THz cyclotron resonance of a 2D hole gas in a GaN/AlN heterostructure Open Access

Gallium nitride (GaN) has emerged as one of the most impactful semiconductors in modern materials science, due to its wide bandgap, exceptional thermal stability, and high electron mobility. These properties have made GaN as a cornerstone material in a broad range of applications, ranging from light-emitting devices to high speed transistors to power electronics.^1–3 Spontaneous polarization effects at GaN-based heterointerfaces can create high-mobility two-dimensional electron gases (2DEGs),^4,5 enabling, for example, field-effect transistors operating at microwave frequencies.⁶ However, the advance of GaN-based electronics has been limited by the challenge of realizing the analogous hole-doped (p-type) counterpart of such heterostructures, which is crucial to achieving integrated complementary electronics. The difficulty arises from GaN's wide bandgap, which results in valence bands with large masses lying deep in energy, making them difficult to chemically dope and inherently low in mobility.⁷ 

Recently, however, high-density 2D hole gases (2DHG) were realized in epitaxially grown GaN/AlN heterostructures without any doping.⁸ The 2DHG carrier density is one of the largest among all known semiconductors, leading to high sheet conductance despite low carrier mobilities. While theoretical modeling suggests the splittings and dispersions of the valence bands in GaN/AlN structures, experimental measurements of the 2DHG properties are still at an early stage. In particular, one of the most fundamental material parameters relevant to any 2D system of mobile carriers is their effective mass $meff$⁠, which in GaN-based 2DHGs is typically assumed from theoretical modeling,⁹ and was only very recently inferred from Shubnikov–de Haas (SdH) oscillations in electrical resistance.¹⁰ 

Arguably the most direct experimental measure of $meff$ arises from studies of cyclotron resonance (CR),¹¹ where the frequency of cyclotron motion of the mobile carriers, $ωc$⁠, in an applied magnetic field B directly reveals their mass via $ωc=eB/meff$⁠. Widely employed for decades in narrower-gap semiconductors using small B and microwave (GHz) frequencies,¹² CR is a contactless measurement amenable for rapid feedback on the quality and uniformity of epitaxially grown wafers. However, resolving the CR of holes in GaN is challenging because it requires both large (THz) frequencies and very large B (many tens of teslas). This is because holes in GaN have short scattering times (⁠ $∼$ 1 ps), such that the linewidth of their CR peak in the frequency domain is quite broad (⁠ $∼$ 1 THz), and also because holes in GaN are expected to have large masses, so that large B is required to shift the CR peak by an appreciable fraction of its linewidth so that the cyclotron shift can be resolved. Moreover, the 2DHGs in GaN populate both heavy-hole (hh) and light-hole (lh) valence bands, meaning once again that large B are needed to spectrally distinguish the CR peaks from the two hole species. These challenges stand in contrast to CR studies of high-mobility lighter-mass 2D holes in GaAs¹³ or Ge,¹⁴ for which low fields suffice.

Finally, we emphasize that the properties of mobile hh and lh carriers, such as their precise masses, densities, and scattering times, depend sensitively on the band splitting and 2DHG Fermi energy, and therefore on the details of the heterostructure layers and growth parameters. Thus, it is desirable to measure these key parameters in a direct way, and CR in high B provides such a route. To this end, we perform time-domain THz spectroscopy of a GaN/AlN 2DHG in pulsed magnetic fields up to 31 T and resolve the CR of both heavy and light holes. These data reveal the effective masses, densities, scattering times, and mobilities of the mobile holes. Our results complement and are compared to values recently inferred from electrical transport studies of related 2DHGs.^10,15

Figure 1(a) depicts the GaN/AlN 2DHG structure grown by molecular-beam epitaxy on an AlN substrate.⁸ It comprises an AlN buffer layer (700 nm thick), followed by an undoped 8.2 nm GaN layer. A high density 2D sheet of mobile holes is created by the large intrinsic and strain-induced polarization at the GaN/AlN interface.^4,5,16 Figure 1(b) shows the calculated splitting and dispersion of the GaN valence bands near the interface,¹⁷ indicating that the 2DHG populates both hh and lh valence bands.

The experimental setup is shown in Fig. 1(c). The samples are mounted in vacuum on the sapphire cold finger of a small helium cryostat, which is positioned within the 15 mm bore of a small pulsed electromagnet. The magnet consists of 144 turns of high-strength copper–silver wire (1 mm diameter), and is powered by a purpose-built 4.4 mF capacitor bank.^18,19 The commercial (Toptica) THz system is based on electronically controlled optical sampling (ECOPS), which permits detection of an entire THz pulse waveform, $E(t)$⁠, in tens-of-microseconds timescales, and at repetition rates up to 1.6 kHz.^20,21 Picosecond pulses of free-space THz radiation are generated in a fiber-coupled photoconducting antenna, then linearly polarized (vertically, at $0°$⁠) and focused through the sample. The transmitted THz pulses then pass through a second linear polarizer (oriented at $±45°$⁠), and their final electric field waveform $E(t)$ is detected by a photoconducting receiver. Measurements are performed in the Faraday geometry, i.e., with B applied along the direction of light propagation and perpendicular to the sample surface.

Figure 1(d) shows a typical field profile of a 31 T magnet pulse, along with the THz signal. An expanded view of a measured THz waveform $E(t)$ at peak field is shown in Fig. 1(e). The usable bandwidth of the THz pulses spans approximately 0.4–1.6 THz. To improve signal-to-noise, the THz waveforms are typically averaged over 10–20 magnet pulses.

It is essential in these studies to measure the sample's optical conductivity, $σ(ω)$⁠, in a basis of circular optical polarization, as it is the natural eigenbasis of cyclotron motion. We therefore measure the complex (real and imaginary) THz transmission spectrum, $T(ω)$⁠, in a $±45°$ linearly polarized basis (by rotating P2), and then transform into a left- and right-circular basis. First, the normalized THz transmission is obtained as $T(ω,B)/T(ω,0)=F[E(t,B)]/F[E(t,0)]$⁠, where $F[E(t,B)]$ and $F[E(t,0)]$ are the (complex-valued) Fourier transforms of the THz waveforms measured in applied field B and in B = 0, respectively. This is performed with P2 oriented at $+45°$ and then at $−45°$⁠. Then, the diagonal and off diagonal components of the sample's linear transmission matrix are calculated via $T±45°=(Txx±Txy)/2$⁠. Right- and left-circular transmission spectra are then obtained using $Tr,l(ω)=Txx(ω)±iTxy(ω)$⁠, from which the right- and left-circular optical conductivity, $σr(ω)$ and $σl(ω)$⁠, are determined (described in more detail below).

Cyclotron resonance is most clearly represented when $σr(ω)$ and $σl(ω)$ are plotted together vs positive and negative frequencies, respectively.^19,22,23 In the simplest model of CR, each mobile carrier species contributes a Drude-like conductivity peak centered at $ω$ = 0 at B = 0. This Drude peak shifts to higher frequency with increasing B at a rate that is inversely proportional to its effective mass $meff$⁠, a consequence of the 2D electronic density of states breaking up into equally spaced Landau levels that are separated by the cyclotron frequency. At high B, CR from carriers with different masses can therefore be spectrally distinguished and accurately analyzed.

Figure 2(a) shows $σ(ω)$ for a model system containing both heavy and light holes, using $mlh$ = 0.5  $me$ and $mhh$ = 2.5  $me$ (values estimated from band structure calculations of GaN). For simplicity, both carrier species are assigned the same spectral weight and equal scattering times $τ$ = 0.4 ps (estimated from Hall mobility studies). At $B=0$⁠, the Drude conductivity peaks of both species are centered at zero frequency (⁠ $ωc=0$⁠), forming a single peak. However, when $B>0$⁠, the two Drude peaks shift in frequency and separate due to their different masses. In this example, the two conductivity peaks become well separated and spectrally resolvable in high fields $B≳20$ T, enabling the characterization of both carrier masses, as well as their spectral weights and scattering times.

The experimentally measured optical conductivity from the 2DHG structure is shown in Fig. 2(b). Both Re[ $σ$] and Im[ $σ$] at high magnetic fields closely follow the trends anticipated by the minimal model described earlier [cf. Fig. 2(a)]. The data clearly show the emergence and separation of two distinct conductivity peaks shifting toward positive frequencies, confirming the presence of both heavy and light holes in the 2DHG. The dashed lines show fits to the data based on the minimal Drude model discussed earlier.

The fitted CR frequencies, shown in Fig. 3, increase linearly with B, with slope that reveals $meff$⁠. We find that $mhh=2.6me$ and $mlh=0.57me$⁠. We emphasize that these data represent the first measurement of hole CR in GaN-based semiconductors, which is made experimentally challenging by the larger masses and lower mobilities of holes (compared, e.g., to mobile electrons in GaN, or to carriers in narrower-gap semiconductors), and the consequent need for both high B and broad (THz) detection bandwidths to spectrally distinguish their broad Drude conductivity peaks.

The ability to extract all six parameters from a single fit is aided by two noteworthy advantages of the THz experiment: First, large B forces the spectral separation of the hh and lh CR peaks, which can then be fit essentially independently. For 2DHGs in GaN, fields of order 30 T are needed because both carriers have relatively large mass and short scattering times that lead to broad CR peaks. By forcing the spectral separation of the CR peaks, not only $meff$ but also $τ$ and N (via the spectral weight) can be more accurately determined for each hole species. We emphasize that without independent knowledge of $meff$ that is provided by the cyclotron shift, the spectral weight reveals only the ratio $N/meff$⁠.

Second, we measure the complex-valued THz transmission in a circular polarization basis. Simultaneously fitting both the real and imaginary transmission spectrum naturally puts additional constraints on the fit and improves its accuracy. This is possible in time-domain THz studies because both amplitude and phase are encoded in the measured electric field $E(t)$⁠, in contrast to phase-insensitive methods that detect only optical intensities (⁠ $∝EE*$⁠).

Table I shows the material parameters of these 2D holes, as determined by our CR experiments. Mobilities are calculated from the measured scattering times via $μ=eτ/meff$⁠. We note that the larger uncertainty in the parameters $mhh$ and $Nhh$ is due to the fact that the peak of the heavy-hole CR does not quite fall within the detection range [see Fig. 2(b)], which makes it more difficult to independently fit the peak's amplitude and central frequency. To improve accuracy, we measured the low-temperature THz optical conductivity of the GaN/AlN 2DHG structure at B = 0, referenced to a bare AlN substrate, and extracted the total spectral weight from both carrier species by fitting to a single Drude peak. This is a reasonable approach for a two-carrier system when both species have similar scattering rates. The total spectral weight is then used as an additional constraint, which reduces uncertainty in $Nhh$ and improves the fitting of $mhh$⁠.

For comparison, Table I also shows the corresponding heavy- and light-hole parameters reported very recently in two electrical transport studies of similar GaN/AlN 2DHGs grown in the same MBE chamber^10,15 (note, however, that the layer thicknesses were not exactly the same, and some structures used in these transport studies also incorporated an additional Mg-doped p-type GaN capping layer to improve electrical contact to the 2DHG). In Ref. 15, magneto-transport measurements up to 9 T were fit to a two-carrier model of the Hall effect, from which Hall mobilities and carrier densities were inferred. In Ref. 10, Shubnikov–de Haas (SdH) quantum oscillations from both heavy and light 2D holes were measured in pulsed magnetic fields to 72 T, providing a measure of carrier densities (via the SdH oscillation frequency) and masses (via the temperature-dependence of the SdH amplitude), along with an estimate of the quantum scattering time $τq$ (via the field-dependence of the SdH oscillations). Note that $τq$ is generally smaller than transport scattering times $τt$⁠, determined by, e.g., Hall effects, because quantum oscillations are sensitive to all scattering mechanisms and sample inhomogeneities.^25–28 

Overall, Table I shows a general agreement between the parameters inferred from these different measurement techniques on these different 2DHG samples. Light hole masses and carrier densities are in reasonable agreement. Low-temperature transport mobilities are in the same general range (⁠ $≈$ 300 cm²/V $·$ s for heavy holes, and on the order of 1500 cm²/V s for light holes), with variations likely arising from differences in sample quality. As expected, the quantum scattering times inferred from SdH studies are shorter than scattering times determined by transport or CR. Where the CR and SdH experiments differ most noticeably is in the determination of the heavy-hole mass $mhh$ (2.6  $me$ vs 1.92  $me$⁠, respectively). In general, masses measured by SdH oscillations reflect the quasiparticle dispersion renormalized by both electron–phonon and electron–electron interactions, and are therefore often found to be heavier than the bare band mass.²⁹ However, Kohn's theorem states that in a Galilean-invariant system, the cyclotron frequency $ωc$ is unaffected by electron–electron interactions.³⁰ Although electron–electron interactions can enhance the cyclotron mass in high-density systems where Galilean invariance no longer holds,³¹ or in systems that have non-parabolic bands³² (in GaN, the bands are not perfectly parabolic), it is still somewhat unexpected to infer a smaller mass from SdH quantum oscillations in comparison with a cyclotron resonance measurement. Despite the increased uncertainty in our determination of $mhh$ by CR (discussed above), a 30% discrepancy seems to be in excess of experimental error and may hint at interesting underlying physics. Further investigations of CR in higher magnetic fields, where the heavy-hole CR peak can be more completely resolved, are needed to resolve this puzzle.

In summary, we directly measured the cyclotron resonance of a 2D hole gas in a GaN/AlN heterojunction by THz spectroscopy in pulsed magnetic fields up to 31 T and revealed the effective masses, densities, scattering times, and mobilities of both heavy and light 2D holes. These are important material parameters for the design and engineering of future complementary GaN-based electronics and photonics. Our results are in broad agreement with parameters inferred from recent electrical transport studies, with the exception of an intriguing difference in $mhh$ that invites further exploration of this system. This experimental capability demonstrates a viable method to characterize wide-bandgap semiconductors, such as p-type ZnO and Ga₂O₃, which have heavier carrier masses, and a path to study the electron–electron interactions in correlated systems through the direct evaluation of cyclotron mass.

S.A.C. and J.W. gratefully acknowledge support from the U.S. Department of Energy (DOE) through the Basic Energy Sciences “Science of 100 T” program. The National High Magnetic Field Laboratory is supported by the National Science Foundation (NSF) under Grant No. DMR-2128556, the State of Florida, and the U.S. DOE. We thank Vladimir Protasenko for help with the operation of the molecular beam epitaxy system. Work at Cornell University was supported in part by SUPREME, one of the seven centers in JUMP 2.0, a Semiconductor Research Corporation (SRC) program sponsored by DARPA. We also made use of the Cornell Center for Materials Research Shared Facilities (Grant No. DMR-1719875).