Abstract
Nanostructures based on buried interfaces and heterostructures are at the heart of modern semiconductor electronics as well as future devices utilizing spintronics, multiferroics, topological effects, and other novel operational principles. Knowledge of electronic structure of these systems resolved in electron momentum k delivers unprecedented insights into their physics. Here we explore 2D electron gas formed in GaN/AlGaN highelectronmobility transistor heterostructures with an ultrathin barrier layer, key elements in current highfrequency and highpower electronics. Its electronic structure is accessed with angleresolved photoelectron spectroscopy whose probing depth is pushed to a few nanometers using softXray synchrotron radiation. The experiment yields direct kspace images of the electronic structure fundamentals of this system—the Fermi surface, band dispersions and occupancy, and the Fourier composition of wavefunctions encoded in the kdependent photoemission intensity. We discover significant planar anisotropy of the electron Fermi surface and effective mass connected with relaxation of the interfacial atomic positions, which translates into nonlinear (highfield) transport properties of the GaN/AlGaN heterostructures as an anisotropy of the saturation drift velocity of the 2D electrons.
Introduction
The concept of highelectronmobility transistors (HEMTs) advanced by Mimura^{1} in the early 80s for GaAs/GaAlAs heterostructures revolutionized the field of highfrequency semiconductor electronics. It exploits an idea of polarization engineering when a large band offset between an intrinsic semiconductor and a doped barrier layer forms a quantum well (QW) at the interface that confines a mobile twodimensional electron gas (2DEG) on the intrinsic semiconductor side. Its spatial separation from defects in the doped barrier layer and in the interface region—in contrast to conventional transistor structures where the 2DEG is formed by doping—allows the electrons to escape defect scattering and dramatically increase their mobility \(\mu _{\mathrm{e}}\), limited then only by phonon scattering. This fundamental operational principle of HEMTs boosts their highfrequency performance, which is exploited in a wide spectrum of applications such as cell phones.
A characteristic property of HEMTs based on wurtzite GaN/AlGaN heterostructures is the accumulation of large sheet carrier concentrations n_{s} ~ 10^{13} cm^{−2}—about one order of magnitude higher compared with other III–V or elementary semiconductors^{2,3}—without intentional doping of the barrier layer. This property is attributed to the formation of a deep spikeshaped QW at the heterojunction, where a large conduction band offset coexists with large piezoelectric and spontaneous polarization^{4,5}. Although \(\mu _{\mathrm{e}}\) in GaNHEMTs is limited by a relatively large electron effective mass m* ~ 0.22 in bulk GaN, nearly three times larger than in GaAs, these devices demonstrate an advantageous combination of sufficiently high operating frequency with exceptionally high current density, resulting from the large n_{s}, saturation drift velocity, operating temperature, and breakdown voltage. These advantages make the GaNHEMTs indispensable components of highpower amplifiers for microwave communication and radar systems. Recently, the ideas of creating mobile 2DEGs using spontaneous and straininduced polarization at the interface have been extended to oxide systems such as the binary Mg_{x}Zn_{1x}O/ZnO,^{6} LaAlO_{3}/SrTiO_{3}^{7}, and CaZrO_{3}/SrTiO_{3}^{8} heterostructures that typically embed orders of magnitude larger n_{s}.
The stateofart GaNHEMTs operate nowadays at the edge of their physical limitations, which remain far from complete understanding. Development of new strategies to improve their performance and conquer the nearTHz operational range^{9} needs qualitatively new experimental knowledge about the physics of these devices. Particularly important is kspace information about the Fermi surface (FS), band dispersions, and wavefunctions of the embedded 2DEG. These fundamental electronic structure characteristics, only indirectly accessible in optics and magnetotransport experiments such as the Hall effect, Shubnikovde Haas oscillations, cyclotron resonance, etc.^{10,11,12}, can be directly probed using the kresolving technique of angleresolved photoelectron spectroscopy (ARPES). However, the small photoelectron mean free path λ_{PE} in conventional ARPES with photon energies hv around 100 eV limits its depth sensitivity to ~ 0.5 nm. Access to buried electron systems such as the HEMTs requires pushing this technique to the softXray energy range (SXARPES, see a recent review ref. 13) with hv around 1 keV and higher, where λ_{PE} grows with photoelectron kinetic energy as ~ E_{k}^{3/4} ^{14,15}. For GaN in particular, elasticpeak electron spectroscopy measurements^{16} show that an increase of E_{k} from 200 eV to 2 keV results in an increase of λ_{PE} from ~ 0.5 to 4 nm. An added virtue of SXARPES, still largely overlooked in applications to 2D systems such as QW states (QWSs), is a significantly sharper definition of the outofplane component K_{z} of the finalstate momentum K. This fact results from the larger λ_{PE} translating, via the Heisenberg uncertainty principle, to a sharper definition of K_{z}^{17,18}. As we will see below, in this case the ARPES signal provides the Fourier composition of the 2DEG wavefunctions. SXARPES on buried systems, challenged by photoelectron attenuation in the overlayers as well as a progressive reduction of photoexcitation crosssection of the valence band (VB) states with hv^{19}, requires advanced synchrotron radiation sources delivering high photon flux (see Methods).
On the sample fabrication side, the 2DEG in GaNHEMT heterostructures has until recently remained inaccessible to SXARPES because of the prohibitively large—of the order of 20–30 nm—depth of the AlGaN barrier layers. However, recent progress in molecular beam epitaxy (MBE) technology, in pursuit of yet higher operation frequencies of these devices, has allowed fabrication of heterostructures with ultrathin barrier layers of 3–4 nm^{20,21,22}, which make them ideally suited to SXARPES. This has allowed direct kspace imaging of the fundamental electronic structure characteristics—the FS, electron dispersions, and the Fourier composition of wavefunctions—of the interfacial 2DEG in such heterostructures.
Results
Fabrication and basic electronic properties of GaNHEMT heterostructures
Our samples were grown on coriented sapphire substrate (see Methods). The 500 nmthick Gapolar GaN layer was grown on top of an AlGaN buffer layer required to suppress the crystal defects and promote growth of a smooth uniform film^{23}. The GaN layer was overgrown by a barrier layer consisting of 2 nm of AlN and 1 nm of Al_{0.5}Ga_{0.5}N, see Fig. 1a. Halleffect measurements on our samples, 1b, have found n_{s} ~ 8.2 × 10^{12} cm^{−2} almost constant through the temperature range T = 5300 K that confirms the high quality of the fabricated structures.
A simplified electronic structure model of our GaNHEMTs is shown in Fig. 1b. It was evaluated within the conventional envelope function approach (neglecting atomic corrugation) by selfconsistent solutions of the onedimensional (1D) Poisson–Schrödinger equations with the Dirichlet boundary conditions adjusted to reproduce the experimental n_{s} (for details see Supplementary Note 1). The effective 1D interfacial potential V(z) as a function of outofplane coordinate z confines two QWSs with different spatial localization. The QWS_{1} embeds larger partial n_{s} and is localized closer to the interface compared to the QWS_{2}, which is shifted into the V(z) saturation region.
FS: anisotropy and carrier concentration
A scheme of our SXARPES experiment is presented in Fig. 1a. The data were acquired in the hv region between 800 and 1300 eV where the interplay of photoelectron transmission through the barrier overlayer increasing with hv and photoexcitation crosssection decreasing with hv maximizes the QWS signal. In view of the even QWS wavefunction symmetry, we used ppolarization of the incident Xrays, which minimizes the geometry and polarizationrelated matrix element effects, which distort the direct relation between ARPES intensities and Fourier composition of the QWS wavefunctions (see below).
The experimental FS map measured as a function of inplane momentum K_{xy} = (K_{x},K_{y}) at hv = 1057 eV (maximizing the QWS signal, see below) is presented in Fig. 2a. The FS formed by the QWSs in the HEMT channel appears as tiny circles whose k_{//} are located around the \({\bar{\mathrm \Gamma }}\)points of the heterostructure’s 2D Brillouin zone (BZ) shown in Fig. 1c. The location of the FS pockets coincides with the VB maxima (VBM), as seen in an isoE_{B} map of the VB shown in Fig. 2b. In the direct band gap GaN, this location in k_{//} is consistent with the conduction band minimum (CBM)derived character of the QWSs.
A highresolution cut of the FS in Fig. 2c displays an external contour with a diameter of ~ 0.2 Å^{−1} manifesting the QWS_{1} and finite spectral weight in the middle, suggesting the presence of QWS_{2}. The latter is confirmed by the momentumdistribution curves (MDCs) of the Fermi intensity around the \({\bar{\mathrm \Gamma }}_{10}\)points (Fig. 2d) where a weak structure in the middle signals the QWS_{2}. Our observation of QWS_{2} is consistent with its recent detection via magnetotransport spectroscopy^{24}. The presence of this state centered further away from the interface compared with the QWS_{1} indicates that V(z) in the GaNHEMTs has a longrange saturated shape (see the model in Fig. 1b). The external Fermi intensity peaks in the E_{F}MDCs directly identify the Fermi momenta k_{F} of the QWS_{1}.
Such a comprehensive kspace view of the buried QWSs, achieved with SXARPES, delivers two important observations. First, the experimental E_{F}MDCs reveal significant anisotropy of the FS characterized by a difference of the k_{F} values of 0.095 ± 0.006 Å^{−1} along the \(\overline {{\mathrm{\Gamma M}}}\) azimuth and 0.085 ± 0.004 Å^{−1} along \(\overline {{\mathrm{\Gamma K}}}\).These values were determined from the maximal gradient of the Fermi intensity^{25}, which delivers accurate k_{F}values even when the bandwidth is comparable with the experimental resolution. The indicated statistical errors are the standard deviations of the k_{F} values over the measurement series and scaled with the Student’s tdistribution coefficients for a confidence interval of 76%. The corresponding k_{F} anisotropy factor \(A_{\mathrm{F}} = \frac{{k_{\mathrm{F}}^{\overline {{\mathrm{\Gamma K}}} }  k_{\mathrm{F}}^{\overline {{\mathrm{\Gamma M}}} }}}{\langle{k_{\mathrm{F}}}\rangle}\), where \(\langle k_{\mathrm{F}}\rangle = \left( {k_{\mathrm{F}}^{\overline {{\mathrm{\Gamma K}}} } + k_{\mathrm{F}}^{\overline {{\mathrm{\Gamma M}}} }} \right){\mathrm{/}}2\), measures in our case ~ 11%. Such a considerable FS anisotropy has been completely overlooked in previous macroscopic experimental studies without kresolution.
Why does the 2DEG show such a pronounced planar anisotropy? Our simulations of electronic structure of the GaNHEMTs (see Methods) used the standard densityfunctional theory (DFT), which is known to well describe excitation energies in GaN apart from the electron exchangecorrelation discontinuity across the band gap approximated by the socalled scissors operator^{26}. As the QWSs inherit their wavefunctions from the CBM of the bulk GaN (see below), we have started our analysis from band structure. The corresponding A_{F} in Fig. 2e is however negligible through the whole range of < k_{F} > defining the band filling. Next, we approached the electronic structure of our heterostructure system using GaN/AlN slab calculations with the unit cell (u.c.) shown in Fig. 2g. The atomic positions were relaxed under the constraint of the bulk GaN lateral lattice constants and symmetry. The resulting coriented GaN bond length d_{v} in Fig. 2f shows a significant increase toward the interface relative to the bulk value. The corresponding atomic displacement contributes to the piezoelectric polarization at the GaN/AlN interface. The layerresolved electronic structure of this system was characterized by the k_{//}resolved layer density of states (LDOS) defined as, \(\rho _z({\bf{k}}_{//},E) = \mathop {\int}\limits_\Omega {\mathrm{d}}x{\mathrm{d}}y{\mathop {\sum}\limits_n {\left {\psi _n({\mathbf{r}},{\bf{k}}_{//},E)} \right^2} }\), where r = (x,y,z), the summation includes all nth electron states with wavefunctions \(\psi _n\) available for given k_{//} and E, and the integration runs over the lateral unit cell Ω. Figure 2h shows the LDOS calculated near the CBM (the VB results are given in Supplementary Note 2) for the interface, subinterface, and deep bulklike GaN layers, where the bottom of the LDOS continuum corresponds to the QWS_{1}. The corresponding A_{F} plots in 2e now show significant anisotropy, increasing toward the interface. At the experimental < k_{F} > the interface layer A_{F} is ~ 7%, which falls almost within the error bars of the experimental value. Finally, we performed the same LDOS calculations with the atomic coordinates in the slab fixed at the bulk values (without relaxation). A_{F} immediately returned to the negligible bulk values. This analysis suggests thus that the discovered 2DEG anisotropy in GaNHEMTs is a purely interface effect caused by relaxation of atomic position near the GaN/AlN interface.
We note however that the predicted atomic relaxation is restricted to a few atomic layers next to the interface, and it is not clear why it should significantly affect the 2DEG, whose maximal density is located ~ 3 nm away from this region (Fig. 1b). In fact, even the stateofart growth methods leave significant intermixing of Ga and Al atoms at the GaN/AlN interface^{21}, resulting in a gradual variation of the lattice parameters over 1–3 nm from the interface. In the spirit of the entanglement between atomic relaxation and LDOS anisotropy, revealed by our computational analysis, this lattice distortion may cause significant electronic structure anisotropy extending into the 2DEG localization region.
Another important characteristic of the buried 2DEG is the experimental FS area which, by the Luttinger theorem^{27}, is directly related to the n_{s} sheet carrier concentration. In our case, the area of the external QWS_{1} contour translates into partial \(n_{\mathrm {s}}^{(1)}\) = (12.8 ± 1.4) × 10^{12} cm^{−2}, and that of the internal QWS_{2} contour into \(n_{\mathrm {s}}^{(2)}\) = (0.5 ± 0.4) × 10^{12} cm^{−2}. We note that the emergence of two QWSs goes together with large n_{s} formed in the anomalously deep QW of the GaNHEMTs. In our case, the QWS_{2} contributes only ~ 4% of the total n_{s} dominated by the QWS_{1}. A significant difference of this ratio to that of 17% found in a cyclotron resonance study^{28} is explained by extremely high sensitivity of the QWS_{2} population to the interfacial QW depth in different samples. The total n_{s} in our case amounts to (13.3 ± 1.8) × 10^{12} cm^{−2}, which is in fair agreement with n_{s} = 8.2 × 10^{12} cm^{−2} obtained by our Hall characterization, see Fig. 1d, in particular taking into account a small systematic overestimate of k_{F} introduced by the gradient method^{25}.
Momentum dependence of ARPES intensity: wavefunction character
Within the onestep theory of photoemission—see, e.g., ref. 29—the ARPES intensity is found as \(I_{{\mathrm{PE}}} \propto \left {\left\langle {f{\mathrm{}}{\mathbf{A}} \cdot {\mathbf{p}}{\mathrm{}}i} \right\rangle } \right^2\), where \(\left\langle {f} \right.\) is the final and \(\left. {i} \right\rangle\) the initial states coupled through the vector potential A of the incident electromagnetic field and momentum operator p. Neglecting the experimental geometry and polarization effects, this expression simplifies to the scalar product \(I_{{\mathrm{PE}}} \propto \left {\left\langle {f{\mathrm{}}i} \right\rangle } \right^2\). For sufficiently high photon energies, \(\left\langle {f} \right.\) approximates a plane wave \(e^{i{\mathbf{Kr}}}\) periodic in the inplane xy direction and damped in the outofplane direction z, and the ARPES intensity appears^{30,31} as \(I_{{\mathrm{PE}}} \propto \left {\left\langle {e^{i{\mathbf{Kr}}}{\mathrm{}}i} \right\rangle } \right^2\). We will now apply this formalism to the QWS wavefunctions \(\psi _{{\mathrm{QWS}}}\).
We will first analyze I_{PE} as a function of photoelectron K_{xy} inplane momentum. If we represent \(\psi _{{\mathrm{QWS}}}\) by Fourier expansion over 2D reciprocal vectors g as \(\psi _{{\mathrm{QWS}}}\left( {\mathbf{r}} \right) = \mathop {\sum}\limits_{\mathbf{g}} {A_{{\mathbf{k}}_{xy} + {\mathbf{g}}}e^{i\left( {{\mathbf{k}}_{xy} + {\mathbf{g}}} \right){\mathbf{r}}_{xy}}}\), the planewave orthogonality will select from the sum only the component whose inplane momentum k_{xy}+g matches K_{xy} (corrected for the photon momentum p = hv/c), i.e., \(I_{{\mathrm{PE}}}\left( {{\mathbf{K}}_{xy}} \right) \propto \left {A_{{\mathbf{K}}_{{\boldsymbol{xy}}}}} \right^2\). Therefore, the (K_{x},K_{y})dependent ARPES maps in Fig. 2a–c visualize essentially the 2D Fourier expansion of \(\psi _{{\mathrm{QWS}}}\)^{30,31}_{.}
We will now analyze I_{PE} as a function of photoelectron K_{z} outofplane momentum varied in the experiment through hv. The corresponding isoE_{B} map in (K_{x},K_{z}) coordinates near the VBM is displayed in Fig. 3a. Its K_{z}dispersive contours demonstrate the threedimensional (3D) character of the VB states inherited from bulk GaN. The FS in Fig. 3b formed by the QWSs demonstrates a different behavior however: the ARPES signal sharply increases whenever K_{z} approaches values of integer G_{z}—corresponding to the Γpoints of the bulk BZ, but without any sign of K_{z} dispersion. The latter is emphasized by the zooms in Fig. 3d,e, where the QWSs form segments straight in the K_{z} direction. This pattern is characteristic of the 2D nature of the QWSs. The corresponding Fermi intensity is represented by E_{F}MDCs in Fig. 3c that show periodic oscillations peaked where K_{z} matches the Γpoints.
Why do the QWSs display such an ARPES response, periodically oscillating as a function of hv? The dependence of \(\psi _{{\mathrm{QWS}}}\) on z for given K_{xy} is represented as \(\psi _{{\mathrm{QWS}}}\left( z \right) = E\left( z \right) \cdot B_{k_z}\left( z \right)\), where the envelope function E(z) confines the oscillating Bloch wave \(B_{k_z}\left( z \right)\) whose k_{z} momentum is adapted to the 3D crystal potential^{32}. In our case, the first term is an Airylike function embedded in the approximately triangular V(z) and the second one derives from bulk states of the GaN host. Inheriting ideas of early ARPES studies on surface states^{32,33}, it can be shown^{34} that if we expand \(B_{k_z}\left( z \right)\) over outofplane reciprocal vectors G_{z} of the 3D host lattice as \(B_{{{k}}_z}(z) = \mathop {\sum}\limits_{G_z} {C_{G_z}e^{i\left( {k_z + G_z} \right)z}}\), then the K_{z} dependence of ARPES intensity for given K_{xy} appears as a sequence \(I_{{\mathrm{PE}}}\left( {K_z} \right) \propto \mathop {\sum}\limits_{G_z} {\left {C_{G_z}} \right^2P\left( {K_z  (k_z + G_z)} \right)}\) of peaks \(P\left( {K_z  (k_z + G_z)} \right)\) centered at K_{z} = k_{z} + G_{z}. Physically, the ARPES intensity blows up whenever the photoelectron K_{z} hits a k_{z} + G_{z} harmonic of \(\psi _{{\mathrm{QWS}}}\) to maximize the \(\left\langle {f} \right.\) and \(\left i \right\rangle\) scalar product. Amplitudes of the \(I_{{\mathrm{PE}}}\left( {K_z} \right)\) peaks \(\propto \left {C_{G_z}} \right^2\) image Fourier composition of the oscillating \(B_{k_z}\left( z \right)\) term of \(\psi _{{\mathrm{QWS}}}\) (modulated by hvdependent photoelectron transmission through the AlGaN/AlN layer) and the peak shapes are related to the Fourier transform of the E(z) term weighted by \(e^{  \lambda _{{\mathrm{PE}}}z}\) ^{34}.
Importantly, the experimental K_{z} dependence of the QWS signal in Fig. 3b exhibits peaks exactly at k_{z} + G_{z}, corresponding to the Γpoint of bulk GaN. In combination with the (K_{x},K_{y}) image in Fig. 2, where the QWS signal corresponds to the same Γpoint, this fact confirms that the \(\psi _{{\mathrm{QWS}}}\)’s are derived from the CBM states of bulk GaN. In a methodological perspective, such identification of the \(\psi _{{\mathrm{QWS}}}\) character can be essential, e.g., for heterostructures of layered transition metal dichalcogenides, where the CB can include two or more valleys almost degenerate in energy but separated in k. The knowledge of the QWS character will then allow the predictive manipulation of the valley degree of freedom for potential valleytronics devices^{35}.
We note that the common models of QWSs based on the 1D potential V(z), such as that in Fig. 1b, imply that their inplane behavior \(\psi _{{\mathrm{QWS}}}({\mathbf{r}}_{xy})\) is described by one single plane wave \(e^{i{\mathbf{K}}_{xy}{\boldsymbol{r}}_{xy}}\) (i.e., one nonzero \(A_{{\mathbf{K}}_{xy}}\) component) and outofplane behavior \(\psi _{{\mathrm{QWS}}}(z)\) is identical to the smooth E(z) function. However, the experimental FS maps in Fig. 2a,c reveal numerous nonzero \(A_{{\mathbf{K}}_{{\mathrm{xy}}}}\) spread through kspace and the \(I_{PE}\left( {K_z} \right)\) oscillations numerous k_{z} + G_{z} harmonics. Accurate QWS models should therefore incorporate atomic corrugation of the interfacial potential in the inplane and outofplane directions to form \(\psi _{{\mathrm{QWS}}}\) as a confined Bloch wave.
The experimental distribution of highenergy ARPES intensity over sufficiently large volume of the (K_{x},K_{y},K_{z}) space will in principle allow, notwithstanding the phase problem, a full reconstruction of \(\Phi _{{\mathrm{QWS}}}(x,y,z)\) in all three spatial coordinates, similar to the reconstruction of molecular orbitals (see refs. 30,31, and references therein). This reconstruction will naturally incorporate full \(\psi _{{\mathrm{QWS}}}\) including the envelope and Blochwave terms that goes beyond the common 1D models such as in Fig. 1b, describing the QWSs as free 2D electrons with empirical m* confined in the z direction. More accurate models of the GaNHEMTs should replace free electrons by Bloch ones, naturally incorporating atomic corrugation.
Band dispersions: effective mass
Experimental band dispersions in GaNHEMTs shown in Fig. 4 were measured along \(\overline {{\mathrm{\Gamma}}{\mathrm{M}}}\) (a) and \(\overline {{\mathrm{\Gamma}}{\mathrm{K}}}\) (b) at hv = 1066 eV bringing K_{z} to the Γpoint of the bulk BZ. Nondispersive ARPES intensity coming from the AlN and AlGaN layers is suppressed in these plots by subtracting the angleintegrated spectral component. The CBMderived QWSs appear as tiny electron pockets above the VB dispersions of GaN. Their energy separation from the VBM is consistent with the GaN fundamental band gap of ~ 3.3 eV. Whereas the VB dispersions are broadened in E_{B} primarily because of band bending in the QW region, the QWS dispersions stay sharp. This confirms their 2D nature insensitive to band bending as well as their localization in the deep defectfree region in GaN, spatially separated from the defectrich GaN/AlN interface region, the fundamental operational principle of the HEMTs delivering high \(\mu _{\mathrm {e}}\). Fig. 4c,f,i show the experimental dispersions as a function of k_{z}. Whereas clear dispersion of the VB states manifests their 3D character, the QWS are flat in K_{z}.
A zoomin of the QWS dispersions along the \(\overline {{\mathrm{\Gamma M}}}\) and \(\overline {{\mathrm{\Gamma K}}}\) azimuths is shown in Fig. 4d,e with the corresponding MDC in Fig. 4g,h. Whereas the outer contour of these dispersions corresponds to the QWS_{1}, the significant spectral weight in the middle is due to the QWS_{2}. Following the k_{F} anisotropy discussed above, a parabolic fit of the QWS_{1} dispersions yields m* values of (0.16 ± 0.03)m_{0} along the \(\overline {{\mathrm{\Gamma M}}}\) azimuths and (0.13 ± 0.02)m_{0} along \(\overline {{\mathrm{\Gamma K}}}\) (m_{0} is the freeelectron mass), which thus differ from each other by ~ 22%.
Our SXARPES experiment presents thus a direct evidence of the planar FS and m* anisotropy in GaNHEMTs. This effect was overlooked in previous studies, because the optics methods are kintegrating and quantum oscillation techniques lose their kresolution in the 2D case. Magnetotransport experiments give only an indirect information on the 2DEG’s m*^{10,11,12}, which is conventionally^{36,37} assumed to be isotropic. We conjecture that further progress of SXARPES on energy resolution will push data analysis from merely band structure to oneelectron spectral function A(ω,k), which will inform, e.g., the interaction of electrons with other quasiparticles such as the phononplasmoncoupled modes.^{38}
Implications for the transport properties
How will the observed lateral anisotropy of the 2DEG electronic structure affect the transport properties? Naively, one might think that it would directly translate into an anisotropy of the electrical conductivity. However, fundamental linear response considerations attest that any physical properties such as conductivity described by a secondorder tensor with C_{6v} symmetry must be scalar, i.e., in the linear (lowfield) regime, conductivity in the hexagonal lattice of GaN must be isotropic (Supplementary Note 3). On the other hand, this restriction is lifted for the nonlinear (highfield) regime where conductivity can become anisotropic. A canonical example of such a crossover is ndoped Ge.^{39,40} Although its FS is anisotropic, cubic symmetry of the Ge lattice results in isotropic lowfield conductivity. However, with an increase of the electric field, conductivity along the \(\left\langle {001} \right\rangle\), \(\left\langle {011} \right\rangle\), and \(\left\langle {111} \right\rangle\) crystallographic directions develops differently. The GaNHEMTs can be easily driven into the nonlinear regime where electronic current saturates due to electron scattering on longitudinal optical (LO) phonons^{41,42}. To reach the LO phonon energy, lighter electrons should gain a larger drift velocity V_{sat}. Therefore, larger V_{sat} and thus saturation current I_{sat} should be expected in the directions of smaller m*.
We have examined low and highfield conductivity in our GaNHEMT heterostructures using samples essentially identical to the ARPES ones, but with the Al_{0.45}Ga_{0.55}N layer thickness increased to 15 nm, to prevent a 2DEG degradation during longer sample handling in air. Hall measurements showed n_{s} = 2 × 10^{13} cm^{−2}, μ_{e} = 1150 cm^{2} V^{−1} s, and sheet resistance R_{s} = 240 Ω sq^{−1} for these samples. The fabricated test modules were oriented at four different angles (0°, 30°, 60°, 90°) with respect to the substrate to promote current flow along the \(\overline {{\mathrm{\Gamma M}}}\) and \(\overline {{\mathrm{\Gamma K}}}\) azimuths (Fig. 5a,b). Results of the transport measurements presented in Fig. 5c show, as expected, isotropic lowfield R_{s}. However, the I_{sat} characteristic of the highfield regime is clearly anisotropic: 1.53 ± 0.01 A × mm^{−1} along \(\overline {{\mathrm{\Gamma K}}}\) and 1.46 ± 0.01 A × mm^{1} along \(\overline {{\mathrm{\Gamma M}}}\) (Fig. 5c). As a consistency check, the modules rotated by 60° with respect to each other showed the same I_{sat} values, in accordance with hexagonal symmetry of the GaN electronic structure. These results on the previously overlooked I_{sat} anisotropy demonstrate that m* along \(\overline {{\mathrm{\Gamma K}}}\) is smaller compared to \(\overline {{\mathrm{\Gamma M}}}\), as predicted by our ARPES results.
Discussion
Our direct kspace imaging of the fundamental electronic structure characteristics—FS, band dispersions and occupancy, and Fourier composition of wavefunctions—of the 2DEG formed in highfrequency GaNHEMTs with ultrathin barrier layer makes a quantitative step compared to conventional optics and magnetotransport experiments. We discover, in particular, significant planar anisotropy of the 2DEG band dispersions caused by piezoelectrically active relaxation of atomic position near the GaN/AlN interface. This effect is found to manifest itself in nonlinear electron transport properties as anisotropy of the saturation drift velocity and current. Our findings suggest a positive use of the crystallographic orientation to improve these highpower characteristics of GaNHEMTs. Furthermore, our kspace image of the Fourier composition of the 2DEG wavefunctions calls for extension of the conventional 1D models of semiconductor interfaces to 3D ones based on the Blochwave description naturally incorporating atomic corrugation. The fundamental knowledge about GaNHEMTs achieved in our work as well as new device simulation methods can clarify the physical limits of these heterostructures, and finally push their reliable operation into the nearTHz range. Methodologically, we have demonstrated the power of the synchrotron radiation based technique of SXARPES with its enhanced probing depth and sharp definition of the full 3D k for the discovery of previously obscured properties of semiconductor heterostructures. Our results complement previous applications of SXARPES to buried oxide interfaces^{7} and magnetic impurities in semiconductors^{43} and topological insulators^{44}, which used elemental and chemicalstate specificity of this technique achieved with resonant photoemission. In a broader perspective, our methodology arms the heterostructure growth technology with means to directly control the fundamental kspace parameters of the electronic structure, thereby delivering optimal transport and optical properties of the fabricated devices. Complementary to imaging of nonequilibrium electron motion in spatial coordinates,^{45} we can envisage an extension of our experimental methodology to pump–probe experiments using Xray freeelectron laser sources, which will image the electron system evolution in kspace during transient processes in electronic devices.
Methods
Sample fabrication
The GaNHEMT heterostructures embedding a 2DEG were grown on coriented sapphire substrate in a SemiTeq STE3N MBEsystem equipped with an ammonia (NH_{3}) nitrogen source. The buffer layer growth adopted the procedure described in ref. 23. Before deposition, the substrate was annealed during 1 h and then nitrided for 40 min with 30 sccm NH_{3} at 850 °C. The following growth was carried out with 200 sccm NH_{3}. Deposition started with 20 nm AlN layer, grown at 1050 °C. The following 200 nm AlN layer was grown at 1120 °C with Ga used as a surfactant. Then a gradient junction to Al_{0.43}Ga_{0.57}N with a thickness of ~ 250 nm was achieved by a gradual decrease of the substrate temperature down to T = 830 °C, followed by 140 nm of growth at constant T. Next, a second gradient junction to Al_{0.1}Ga_{0.9}N with a thickness of ~ 140 nm was formed by reducing T of the Al effusion cell. Then a 500 nm GaN layer was grown. The growth was finished by deposition of a barrier layer consisting of 2 nm AlN and 1 nm Al_{0.5}Ga_{0.5}N for the ARPES experiments, and 1 nm AlN and 15 nm Al_{0.45}Ga_{0.55}N for measurements of transport properties. Hall effect characterization was carried out in magnetic fields up to 40 kG. The magneticfield dependences were measured in both the standard Hall and van der Pauw geometries. All transport measurements were carried out at the Resource Center of Electrophysical Methods (Complex of NBICSTechnologies of Kurchatov Institute).
SXARPES experiments
Raw SXARPES data were generated at the Swiss Light Source synchrotron radiation facility (Paul Scherrer Institute, Switzerland). The experiments have been carried out at the SXARPES endstation^{46} of the ADRESS beamline^{47}, delivering high photon fluxes up to 10^{13} photons × s^{−1} × (0.01% BW)^{−1}. With the actual experimental geometry, ppolarized incident Xrays selected electron states symmetric relative to the \(\overline {{\mathrm{\Gamma M}}}\) and \(\overline {{\mathrm{\Gamma K}}}\) azimuths. The projection K_{x} of the photoelectron momentum was directly measured through the emission angle along the analyser slit, K_{y} is varied through tilt rotation of the sample, and k_{z} through hv. The experiments were carried out at 12 K to quench the thermal effects reducing the coherent kresolved spectral component at high photoelectron energies^{48}. The combined (beamline and analyzer) energy resolution was ~ 150 meV and the angular resolution of the PHOIBOS150 analyzer was 0.07°. The Xray spot size in projection on the sample was 30 × 75 μm^{2}, which allowed us to control spatial homogeneity of our samples. Charging effects were not detected due to the small thickness of the AlGaN barrier layer.
Electronic structure calculations
Firstprinciples calculations for bulk GaN have been carried out in the DFT framework as implemented in the pseudopotential Quantum Espresso code^{49} using ultrasoft pseudopotentials. The electron exchangecorrelation term was treated within the Generalized Gradient Approximation using the Perdew–Burke–Ernzerhof functional. Selfconsistent calculations for bulk GaN were performed with the planewave cutoff energy 60 Ry and kspace sampling over a grid of 10 × 10 × 5 points in the BZ and corrected with the scissors operator to reproduce the experimental band gap. Calculations for the GaNHEMT heterostructure used a 1 × 1 slab geometry with the supercell including 18 u.c. of GaN in the middle between 3 u.c of AlN at each end, Fig. 2g. Atomic coordinates in the supercell were relaxed, but imposing the lateral u.c. of bulk GaN until the Hellmann–Feynman forces on each atom were < 30 meV Å^{−1}. The planewave cutoff energy was 25 Ry with a kgrid of 10 × 10 × 1 points. The Gaussian window for LDOS calculations was set to 0.05 eV.
Transport measurements
Raw transport data were generated at the Kurchatov Institute. To measure linear and nonlinear transport properties of the GaNHEMT heterostructures, two types of test modules with lowresistance regrown ohmic contacts^{50,51} were formed. Details on processing can be found in ref. 51. The first type modules were conventional transmission line measurement (TLM) modules with a channel width of 20 μm and channel lengths of 2.5, 10, 20, and 40 μm (marked 1 in Fig. 5a). These modules were used to determine contact resistance, which was found to be 0.15 Ω mm. Also I–V curves were measured at the smallest gaps (2.5 μm length channels, see Fig. 5b) of 24 such TLM modules (6 modules per each of four directions) in DC mode. The voltage sweep time (1 ms per point with a voltage step of 0.5 V) was chosen to be small enough to suppress sample heating effects as judged by the absence of hysteresis in the forward and backward voltage scans as well as repeatability of the I–V curves with a sweep time reduction. The second type “resistor” modules (marked 2 in Fig. 5a) were arrays of 1 mm long and 20 μm wide stripes (25 stripes per module separated by 20 μm mesa isolation) with contact pads on each side. These modules had negligible contact resistance and were used for precise measurement of the 2DEG lowfield conductivity in different directions.
Data availability
Derived data supporting the findings of this study are available from the corresponding author on request. The SXARPES data were processed using the custom package MATools available at https://www.psi.ch/sls/adress/manuals.
References
 1.
Mimura, T. The early history of the high electron mobility transistor (HEMT). IEEE Trans. Microw. Theory Techn. 50, 780–782 (2002).
 2.
Gurusinghe, M. N., Davidsson, S. K. & Andersson, T. G. Twodimensional electron mobility limitation mechanisms in Al_{x}Ga_{1−x}N∕GaN heterostructures. Phys. Rev. B 72, 045316 (2005).
 3.
Medjdoub, F. & Iniewski, K. (eds). Gallium Nitride (GaN): Physics, Devices, and Technology (CRC Press, Taylor & Francis Group, Boca Raton, 2016).
 4.
Yu, E. T. et al. Measurement of piezoelectrically induced charge in GaN/AlGaN heterostructure fieldeffect transistors. Appl. Phys. Lett. 71, 2794–2796 (1997).
 5.
Gaska, R. et al. The influence of the deformation on the twodimensional electron gas density in GaN–AlGaN heterostructures. Appl. Phys. Lett. 72, 64–66 (1998).
 6.
Tsukazaki, A., Ohtomo, A. & Kita, T. Quantum Hall effect in polar oxide heterostructures. Science 315, 1388–1391 (2007).
 7.
Cancellieri, C. et al. Polaronic metal state at the LaAlO_{3}/SrTiO_{3} interface. Nat. Commun. 7, 10386 (2016).
 8.
Chen, Y. et al. Creation of high mobility twodimensional electron gases via strain induced polarization at an otherwise nonpolar complex oxide interface. Nano Lett. 15, 1849–1854 (2015).
 9.
Nagatsuma, T., Ducournau, G. & Renaud, C. C. Advances in terahertz communications accelerated by photonics. Nat. Photonics 10, 371–379 (2016).
 10.
Shockley, W. Cyclotron resonances, magnetoresistance, and Brillouin zones in semiconductors. Phys. Rev. 90, 491 (1953).
 11.
Krieger, J. B., Meeks, T. & Esposito, E. Conductivity in anisotropic semiconductors: application to longitudinal resistivity and Hall effect in saturationstressed degenerately doped ntype germanium. Phys. Rev. B 5, 1499–1504 (1972).
 12.
Brandt, N. B. & Chudinov, S. M. Shubnikovde Haas effect and its application to investigation of the energy spectrum of metals, semimetals, and semiconductors. Sov. Phys. Usp. 25, 518–529 (1982).
 13.
Strocov, V. N. et al. SoftXray ARPES at the Swiss Light Source: From 3D materials to buried interfaces and impurities. Synchrotron Radiat. News 27, 31–40 (2014).
 14.
Powell, C. J., Jablonski, A., Tilinin, I. S., Tanuma, S. & Penn, D. R. Surface sensitivity of Augerelectron spectroscopy and Xray photoelectron spectroscopy. J. Electron Spectrosc. Relat. Phenom. 98/99, 1–15 (1999).
 15.
Gray, A. X. et al. Bulk electronic structure of the dilute magnetic semiconductor Ga_{1−x}Mn_{x}As through hard Xray angleresolved photoemission. Nat. Mater. 11, 957–962 (2012).
 16.
Krawczyk, M., Zommer, L., Jablonski, A., Grzegory, I. & Bockowski, M. Energy dependence of electron inelastic mean free paths in bulk GaN crystals. Surf. Sci. 566–568, 1234–1239 (2004).
 17.
Strocov, V. N. Intrinsic accuracy in 3dimensional photoemission band mapping. J. Electron Spectros. Relat. Phenom. 130, 65–78 (2003).
 18.
Strocov, V. N. et al. Threedimensional electron realm in VSe_{2} by softXray photoelectron spectroscopy: origin of chargedensity waves. Phys. Rev. Lett. 109, 086401 (2012).
 19.
Yeh, J. J. & Lindau, I. Atomic subshell photoionization cross sections and asymmetry parameters: 1≤ Z≤ 103. At. data Nucl. data Tables 32, 1–155 (1985).
 20.
Cao, Y. et al. MBE growth of high conductivity single and multiple AlN/GaN heterojunctions. J. Cryst. Growth 323, 529–533 (2011).
 21.
Kaun, S. W. et al. Pure AlN layers in metalpolar AlGaN/AlN/GaN and AlN/GaN heterostructures grown by lowtemperature ammoniabased molecular beam epitaxy. Semicond. Sci. Technol. 30, 055010 (2015).
 22.
Guo, L. et al. Hot electron induced nonsaturation current behavior at high electric field in InAlN/GaN heterostructures with ultrathin barrier. Sci. Rep. 6, 37415 (2016).
 23.
Mayboroda, I. et al. Growth of AlGaN under the conditions of significant gallium evaporation: phase separation and enhanced lateral growth. J. Appl. Phys. 122, 105305 (2017).
 24.
Lu, F. et al. Magnetotransport spectroscopy of the first and second twodimensional subbands in Al_{0.25}Ga_{0.75}N/GaN quantum point contacts. Sci. Rep. 7, 42974 (2017).
 25.
Straub, Th et al. Manybody definition of a Fermi surface: application to angleresolved photoemission. Phys. Rev. B 55, 13473–13478 (1997).
 26.
Svane, A. et al. Quasiparticle selfconsistent GW theory of IIIV nitride semiconductors: bands, gap bowing, and effective masses. Phys. Rev. B 82, 115102 (2010).
 27.
Luttinger, J. M. Fermi surface and some simple equilibrium properties of a system of interacting fermions. Phys. Rev. 119, 1153–1163 (1960).
 28.
Mishra, M. K. et al. Comprehensive magnetotransport characterization of two dimensional electron gas in AlGaN/GaN high electron mobility transistor structures leading to the assessment of interface roughness. AIP Adv. 4, 097124 (2014).
 29.
Feibelman, P. J. & Eastman, D. E. Photoemission spectroscopy – correspondence between quantum theory and experimental phenomenology. Phys. Rev. B 10, 4932–4947 (1974).
 30.
Puschnig, P. et al. Reconstruction of molecular orbital densities from photoemission data. Science 326, 702–706 (2009).
 31.
Kliuiev, P., Latychevskaia, T., Osterwalder, J., Hengsberger, M. & Castiglioni, L. Application of iterative phaseretrieval algorithms to ARPES orbital tomography. New. J. Phys. 18, 093041 (2016).
 32.
Ortega, J. E., Himpsel, F. J., Mankey, G. J. & Willis, R. F. Quantumwell states and magnetic coupling between ferromagnets through a noblemetal layer. Phys. Rev. B 47, 1540–1552 (1993).
 33.
Louie, S. G. et al. Periodic oscillations of the frequencydependent photoelectric cross sections of surface states: theory and experiment. Phys. Rev. Lett. 44, 549–553 (1980).
 34.
Strocov, V. N. Photoemission response of 2D states. Preprint at http://arXiv.org/abs/1801.07505 (2018).
 35.
Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).
 36.
Knap, W. et al. Effective g* factor of twodimensional electrons in GaN/AlGaN heterojunctions. Appl. Phys. Lett. 75, 3156–3158 (1999).
 37.
Syed, S. et al. Nonparabolicity of the conduction band of wurtzite GaN. Appl. Phys. Lett. 83, 4553–4555 (2003).
 38.
Talwar, D. N. Direct evidence of LO phononplasmon coupled modes in nGaN. Appl. Phys. Lett. 97, 051902 (2010).
 39.
Schweitzer, D. & Seeger, K. The anisotropy of conductivity of ntype germanium in strong D.C. fields. Z. für Phys. 183, 207–216 (1965).
 40.
Nathan, M. I. Anisotropy of the conductivity of ntype germanium at high electric fields. Phys. Rev. 130, 2201–2204 (1963).
 41.
Fang, T. et al. Effect of optical phonon scattering on the performance of GaN transistors. IEEE Electron Dev. Lett. 33, 709–711 (2012).
 42.
Gonschorek, M. et al. Self heating in AlInN/AlN/GaN high power devices: origin and impact on contact breakdown and IV characteristics. J. Appl. Phys. 109, 063720 (2011).
 43.
Kobayashi, M. et al. Unveiling the impurity band induced ferromagnetism in the magnetic semiconductor (Ga,Mn)As. Phys. Rev. B 89, 205204 (2014).
 44.
Krieger, J. A. et al. Spectroscopic perspective on the interplay between electronic and magnetic properties of magnetically doped topological insulators. Phys. Rev. B 96, 184402 (2017).
 45.
Man, M. K. L. et al. Imaging the motion of electrons across semiconductor heterojunctions. Nat. Nanotech. 12, 36–40 (2017).
 46.
Strocov, V. N. et al. SoftXray ARPES facility at the ADRESS beamline of the SLS: concepts, technical realisation and scientific applications. J. Synchrotron Rad. 21, 32–44 (2014).
 47.
Strocov, V. N. et al. Highresolution soft Xray beamline ADRESS at the Swiss Light Source for resonant inelastic Xray scattering and angleresolved photoelectron spectroscopies. J. Synchrotron Rad. 17, 631–643 (2010).
 48.
Braun, J. et al. Exploring the XPS limit in soft and hard xray angleresolved photoemission using a temperaturedependent onestep theory. Phys. Rev. B 88, 205409 (2013).
 49.
Giannozzi, P. et al. Quantum expresso: a modular and opensource software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
 50.
Yue, Y. et al. InAlN/AlN/GaN HEMTs with regrown ohmic contacts and f _{T} of 370 GHz. IEEE Electron Dev. Lett. 33, 988–990 (2012).
 51.
Mayboroda, I. et al. Selective MBE growth of nonalloyed ohmic contacts to 2D electron gas in highelectronmobility transistors based on GaN/AlGaN heterojunctions. Tech. Phys. Lett. 40, 488–490 (2014).
Acknowledgements
We thank M.B. Tsetlin, V.G. Nazin, E.E. Krasovskii, and G. Aeppli for fruitful discussions. M.A.H. was supported by the Swiss Excellence Scholarship grant ESKASno. 2015.0257. N.K.C. was partly supported by the Grant RFBR 160701188.
Author information
Affiliations
Contributions
L.L.L. and V.N.S. conceived the SXARPES research at the Swiss Light Source. L.L.L., M.A.H., and V.N.S. performed the SXARPES experiment supported by X.W., B.T., and T.S. L.L.L. and V.N.S. processed and interpreted the data. V.N.S. set theoretical description of the ARPES response. M.A.H. performed the DFT calculations. M.L.Z., I.O.M., E.S.G., and V.G.V. conceived the GaNHEMT project at the Kurchatov Institute. I.O.M. and I.S.E. grew the samples supported by M.L.Z. N.K.C. performed Hall characterization. E.S.G. implemented the 2DEG numerical model. I.O.M., I.A.C., and M.L.Z. fabricated the TLM modules and performed transport measurements. V.N.S. wrote the manuscript with contributions of L.L.L., V.G.V., E.S.G., and I.O.M. All authors discussed the results, interpretations, and scientific concepts.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lev, L.L., Maiboroda, I.O., Husanu, MA. et al. kspace imaging of anisotropic 2D electron gas in GaN/GaAlN highelectronmobility transistor heterostructures. Nat Commun 9, 2653 (2018). https://doi.org/10.1038/s4146701804354x
Received:
Accepted:
Published:
Further reading

Extremely lowenergy ARPES of quantum well states in cubicGaN/AlN and GaAs/AlGaAs heterostructures
Scientific Reports (2021)

Mechanochemical reactions of GaNAl2O3 interface at the nanoasperity contact: Roles of crystallographic polarity and ambient humidity
Friction (2021)

Straincontrolled power devices as inspired by human reflex
Nature Communications (2020)

Highaccuracy bulk electronic bandmapping with eliminated diffraction effects using hard Xray photoelectron momentum microscopy
Communications Physics (2019)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.