DAÑOS A LA AERONAVE

The electron wave vector, k, denotes the expected value for an electron wave packet and can be represented as a point in reciprocal space. The boundary conditions imposed by the periodicity of the reciprocal lattice in a crystalline solid require values of k to become quantised, with all unique values located within one reciprocal lattice unit cell [29]. The

x y z Γ A K H M L Figure 2.5

An illustration of the first Brillouin zone in 4H–SiC. Grey spheres denote reciprocal lattice points with nearest neighbours joined by dashed lines. Purple lines represent the Wigner-Seitz cell of the reciprocal lattice, termed the first Brillouin zone. Green spheres represent high symmetry points in the Brillouin zone, whose bounding volume denotes the

reduced zone containing all unique states.

energy associated with each k deviates from that of a free electron and includes large regions of forbidden energy. The highest band of closely spaced levels that is fully occupied at absolute zero temperature is referred to as the first valence band with the next unoccupied band called the first conduction band. The region of energy between the bottom of the first conduction band, Ec,1, and the top of the first valence band, Ev,1is termed the fundamental band gap [29]

E_g = E_c,1− E_v,1. (2.5)

The reciprocal lattice of a hexagonal direct lattice is itself hexagonal∗ _{with corresponding} lattice constants a∗_{= 4π/(}√_{3a)and c}∗ _{= 2π/c. The Wigner-Seitz cell represents a valid unit} cell of a reciprocal lattice and is termed the first Brillouin zone. For a hexagonal reciprocal lattice the Brillouin zone is also hexagonal, as illustrated in Figure 2.5. An important consequence of the symmetry in the Brillouin zone is that wave vectors at equivalent points have identical corresponding electronic energies. Unlike the lowest three valence bands which are located at the Γ–point, the minima of the two lowest conduction bands of interest are located at each of the six equivalent M–points, located at the Brillouin zone boundary [30].

In quantum mechanics a particle is described by its wave function. Solving the Schrödinger equation for a single electron in a periodic potential (a Bloch electron) results in a wave func- tion similar to that of a free electron but multiplied by a function with equal periodicity as the Bravais lattice [16]. It is therefore often justifiable to characterise the behaviour of such electrons using a semiclassical model wherein their masses are modified to account for the interaction with the crystal potential [31]. The absence of an electron may equally well be represented by a quasi-particle called a hole exhibiting a positive charge and an appropriate effective mass [12].

Cyclotron resonance (CR) provides an accurate and direct means of determining the

effective masses of carriers in semiconductors about lowest energy band extrema provided sufficiently high quality samples can be obtained [21]. Volm et al. performed optically de- tected cyclotron resonance (ODCR) measurements∗ _{at 1.6 K and 36 GHz at magnetics fields} up to 4 T to determine the individual components of the electron effective mass tensor at the bottom of the first conduction band of 4H–SiC, obtaining values of 0.33(1), 0.58(1) and 0.31(1) in units of free electron mass in the M–L, M–Γ and M–K directions, respec- tively [21]. Son et al., using ODCR performed at 9.23 GHz and 4.4 K, were able to identify the transverse† _{and longitudinal hole effective mass relative to the principle axis about the} extremum of the first valence band as 0.66(2) and 1.75(2), respectively, in units of free elec- tron mass [32]. The slight underestimates of the effective masses determined via theoretical calculations by groups such as Persson et al. [33, 34] can be attributed to the neglecting of the polaron effect‡ _{as well as the assumptions made in their calculations [21,32].}

For small changes in k around a conduction band minimum the increase in energy is approximately quadratic, as it is for free electron [12]; however, for larger values an electron wave will interact more strongly with the lattice causing the effective mass to increase [9]. A mobile electron in a practical bulk semiconductor in the presence of an electric field will be scattered after only a small change in k, meaning only a narrow range of energy need be considered§ _{being dependent on the maximum temperature and doping concentration of} interest [9].

k.p theory¶ allows approximate analytical expressions for band dispersion around high symmetry points to be defined. From density functional theory (DFT) band theory calcu- lations under the local density approximation (LDA), Wellenhofer and Rössler discovered that the dispersion in the M–K and M–Γ directions for the lowest two conduction bands are essentially parabolic, and that the hyperbolic model given by‖ _[30]

E(kz) = Ec− cz 2 +  c2 z 4 + cz~2kz2 2m∗ zz 1/2 (2.6) could accurately account for the dispersion along the M–L direction up to 300 meV above the first conduction band minimum, wherein Ec is the energy of the respective conduction band minimum and kz, m∗zz and cz are, respectively, the component of k, the effective mass and the first-order k.p dispersion parameter, in the M–L direction in reciprocal space. It will later be shown that this represents the appropriate energy range of interest for high temperature 4H–SiC JFETs. A comparison of this model with the parabolic band model and the theoretically calculated values of Wellenhofer and Rössler using the DFT-LDA method is given in Figure 2.6a.

As the rate of increase of energy becomes progressively smaller than the square of the

∗_{The short carrier scattering times, particularly for holes, for samples produced during this period de-}

manded high frequencies and the photo-neutralisation of ionised impurities provided by ODCR for successful cyclotron resonance CR measurements [32].

†_{The transverse value is an appropriate average of the two basal plane principle tensor components.} ‡_{A polaron describes an electron or hole coupled to an optical phonon in polar semiconductor.} §_{This is not the case when ballistic effects or injection of high energy electrons are present.}

¶_{The k.p relation is an approximate solution of the one electron wave equation for a Block wave [35].} ‖_{A derivation is given by Smith [36].}

−0.2 0 0.2 0.4 0.6 0.8 L M →K E − Ec ,1 / eV k / m−1

(a) Conduction band dispersion

0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 N (E ) / 10 21 eV − 1cm − 3 E − Ec,1/ eV parabolic hyperbolic

(b) Conduction band density of states Figure 2.6

A comparison between (a) the DFT-LDA calculations (points), parabolic dispersion approximation (dashed lines) and hyperbolic k.p dispersion model (solid lines), for the first

two conduction bands in 4H–SiC along M–L and M–K close to the band minima and (b) the combined density of states from each bands determined using the parabolic (dashed

lines) and hyperbolic k.p (solid lines) models. Reprinted [adapted] from [30].

wave number the effective mass increases, resulting in a greater number of k–space states within a given energy interval [35, 37], as illustrated by the density of states (DOS) plot presented in Figure 2.6b. These theoretical results have been confirmed by ballistic-electron emission microscopy (BEEM) experiments [38].

The difference in energy between the first and second set of equivalent conduction bands is small enough to cause appreciable occupation of the second set at modest temperature and thereby influence the electron transport properties. As the DFT-LDA effective masses for the first set agree well with experiment, the values determined for the second set are also likely to be accurate [30, 39]. This information can thus be used within phenomenological models and applied to problems concerned with carrier statistics and transport theory [36]. Spin-orbit interaction∗ _{and the internal crystal field result in three closely spaced two-} fold spin degenerate valence bands each located at the Γ point in the Brillouin zone [38]. The two uppermost (heavy-hole and light-hole) valence band edges are split due to spin-orbit interaction, by an amount Eso = 8.6 meV. The third band is separated by the crystal field by an amount Ecf= 73 meV. While the third band has been found to be essentially parabolic the lowest two interact rather strongly [30]. Appropriate k.p theory models for their dispersions have been reported and represented as corrections to the parabolic DOS [40]; however, while the individual valence bands are not parabolic, the total DOS provided by all three can be closely approximated by three appropriately chosen parabolic valence bands [30].

∗_{The spin-orbit interaction results from the coupling between the magnetic dipole field of a spinning}