Transport effects in remote-doped InSb/AlxIn1-xSb heterostructures

Low- and high-field magnetotransport measurements on two 30 nm δ-doped InSb/AlInSb quantum wells (QWs) with different doping densities are reported. The QW two-dimensional electron gas (2DEG) carrier densities and mobilities were extracted by analysis of the Hall and quantum Hall data, mobility spectra and Shubnikov-de Haas oscillations. 2DEG channel mobilities of up to 324 000 cm2 V-1 s−1 (T=2 K) and 44 000 cm2 V- 1 s−1 (T=300 K) are extracted. Carrier densities and mobilities for transport parallel to the 2DEG layer are also deduced where observable. The importance of thermally generated carriers in the lower AlInSb barrier material and the role of transport within the δ-doping plane is considered and the total carrier population as a function of temperature of the two samples is deduced, which is in excellent agreement with experimental observation.


Introduction
Narrow band gap semiconductors are exciting candidates for high-speed electronics [1,2], quantum devices [3,4], ballistic transport [5] and spintronic applications [6]. Modulation-doped heterostructures provide the high-mobility samples that are required for these applications. Of the III-V semiconductors, InSb has the lightest effective mass (0.014m e ) and narrowest band gap (170 meV at room temperature) along with a large dielectric constant (16.8 0 ) and an exceptionally large Landé g-factor (∼−51) [7]. This makes the InSb/AlInSb system a particularly interesting candidate for these applications. The various challenges faced in heavily mismatched epitaxial growth have meant that development in the understanding of these systems is still at a relatively immature stage compared to wider gap III-V semiconductor systems such as GaAs or GaN.
In this paper, we present the growth and magnetotransport measurements of two highquality InSb/Al x In 1−x Sb modulation-doped heterostructures with different doping densities. The lower-doped sample is shown to exhibit single carrier conduction at low temperature (2 K), with the onset of parallel conduction at higher temperature, while the higher-doped sample shows a parallel conduction at low and high temperatures. In both cases, the mobility of the 2D carrier is extracted.

Sample preparation and measurements
Two InSb/Al x In 1−x Sb heterostructures of identical compositional design (shown schematically in figure 1) but different doping densities were grown by molecular beam epitaxy (MBE) using standard effusion cells for the In, Ga and Al and cracker cells for the Sb and As, at a nominal growth rate of 1 µm h −1 . The 30 nm wide InSb quantum wells (QWs) were surrounded by an AlInSb matrix. The top 50 nm barrier of Al 0.15 In 0.85 Sb contained the Te delta-doping plane. The lower Al 0.10 In 0.9 Sb barrier was 3 µm thick and grown with a nominal n-type background doping of 1 × 10 15 cm −3 at room temperature. This was separated from the GaAs cleaned and buffered substrate surface by 200 nm AlSb. Secondary Ion Mass Spectrometry (SIMS) studies on similar layers to these have shown the Te dopant ions to be distributed through the top cap above the nominal point of introduction [8]. This is illustrated in figure 1 by the solid points.

Figures 2(a) and (b)
show the high-field magnetoresistivity (ρ x x ) and Hall resistivity (ρ x y ) of samples 1 and 2, respectively, measured at 2 K. Sample 2 shows distinct, single-period Shubnikov de-Haas (SdH) oscillations in ρ x x and well-developed quantum Hall plateaux in ρ x y , which have previously been observed in samples of this type [9,10]. In stark contrast, ρ x x data for sample 1 show SdH oscillations superposed on a significant background signal with negligible structure observed in the ρ x y data. It is noteworthy that the SdH oscillations remain single period in 1/B. The background contribution to ρ x x is a clear indication of significant parallel charge transport elsewhere in the device. Assuming that the SdH oscillations originate from the ground state subband in the QW, the periodicity in 1/B can be used to determine the carrier density of the two-dimensional electron gas (n 2DEG ). For sample 1, this was 3.5 × 10 11 cm −2 and for sample 2, 1.9 × 10 11 cm −2 .
The gradient of the slope of ρ x y at high field, bisecting quantum Hall plateaux where they exist, gives a good approximation to n tot [11]. In sample 2 at low temperature, this is in excellent agreement with the concentration calculated by the period of the SdH oscillations. However, in sample 1, the total carrier density is significantly greater than n 2DEG . This can be used to calculate the density of the parallel carrier (n ) by For sample 1, the mobilities of the individual carriers were calculated from fits of the ρ x y data to a two-carrier model [12]. The model provides an excellent fit to the experimental data for ρ x y , as shown by figure 3(a) for sample 1. At 2 K, the mobilities of the two-carriers in sample 1 were found to be 324 000 cm 2 V −1 s −1 for the 2D carrier contribution and 4000 cm 2 V −1 s −1 for the parallel carrier contribution. The second carrier has a concentration of 11.7 × 10 11 cm −2 .
The effective average mobility extracted from the model was 74 500 cm 2 V −1 s −1 . This is in good agreement with the measured mobility of 74 000 cm 2 V −1 s −1 , extracted from the high-field To give further confidence in this analysis, we have used mobility spectra analysis on low-field data. The resistivity data were corrected for the measured length/width ratio of the Hall bars. The mobility spectrum from the magnetoresistance and Hall coefficient data, after conversion to the conductivity tensor components, were calculated using a matrix inversion approach similar to the formalism of Beck and Anderson [13]. At low temperature the magnetic fields used were limited to those below which the quantum Hall effect might produce small but significant (when looking for a second carrier) oscillations in the magnetoresistance or Hall coefficient ( 0.4 T). The matrix inversion approach is known to be susceptible to ill-conditioned matrices. The robustness of the spectra produced were checked by permuting a selected set of experimental data points and removing data permutations producing solutions that were inconsistent with the most common number and nature of carriers (i.e. n-type). Data point permutations that produced peaks at zero mobility, or balanced mobility peaks at unphysical, large, positive and negative values (unphysical compensation) were ignored. Generally, the mobility spectra in these samples yielded two well-resolved n-type mobility peaks. In the fitting procedure, the mobility of the 2nd (high mobility) carrier peak in the spectrum was initially fixed. The remaining parameters (peak conductivity of 1st and 2nd peaks and mobility of the 1st peak) were used as initial values in a two-carrier fit to the magnetoresistance and Hall coefficient data. The mobility of the high mobility carrier was then allowed to vary simultaneously with the other three fitting parameters to see if a true local minimum in the fitting function had been achieved. This produced little mobility change from the original mobility spectrum. The resulting mobility spectrum of sample 1 at 2 K is shown in figure 3(b). The mobilities calculated by the analytical two-carrier model are also shown in the figure as dashed lines and show that there is extremely good agreement between the two methods of analysis. Figure 4(a) shows a self-consistent Schrödinger-Poisson (SP) solution [14] for the conduction band of sample 1 at low temperature. This sample is heavily doped with n tot = 15.6 × 10 11 cm −2 . Figure 4(b) shows the SP solution for sample 2, which is more lightly doped, with n tot = 1.9 × 10 11 cm −2 . The SP solution indicates that sample 1 has four occupied levels, with the majority of carriers confined in the delta-doping region and not the QW. This is in good agreement with the multi-carrier behaviour observed in experimental data. Sample 2, has a single occupied level, which is also in good agreement with the observed magnetotransport. As the temperature is increased, the ρ x y of sample 2 shows a second carrier above 140 K. At the same temperature, the SP solution shows the occupation of energy levels in the delta-doping energy minimum of the conduction band profile. Figures 5(a) and (b) show the temperature variation of the measured mobility (lines) and the mobility of the 2D carrier (open circles) for samples 1 and 2, respectively. In both samples the mobility of the 2D carrier behaves as expected from analysis of the various scattering mechanisms present [8]. The mobility of the 2D carrier in sample 1 is relatively stable at 324 000 cm 2 V −1 s −1 until the temperature reaches 40 K, where phonon scattering becomes significant and the mobility reduces to 42 000 cm 2 V −1 s −1 by 290 K. The 2D carrier in sample 2 has a mobility of 84 000 cm 2 V −1 s −1 at low temperatures, which reduces to 44 000 cm 2 V −1 s −1 at 290 K as a result of phonon scattering. Figure 5 also shows the mobility of the parallel carriers of (a) sample 1 (open diamonds) and (b) sample 2 (filled diamonds). In sample 1, the mobility of the second (parallel) carrier is relatively temperature invariant. This suggests that it is limited  by a weakly temperature-dependent scattering mechanism, consistent with impurity scattering, supporting the conjecture that the parallel carrier is localized in the delta-doping plane, where the greatest density of scattering centres is located and not below the well. Figure 6 shows the total carrier density of samples 1 and 2 (bold solid lines) as a function of temperature. In addition, for sample 2 we have included n 2DEG (open circles) and n (open diamonds), extracted from the analytical two-carrier model and the carrier density in the well (n QW ) and delta-doping plane (n doping ) extracted from the SP model (feint solid lines). It can be seen that n QW provides excellent agreement with n 2DEG at all temperatures. However, when the second carrier is present, there is a large discrepancy between n doping and n . The difference between the two-carriers varies approximately exponentially with temperature, suggesting that thermally generated carriers (which are not considered in the SP model) strongly contribute to parallel conduction. The approximate intrinsic carrier density (n barrier ) was calculated for the 10% Al lower barrier. Assuming a background doping level of 1 × 10 15 cm −3 , the Fermi level was found to lie 115 meV below the conduction band. Using this Fermi level, n barrier was then found by [15] where N c = 2.4 × 10 13 T 3/2 cm −3 is the effective density of states in the conduction band and n background is the electron density resulting from background doping. After conversion to a sheet density, n barrier was estimated to vary between 0.6 × 10 11 cm −2 at 140 K and 5.8 × 10 11 cm −2   figure 6. It can be seen that at lower temperatures, thermally excited carriers are a much smaller proportion of the total carrier density. n from the analytical two-carrier model is in good agreement with the sum of n doping from the SP model and n barrier . When added to n QW from the SP model, this gives good agreement with the measured total carrier density.
The same contributions (n 2DEG , n , n QW and n doping ) can be calculated for sample 1. Assuming the same growth conditions, n barrier is the same in samples 1 and 2, therefore the lower barrier contribution should be identical for the two samples. As can be seen in figure 6, the effect of intrinsic carriers in the lower barrier in the highly doped sample is less significant than in the lower-doped sample at high temperatures, where it is a significant proportion of the total number of carriers. These results are consistent with the entirely different temperature dependence of the mobility of the parallel carrier from samples 1 and 2.

Conclusion
Growth of high-quality InSb/AlInSb heterostructures has been demonstrated, with high room temperature (44 000 cm 2 V −1 s −1 ) and low temperature (324 000 cm 2 V −1 s −1 ) 2DEG mobility. Samples with both single-and two-carrier transport at low temperature have been measured. Sample 2, whilst single carrier at low temperature, becomes multiple carrier by ∼140 K. The large increase in the carrier concentration of this sample from 1.9 × 10 11 to 10.4 × 10 11 cm −2 is explained by the presence of thermally generated carriers in the lower barrier, which accounts for the parallel conduction. In higher-doped samples, it is concluded that parallel conduction originates from carriers occupying the delta-doping plane.