Question 5 of 7: Band Filling, Metals/Insulators/Semiconductors, and Intrinsic Silicon Carrier Density

Question 5: Band Filling, Metals/Insulators/Semiconductors, and Intrinsic Silicon Carrier Density (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Five stacked band-filling diagrams (Fig. P5), each showing a lower band, a middle band and an upper band with varying grey (filled) fractions. For (b): $E_g=1.08\ \text{eV}$, $m_e^*=1.1m$, $m_h^*=0.56m$, $T=300\ \text{K}$.

Find. (a) classify each of the five band-filling cases; (b) $n_i=p_i$ for intrinsic Si at 300 K; (c) qualitative shift of $E_F$ under heavy donor doping.

Approach. (a) apply the standard classification rule (a partially-filled band, or two bands overlapping in energy so that carriers spill between them, both conduct → metal/semimetal; two completely full bands separated by a real gap from an empty band → insulator/semiconductor, distinguished by gap size and any trace thermal population). (b) use eq. (15), the intrinsic-carrier formula, directly with the given effective masses and gap. (c) reason from Fermi-level pinning near the donor level.

1. Part (a) — classify the five cases. In every case the lowest band is completely filled (inert core levels) — the classification hinges on the two upper bands:
   • Case 1: the middle band is drawn completely filled, with a clear energy gap to an empty top band → two full bands under a gap → insulator.
   • Case 2: the middle band is only about half filled → a partially-occupied band has states immediately above $E_F$ within the same band → metal.
   • Case 3: a full band directly overlaps in energy with the bottom of a second, mostly-empty band (no true gap between them) → carriers spill from the top of the lower band into the bottom of the upper one in small numbers → semimetal.
   • Case 4: the middle band is almost entirely empty, with only a thin sliver of population at its very bottom edge, separated from the full band below by a small gap → a nearly-empty conduction band with a trace thermal population across a narrow gap → semiconductor.
   • Case 5: the middle band is almost completely filled (only a thin unfilled sliver at the top), with a gap to an empty top band → a nearly-full band still has adjacent empty states (equivalently, a small hole population) → metal.
   Check: the shading levels for cases 3 and 4 were read from the printed figure (Fig. P5); the classification RULE above is unambiguous, but a reader with the original exam sheet should confirm the fill fractions match this reading before treating the semimetal/semiconductor labels as final.
2. Part (b) — intrinsic carrier concentration of Si. Using eq. (15), $n_i=p_i=2\left(\dfrac{k_BT}{2\pi\hbar^2}\right)^{3/2}(m_em_h)^{3/4}\exp\!\left(\dfrac{-E_g}{2k_BT}\right)$, with $m_e=1.1m$, $m_h=0.56m$ ($m=9.10956\times10^{-31}\ \text{kg}$), $E_g=1.08\ \text{eV}=1.729\times10^{-19}\ \text{J}$, $T=300\ \text{K}$ (SI, $\hbar=1.05459\times10^{-34}\ \text{J}\cdot\text{s}$, $k_B=1.38062\times10^{-23}\ \text{J/K}$): $$\frac{E_g}{2k_BT}=\frac{1.729\times10^{-19}}{2(1.38062\times10^{-23})(300)}=20.89$$ $$n_i=2\left(\frac{(1.38062\times10^{-23})(300)}{2\pi(1.05459\times10^{-34})^2}\right)^{3/2}\bigl[(1.1m)(0.56m)\bigr]^{3/4}e^{-20.89}\ \text{m}^{-3}$$ $\boxed{n_i=p_i\approx1.48\times10^{10}\ \text{cm}^{-3}}$ — this lands very close to silicon's accepted room-temperature intrinsic carrier concentration ($\approx1.5\times10^{10}\ \text{cm}^{-3}$ for the real $E_g=1.12\ \text{eV}$), a strong check on the arithmetic given the paper's slightly different assumed gap.
3. Part (c) — Fermi level under heavy donor doping. Each donor level sits energetically just below the conduction-band edge $E_c$ and, at room temperature, is essentially fully ionized, flooding the conduction band with electrons. To keep the electron population consistent with Fermi-Dirac statistics, the Fermi level must rise from mid-gap toward (and, for very heavy doping, potentially into) the conduction band, $E_F\to E_c$ — i.e. the crystal becomes an n-type extrinsic semiconductor with $E_F$ shifted up from mid-gap toward the donor/conduction-band side.