Locally‐Strain‐Induced Heavy‐Hole‐Band Splitting Observed in Mobility Spectrum of p‐Type InAs Grown on GaAs

High‐quality Be‐doped InAs layer grown by molecular beam epitaxy on GaAs substrate has been examined via magnetotransport measurements and high‐resolution quantitative mobility spectrum analysis (HR‐QMSA) in the range of 5–300 K and up to 15 T magnetic field. The results show four‐channel conductivity and essential splitting of the most populated hole‐like channel below 55 K. It is concluded that origin of such effect results from the locally strain‐induced interlayer, which direct observation is difficult or impossible via alternative techniques. Based on the magnetotransport data analysis, the multilayer model is proposed, which is implemented into nextnano simulation, giving the proof of the argumentation correctness. These results indicate potential usefulness of HR‐QMSA technique even in the degeneration statistic regime.

to four distinct conductance channels at low temperatures, one related to electrons, second one to light holes, and two to the metallic transport in the heavy-hole band. The last observation has been related to the gradual reduction of strain caused by the lattice mismatch on the InAs/GaAs interface. This interpretation has been supported by the comparison of experimental data with the numerical simulations performed for different temperatures, using the Nextnano code.
The 2 μm p-type InAs epilayer was grown on (001)-oriented semi-insulating epi-ready GaAs substrate via RIBER Compact 21-DZ solid-source molecular beam epitaxy (MBE) system. The layer was deposited on 250 nm-thick GaAs smoothing buffer and high quality of the growth with minimal residual strain has been indicated by X-ray diffraction (XRD) measurement and Raman spectroscopy analysis. Details of the growth procedure and basic characterization have been published elsewhere. [18] The InAs layer studied here was doped with Be-dopant of concentration %2 Â 10 18 cm À3 and the doping uniformity has been checked via SIMS analysis using CAMECA SC Ultra system, with depth resolution below 1 nm. The Ga, In, C, O, and Be intensities were recorded versus measurement time. Knowing the thickness of the InAs layer from HR-SEM results, [18] we determined the sputter rate and depth profile. The results are shown in Figure 1a, in which the intensity signal is converted into the volume concentration as indicated on the right-hand axis.
Data show that the sample is uniformly doped with beryllium. Be, as a light element, can be mixed into the sample during the ion bombardment more effectively than heavier elements such as In. Note that the Be decay length is quite large. However, it depends strongly on the impact energy, so the observed uniformity of distribution is not a SIMS-related artifact. The concentration of carbon and oxygen atoms inside the layers was found to be below the detection limit (10 16 atoms cm À3 and 3 Â 10 16 atoms cm À3 , respectively). Nevertheless, monitoring of these signals allowed us to localize the buffer layer position. Obtained results confirm the assumed architecture of our sample and proof the high homogeneity of the Be-dopant profile, which is important for the MSA.
The application of the HR-QMSA method requires very precise Hall-effect measurements; therefore, the special attention has been devoted to the fabrication of highly symmetrical samples in the van der Pauw (vdP) geometry ( Figure 1b). [19] Moreover, it guarantees subtraction of most parasitic effects added to the Hall effect, and ensures better data symmetrization than, for example, "hall bar" geometry. In our vdP case, the four-terminal devices have been defined lithographically using AZ 4533 photoresist and NaOH water solution as a developer. The gold contacts were made by electrolytic deposition using the 1.2% KAu(CN) 2 water solution under 350-375 μA cm À2 current density. In the following step, the target cloverleaf shape has been obtained in wet etching process. The surface outside the covered shape was etched down to the GaAs substrate using the orthophosphoric acid:citric acid: hydrogen peroxide:water (molar ratio: 1:1:4:16) solution and 0.4 mol dm À3 hydrochloric acid water solution. Finally, gold wires have been bonded to contact areas. The photoresist has not been washed out after the etching stage, to reduce aging effects by reducing the rate of uncontrolled oxidation. [20] For the MSA, not only the shape of a sample is important but also the linearity of electrical response. Therefore, our devices have been electrically pre-examined in 300 K to check the symmetry and the electrical contact quality by current-voltage linearity test in the voltage range from À150 to þ150 mV. All of the contact pairs gave linear response on voltage bias, where determinacy coefficient for linear function was better than R > 0.999. Using excitation direct current I exc. ¼ 500 μA, the dissipated power was lower than 0.1 mW, as required. [21] Sample resistivity and Hall-effect measurements were performed using a Keithley Source Meter 2400 and Keithley 2182A for voltage measurements. Equally important is the homogeneity of magnetic field. Therefore, for the purposes of this study, the superconducting 16T Cryogen-Free Magnet System (CFMS16T), made by Cryogenic Ltd., has been used in which magnetic-field homogeneity ensures ≤0.1% total variation over 10 mm sphere diameter. [22] In this system, the Hall sample is placed on the special holder inside the variable temperature insert (VTI), directly in the circulating high-purity helium, which is kept at a constant pressure. Such environment ensures very good thermal conductivity and in consequence the required temperature stabilization (2σ in the order of 6-20 mK, for 5-300 K), monitored by CERNOX sensor. Our measurements have been performed over the AE15 T range for each temperature in the so-called Step Scan mode. To assure the proper sample temperature, before each magnetic field scan sequence, at least 20 min-long stabilization www.advancedsciencenews.com www.pss-rapid.com have been applied. The ΔB steps have been chosen properly for uniform adjustment into log 10 scale, namely ten points on each decade. For being ensured that the magnetic field is stable, we used 90 s stabilization time before each data collection of all 12 configurations in the vdP method. The results for two of these resistances are shown in Figure 2.
As it is seen, the R 4-point and the R H(ik) (B) Â B curve families exhibit a high smoothness in the whole ranges of magnetic field and temperature. Moreover, the corresponding surfaces obtained for the opposite magnetic fields direction (data not shown) behaved in the highly symmetrical and antisymmetrical manner, as required by vdP method and what is important in mobility spectra calculation step. Therefore, we conclude that the sample was electrically homogeneous in agreement with SIMS analysis, as discussed earlier. Most importantly, the R 4-point resistance is clearly nonmonotonic as a function of magnetic field and Hall resistances reveal strong nonlinearities. It is characteristic for the multicarrier transport in semiconductors and can be analyzed by the mobility spectrum method.
The main concept of mobility spectrum relies on the transformation of measurement data from the magnetic field (B) domain into the mobility (μ) domain. The general idea of such calculations is to obtain the conductivity tensor components σ xx ðBÞ and σ xy ðBÞ using experimental Hall constant R H ðBÞ and sheet resistance R s ðBÞ values, according to the following relations [15] These coupled expressions contain information about all carriers present in the sample according to the discrete mobility transform equations [23] where S p ðμÞ ¼ eμpðμÞ and S n ðμÞ ¼ eμnðμÞ are the hole and electron mobility spectra (i.e., hole and electron conductivities in the mobility domain). The symbols p s ðμÞ and n s ðμÞ are hole and electron sheet densities, respectively, and e is the electronic charge. Therefore, to identify each type of the multiple carriers contributing to transport, we have to obtain mobility distributions S p ðμ j Þ or S n ðμ j Þ, using the appropriate numerical methods.
Here, μ is negative for electrons (e < 0) and positive for holes (e > 0) in accordance with the generally accepted convention. The basic algorithms for that purpose have been described by Beck and Anderson; [23] however, they might be insufficient for a proper recognition of multichannel conductivity in semiconductor devices. [24] Fortunately, a significant progress has been made in recent years and new, improved methods have been successfully tested on many different material systems. [25,26] Here, we applied HR-QMSA, [15] which previously gave excellent results for proper recognition of carrier transport through VBs in nonintentionally doped GaSb substrate. [27] Results of HR-QMSA method applied to our data are shown in Figure 3.
Clearly, we identify the four distinct mobility spectra peaks, which contribute to the total conductivity tensor. One is electronlike (denoted E 1 ), and three others are hole-like (denoted H 1 , H 1 0 , and H 2 ). Note that H 1 0 peak disappears for higher temperatures T > 55 K or rather merges with stronger H 1 contribution. Note also that H 2 feature, which corresponds to carriers with the highest mobility, is rather weak. From the obtained mobility spectra, we calculated the partial conductivity σ i , mean Hall mobility μ H i , and sheet carrier concentration N i associated with the ith conductivity peak, using the following formulas Results, as a function of temperature, are shown in Figure 4. The densities N j and corresponding mobilities μ H j are collected on the left-hand side, and the stack area plot of partial conductivities σ j is presented on the right-hand side. www.advancedsciencenews.com www.pss-rapid.com Both plots reveal at least four characteristic temperature sets. At T < 30 K (Set I) and for T > 150 K (Set IV), all conductivities, despite small fluctuations, decrease monotonically and the mobility spectra peaks become wider with increasing T. This is not the case for the intermediate range of temperatures (Sets II and III), in which the partial conductivities fluctuate very strongly. At the same time, the width of dominant H 1 contribution decreases, somehow "at the cost" of electron-like peak E 1 , which becomes dominant at T ¼ 80 K.
We believe that E 1 peak, detected at low temperatures, is associated with the presence of a surface inversion layer, observed on p-type InAs samples since 1970s. [28] Later, the direct evidence for the existence of charge accumulation on free and clean InAs surfaces were reported, independently on type and doping level. [29,30] Currently, it is widely believed that InAs has naturally high concentration of surface electrons, which might be, e.g., the main source of anomalous effects in temperature behavior of Seebeck coefficient. [31] For example, Olsson et al. [30] estimated electron surface concentration on about 10 12 cm À2 , which compares well with our estimation of n Set I % 3 Â 10 12 cm À2 , at the lowest temperatures. Similar results have been obtained at T ¼ 300 K by Lin et al. [32] by QMSA method for the surface electrons on n-type sample, which also supports our identification of E 1 peak, the same as the fact that frequently published bulk electron mobilities are essentially higher. For high-quality samples μ e > 1 Â 10 5 cm 2 (V s) À1 at 77 K. [33] It is unlikely, however, that  H 1 0 , and H 2 ) have been recognized in low-temperature regime. The arrow represents the H 1 0 peak position trace, which becomes the H 1 peak position for higher temperatures. The increase in ΔT is nonlinear. It follows constant step in log 10 T scale. b) Electron mobility spectra S n ðμ j Þ versus temperature, here μ > 0 for clarity. Only the single conductivity channel is visible, whose influence on total conductivity is less than 3.5% for all temperatures below 30 K. c) The partial contributions to the total conductivity are shown on the right-hand side as the stacked area plot. The whole temperature range is divided into characteristic sets, numbered from I to IV.
www.advancedsciencenews.com www.pss-rapid.com the surface electron concentration reaches the extremely high value of 6.55 Â 10 14 cm À2 , at T ¼ 80 K, as can be observed in Figure 4b. Therefore, the unexpected behavior of E 1 peak at the temperature Sets II and III might be an artifact of QMSA method, as we showed by the model calculations, described later. Clearly, such artifacts related to electrons may also affect the identification of a hole-like mobility spectra. Fortunately, it is not the case for T < 30 K and T > 200 K; therefore, in the following, we concentrate on the results obtained at low (Set I) and high (Set IV) temperatures. Judging from the partial concentrations and mobilities, H 1 spectrum is associated with heavy holes, whereas H 2 peaks originate from the presence of light holes, in agreement with the recent results of Casias et al. [34] for very similar 2 μm-thick InAs 0.91 Sb 0.09 layer, acceptor doped at 3 Â 10 18 cm À3 obtained at T ¼ 300 K. However, it is not clear what is the origin of the strong splitting of heavy-hole spectra, observed at low temperatures, which gradually disappears when peaks broaden and start to overlap.
To answer this question, we first assumed that the presence of the additional H 1 peak is not related to the inhomogeneity of our sample. This risk can lead to a wrong number of carriers species obtained by the MSA, as it was shown for two-carrier transport in a Hall-bar device. [35] However, the SIMS results (Figure 1a) show the excellent uniformity of both parent atoms and Be-dopant. Moreover, we believe that the application of vdP method effectively averages any residual inhomogeneities of our sample.
Second, we considered the anisotropy (so-called warping) of VB, which is characteristic for all A III B V materials, as a possible source of peak splitting. We calculated the effective mass tensor for heavy holes using the InAs parameters and results showed that σ xx indeed deviates from the Drude formula. If a spherical Fermi surface and degenerate statistics are assumed, warping leads only to a broadening of mobility spectrum. In reality, Fermi momentum k F depends on crystallographic directions, so the noncircular shape of cyclotron orbits should be considered in a more detailed calculation. Also, in that case, however, no splitting but rather electron-like contribution is expected. [36,37] Therefore, we assumed that the additional H 1 0 peak is not related to inhomogeneities, QMSA artifact, or warping, but to an additional conduction channel in the degenerate VB, as explained later.
Degenerate statistics is suggested by a very weak temperature dependence of partial conductivities, observed up to T % 10 K, which is characteristic for metallic transport. For higher temperatures, the H 1 0 contribution increases, which may suggest hopping-like conduction, which is rather typical for an amorphous films. [38] In our case, the H 1 0 conductivity does not follow Mott or Efros-Shklovskii laws [39] and the total contribution H 1 þ H 1 0 is practically temperature independent up to 30 K (region I). The above conclusions led us to propose an alternative explanation of the splitting. We inferred that such effect is a result of strained InAs interlayer presence between GaAs substrate and fully relaxed InAs layer. This assumption has been verified via numerical calculations performed in the nextnano software. [40] The assumptions about geometry do not differ much from Figure 1b, except additional 20 nm surface cap layer and 200 nm strained interlayer. The details have been shown in Figure 5. Beginning from the top part, we have 20 nm surface states that correspond to E 1 channel. Unfortunately, proper modeling of the surface effects (especially for InAs) is a difficult challenge. To avoid this inconvenience, we proposed two numerical treatments. First, we use higher than "standard" effective densityof-states electron mass ðm Ã e DOS ¼ 0.25m 0 Þ, which corresponds www.advancedsciencenews.com www.pss-rapid.com in our model to electron state from asymmetric near-surface quantum well. Second, we added two extra donor states to obtain the effect of electron concentration increasing above 30 K. Their concentrations have been n d ¼ 3.0 Â 10 19 cm À3 and n d ¼ 5.0 Â 10 20 cm À3 , with activation energies E D1 ¼ À15 meV and E D2 ¼ À5 meV, respectively. Numerical procedures for obtaining desired near surface concentration, such as increasing the m Ã e DOS value, are rather typical treatment. In this case, however, more detailed calculations would be needed.
Due to the expected strain effects, we divided InAs layer (except surface) into two regions. We assumed that the first 1780 nm layer is practically relaxed, with ε xx ¼ ε jj ¼ À0.8 Â 10 À3 and ε zz ¼ ε ┴ ¼ 0.9 Â 10 À3 . It is similar to averaged XRD and Raman measurement results (from this growth) published earlier, in which the parallel and perpendicular residual strain have been determined to be -1.17 Â 10 À3 and 1.12 Â 10 À3 , respectively. [18] The second 200 nm layer, close to the GaAs interface, has been assumed as relatively heavy stained with the parallel and perpendicular residual strain equal to ε xx ¼ ε jj ¼ À0.0181 and ε zz ¼ ε ┴ ¼ 0.0197, respectively. For both layers, the concentration of Be acceptors was n A ¼ 3.2 Â 10 18 cm À3 , with different acceptor ionization energies equal to E a ¼ À14 meV in the first layer and E a ¼ À8 meV in the second one, counted from the top of the heavy-hole band. The effective density-of-states mass of electrons and all VBs, that is, heavy-hole (m Ã hh DOS ), light-hole (m Ã lh DOS ), and split-off band (m Ã so DOS ), have been used the same as suggested in the literature, namely 0.026 0.41, 0.026, and 0.14 m 0 , respectively. [17] It should be noted that the model did not require any special assumptions about the InAs/GaAs interface. GaAs "epi-ready" substrate before InAs deposition has been capped by the 0.25 μm GaAs layer, which effectively covered leftovers of impurities. Thus, we did not expect additional highly conductive interlayer.
In Figure 5b, the most interesting parts are placed between 500 and 700 nm and close to the surface region. For the first area, VB splitting on over 100 meV can be observed. In our opinion, despite relatively low thickness, this layer is the most probably source of the mobility spectra splitting (Figure 3a). We interpret it as the result of spatial change of shear strain in the InAs layer.
This type of strain causes splitting of heavy-and light-hole bands (see, e.g., ref. [41]), which can be observed in the strained 200 nm-thick interlayer, as in our case. Such splitting is sometimes intentionally used to suppress the Auger phenomenon in strained-layer superlattices. [42] This effect makes that the light-hole band moves down in the energy scale, away from the Fermi level. Therefore, we cannot observe light holes from that interlayer and for the same reason we cannot observe electrons from the bulk part of the InAs layer. Thus, we rejected here the intrinsic sheet concentration of electrons, as a reliable explanation of the origin of the E 1 , because even for 300 K, this channel is negligibly small (%1 Â 10 8 cm À2 ). We conclude that conducting electrons come exclusively from the surface of our sample.
Moreover, we performed qualitative verification of calculations correctness. Namely, the relative Fermi-level position in the middle of unstrained InAs layer versus temperature has been checked to confirm occurring of the metal-semiconductor transition (right side of the Figure 5). The plot crosses the zero value, which means the Fermi level crossing the top of the heavy-hole band. This process takes place not exactly for 55 K, as in QMSA results, but for about 105 K. Presented results qualitatively confirm our interpretation related to Set II in Figure 4a,b. It is clear that the rapid changes of concentration and mobility are related to metalsemiconductor transition of InAs layer. To obtain the quantitative agreement, a more detailed fitting procedure would be necessary.
It is worth noting that negative ionization energies for acceptors have been used intentionally to simulate metallic conduction. Namely, E a ¼ À14 meV in the first layer and E a ¼ À8 meV in the second one, counted from the top of the heavy-hole band. These values are different from the literature data of E a % 20 meV. [34] However, it is well known that as the dopant concentration increases, the dopants start to interact and form an impurity band. [43][44][45] The increase in the width of this band decreases the ionization energy. [43][44][45][46] For sufficiently high-dopant concentration, the associated broad impurity band and the conduction (valence) band edge merge leading to the negative ionization energies.
In the final step, the comparison between presented nextnano simulation and HR-QMSA analysis has been performed. The results are shown in Figure 6a,b. Except transitional regime www.advancedsciencenews.com www.pss-rapid.com and the results for channel H 1 0 , Figure 6a shows a good agreement between two methods. However, the obtained numerical curves have no additional peaks for 30 K < T < 110 K and they behave smoother than the experimental ones. In our opinion, this comparison also suggests that unexpected appearance of E 1 peak might be an artifact of QMSA method.
In Figure 6a, normalization to the depth profile has not been applied. It has been made in the next step, in which normalization to the assumed layers' thicknesses and changing units into cubic centimeters show that volume concentration of both 1.8 μm relaxed and 0.2 μm strained are nearly the same in the Boltzmann statistics regime. We infer here that the same volume concentrations might be the reason of disappearing of H 1 0 as a separate channel for T > 55 K.
It should be noted that Hall effect measurements for any configuration similar to that presented in Figure 1b has fundamental limitation-it has 2D character. Therefore, rescaling procedure of carrier concentration from 2D to 3D system is rather arbitrary. During nextnano calculation, we noted that rescaling H 1 0 channel, assuming a thickness of 200 nm, [47] gives the volume concentration for T > 55 K nearly the same value as H 1 for the rest of InAs layer. Therefore, channels H 1 and H 1 0 are indistinguishable via magnetotransport measurements above the temperature of metal-semiconductor transition. In our opinion, this is the most plausible explanation of the observed heavy-hole band splitting via HR-QMSA method.
The set of arguments have been presented that observed splitting of the heavy-hole-band mobility spectra origin from the thin, strained InAs interlayer between GaAs substrate and fully relaxed InAs layer despite its very high quality confirmed via multiple characterization techniques. Apart from the heavy holes, we also identified two additional channels: one related to surfaceoriginated electrons and second one to light holes in the whole temperature range of 5-300 K. These results confirm the usefulness of the HR-QMSA technique even for the degenerate statistic regime. This interpretation has been supported by the comparison of experimental data with the numerical simulations performed for different temperatures, using the nextnano code.