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Quantum Hall Effect in Polar Oxide Heterostructures

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Science  09 Mar 2007:
Vol. 315, Issue 5817, pp. 1388-1391
DOI: 10.1126/science.1137430

Abstract

We observed Shubnikov–de Haas oscillation and the quantum Hall effect in a high-mobility two-dimensional electron gas in polar ZnO/MgxZn1–xO heterostructures grown by laser molecular beam epitaxy. The electron density could be controlled in a range of 0.7 × 1012 to 3.7 × 1012 per square centimeter by tuning the magnesium content in the barriers and the growth polarity. From the temperature dependence of the oscillation amplitude, the effective mass of the two-dimensional electrons was derived as 0.32 ± 0.03 times the free electron mass. Demonstration of the quantum Hall effect in an oxide heterostructure presents the possibility of combining quantum Hall physics with the versatile functionality of metal oxides in complex heterostructures.

Zinc oxide (ZnO), a wide–band gap semiconductor, is of growing importance in advanced electronics, and its potential applications include transparent conducting oxide layers for flat-panel displays and transparent field-effect transistors (1). Research focused on the epitaxial growth of ZnO, particularly in terms of its novel excitonic properties, has led to the recent realization of homostructural light-emitting diodes (2). Studies of the intrinsic properties of ZnO have yielded a recipe for the preparation of high-quality epilayers having high mobility and excitonic luminescence with high quantum efficiency (3, 4).

Certain aspects of two-dimensional electron gas (2DEG) behavior in semiconductor heterostructures have been studied by observing the quantum Hall effect (QHE)—a quantized magnetotransport accompanied by Shubnikov–de Haas (SdH) oscillations in the longitudinal resistivity ρxx and Landau plateaus in the Hall resistivity ρxy (5). Early results were obtained in the material systems of Si/SiO2 or GaAs/AlGaAs (6, 7). However, after discovery of the fractional QHE (8, 9), the focus has been extended to a variety of other material systems, such as III-nitrides (10) and graphene (11). The observation of SdH oscillation requires conditions such as ωcτ >1 and ħωc > kBT, where ωc is the cyclotron frequency equal to eB/m* (where e is the charge on the electron, B is magnetic field, and m* is the electron effective mass), τ is the carrier relaxation time, ħ is Planck's constant divided by 2π, kB is Boltzmann's constant, and T is absolute temperature. Although several epitaxial oxide heterostructures have satisfied these conditions (12), the QHE has not been observed in those materials. [However, a QHE-like state was seen in a quasi-2D crystal of bulk η-Mo4O14 (13).]

In our study, (0001)-oriented ZnO/MgxZn1–xO heterostructures were grown by laser molecular beam epitaxy with the use of a semiconductor-laser heating system (3). The MgxZn1–xO layer acts as a potential barrier for the 2DEG in the adjacent ZnO layer (14). We used a temperature gradient method that allowed us to grow the films over a wide range of temperatures on a single substrate (3). The three samples (samples A, B, and C) discussed here were of such high quality that we were able to look at the effects of growth temperature (Tg) of the ZnO layers and Mg content x in the barrier layers (Table 1) [see (15) for sample preparation and characterizations]. Sample A was selected from the highest-Tg region of a film with an x = 0.15 barrier, whereas samples B and C were selected from lower- and intermediate-Tg regions, respectively, of another film with an x = 0.2 barrier. From the low-field Hall coefficient RH, carrier densities n = –1/(RHe) and mobilities μ = –RHxx at 1 K were evaluated to be 0.66 × 1012 to 3.7 × 1012 cm–2 and 2700 to 5500 cm2 V–1 s–1, respectively. These values were nearly independent of temperature below 1 K, but with increasing temperature, n increased while μ decreased, eventually reaching the room-temperature values also listed in Table 1 (see fig. S4).

Table 1.

Growth and electronic parameters of ZnO/MgxZn1–xO heterostructures. Electron densities were obtained independently from the low-field Hall coefficient RH and the low- and high-field slopes SL and SH, respectively (see Fig. 2C).

Electron density (1012 cm-2)Mobility μ (cm2 V-1 s-1)
SampleThickness of ZnO (μm)Growth temp. (°C)Mg content x n 2D n = 1/RHeRHxx
eSL/heSH/h
45 mK (A), 1 K (B, C)1 K300 K1 K300 K
A 1.1 1020 0.15 0.2 0.6 0.66 39 5500 150
B 0.4 900 0.20 0.4 1.2 1.8 5.0 4900 160
C 0.4 1000 0.20 0.9 1.8 3.7 11 2700 160

We found that n systematically depended on x and Tg for the ZnO layers. Our previous study showed that Tg was a good parameter to control the free carrier density; the intrinsic donor concentration increases with increasing Tg (3). To investigate the dependence on x, it is necessary to consider spontaneous and piezoelectric polarization effects along the polar (0001) orientation of wurtzite ZnO (16, 17). These polarizations induce surface charges in individual layers, resulting in accumulation or depletion of free electrons at the heterointer-faces (18). Because the growth direction of our samples was identified to be the O-face (2), the direction of spontaneous polarization was upward toward the surface (Fig. 1A). From high-resolution x-ray diffraction analysis (fig. S3), the piezoelectric effect vanishes in unstrained MgxZn1–xO layers, whereas upward piezoelectric polarization should arise in ZnO layers from tensile strain (14). Thus, the total polarization is defined by Math(1) Math where Psp(x) is spontaneous polarization in the MgxZn1–xO layer, and Ppe(x) is piezoelectric polarization in the strained ZnO layer on unstrained MgxZn1–xO. Note that positive σ means that free electrons are accumulated at the heterointerface, and negative σ means that free electrons are accumulated at the ZnO surface.

Fig. 1.

(A) Schematic of the ZnO/MgxZn1–xO heterostructures grown on ScAlMgO4 substrates. Depending on the sign of |ΔPsp(x)| – |Ppe(x)|, an accumulation layer represented by broken lines is formed either at the surface or in the interface. (B) Calculated (gray and blue shaded regions) and measured (solid circles) n as a function of Mg content x in the barrier. Red shaded region [|Ppe(x)/e|] and solid orange curve [|ΔPsp(x)/e|] were calculated by using theoretical values listed in tables S1 and S2. (C) Potential diagram near the heterointerfaces. Calculated energy parameters are listed in the upper panel, where EF is the Fermi energy, E0 and E1 are the first and second subband energies with respect to the bottom of the conduction band in the wells, respectively, and ΔEC is the conduction band offset. The colors of the potential profile (solid line) and position of EF (broken lines) in the middle panel correspond to those of the upper panel, representing samples A, B, and C, respectively.

We calculated σ as a function of x (Fig. 1B). When x is higher than ∼0.1, polarization-induced positive charges |σ/e| (blue shading) are formed at the heterointerfaces, and their density increases as x increases. There is good agreement between experimental n (solid circles) and estimated |σ/e|, despite the ambiguity in Ppe(x), a range of the theoretical values representing all of the shaded regions, and the lack of consideration of charge compensation with free electrons. Using the aforementioned polarization charges and band offset (19), we can construct the potential diagram near the heterointerfaces (Fig. 1C). The subband energies are estimated using a triangular-potential approximation (20) and are listed in the upper panel. The energy separations between the two lowest subband levels are greater than the Fermi energy, which suggests that in the temperature range of the experiments described below, carrier occupation in the second subband is negligible.

We measured the magnetotransport properties (Fig. 2, A and B) using a standard lock-in technique with ac excitation (10 nA, 19 Hz). At low field, all the samples exhibited negative magnetoresistance, presumably because of weakly localized carriers. Above ∼2 T, clear ρxx oscillations that were periodic in 1/B appeared, and their amplitudes increased with increasing B. Although the zero-resistance state was absent because of the large scattering rate, each minimum of ρxx coincided with the quantized ρxy plateaus equal to h/(νe2), where ν is the Landau filling index. These observations confirmed the existence of the QHE in our samples and allowed direct determination of the 2DEG density (n2D). Note that in sample A, the odd states, such as ν = 3 and 5, had much wider Hall plateaus and larger amplitudes of ρxx minima relative to those of the even states (Fig. 2A). Consequently, the even states, such as ν = 4, were barely observable in dρxy/dB (see circle superimposed on red broken line). These features were well preserved at elevated temperatures (see Fig. 2A and upper panel of Fig. 2D).

Fig. 2.

(A) Longitudinal resistivity ρxx, Hall resistivity ρxy, and differential Hall resistivity dρxy/dB versus B measured at 45 mK for sample A. Integers on the horizontal tick marks are the Landau level filling factors defined as ν = h/(ρxye2). (B) ρxx and ρxy versus B measured at 1 K for samples B (top) and C (bottom). (C) Standard fan diagrams extracted from the data shown in (A) and (B). The symbols SL and SH refer to the low- and high-field slopes, respectively. Colors are consistent with those used for ρxy(B) in (A) and (B). Open symbols were evaluated from data of sample C at 2 K and θ = 30° shown in (D). Note that each increment of index of extrema is set to 0.5. (D) Angular dependence of normalized magnetoresistance measured at 2 K for sample A (top) and sample C (bottom). Inset depicts the measurement configuration.

Standard analyses were performed using fan diagrams (Fig. 2C), where indices of extrema in ρxx and/or dρxy/dB are plotted as a function of 1/B. Taking low- and high-field slopes (SL and SH), we independently evaluated n2D as eSL/h and eSH/h, respectively, and compared these values with n (Table 1), where eSH/h generally corresponds to the 2DEG density contributing to the QHE. The obtained eSH/h values were systematically smaller than n by 9% to 51%, apparently indicating that 2D confinement becomes weaker with increasing n. To investigate carrier dimensionality, we further studied the B orientation dependence of the SdH effect under the configuration shown in the inset of Fig. 2D. Sample A exhibited weak oscillation, even with B parallel to the interface (θ = 90°), and similar behavior was obtained for sample B (fig. S6). In contrast, sample C, which was expected to have more unconfined free carriers, showed vanishing oscillations with θ approaching 90°. The fact that the oscillation periods depended on the perpendicular B component (B cos θ)–1, as evidenced by the data at θ = 30° (open circles in Fig. 2C), is consistent with a 2D character of the electron gas. In this regard, however, we found inconsistent results between n2D extracted from fan diagrams and carrier dimensionality evidenced by angular dependencies of the SdH oscillations (see below).

Having established the presence of SdH oscillations, we extracted the value of m* from the temperature dependence of the SdH oscillation amplitude. The resulting Dingle plot for sample C is shown in the inset of Fig. 3. We subtracted the ordinary magnetoresistance from the raw ρxx(B) data and normalized it through [ρxx(B) – ρxx(0)]/ρxx(0) (21). The slope of a linear fit to the plot at B ∼3.9 T gave m* = 0.32 ± 0.03 m0 (where m0 is the free electron mass), and similar values were obtained for B ranging from 2.6 to 4.5 T. This m* value is somewhat heavier than the bulk polaron mass (0.28 m0) estimated by cyclotron resonance (22), giving rise to the possibility of mass enhancement with 2D polaronic correction (23, 24).

Fig. 3.

Normalized magnetoresistivity versus 1/B recorded at various temperatures for sample C. Inset depicts temperature dependence of the logarithmic amplitude of ρxx peak at 3.86 T (indicated by arrow).

We next discuss anomalous periodicities of the SdH oscillations. The SH/SL ratios were found to be ∼3 for samples A and B, and ∼2 for sample C. These results suggest that an internal electric field arising from the asymmetric triangular well substantially removed spin degeneracy, even in zero magnetic field. In this case, the ratio of the slopes directly gives the carrier populations in two spin-split subbands, n+ and n, as SH/SL = 1 + n+/n. When applied to sample A, this analysis gives a carrier density in one band twice as large as that in the other band at low fields and total carrier density n2D = n+ + n = 6×1011 cm–2 (25, 26). As for sample C, we obtained clear evidence of the 2D nature, and hence we expected that n2D would be equal to n, but in fact we found that n2D was about half of n. Together with the approximately linear slopes, this finding implies that the missed ρxx extrema occurred in a systematic fashion. In this situation, the values of n2D and SH/SL are no longer reliable. Because of the stronger polarization fields in samples B and C, rather large spin splitting would be expected, making it difficult to resolve all Landau states (23). Relatively lower μ and higher ν may further reduce the ability to discriminate the Landau levels. In fact, even states with ν < 10 tended to disappear in sample A, which has the highest μ. In the above discussion, we ignored the possibility of valley splitting because there was no evidence that either ZnO or MgxZn1–xO has an indirect band structure. Whether or not the spin-splitting scenario is likely in our samples, it is an interesting problem to be studied in future experiments.

Despite this open question, the observation of the ν = 2 state is remarkable, particularly when compared with isostructural AlxGa1–xN/GaN heterostructures. It is difficult to obtain an n2D value lower than 1012 cm–2 in nitrides having AlxGa1–xN lattice constants smaller than GaN (18). A Ppe direction opposite to that in ZnO results in piling up of charges induced by both polarizations. Thus, our results imply the exciting possibility of realizing the fractional QHE in the present polar heterostructure if the carrier mobility can be improved. In addition, given chemical compatibility with certain other classes of oxides, the quantized Hall state may be combined with a broad range of physical properties in a complex oxide heterostructure.

Supporting Online Material

www.sciencemag.org/cgi/content/full/1137430/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 and S2

References

References and Notes

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