Thermal evolution of silicon carbide electronic bands

TL;DR

This study uses ab initio calculations to analyze how the electronic band structure of silicon carbide changes with temperature, providing insights crucial for high-temperature electronics and quantum defect applications.

Contribution

It applies many-body perturbation theory to determine the temperature dependence of SiC bands, a novel approach for this material.

Findings

Significant electron-phonon renormalization of the band gap at 0 K

Different temperature behaviors of conduction and valence bands

Theoretical results align with experimental observations

Abstract

Direct observation of temperature dependence of individual bands of semiconductors for a wide temperature region is not straightforward, in particular. However, this fundamental property is a prerequisite in understanding the electron-phonon coupling of semiconductors. Here we apply \emph{ab initio} many body perturbation theory to the electron-phonon coupling on hexagonal silicon carbide (SiC) crystals and determine the temperature dependence of the bands. We find a significant electron-phonon renormalization of the band gap at 0~K. Both the conduction and valence bands shift at elevated temperatures exhibiting a different behavior. We compare our theoretical results with the observed thermal evolution of SiC band edges, and discuss our findings in the light of high temperature SiC electronics and defect qubits operation.

Figures

Tables

Table 1. Table SI: 4 ​ H 4 𝐻 4H - and 6 ​ H 6 𝐻 6H -SiC optic (Opt.) and low energy acoustic (Ac.) phonon frequencies at Γ Γ \Gamma point (in cm -1 ) either in the basal plane (B.) or along c-axis (Ax.) and their corresponding irreducible representation (Irrep). B 1 subscript 𝐵 1 B_{1} modes are Raman and infrared (IR) forbidden. There are modes for both compounds that were not observed (N.O.).

Irrep	DFPT-LDA	Exp.111Refs. Feldman et al.,1968a; Nakashima and Harima,1997	Exp.222Ref. Bluet et al.,1999	Branch
	freq.	Raman	IR reflectivity
$E_2$	192.3	196.0	-	B.Ac.
$E_2$	200.3	204.0	-	B.Ac.
$E_1$	259.2	266.0		B.Ac.
$B_1$	401.1	-	-	Ax.Ac.
$B_1$	409.2	-	-	Ax.Ac.
$A_1$	608	610.0	610.5	Ax.Ac.
$E_1$	763.9	N.O.		B.Opt.
$E_2$	771.4	N.O.	-	B.Opt.
$E_2$	779.7	776.0	-	B.Opt.
$E_1$	790.2	797		B.Opt.
$A_1$	838.7	838	839	Ax.Opt.
$B_1$	910.6	-	-	Ax.Opt.
$B_1$	920.1	-	-	Ax.Opt.
$A_1$	958.8	964		Ax.Opt.

Table 2. Table SII: 4 ​ H 4 𝐻 4H SiC effective masses of electrons in the conduction band minima in units of free electron mass m 0 subscript 𝑚 0 m_{0} . Experimental data (exp.) are taken from Ref.  Son  et al. , 1995 .

Effective mass	direction	$M_1$	$M_1$ (exp.)	$M_2$
$m_{\perp }$	$m_{\Gamma M}$	0.48	0.42	0.67
$m_{\left|\right|}$	$m_{LM}$	0.27	0.29	0.61
	$m_{KM}$	0.28		0.16

Equations

$$k_B{T}_d={ϵ}_d$$

$${\sigma }_{imp}\sim (\sqrt{N_c(T)\frac{N_d}{2}}{e}^{-{ϵ}_d/2k_{B}T})\cdot e\cdot \frac{e}{m_c^{\ast }}{T}^{3/2},$$

$$\sigma \sim N_{d}e\frac{e}{m_c^{\ast }}{T}^{3/2}.$$

Full text

Supplemental Materials

for the paper entitled

Thermal evolution of silicon carbide electronic bands

E. Cannuccia

Aix-Marseille Université , Laboratoire de Physique des Interactions Ioniques et Moléculaires (PIIM), UMR CNRS 7345, F-13397 Marseille, France

Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1,I–00133 Roma, Italy

  

A. Gali

Wigner Research Centre for Physics, PO. Box 49, H-1525, Budapest, Hungary

Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki út 8., H-1111 Budapest, Hungary

I Description of the hexagonal silicon carbide crystals

Silicon carbide (SiC) has over 250 polytypes which share the same hexogonal lattice in the basal plane and different stacking sequences of Si-C bilayers perpendicular to the basal plane. The most important polytypes are the so-called $4H$ and $6H$ SiC as shown in Fig. S1(a) and Fig.1(a) in the main text. They consist of 4 and 6 Si-C double layers having a close-packed hexagonal arrangement. There are three types of hexagonal close packing (A, B and C) in arranging the Si-C double atomic layers. In $4H$ and $6H$ the stacking sequences are ABCB$|$ABCB$|\dots$ and ABCACB$|$ABCACB$|\dots$, respectively. Each atom (Si or C) has a local quasicubic ($k$) or hexagonal ($h$) environment with respect to the immediate neighbors. The experimental lattice constants are a = 3.08 Å  with c = 10.08 Å  Stockmeier et al. (2009) and a = 3.08 Å  with c = 15.12 Å  Stockmeier et al. (2009) for $4H$ and $6H$ SiC, respectively, and the ideal values for the internal parameters follow from the space group symmetry $C^{4}_{6v}$Käckell et al. (1994).

II Electronic structure of hexagonal silicon carbide crystals

We applied density functional theory (DFT) within local density approximation (LDA) as reported in the main text. The calculation of the phonon dispersion curves required the optimization of the lattice constants. This results into a = 3.07 Å  with c = 10.0 Å  Stockmeier et al. (2009) and a = 3.07 Å  with c = 15.0 Å  for $4H$ and $6H$ SiC, respectively, that are close to the experimental values. The optimized internal parameters remain close to the ideal ones Käckell et al. (1994).

$4H$ and $6H$ SiC are indirect semiconductors with relatively large band gaps at room temperature, $3.27$ and $3.02$ eV, respectively Hellwege et al. (1982). The calculated electronic band structures are shown in Fig. S1(c) and (d) versus high symmetry lines $A$-$\Gamma$-$M$-$L$-$A$ within the Brillouin-zone (BZ) of the hexagonal system (see Fig.S1(b)). The overall features show the agreement with previous calculations Käckell et al. (1994); Park et al. (1994); Wenzien et al. (1995). The top of the occupied valence-band states is located at $\Gamma$ in both polytypes. As reported by Käckell et al. (1994); Lambrecht et al. (2001) the exact position of conduction band minimum depends on the details of the calculation, on the ratio $c/a$ of the hexagonal lattice constants, as well as on the atomic positions inside the hexagonal unit cell. We find the conduction band minimum at $M$ for $4H$ SiC and at about $0.5$ $M-L$ for $6H$ SiC. However, the lowest conduction band is rather flat between $L$ and $M$ in $6H$ SiC case. For this reason, we assume that the bottom of the conduction band occurs at the $M$ point of the BZ, yielding just a minor error in the calculated LDA fundamental gap for $6H$ SiC. Moreover, we observe that at the CBM two bands are very close in energy. We will label them in the follow as $M_{1}$ and $M_{2}$.

At DFT-LDA level we obtain 2.25 eV for $4H$ SiC and 2.04 eV for $6H$ SiC, about 1.0 eV lower than the corresponding measured values, and in agreement with other theoretical works Wenzien et al. (1995); Käckell et al. (1994). The discrepancy in the absolute value of the indirect band gaps comes from the well-known self-interaction error of the LDA but the curvature of the individual bands are well reproduced by this approximation. As a consequence, the effective masses of the electrons, i.e., the inverse of the second derivatives of the energies around the $M$ point for the conduction band minimum, can be accurately calculated by DFT-LDA.

III Quasiparticle correction in the electronic structure of hexagonal silicon carbides

Assuming DFT-LDA orbitals as a good approximation to the quasi-particle (QP) orbitals, the first-order correction to the Kohn-Sham eigenenergies is obtained within a single iteration of Hedin’s GW approximation Hedin (1965); Hedin and Lindqvist (1969). For $6H$ SiC, the exact position of the conduction band minimum (CBM) on the $L-M$ line has been under discussion Park et al. (1994); Käckell et al. (1994); Lambrecht et al. (2001). Wenzien et al. Wenzien et al. (1995) found a tendency of CBM of $6H$-SiC to move towards $M$ point after the inclusion of QP effects. They used a simplified GW method, meaning a model dielectric function and an approximate treatment of the local filed effects. Ummels et al. Ummels et al. (1998) performed GW calculations for the $2H$, $4H$, and $6H$ SiC polytypes using the Engel and FaridEngel and Farid (1993) plasmon pole model for the description of the energy dependence of the screened interaction. They obtained G0W0 indirect band gaps of $3.35$ eV and $3.24$ eV for $4H$ and $6H$ SiC, respectively (see Table I in the main manuscript). Our correction obtained with Godby-Needs plasmon-pole model Godby and Needs (1989); Hedin (1965); Hedin and Lindqvist (1969); Bockstedte et al. (2010) as implemented in Yambo Sangalli et al. (2019) code, results to be lower with respect to those calculated by Ummels et al.Ummels et al. (1998). This is due to the different plasmon pole model (PPM) used. Godby-Needs PPM best reproduces the model-free contour deformation (CD) resultsFriedrich et al. (2011), when other PP models are applied the gap obtained is significantly higher and in particular Engel and Farid PPM gives for ZnO a GW correction $0.3$-$0.4$ eV larger than the CD one Stankovski et al. (2011).

In our study, we have to rely on the DFT-LDA geometries as we have to calculate phonon bands which assumes to stay close to the global energy minimum of the adiabatic potential energy surface. Our GW calculations justify to apply a simple scissor correction on the DFT-LDA band gaps, in order to reproduce the experimental ones.

IV Phonons of hexagonal silicon carbides

In view of the calculation of the temperature dependent renormalization of the electronic gap, the phonon dispersion curve is an essential input for phonon-related properties. Besides the fact that the phonon dispersion curve reflects the structure of a material and its symmetries, phonon spectra are one of the key ingredients that help in understanding the coupling between electrons and lattice vibrations. Symmetry analysis shows that phonon modes can be decomposed into $N(A_{1}\oplus B_{1}\oplus E_{1}\oplus E_{2})$ modes, where $N=4,6$ for $4H$ SiC and $6H$ SiC, respectively. Whose modes $A_{1}\oplus E_{1}$ are acoustic and $(N-1)A_{1}\oplus NB_{1}\oplus(N-1)E_{1}\oplus NE_{2}$ are optic modes. Except for $B_{1}$ modes which are infrared and Raman forbidden, all the other are Raman active modes. Out of them $A_{1}$ and $E_{1}$ modes are also infrared active.

The calculated phonon dispersion curves are shown in Fig. S2. Table SI reports the theoretical and experimental phonon frequencies at $\Gamma$ point for $4H$ and $6H$ SiC. All the available experimental modes were readily assigned. The phonon modes with longer wavelength are divided into axial or basal plane modes, depending if the atoms displace parallel or perpendicular to the $c$-axis, as explained in Ref. Patrick (1968). The planar modes consist of Raman and IR active $E_{1}$ modes and Raman active $E_{2}$ modes. The axial modes consist of Raman active $A_{1}$ and Raman inactive $B_{1}$ modes. Our vibrational frequencies at $\Gamma$ point are in excellent agreement with previously published calculations and with experimental phonon dispersion curves obtained by infrared and Raman spectroscopies Feldman et al. (1968a, b); Nienhaus et al. (1995); Nakashima and Harima (1997); Bluet et al. (1999), being the average error smaller than 4 cm*-1*. This supports our choice of using DFT-LDA eigenergies and DFPT-LDA phonon frequencies as key ingredients for the calculation of the electron-phonon interaction.

V Additional data on the calculated electron-phonon interaction

This section provides details about the convergence of the temperature dependent electron-phonon coupling of the valence bands as well as the the calculated temperature shifts for the two lowest energy conduction bands around $M$ point.

We find that the calculated temperature shifts of the valence bands depends on the number of conduction bands used in the calculation of electron-phonon coupling whereas the temperature shifts of the conduction bands converge much faster with the number of conduction bands as shown in Fig. S3(c). As can be inferred in Fig. S3(a), the calculated temperature shifts in the valence band and the indirect band gap are absolutely convergent by using 100 bands in the calculation of the electron-phonon coupling for $4H$ SiC, as the 90 bands and 100 bands data are practically identical. This corresponds to about $6\times$ of the number of valence bands ($16$ bands without core electrons). As a consequence, the convergent number of bands should be around 150 for $6H$ SiC that has $24$ valence bands. However, it is technically prohibitive to carry out such calculations for $6H$ SiC. Thus, we carried out electron-phonon coupling calculations with non-converged 120 bands, which is only $5\times$ of the number of valence bands. We estimated the converged data by taking the difference in the calculated 80 bands data ($5\times$ of the number of valence bands in $4H$ SiC) and the convergent 100 bands data of $4H$ SiC. Then we extrapolated the convergent 150 bands numerical data by these differences at the given temperature in $6H$ SiC obtaining the thermal evolution of the indirect band gap and of VBM and CBM as shown in Fig. S3(b)-(d).

Next we turn to the calculated temperature shifts for the two lowest energy conduction bands around $M$ point. We found an energy separation of $89$ meV at DFT level between the two lowest conduction bands of $4H$-SiC, as reported in Table I of the main manuscript. QP corrections shift down the second conduction band with respect to the first one of $3$ meV. The final inclusion of the electron-phonon interaction leads to a positive zero point energy correction of $2$ meV and $6$ meV at $800$ K. For $6H$ SiC the energy separation is $170$ meV at DFT level, it increases up to $180$ meV by the inclusion of QP effects, and then it gets to $183$ meV with zero point energy correction.

VI Additional data about conductivity of electrons in $4H$ SiC

In extrinsic semiconductors, containing donor impurities, the presence of donor levels shifts the Fermi level from the middle of the energy gap toward the edge of the conduction band. Let us define the ionization temperature of the donor levels $T_{d}$ as

[TABLE]

where $\epsilon_{d}$ is the binding energy of the donor levels, $k_{B}$ is the Boltzmann constant and the density $N_{d}$ of donor impurities is supposed to be uniform in the sample. Within the perspective of employing $4H$ SiC in devices operating at temperature of around 800 K, we expect that all the donor levels at $120$ meV below the bottom of the conduction band, are occupied and the chemical potential must be located in the energy range $E_{d}<\mu(T)<E_{c}$, where $E_{d}$ is the donor level and $E_{c}$ is the lowest conduction band position. In the low temperature regime, $T\ll T_{d}$ the temperature dependence of carrier conductivity is given by Refs. Grosso and Pastori Parravicini, 2000; Yepifanov, 1974

[TABLE]

being inversely proportional to the effective mass of the electron, which is the inverse of the second derivative of the conduction band along a given direction in the Brillouin-zone. The effective masses of the second conduction band ($M_{2}$) is generally greater (except along $K-M$ line) than of the first conduction band ($M_{1}$) (see Table SII). $N_{d}$ is the donor impurities density $\sim 10^{18}-10^{19}\frac{\#donors}{cm^{3}}$ and electronic charge $e=1.6\times 10^{-19}C$. The carrier conductivity as a function of temperature for binding energy donor levels equal to 120 meV in $4H$ SiC is represented in Fig. S5.

On the other hand for binding energy donor levels equal to 60 meV the carrier conductivity as a function of temperature is given by Refs. Grosso and Pastori Parravicini, 2000; Yepifanov, 1974

[TABLE]