Effective masses and electronic structure of diamond including electron correlation effects in first principles calculations using the GW-approximation

Diamond has many properties that make it an advantageous alternative to more conventional group IV, III−V and II−VI semiconductors in many high performance electronic applications. Examples of such attractive material properties include: the highest thermal conductivity, the highest electrical breakdown field and high carrier mobilities. In addition, diamond exhibits unique mechanical, chemical and optical properties, including exceptional hardness, chemical stability and a wide electromagnetic radiation transparency range, ranging from deep UV to the far infrared. Although the development is still at a very early stage, these properties together with recent advances in chemical vapour deposition (CVD) diamond growth has enabled the demonstration of some devices with promising future prospects, e.g. : high voltage¹ and high temperature² diodes, x-ray sensors³ transfer-doped^4,5 and delta-doped⁶ field effect transistors. The utilization of spin states in diamond defects for quantum computing⁷ and magnetic field sensors⁸ are also topics that have recently attracted strong attention.

In comparison with more mature semiconductor materials such as Si or GaAs, the understanding of many electronic properties of diamond is comparatively poor, although considerable experimental and theoretical work has been performed. This lack of knowledge can be attributed to the difficulty in growing high-quality single crystals, the wide energy gap and difficulties in achieving effective n or p doping. Basic parameters, such as the values of carrier effective masses are still uncertain. Conduction band masses, in particular, are difficult to obtain experimentally due to the low activation of n type dopants in diamond. In applications, particular when working with doped diamond, the effective masses are of great interest when estimating mobility and diffusion coefficients. Earlier work⁹ using the linear muffin-tin orbital method (LMTO) in the local density approximation (LDA) with the atomic spheres approximation (ASA) have been done to theoretically estimate the effective masses via k · p-pertubation theory, but there still exist large uncertainties in the results.

In the present work, we use state-of-the-art density functional theory (DFT) methods to accurately calculate the band-structure of pristine diamond. The accurate band-structure enables us to determine the effective hole- and electron conduction-masses along the high symmetry directions in the crystal.

The calculations were performed using state-of-the-art DFT-GGA/LDA, G₀W₀ and self-consistent GW methods. To model the core electrons we use norm-conserving pseudo potentials (nc-PP) as implemented in ABINIT,^10,11 with an energy cutoff of 48 Ha for the plane wave basis set. For details about the GW-implemetation see.^12,13 First, the one-shot approximation G₀W₀ is used where the GW self-energy is approximated as

$Σ≈iGGGAWGGA$⁠, using the GGA-eigenfunctions. To evaluate the screening, W, we use both a plasmon-pole model (PPM)¹⁴ and numerical integration using a contour deformation method.¹⁵ The PPM is computionally cheaper, but is known only to be valid near the Fermi level. Hence the deep bands are not well described by this method but should be better described by the numerical integration. For all the calculations a Monhorst-Pack 8x8x8 k-point grid in the full Brillouin zone (BZ) (512 k-points) including the Γ-point is used for the integrations, which corresponds to 29 k-points in the irreducible BZ when using symmetries, the GW-calculations are performed on all k-points starting from the well-converged GGA-eigenfunctions obtained in the nc-PP formalism. To correct the G₀W₀ scheme another route to follow was proposed by Faleev¹³ where self-consistency is obtained by iteration over the obtained quasiparticle energies and wavefunctions to calculate a new self-energy, until convergence or the desired accuracy is reached (QSGW). The QSGW calculations were performed starting from a ground state calculated in the GGA (nc-PP) formalism using the same parameters as before. To obtain the band structure for the full BZ the quasiparticle wavefunctions are interpolated using maximally-localized Wannier functions.¹⁶ 

The unit cell used in all calculations consists of two atoms, and since the experimental and DFT geometry differs with less than 1% the experimental lattice parameters have been used in the calculations. This had a neglible effect on the band structure in our calculations.

The effective masses for the valence band and conduction band are also calculated, for the valence band we can use the standard expression

while in the case for the conduction band it is necessary to use the expression for a sligthly non-parabolic band

where α is a small parameter (∼0.1).

The band structure of diamond calculated within the G₀W₀-approximation including the full frequency dependence of the screening using numerical integration (NI), is shown in Fig. 1. The band structure from DFT-LDA including spin-orbit (SO)-coupling (PPs parametrized in the HGH²¹-form) is shown as dashed line to illustrate the bandgap opening. Including SO-coupling gives a split-off energy (Δ₀) of 0.014 eV at Γ, but leads to no significant change in the shape of the bands. The resulting bandgap together with some specific transitions in diamond are shown in Table I, where the bandgap from our G₀W₀ calculation is close to the experimental gap (if the experimental gap is “adjusted” by 0.37eV, the zero-point motion obtained from Table II in Ref. 22).²³ The inclusion of the full frequency dependence in the screening does not change the results greatly, as seen in the table, hence the PPM works very well in this case. For computational reasons the PPM was therefore used in the QSGW calculations. Investigating other transitions, for example the direct gap at the Γ-point of roughly 7.3 eV, we observe good agreement between our G₀W₀-results and experiments. The QSGW method yields consistently larger values, about 10% larger than G₀W₀. It seems that the G₀W₀-method reproduces the experimental results better close to the Fermi-level, while the results from the QSGW-method is closer to experimental results for transitions further away from the Fermi-level. In Ref. 24, Hybertsen et al. showed that for the valence bandwidth at least two iterations of the GW-scheme are needed for an accurate result. Here our QSGW gives a valence bandwidth of 23.1 eV, in very good agreement with experimental results.

To further see how the curvature of the bands are affected by including the electron correlations in the GW-approximation we calculate the effective masses, both for conduction electrons at the conduction band minima, located in the BZ at

$2πa(0.76,0,0)$⁠, and holes in different direction at the valence band maximum at the Γ-point. The results for the effective masses are shown in Table II where also the previous LMTO(ASA-LDA)-effective-masses by Ref. 9 are given. The results shows that for the first GW-iteration we see some change in the curvature of the bands, i.e. some effective masses are affected. When going towards self-consistency (QSGW) there are only small changes. The masses calculated via the LMTO-method are consistently larger. For the longitudinal conduction mass the uncertainty is larger due to the strong non-parabolic dispersion along the band at the conduction band minimum. Effective masses are difficult to measure experimentally and therefore the full comparison between our results and experiments is not possible. From experiments values of 1.4m₀ and 0.36m₀²⁵ for m_l respectively m_t have been found. We also expect a ratio of the effective conduction electron masses as

$m_l^*/m_t^*=5.2\pm 0.1$

$ml*/mt*=5.2±0.1$⁠,²⁶ which is in agreement with our results.

We have calculated the bandstructure and effective masses for the diamond crystal using standard DFT-methods as well as different approximations using the GW-formalism. Our results show that the band gap in standard DFT is underestimated and can be corrected via GW. The results further show that already the first iteration of the GW-scheme gives good agreement with experimental results, but when trying to reach self-consistency the agreement can worsen, especially for bands close to the gap.

Our calculated effective masses for both n- and p-type carriers are generally somewhat smaller than previous reported values via the LMTO-method. There are still large uncertainties in the experimentally measured values so the full comparison is not possible.

Carl Tryggers Stiftelse för Vetenskaplig Forskning, the Swedish Energy Agency and the Uppsala University Unimolecular Electronics Center (U3MEC) are acknowledged for financial support. The calculations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at C3SE and NSC.