Bloch's theorem states that an eigenfunction in a periodic potential can be written ${\psi }_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{\mathbf{k}}(\mathbf{r})$, with $u_{\mathbf{k}}$ lattice-periodic. In the orthogonal tight-binding approximation, choose localized orthonormal orbitals $|j⟩$ with on-site matrix element $E_0$ and real nearest-neighbour hopping integral $t$. The Bloch sum $|k⟩=N^{-1/2}{\sum }_j{e}^{ikja}|j⟩$ is an eigenstate because each neighbour shift contributes a phase $e^{\pm ika}$. ThusThe simple-cubic nearest-neighbour tight-binding band has six neighbours, giving $E(\mathbf{k})=E_0+2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. The cosine sum ranges from $-3$ to $3$, so the positive bandwidth is $W=12|t|$. The stated hopping formula is negative when $E_1>0$, so the printed $12t$ must be read as a bandwidth magnitude.

Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$. To leading order the constant-energy surfaces are spheres, and comparison with $E_{\Gamma }+{\hslash }^2{k}^2/(2m^{\ast })$ givesThis is the effective mass for overlap-weighted exponential hopping. With $s=a/a_1$, its logarithmic derivative is $d\ln m^{\ast }/ds=1-2/s-1/(1+s)$. The positive stationary point is $s=1+\sqrt{3}$, and it is a minimum. The formula grows at both very small and very large $s$; the small-separation divergence is a mathematical extrapolation into a regime where localized tight binding itself becomes unreliable.

For the bandwidth, $W=24E_1(1+s)e^{-s}$ and $d\ln W/d\ln a=-s^2/(1+s)$. At $s=1$, a fractional spacing change $-0.01$ therefore givesThe exact ratio $W(0.99a_1)/W(a_1)=(1.99/2)e^{0.01}=1.004999916\dots$ checks the first-order result. Compression strengthens the hopping and widens the band.

A direct band gap has its conduction minimum and valence maximum at the same crystal momentum. A photon can then connect the band-edge states with negligible momentum change. An indirect band gap requires an additional phonon to supply or remove momentum, making radiative recombination much less probable. Direct-gap materials such as gallium arsenide or gallium nitride are therefore usually better for a light-emitting diode than silicon.

For the graded junction, let $q=|e|>0$, ${\epsilon }_s={\epsilon }_0{\epsilon }_r$, and use electrostatic potential $\phi$ with $E_x=-{\phi }^{\prime }$. Fully ionized dopants have charge density $\rho =q(N_D-N_A)=q\alpha x$, so Poisson's equation is ${\phi }^{\prime \prime }=-q\alpha x/{\epsilon }_s$. The paper's displayed plus sign corresponds to using the negative potential $V=-\phi$, or a signed electron charge in that formula. The positive barrier magnitude and charge magnitude must be used consistently in the width below.

The intended compensation point is $x=0$, where $N_D=N_A$ and the net doping changes sign. A locally neutral model with comparable electron and hole mobilities has its lowest carrier conductivity near there. The depletion approximation neglects mobile-carrier conductivity throughout the layer, so it does not by itself locate an exact conductivity minimum; unequal mobilities or asymmetric carrier profiles can shift it.

Let the depletion edges be $x=\pm d$. With zero exterior field, integrating $E_x^{\prime }=\rho /{\epsilon }_s$ and imposing $E_x(\pm d)=0$ gives the linearly graded p-n junction fieldIts integral gives $\phi (x)=\phi (0)+q\alpha (d^{2}x-x^3/3)/(2{\epsilon }_s)$, so the electric potential increases toward the n-type side while the electron band edges $E_c,E_v$ bend downward together. Their separation is the local band gap, and the equilibrium electrochemical potential is flat. The built-in potential arises when carrier diffusion exposes ionized dopants until the opposing field balances further diffusion.

Finally the positive barrier isWith total width $w=2d$, this yields the linearly graded junction depletion widthHere $\alpha >0$ labels p-to-n progression with increasing $x$; otherwise use its magnitude in the width formula.