Spin polarisation and non-isotropic effective mass in the conduction band of GdN

Density functional theory (DFT) based calculations were undertaken using Quantum Espresso Giannozzi et al. (2009); Cococcioni and de Gironcoli (2005) and rare earth pseudo-potentials developed using the rare-earth nitrides Topsakal and Wentzcovitch (2014). Self-consistent calculations on the primitive cell were completed using a $k$-mesh with $10\times 10\times 10$ divisions, while super-cell calculations were on a $4\times 4\times 4$ division k-mesh. The wave function and charge density cut-off energies were 50 Ry and 200 Ry respectively for all calculations. Following our DFT calculations the output from Quantum Espresso was used to generate maximally localised Wannier functions using Wannier 90 Mostofi et al. (2014); Marzari and Vanderbilt (1997); Souza et al. (2001). The resulting Wannier functions were then used to calculate the electronic properties on denser k meshes, a $25\times 25\times 25$ grid for calculation of the density of states and Fermi level) and a $125\times 125\times 125$ grid for the optical conductivity. The effective mass was determined using conventional DFT based on 400 k along each high symmetry direction.

The 4$f$ electrons of the LnN series are strongly correlated and thus require careful treatment beyond the traditional DFT methods Larson and Lambrecht (2006); Larson et al. (2007). In the basic DFT the 4$f$ states are found at or near the Fermi level for most of the stoichiometric rare earth nitrides, when in reality this is not the case. The strongly correlated nature of these electrons pushes the filled states below and unfilled states above the Fermi level. This physics can be approximated using the DFT+$U$ method where the behaviour of the correlated orbitals is determined by an adjustable parameter $U$. In the present study two $U$ parameters are used, as described first in reference Larson and Lambrecht (2006). One to account for the strongly correlated 4$f$ states ($U_{f}$), and a second applied to the 5$d$ states ($U_{d}$) to adjust to the experimental optical band gap in the ferromagnetic phase. The process is discussed further in references Holmes-Hewett (2021) and Holmes-Hewett et al. (2023).

GdN thin films on the order of 100 nm were prepared in a Thermionics ultrahigh vacuum chamber with a base pressure of $10^{-9}$ mbar. Gd metal was evaporated by an electron gun in a N₂ partial pressure of $1\times 10^{-4}$ mbar at a rate of $\sim 1$ Å s^-1. Varying the N₂ growth pressure acts to control the concentration of nitrogen vacancies (V_N) in the films. The films were capped with $\sim 50$ nm insulating AlN to minimise deterioration when exposed atmospheric water vapour and oxygen. The films were simultaneously deposited at ambient temperature on Si and Al₂O₃ substrates to suit the required measurements.

Samples for electrical transport measurements were deposited on $10~{}\times~{}10$ mm² Al₂O₃ substrates predeposited with Cr/Au contacts in the van der Pauw configuration. Electrical measurements were conducted in a Janis closed-cycle helium cryostat and a Quantum Design Physical Properties System at temperatures from 300 to 4 K. Reflection and transmission measurements were performed with a Bruker Vertex 80v Fourier transform spectrometer in the 0.012-3 eV spectral range using films deposited on both Si and Al₂O₃ substrates. The reflectivity was measured relative to a 200 nm Al film and then corrected for the finite reflectivity of Al using data from Ehrenreich Ehrenreich et al. (1963). The reflectivity and transmission spectra were simultaneously modelled with refFIT software Kuzmenko (2005) using a Kramers-Kronig consistent consistent sum of Drude-Lorentzians to calculate the energy-dependent dielectric function from which the presented optical conductivity data was generated.

As is commonly reported, GdN is grown with varying nitrogen vacancy doping to control the carrier concentration. In the doped crystal the Gd ions in the lattice retain their 3+ oxidation state, so to maintain charge neutrality the three electrons formerly on the nitrogen ion must be accounted for. Two electrons occupy defect states which localise to the vacancy site, while the third electron finds a higher energy defect state, drawing the Fermi level up into the extended state Gd 5d conduction band. This one mobile electron per nitrogen vacancy is responsible for electron transport. The Fermi surface is then a prolate spheroid with its centre at the X point, major axis pointing along X$\Gamma$ and minor axes in the XW and XU plane. The first Brillouin zone with the Fermi surface highlighted is shown in the inset to Figure 2. It is then the dispersion along these three high symmetry directions which will dictate the transport properties of the material.

To represent the above description Figure 2 shows the calculated band structure of nitrogen vacancy doped GdN along these high symmetry directions. It is clear that the curvature of the bands contrasts significantly, resulting in a differing effective mass, and ultimately transport properties along different high symmetry directions. The figure shows the calculated wave-functions projected onto the atomic orbitals (colours noted in caption), in addition we have calculated the dispersion for the stoichiometric Gd 5d conduction band along the same high symmetry directions, shown for the majority spin as black dashed lines in the left hand panel of Figure 2. It is significant that the stoichiometric and nitrogen vacancy doped conduction bands are essentially superimposed on one another, fitting the picture that the electrons donated by nitrogen vacancy doping are simply lifting the Fermi level into the intrinsic conduction band. Although the shape of the bands is retained between the stoichiometric and doped calculations the gap between the Gd 5d and N 2p states at the X point has reduced from $\sim$ 0.9 eV in the stoichiometric case to $\sim$ 0.6 eV in the doped case, the renormalzation likely a result of screening from the delocalised electrons. The onset of optical absorption is still near $\sim$ 0.9 eV, however now between $\Gamma$ and X, and expected to be much weaker as the initial states are the occupied defect states, here with a concentration of only 1/27 to that of the N 2p. The intrinsic majority spin optical band gap (i.e. not involving defect states) increases to $\sim$ 1.2 eV ($\sim$ 0.9 eV in the stoichiometric crystal) in a Moss-Berstein like shift.

The minority spin states are shown in the right hand panel of Figure 2. The Gd 5d states are again very similar to the stoichiometric crystal. However, the presence of the occupied defect states in the gap, below the Fermi level, effectively reduces minority spin optical gap from 2.5 eV in the stoichiometric to 1.8 eV in the doped crystal. The average absorption onset of the majority and minority spin gaps in the doped crystal (both with initial impurity states and Gd 5d final states) is now $\sim$ 1.35 eV (1.75 eV in the stoichiometric crystal). This is now less than is commonly reported for the paramagnetic band gap.

Using the present results from the doped crystal the spin splitting can be reported in various ways. The most appropriate for electrical transport (i.e. as in a spin filter) is the minimum energy difference between available (hole or electron) majority and minority spin states. As Fermi level lies in the majority spin Gd 5d band, for electron carriers this is the energy difference between the Fermi level and the bottom of the minority spin conduction band, roughly 1.1 eV, while for hole carriers this is the energy difference between the Fermi level and the top of the minority spin valance band, roughly 0.6 eV. For optical studies the most relevant spin splitting is the difference between the majority spin and minority spin optical gaps, which now $\sim$0.9 eV. This is closer to the experimental reports, yet still beyond the maximum 0.6 eV reported in reference Vilela et al. (2024).

As an initial comparison of our computational results to experiment we have calculated the optical conductivity for both the pristine and doped cells discussed above. These are displayed in Figure 3. The top panel of Figure 3 shows the calculated optical conductivity for stoichiometric GdN. The onset of optical absorption is in the majority spin states at $\sim$ 0.9 eV and initially shows an increase with a near $A+(\epsilon-E_{g})^{1/2}$ form, which is expected from the Van Hove singularity at the conduction band minimum. Above 2 eV the gradient increases relating to the flattening of both the 5d and the N 2p bands near +2 eV and -2 eV respectively. At roughly 2.5 eV there is a similar onset of absorption in the minority spin states. The inset show the spectra between 0.8 eV and 3 eV now with experimental data (blue) measured at 8 K. Although the onset of absorption is a roughly the same energy in both spectra the form is very different, with the experiment not taking the typical shape expected for the onset of optical transitions in a parabolic band.

The lower panel of Figure 3 shows the calculated optical conductivity for the nitrogen vacancy doped super-cell. Here, as discussed in section III.1, the absorption onset is at roughly the same energy (0.9 eV) as the pristine cell. For the doped material the form of the optical conductivity differs greatly from the simple $\epsilon^{1/2}$ shape as the Fermi level now lies within the conduction band. The inset again shows the comparison between the calculation and experiment with a clear qualitative similarity between the two (the calculation has been scaled by a factor of 0.2 to best match the experiment). The film used for the measurement was moderately conductive with a resistivity of 0.1 $\Omega$cm at room temperature, increasing to 10 $\Omega$cm at 4 K. Given this resistivity, and based on Hall measurements on other films Lee et al. (2015), one could assume a carrier concentration on the order 10 ¹⁷cm^-3 at 4 K, however the striking resemblance between the optical spectra for this film and the super-cell calculation (carrier concentration $\sim 1~{}\times 10^{21}$cm^-3) may rather imply that in this is incorrect. This film and possibly other nominally undoped rare-earth nitrides may harbour a significant population of localised carriers. These do not contribute to electron transport in the usual extended state manner; however, they do occupy states at the bottom of the conduction band, affecting the optical spectra and other related electrical properties. Similar optical spectra have been seen in other rare-earth nitrides Holmes-Hewett et al. (2019, 2020, 2023) and along with electrical transport measurements were discussed in the context of localised hopping type conductivity and a mobility edge scenario.

Although the details of the spin-splitting and related exchange energy remain to be determined the present data, calculations and existing experimental reports Trodahl et al. (2017) seem to agree that when GdN is doped with electrons via nitrogen vacancies the Fermi level is raised into the Gd 5d band, which is majority spin only at dilute to moderate doping levels. Motivated by these observations, and the similarity between the stoichiometric and nitrogen vacancy doped Gd 5d bands we have used the former to calculate the effective mass along the X$\Gamma$, XW and XU directions, which may be of use in the context of device design. The results are shown in Figure 4 with the dispersion along the X$\Gamma$, XW and XU directions in the top panel, and the effective mass in the bottom panel, both as a function of wave-vector measured relative to the X point at $k=0$.

At the X point the first Brillouin zone of the FCC lattice has four-fold symmetry (see inset of Figure 2), which dictates that along any two directions in the plane of the X-face the effective mass is degenerate at $k=$X. Without loss of generality we choose directions along XU and XW to form our basis, the final perpendicular direction is then along X$\Gamma$ and will have a distinct effective mass at the X point. The $k=$X effective mass tensor is diagonal in this coordinate system and given by

The enhanced curvature along the XU and XW directions results in an order of magnitude effective mass difference between the longitudinal and transverse directions of the Fermi surface. Using these values we can estimate the conductivity effective mass as $m^{*}_{c}=0.25~{}m_{e}$, which is dominated by the low mass components. Given the relative orientation between the equivalent X points in the first Brillouin zone an electrical field along any direction will result in contributions from both the transverse the longitudinal directions. The order of magnitude lower mass in the transverse direction will result in this dominating in any electrical transport measurement.

The effective mass precisely at the X point is useful in the context of tunnelling experiments where the GdN is likely intended to be insulating and close to stoichiometric, in addition the geometry of these experiments is well controlled and for example the common case of NbN/GdN/NbN multi-layers Senapati et al. (2011) results in transport through the GdN along a 001 direction. As discussed above the degeneracy of the X point results in contributions from both the transverse and longitudinal directions; at the X point the combination of these contributions can in principle be easily treated.

As one dopes the crystal with electrons the Fermi surface expands from the X point and the effective mass is a function of wavevector as shown in Figure 4. Along the X$\Gamma$ direction the effective mass is relatively stable with a slight drop before an increase near $k=\pi/a$. As one continues along this direction there is ultimately a sign inversion near the first Brillouin zone boundary as the curvature of the band itself changes sign.

Due to the rapidly changing mass along the two transverse directions it may be difficult to predict the transport properties of even moderately electron doped GdN. The super-cell used in the calculation of Figure 2 was constructed from $3\times 3\times 3$ primitive unit cells with a N atom removed. The single mobile electron in a volume of $\sim 915\mathrm{\mathring{A}}^{3}$ results in a carrier concentration of $\sim$1$\times 10^{21}$cm^-3 and is representative of the heavily doped GdN from experimental reports Maity et al. (2018). The presence of this electron raises the Fermi level roughly 0.6 eV into the conduction band, the resulting Fermi wave vectors are indicated on Figure 4. Far from the X point the effective mass tensor will no longer take the simple diagonal form presented above, but even so the wave vector dependent effective mass shown in Figure 4 gives a qualitative indication of the evolution of the effective mass with electron doping in GdN.