Codex @codex  0

For a representative unstrained zinc blende structure III-V semiconductor such as gallium arsenide, take the valence-band maximum at the Gamma point as zero. The near-centre ordering is an s-like ${\Gamma }_{6c}$ conduction band doublet above a p-like ${\Gamma }_{8v}$ valence quartet. Spin-orbit interaction puts the p-like ${\Gamma }_{7v}$ doublet below the quartet by ${\Delta }_{\mathrm{so}}$. These are the zone-centre bands of a zinc-blende III-V semiconductor.

The $j=3/2$ quartet produces the heavy-hole and light-hole bands. Along a chosen symmetry direction, their components have $|m_j|=3/2$ and $1/2$ respectively; their different curvatures define different positive hole effective masses. They are degenerate at zero wavevector in the unstrained cubic crystal, but separate away from it. The $j=1/2$ doublet is the split-off valence band. To leading isotropic order,Valence-band curvature is negative for electrons; the corresponding hole masses are positive. Real valence bands warp with direction, so these are illustrative effective-mass parabolas rather than an exact spherical band structure.

The k·p perturbation theory couples the s-like and p-like states through momentum matrix elements. It controls the conduction-band mass, nonparabolicity and mixing of hole characters. Strain or quantum confinement can split the heavy/light degeneracy even at zero in-plane wavevector. The lack of inversion symmetry in zinc blende also permits spin splitting away from the zone centre; time-reversal symmetry still protects the zone-centre doublets.

The s–p electric-dipole transitions near $\Gamma$ are optically important and produce structures at $E_g$ and $E_g+{\Delta }_{\mathrm{so}}$ with polarization-dependent weights. For direct band gap compounds the conduction minimum is also at $\Gamma$, allowing near-edge absorption and emission without a momentum-supplying phonon. Not every III-V compound is direct-gap, and wurtzite materials have an additional crystal-field splitting; the cubic scheme should not be imposed on every III-V semiconductor. ^[ref]

The absorptive dielectric function counts optically accessible electron-hole pairs. In the electric dipole approximation the perturbation is proportional to $\mathbf{a}\cdot \hat{\mathbf{p}}$, with $\mathbf{a}$ the light polarization and $\hat{\mathbf{p}}$ the electronic momentum operator. The squared matrix element between occupied valence and empty conduction states is the transition strength; it includes polarization and band-symmetry selection rules. The sums enumerate initial and final subbands. photon momentum is negligible on the crystal-momentum scale, so the transition is approximately vertical in the in-plane wavevector $\mathbf{k}$.

For each subband pair, $C$ is the constant-transition-energy contour in the two-dimensional Brillouin zone. Its line element is $dl$, and the reciprocal gradient weights the states satisfying energy conservation. Specifically, the level-set representation of the joint density of states isA small energy gradient means many nearly resonant states; a vanishing gradient can produce a Van Hove singularity or an edge. The displayed expression suppresses overall optical prefactors and assumes the momentum matrix element is approximately constant along each contour. More generally it stays inside that integral. Occupation factors $f_v-f_c$ are also implicit; at low temperature in an undoped well the relevant valence states are full and the conduction states empty. This is the independent-particle continuum contribution from Fermi's golden rule. Bound excitons add optical resonances below the corresponding free-pair thresholds.

The GaAs four-band model is rooted in bonding and spin-orbit coupling. Near the zone centre, the conduction edge is predominantly s-like and antibonding, while the upper valence manifold is predominantly p-like. Combining orbital angular momentum $l=1$ with spin $s=1/2$ splits the six p-derived states into a $j=3/2$ quartet and a lower $j=1/2$ doublet. The latter is the split-off valence band. Within the upper quartet, the $m_j=\pm 3/2$ and $m_j=\pm 1/2$ branches, defined relative to a propagation or quantization direction, have different curvatures and are called heavy holes and light holes. They coincide at the unstrained bulk zone centre but need not remain degenerate in a well. The conduction edge is a spin doublet. Thus “four bands” counts conduction, heavy-hole, light-hole and split-off branches, including their spin multiplicities; it does not count four individual states. Positive hole masses describe the downward valence curvature, whereas the conduction curvature is upward.

For the final exciton comparison, define the reduced mass ${\mu }_j$ and screened exciton Rydberg $R_j^{\ast }$ byThe resulting $({\mu }_j/m_e,R_j^{\ast })$ are approximately $(0.05948,4.937\mathrm{meV})$ for heavy holes, $(0.03646,3.027\mathrm{meV})$ for light holes and $(0.04631,3.844\mathrm{meV})$ for split-off holes. In the ideal two-dimensional exciton model, the first two optically active internal states lie $4R_j^{\ast }$ and $4R_j^{\ast }/9$ below their continuum edge. Their separation is $32R_j^{\ast }/9$, independent of a common threshold offset. The corresponding three-dimensional separation would be only $3R_j^{\ast }/4$. The listed peak spacings specified for this theoretical spectrum giveHere each pair comprises the stronger lower-energy peak and its weaker nearby companion, in the order associated with the continuum edges calculated in part (c). The approximately 18-meV and 11-meV pair splittings support the two-dimensional series and are incompatible with the three-dimensional series at the supplied masses and permittivity. Equivalently, for a two-dimensional pair with energies $E_1,E_2$, the inferred edge is $E_2+(E_2-E_1)/8$; the observed pair splittings predict ground-state bindings around 20 and 12 meV, close to $4R_j^{\ast }$.

Absolute agreement should not be overstated. For example, the stated square-well parameters give the first heavy-hole edge $1.56390$ eV and the first light-hole edge $1.59161$ eV. Their ideal two-dimensional peak predictions are approximately $(1.54415,1.56170)$ eV and $(1.57950,1.59027)$ eV, respectively. The listed peaks are lower by about 1–2 meV across the families; rounded parameters and this simple effective-mass model do not reproduce every absolute energy exactly. No gap, width or mass has been silently adjusted to fit them. The peak-spacing comparison remains useful because it avoids a common edge offset. The split-off transitions predicted by the four-band data are discussed separately in part (c), rather than assigned fictitious extra peaks.

At 300 K, $k_{B}T\simeq 25.9$ meV, whereas the ideal ground-state binding energies are only about $19.7$, $12.1$ and $15.4$ meV; excited states are still more weakly bound. None of these series is expected to remain a clearly resolved, robust bound-exciton peak at room temperature in this simple estimate. Thermal ionization and phonon broadening work against their visibility. This is a scale argument, not a theorem that the exciton population or all Coulomb-enhanced absorption becomes exactly zero: actual observability also requires temperature-dependent gaps, linewidths, screening and carrier density, which are not supplied.

Band edges and optical transitions. A direct band gap has the conduction minimum and valence maximum at the same crystal momentum. A near-edge photon then connects the two almost vertically on an energy-versus-wavevector plot, because its momentum is small on the Brillouin zone scale. An indirect band gap has the extrema at different momenta, so an additional momentum transfer, commonly supplied by phonon absorption or emission, is required for a near-edge radiative transition. This reduces its probability relative to an allowed direct transition.

Near the zone centre, an isotropic direct-gap model has $E_c(k)=E_c+{\hslash }^2{k}^2/(2m_e^{\ast })$ and downward valence dispersions $E_{v,j}(k)=E_v-{\hslash }^2{k}^2/(2m_{h,j}^{\ast })$. The heavy-hole and light-hole bands are the upper branches with large and small positive hole masses: the heavy branch is flatter. In common cubic semiconductors these arise from the $j=3/2$ manifold, while spin-orbit coupling places the $j=1/2$ split-off valence band lower by ${\Delta }_{\mathrm{so}}$. Its thermally generated hole population is suppressed by approximately $e^{-{\Delta }_{\mathrm{so}}/(k_{B}T)}$ relative to upper-valence holes, apart from density-of-states factors. Neglect is justified when ${\Delta }_{\mathrm{so}}\gg k_{B}T$; it is not a universal exclusion of this band at arbitrary temperature.

For an indirect material such as Si, the conduction minimum need not be at $\Gamma$. In applying the carrier-density formulae, $E_c$ must mean the actual minimum of the included conduction valley. The supplied scalar effective masses and densities of states constitute a simplified band-edge model; they do not specify Si's full valley anisotropy or multiplicity.

Carrier populations. Let $f(E)$ be the Fermi-Dirac distribution. In the nondegenerate conduction tail, $f(E)\simeq e^{-(E-\mu )/(k_{B}T)}$. Set $x=E-E_c$ and integrate the given density of states:The integral is $\Gamma (3/2)(k_{B}T{)}^{3/2}=\sqrt{\pi }(k_{B}T{)}^{3/2}/2$. Defining the effective density of states givesThe factor two is already included in the supplied density of states; it must not be added a second time.

A hole is an unoccupied valence state. For $E$ sufficiently below $\mu$, $1-f(E)\simeq e^{(E-\mu )/(k_{B}T)}$. Substitution of $x=E_v-E$ gives in exactly the same wayTheir product eliminates the chemical potential:This is the semiconductor mass-action law at thermal equilibrium. In an intrinsic semiconductor, charge neutrality additionally givesIn a doped semiconductor the product still applies in this nondegenerate equilibrium model, but the two populations need not be equal.

Intrinsic chemical potential. Taking logarithms of $n=p$ yields the intrinsic-semiconductor chemical potentialFor unequal band degeneracies, use the first form with the full $N_c,N_v$. For the supplied mass ratio $m_h^{\ast }/m_e^{\ast }=1/2$, the shift from midgap at $300\mathrm{K}$ is approximately $-13.4\mathrm{meV}$, toward the valence edge.

Three temperature regimes. The original plot runs from high temperature on the left to low temperature on the right. At high temperature, thermal excitation across the gap dominates and $n\simeq p\simeq n_i$ rises rapidly with temperature. In the middle, donors and acceptors are effectively ionized, the material is extrinsic, and $n\simeq N_D-N_A$ makes the plateau. At low temperature, dopant freeze-out binds increasing numbers of electrons to donor levels, reducing the mobile-electron population as the sample cools. Its activation scale is a donor binding energy, not the full band gap. Compensation and incomplete ionization affect that regime.

Gap estimate from the original figure. On the dashed intrinsic branch, $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. Hence band-gap extraction from intrinsic carrier activation uses the slope of $\ln (n_i/T^{3/2})$ versus $1/T$. With $x=1000\mathrm{K}/T$ and $k_B$ expressed in eV/K, two points giveThe very compressed intrinsic branch is only approximately readable: $n_i\sim 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 1.8$ and $n_i\sim 10^{13}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 2.5$. These give $E_g\sim 1.0\mathrm{eV}$, rather than a precision determination. Neglecting the thermal prefactor gives about $1.13\mathrm{eV}$ from those same rough points. Thus a gap of order $1$–$1.1\mathrm{eV}$ is justified by this schematic plot; its resolution does not support distinguishing a few hundredths of an electronvolt. The declining freeze-out branch must not be used to infer the full gap.

Room-temperature densities. At $300\mathrm{K}$, the supplied masses giveIn the fully ionized extrinsic plateau, put $N=N_D-N_A=10^{15}\mathrm{c}{\mathrm{m}}^{-3}$. Charge neutrality and the mass-action law implySince $n_i\ll N$, $n\simeq 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ is insensitive to the imprecise gap. The minority density is much more sensitive:Using $E_g=1.0\mathrm{eV}$ gives $n_i\simeq 5.8\times 10^{10}\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 3.3\times 10^6\mathrm{c}{\mathrm{m}}^{-3}$. Using $1.1\mathrm{eV}$ instead gives $n_i\simeq 8.3\times 10^9\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 7.0\times 10^4\mathrm{c}{\mathrm{m}}^{-3}$. These are explicitly conditional estimates within the stated mass model. The plot alone cannot fix a precise minority density because the uncertain gap occurs inside an exponential.

A common unstrained cubic III-V band ordering consists of an s-like ${\Gamma }_{6c}$ conduction doublet, a p-like ${\Gamma }_{8v}$ quartet and a lower spin-orbit split-off ${\Gamma }_{7v}$ doublet. In the cubic limit the quartet gives heavy-hole and light-hole bands that meet at the Gamma point; different curvatures and direction-dependent mixing distinguish them away from it. The split-off valence band is separated by ${\Delta }_{\mathrm{so}}$. This common ordering is a model to check against the actual material, not a consequence of chemical stoichiometry alone.