Structural and transport properties of the NbN/GaN heterostructures:
The structural properties of the MBE-grown NbN/GaN heterojunction with a NbN thickness of 5.5 nm are shown in Fig. 1a-c. A high-resolution scanning transmission electron microscopy (STEM) image of the NbN/GaN interface in Fig. 1a is superimposed with a schematic of SX-ARPES, illustrating how this technique probes the interface electronic structure for NbN/GaN. Fig. 1b shows high-angle annular dark field (HAADF-) and annular bright field (ABF-) STEM image of the GaN/NbN interface in atomic scale. The GaN is metal (Ga-) polar and aligned along the  direction. The NbN  axis aligns with the  crystal direction of GaN. Both the Nb and N sublattices can be resolved in Fig. 1b. The atomic arrangement of the two crystalline layers is indicated by the ball-and-stick models in Fig. 1b-c. The NbN/GaN interface is atomically sharp, with no evidence of disorder, contamination, or intermixing across the interface.
Introducing our electronic structure results, Fig. 1d sketches the k-space configuration as a superposition of the surface Brillouin zones (BZs) of (0001) GaN and (111) cubic NbN. Fig. 1e-f shows the essentials of our SX-ARPES results on the k-resolved band structure of the NbN/GaN heterointerface, represented as the experimental band structure of GaN matched to that of the NbN overlayer. In the GaN band structure, the valence-band maximum (VBM) is at the Γ-point, and the dispersions along both the Γ-K-M and the Γ-M directions are clearly resolved. The dashed lines are the energy bands calculated using density-functional theory (DFT), where the unoccupied conduction bands of GaN, inaccessible to SX-ARPES, are shown as a thicker dashed line. For NbN, the experimental band structure shows bands dispersing upwards from the Γ point and crossing the Fermi level (EF) at about halfway along the Γ-K and Γ-M lines. The theoretical bands are superimposed as the dashed lines. One of the key results of our study is that the valence-band offset (the energy difference between the VBM of GaN and EF of NbN) is determined to be 2.49 eV. With the fundamental GaN band gap of 3.42 eV, this figure implies a conduction-band offset of 0.93 eV at the Γ point of the interface. This is summarized in the density-of-states calculations in Fig. 1g. We emphasize that the VBM of GaN is separated in k-space from the superconducting states of NbN at EF by about a quarter of the BZ. We will expand on the SX-ARPES measurements and its implications for the physics of the NbN/GaN heterojunction later on, after describing our measurements of the structural, transport and optical properties of our NbN/GaN samples.
A series of NbN thin films of differing thickness were grown on GaN to ensure that the NbN would retain its structural and electronic properties when the films were scaled to the thicknesses tailored to the probing depth of the SX-ARPES measurements. The sample series covered the NbN film thicknesses of 1.2, 1.5, 2.0, 2.5, and 10 nm. Reflection high energy electron diffraction (RHEED), X-ray diffraction (XRD), and atomic force microscopy (AFM) were also used to confirm that the NbN films grown on GaN at the thickness necessary for SX-ARPES measurements are epitaxial, with uniform thickness and smooth surfaces. Using XRD we measure the lattice constant of the rocksalt cubic NbN to be a = b = c ~ 4.34 Å, while that of the hexagonal GaN is a = b ~ 3.19 Å and c ~ 5.19 Å (crystal structures are shown in Fig. 1c). As a result, the in-plane Nb-Nb spacing is ~ 3.07 Å along (111) orientation, which results in ~3.8 % lattice misfit at the interface. Photoluminescence measurements (Fig. 2a) of a sample with a 5.5 nm NbN film on GaN indicates that the MBE grown GaN underneath the NbN exhibits a photoluminescence peak at 3.42 eV. This value is in good agreement with previously reported values for GaN, and provides confidence that the electronic properties of the MBE grown GaN film are preserved through the growth of the NbN thin film, a process which occurs at high temperature. Additional details of the heterostructure growth and characterization as well as removal of the in-situ indium cap needed to protect sample quality until SX-ARPES measurement is described in the Methods section.
Electronic transport across the NbN/GaN interfaces was studied by fabricating circular Schottky barrier diode devices with a diameter of 50 μm. A current-voltage (IV) measurement of such a device performed at 300 K is shown in Fig. 2c. The diodes exhibit strong rectification, demonstrating exponential increase in the current of 7 orders of magnitude in forward bias. This behavior is modelled well using thermionic-emission theory,
where I is the current, a is the device area, A** is the Richardson constant for GaN, T is the temperature, q is the elementary charge, k is the Boltzmann constant, Φb is effective barrier height, η is the ideality factor, V is the voltage, and R is a resistance in series with the diode. In this model, Φb, η, and R are used as fitting parameters.
Using the GaN effective mass (m*) in the CBM of 0.222 m0 (m0 is the free-electron mass)30 to calculate the Richardson constant for GaN of 26.64 A K-2 cm-2, the effective NbN/GaN barrier height is calculated from the best-fit thermionic emission model to be 0.7660±0.0002 eV. Using capacitance-voltage measurements of the same NbN/GaN diode in reverse bias, we determine the donor concentration in the GaN is determined to be ~2*1017 cm-3. Using this value for the donor concentration in the GaN, the Schottky barrier height lowering due to electric fields within the GaN is calculated31. Thereby we determine the fundamental Schottky barrier height of the NbN/GaN junction to be 0.812±0.001 eV. We see a 0.12 eV difference in the measured barrier height when comparing the transport with the SX-ARPES (0.93 eV barrier height) which may be traced to several of the simplifying assumptions of the thermionic emission model, such as the assumption of a spatially homogeneous Schottky barrier, which ignores the effects of threading dislocations in the semiconductor, and the assumption of an absence of tunneling current. Effects confined to within a few monolayers of the interface, such as strain in the NbN and GaN, and chemical bonding between the GaN and NbN, may also affect the electronic transport across the interface without being observable in the SX-ARPES measurement of electronic states. and the. which may trace back to the neglect of band structure effects in the thermionic-emission theory.
Resistance vs. temperature measurements of a NbN film with a thickness of 2.9 nm exhibit both a low normal state resistivity of 94 μΩ cm and a superconducting critical temperature of 12.8 K, as shown in Fig. 2c. These values compare favorably to other reports of superconducting and normal metal properties of ultra-thin NbN films, which we ascribe to the high quality of the MBE grown interface and films32.
Electronic structure of the NbN/GaN interface:
Comparing the different samples of the NbN/GaN heterostructure, we find that neither the NbN band structure nor that of GaN shows any changes with the film thickness within the experimental resolution. The highest quality SX-ARPES data on NbN was found for the sample with its largest thickness of 10 nm (Figures 1e-f & 3). The GaN band structure could be seen in the hv range of our experiment through a NbN film thickness of less than 2 nm, with the strongest signal recorded in the 1.2 nm thick NbN film (Figures 1e-f & 4).
Our results for NbN, the first direct measurement of the k-resolved electronic structure of this material, are presented in Fig. 3. The in-plane and out-of-plane Fermi surface (FS) maps (Figs. 3b-c, respectively) show large electron pockets of the NbN conduction bands centered at the Γ- and X-points of the extended BZ. The Γ-centered FS pocket in the out-of-plane map has a six-fold symmetry, consistent with the NbN growth direction of . Importantly, there is no spectral weight in the FS of NbN at the Γ-point where the CBM and VBM of GaN are located. The dispersive FS contours in the out-of-plane map identify the three-dimensional (3D) character of the electron states formed in the 10-nm thick NbN film. The broad spectral width of the experimental bands contributing to the FS (Fig. 3f-g) does not directly allow us to distinguish the number of separate bands, but the band structure calculated from DFT for bulk NbN gives a useful starting point for the interpretation of the SX-ARPES dispersions. An electron-like band centered about Γ with a binding energy (EB) of -1.1 eV is expected to be Nb-3dt2g in orbital character. The octahedral crystal field due to the N-atoms is expected to split the Nb-3d orbitals, pushing the eg bands up towards EF. The bands starting near EB = -3.4 eV at the Γ-point are expected to be N-2p in orbital character. DFT underestimated the energy difference between the N-2p and Nb-3dt2g states by ~1 eV, where it is clear that the measured Nb-3dt2g bands are pushed up relative to the N-2p bands. This disagreement could be due to the approximations made in DFT or due to nitrogen vacancies in the NbN film which are known to reduce EF33. The two dispersion branches along Γ-M are consistent between the experiment and theory. The three branches predicted by DFT along Γ-K are not resolved independently in the experimental data, however, the Fermi momentum (halfway the Γ-K line) is consistent between the experiment and theory. The experimental dispersion range of these bands is significantly smaller than that predicted by DFT, which may indicate a strong renormalization of the Nb-3dt2g bands due to yet unknown many-body effects. The increase in the density of states near EF associated with the enhanced m* may contribute to the robust superconductivity and high critical temperature known for NbN. While a dispersive band of Nb-3d character can be seen clearly at the K-point, low intensity at the X-point prevents a direct comparison with theory (Fig. 3e). DFT predicts that this band changes character from Nb-3d to N-2p, becoming degenerate with the N-2p manifold at the X-point (Fig. 3g). Overall, our experiment confirms the characteristic features of the DFT band structure for bulk NbN.
We now focus on the GaN band structure measured by SX-ARPES in the 1.2 nm sample (Fig. 4). We note that zero EB of the heterostructure is defined by EF of the NbN film. Both in-plane and out-of-plane iso-energy maps taken 2.6 eV below EF (Figs. 4b-c) display the expected hole-like pockets of the Γ-centered valence bands of GaN. The VBM of GaN is 2.49 eV below EF at the Γ point probed at hv = 1064 eV (Figs. 4d-k). Interestingly, although the M-Г-M and M-K-Г images taken at hv = 1064 eV (Figs. 4d-g) and hv = 1216 eV (Figs. 4h-j) correspond to kz values different by the reciprocal lattice vector 2π/c (where c is the  lattice constant of GaN) and therefore equivalent, the photoemission dipole selection rules in the non-symmorphic crystal structure of GaN for these kz are different and light up different sets of bands34. Apart from an energy shift, the measured band structure is identical to that observed in other GaN-based heterostructures12. The calculated bulk band structure of GaN is shown overlaid on the experimental dispersion curves in Fig. 4. The heavy-hole m* is estimated from a fit to the SX-ARPES data to be ~1.82 ± 0.04m0 compared to the DFT value 2.1m0 (the light-hole m* can not be evaluated accurately from our SX-ARPES data because this band is much obscured by the heavy-hole band’s spectral weight at the Г-point and, in addition, its apparent dispersion can be flattened by the intrinsic kz broadening of the ARPES final states35). Overall, the agreement between the theoretical and experimental band structure of GaN can be regarded as excellent. Concurrently with the GaN valence bands, our SX-ARPES data resolves NbN conduction bands in vicinity of EF (dashed white lines in Fig. 4d-o). Their difference from the NbN bands in Fig. 3 indicates the 3D character of the electron states formed in NbN. Indeed, the hv values used in Fig. 4 set kz to the Г-points of GaN but miss the Г-points of NbN because of the different c lattice constants of these materials in our heterostructure. Because of large electron concentration and thus small Thomas-Fermi screening length in NbN, only 4.5 monolayers stacked within the 1.2-nm thickness of the NbN film are already sufficient to form its 3D band structure, with quantum-size effects suppressed by an inhomogeneity of the film thickness.
Finally, in order to visualize the k-dependent band alignment, we bring together the NbN experimental band structure measured at hv = 570 eV (Fig. 3) and the GaN one measured at 1064 eV (Fig. 4), both corresponding to kz tuned to the Г-point. The energy scale of these band structures is matched via the EF position. These plots, displayed in Fig. 1e-f, are the key result of our work. We directly observe that the energy separation of the GaN states from EF, where the superconducting states of NbN are located, dramatically varies across the BZ, and attains its minimum value of 2.49 eV at the VBM of GaN. Importantly, the superconducting states of NbN are separated in k-space from this point as much as about a quarter of the BZ. The consequences of these experimental results for physics of the NbN/GaN heterojunction are discussed below.