisms that are most important in MCT: the LO (longitudinal optical) phonon scattering, CI (charged impurities) scattering and EH (electronhole) scattering. The LO phonon scattering is inelastic, while the CI and EH scattering are elastic. In HgTe, longitudinal optical phonons can be considered dispersionless, with the constant energy w 0 = 17 meV (Liubchenko et al. 1984). Differential scattering rates for different scattering mechanisms are additive, thus the total differential scattering rate S k, k ′ can be expressed as: where S a LO k, k ′ and S e LO k, k ′ are the differential scattering rates for phonon absorption and emission, while S CI k, k ′ and S EH k, k ′ are the differential scattering rates for charged impurities and electron-hole scattering. Here θ (E − w 0 − E 0 ) is the Heaviside (unit step) function, which reflects the fact that electrons do not scatter below the bottom of the band.
To obtain the differential scattering rate for charged impurities scattering, we have adopted the approach of Bastard (1988) for nonzero temperatures. According to Szymanska and Dietl (1978), the electron-hole scattering rate can be calculated similarly to the electron scattering rate on CI, but the concentration of charged impurities should be replaced by the effective number of holes. For the LO phonon scattering, the relevant rate for the relaxation of the carriers from the state (k 0 ) is given by Mitin et al. (1999).
Electrons scatter on charged impurities, heavy holes and polar optical phonons by the electrostatic potential. Consequently, this potential is screened simultaneously by the electron and hole subsystems. Such a screening drastically reduces the effectiveness of scattering and it should be taken into account in the mobility calculations. To our knowledge, however, the explicit form of the screening function for this problem has not been established. In our calculations, we have used the two-dimensional screening function for graphene, obtained in Ref. Hwang and Das Sarma (2007). One can argue that since the energy dispersion near the Fermi level is almost linear due to k-p band mixing and strong electron gas degeneration in doped semi-metallic HgCdTe quantum wells, such a screening function should be a good approximation. Recent experiments (Brüne et al. 2014) confirm the applicability of graphene-type screening for HgTe two-dimensional systems. For charged impurities scattering and heavy-hole scattering one should use the static screening function, while for the optical phonon scattering one should use the dynamic screening function at the phonon frequency. Our modeling incorporates screening for all three considered scattering mechanisms.
Substituting (7) into (6), one can obtain: One should note that δ(E + ℏw 0 , E′) and δ(E − ℏw 0 , E′) reflect energy conservation laws that restrict the range of possible values of k′ in (8), since δ(E, E′) is equal to δ(k, k′), where k and k′ are scalars. Thus two-dimensional integrals over the whole 2D k-plane reduce to one-dimensional integrals over the angle ϑ.
To obtain the perturbation function ф(E), we iteratively solve the Eq. (8). The iterative procedure for the calculations is described in Basu and Nag (1980). We use the modified iterative procedure (Melezhik et al. 2015). It allows us to reduce the computational time in several times in comparison with the iterative procedure from Basu and Nag (1980). In our modification of the iterative procedure, the starting values for the procedure consisting of (n + 1) iterations were determined as follows: ф 0 (E − nℏw 0 ) was found from (8) as in the standard iteration procedure (Basu and Nag 1980) (in the first step of this iterative procedure one assumes that the "upper" ф(E + ℏw 0 ) and the "lower" term ф(E − ℏw 0 ) are both equal to zero). Then ф 0 (E + lℏw 0 ) (for l > −n) was found from (8) using ф 0 (E + (l − 1)ℏw 0 ) as the lower term and taking the upper term to be zero. To find n-th order term ф n (E) from (8) one uses φ n (E − w 0 )as the lower term [if it is already found, if not φ n−1 (E − w 0 ) is used and ф n−1 (E + ℏw 0 ) as the upper term]. We repeat these iterations until the difference between ф n−1 (E) and ф n (E) became smaller than 5 %. Such a convergence is usually reached after 3-4 iterations.
As a result of iterative solution of the lBTE, we can find the dependence of the disturbed distribution function ф(E). The electron mobility can be easily calculated from this function.

Results
Under the action of the external dc electric field E ext , the distribution function changes from f 0 (k) to f(k) (4), and the average electron velocity in the 2DEG becomes nonzero. This causes the electric current with the density j, which is found by averaging all possible electron velocities in the QW: The drift mobility, according to its definition µ D = j/(n s F ), can be obtained as follows: The results of our numerical calculations for the electron mobility in HgTe quantum well of 12 nm width at T = 77 K are presented in Fig. 2. Due to the lack of experimental data on electron mobility in considered system at T = 77 K, we compared our numerical results (Fig. 2, curve 2) the experimental mobilities measured by Tkachov et al. (2011) at T = 4.2 K (Fig. 2, curve 1). In Ref. Tkachov et al. (2011), the increase of the electron concentration was achieved dynamically by applying the top-gate bias to the QW.
Comparing our numerical data to the experiment, one can see that the mobility dependencies have several similar features. First, for low electron concentrations, mobility values are fairly low in both curves. Second, such a growth of the electron concentration leads to a sharp increase of the electron mobility. These similarities could be explained by the decrease of the hole concentration and the increase of 2DEG screening while the electron concentration grows. However, experimental mobility at liquid helium temperature from Tkachov et al. (2011) increases faster than our simulated liquid nitrogen temperature mobility. This probably can be explained by a faster decrease of the hole concentration with the Fermi level shift at T = 4.2 K due to the much lower temperature. To compare quantitatively the mobility in Fig. 2 (curves 1 and 2) one should note that in the experiment, the mobility in HgTe at liquid helium temperature is 1.5-2 times larger than the mobility at liquid nitrogen temperature (Pyshkin and Ballato 2013; Meyer et al. 1992).
It is important to note that the above analysis and comparison of mobilities for different temperatures is reasonable because charged centers scattering is usually dominant in semi-metal HgCdTe structures for both liquid helium and liquid nitrogen temperatures (see the discussion below). For liquid helium temperature, this statement can be illustrated by Fig. 2 from Dubowski et al. (1981) and Fig. 5 from Meyer et al. (1992).
We have also modeled the dependence of the electron mobilities in semi-metal type Hg 0.32 Cd 0.68 Te/Hg 0.94 Cd 0.06 Te/Hg 0.32 Cd 0.68 Te QW on the well width, for different electron concentrations. These results are presented in Fig. 3. The increase of mobility with the growing electron concentration can be explained by the hole concentration depletion for EH scattering mechanism and screening effect enhancement, which suppresses all three scattering mechanisms considered. Decrease of the QW width L shifts the system towards the phase transition point, which is located slightly below L = 12 nm in the case under study. It is important to note that the behavior of the electron mobility as a function of the QW width is different at different electron concentrations: For high electron concentrations, the mobility does not change sufficiently, while for low electron concentration the mobility changes by factor of several times. This behavior is explained by the presence of residual charged impurities. Variation of the well width results in the variation of the hole concentration in the well (see Fig. 3). For high electron concentrations, the hole concentration is smaller than the residual impurity concentration, and the variation of holes concentration does not affect the electron mobility. For low electron concentration, the hole concentration is much greater than charged impurities concentration, and the electron mobility is strongly influenced by the change in the hole concentration.
One should note that near the band inversion point one more scattering mechanism becomes important-the scattering on fluctuations of effective mass (Tkachov et al. 2011). However, at liquid nitrogen temperature we cannot use the methodology of Tkachov et al. (2011), because due to higher temperature the simple 4-band approximation is no longer valid. Thus we can expect that near the point L = 12 nm (see Fig. 3), our simulation overestimates the electron mobility.
Consider impacts of each scattering mechanism. Strong dynamical screening leads to a strong suppression of the longitudinal optical phonon scattering. For an intrinsic 20 nm wide quantum well with the composition x = 0, the electron mobility for the LO phonon scattering is about 3.8 × 10 6 cm 2 /(V s), while for n-doped quantum well of the same width with composition x = 0.06 (electron concentration 1.5 × 10 17 cm −3 ), the mobility is about 6.8 × 10 6 cm 2 /(V s). As these mobilities are much greater than the corresponding total mobilities, we can conclude that the main contribution to the total 3)

Fig. 3
Dependence of the electron mobility on the quantum well width L. Curves 1, 2, 3 correspond to 5 × 10 16 , 10 17 , 1.5 × 10 17 cm −3 electron concentrations in the well. The QW composition is x = 0.06. The concentration of residual charged impurities in the well is 10 15 cm −3 mobility comes from the charged impurity scattering and electron-hole scattering. Relative importance of these two scattering mechanisms can be easily seen from the comparison of the hole and charged impurity concentrations presented in Fig. 4. To use semi-metallic HgCdTe quantum well as a channel of THz hot-electron bolometer, one needs to estimate the noise in the channel. There are three main noise generation mechanisms present in this case: thermal (Johnson's) noise, generation-recombination noise and photon noise, which can be calculated using standard formulas ). For such an estimate one needs the bolometer channel resistance which we have extracted from the calculated values of the mobility and the channel dimensions. Further, for the estimate of the generation-recombination noise one needs to know the electron lifetime in such heterostructures. The corresponding experimental data is scarce; however, from Tkachov et al. (2011), Grein et al. (2005), Morozov et al. (2012) this time could be roughly estimated as being about 10 −10 s.
We calculate the noises using the value of bias current of 1 mA from Dobrovolsky et al. (2010), for the quantum well of 20 nm width, with the composition x = 0.06, the electron concentration 1.5 × 10 17 cm −3 and the electron mobility 9 × 10 5 cm 2 /(V s). For those parameters the total voltage noise, which consists of the Johnson-Nyquist, generationrecombination, and photon noises, is about 3 × 10 −10 V. The evaluated noise is almost two orders of magnitude smaller than the corresponding noise for n-type semiconductor HgCdTe bolometers from Dobrovolsky et al. (2010).
An estimate of the resistance in semi-metallic HgCdTe quantum well used as a channel of THz hot-electron bolometer shows that at high electron concentration this resistance is much smaller than 1 kΩ. This resistance is 1-2 orders of magnitude lower than the corresponding resistances in graphene channels [where the typical resistance is of the order of 10 kΩ (Du et al. 2014;Farmer et al. 2009)]. Thus, HgCdTe channels produce smaller thermal noise and, compared to graphene, can provide more efficient coupling to planar antennas in hot-electron bolometer applications.
3) Fig. 4 Dependence of hole concentration on the quantum well width L, for the same electron concentrations as presented in Fig. 3. Curves 1, 2, 3 correspond to the electron concentrations in the well 5 × 10 16 , 10 17 , 1.5 × 10 17 cm −3 respectively. The QW composition is x = 0.06. Dashed horizontal line represents the concentration of residual charged impurities in the well (10 15 cm −3 )