16.07.04 · inorgchem / solid-state

Electronic properties of solids: insulators, semiconductors, conductors, and the band gap

stub3 tiersLean: nonepending prereqs

Anchor (Master): West — Solid State Chemistry and its Applications (1984)

Intuition Beginner

In an isolated atom, electrons occupy discrete energy levels — the 1s, 2s, 2p orbitals you learn about in general chemistry. When billions of atoms come together to form a solid, these discrete levels merge into continuous energy bands. Each band can hold a vast number of electrons, and the gaps between bands are regions where no electron can exist — the band gaps.

Whether a solid conducts electricity depends entirely on how these bands are filled. A conductor (metal) has a partially filled band — electrons can move to nearby energy levels with almost no energy input, so they flow freely through the material. An insulator (like NaCl or diamond) has completely filled bands separated from the next empty band by a large gap (typically $> 3$ eV). Electrons cannot jump the gap, so no current flows.

A semiconductor (like Si or Ge) sits between these extremes. The band gap is small enough (0.5–2.5 eV) that thermal energy at room temperature can promote a few electrons across the gap. These promoted electrons leave behind empty spots called holes in the filled band. Both the electrons in the upper band and the holes in the lower band can carry current — but much less of it than in a metal.

Visual Beginner

Worked example Beginner

Classify copper, diamond, and silicon as conductor, insulator, or semiconductor and explain why.

Copper is a conductor. It has a partially filled 4s band — there are available energy states immediately adjacent to the occupied ones, so electrons move freely with no activation barrier. This is why metals have high electrical conductivity and reflect light (their free electrons oscillate in response to electromagnetic waves).

Diamond is an insulator. Carbon atoms form four strong covalent bonds in the diamond structure, completely filling the valence band. The band gap is 5.5 eV — far too large for thermal energy to promote electrons ( $k_{B} T \approx 0.026$ eV at room temperature). No electrons can reach the empty conduction band, so diamond does not conduct.

Silicon is a semiconductor. It also has four covalent bonds filling the valence band, but the band gap is only 1.1 eV. At room temperature, a small fraction of electrons gain enough thermal energy to cross the gap. The conductivity is intermediate — far below copper but far above diamond — and it increases strongly with temperature.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which statement correctly explains why silicon is a semiconductor while diamond is an insulator?

A. Silicon has more valence electrons than diamond B. Silicon has a smaller band gap than diamond, allowing thermal promotion of electrons C. Silicon has stronger covalent bonds than diamond D. Silicon is a metal and diamond is not

Hint

Both silicon and diamond have completely filled valence bands with four covalent bonds per atom. The key difference is the size of the band gap separating the filled band from the empty conduction band.

Answer

B. Both Si and C(diamond) have filled valence bands from four covalent bonds per atom. Silicon has a band gap of 1.1 eV while diamond has 5.5 eV. The smaller gap allows thermal energy ( $k_{B} T \approx 0.026$ eV) to promote a measurable number of electrons into the conduction band in silicon, but not in diamond.

Exercise (easy, multiple choice).

What happens when an electron is promoted across the band gap of a semiconductor?

A. Only the promoted electron can carry current B. Both the promoted electron and the hole it leaves behind can carry current C. The material becomes an insulator D. The band gap disappears

Hint

When an electron leaves the valence band, it creates an unfilled state (a hole) in an otherwise full band. Can other electrons in that band move into the hole?

Answer

B. The promoted electron enters the previously empty conduction band and can move freely through available states. Simultaneously, the hole left behind in the valence band behaves as a positive charge carrier — neighbouring electrons can shift into the hole, effectively moving the hole through the band. Both carriers contribute to electrical conductivity.

Formal definition Intermediate+

Band theory from molecular orbitals. Consider $N$ identical atoms brought together from infinite separation. Each atomic orbital splits into $N$ closely spaced molecular orbitals, forming an energy band. The width of each band depends on the overlap between neighbouring atomic orbitals: core orbitals (tight binding, small overlap) produce narrow bands, while valence orbitals (diffuse, large overlap) produce wide bands.

The valence band is the highest energy band that is fully or partially occupied at 0 K. The conduction band is the next band above it. The energy separation between the top of the valence band ( $E_{v}$ ) and the bottom of the conduction band ( $E_{c}$ ) is the band gap $E_{g} = E_{c} - E_{v}$ .

Classification by band filling and gap size:

Material type	Band structure	$E_{g}$ (eV)	Conductivity (S/cm)	Examples
Conductor	Partially filled band or overlapping bands	0	$1 0^{4}$ – $1 0^{6}$	Cu, Na, Fe
Semiconductor	Small gap between filled VB and empty CB	0.1–3.5	$1 0^{- 8}$ – $1 0^{2}$	Si (1.1), Ge (0.67), GaAs (1.4)
Insulator	Large gap	$> 3.5$	$< 1 0^{- 10}$	Diamond (5.5), NaCl ( $\sim 8$ ), SiO $_{2}$ ( $\sim 9$ )

The boundaries are approximate. SiC ( $E_{g} = 2.9$ eV) is called a wide-band-gap semiconductor, while some materials with $E_{g} \approx 3$ –4 eV (ZnO, GaN) are semiconductors used in optoelectronics despite being borderline insulators by the table above.

The Fermi level. The Fermi level $E_{F}$ is the energy at which the probability of an electronic state being occupied is exactly 1/2. In an intrinsic semiconductor at $T = 0$ K, $E_{F}$ lies exactly in the middle of the band gap. At finite temperature, $E_{F}$ shifts slightly toward the band with the smaller effective density of states.

The carrier concentrations in an intrinsic semiconductor are:

n_{i} = p_{i} = N_{c} N_{v} exp (- \frac{E _{g}}{2 k _{B} T})

where $N_{c}$ and $N_{v}$ are the effective density of states in the conduction and valence bands:

N_{c} = 2 (\frac{2 π m _{e}^{*} k _{B} T}{h ^{2}})^{3/2}, N_{v} = 2 (\frac{2 π m _{h}^{*} k _{B} T}{h ^{2}})^{3/2}

Here $m_{e}^{*}$ and $m_{h}^{*}$ are the electron and hole effective masses — a measure of how an electron (or hole) responds to applied forces in the periodic lattice potential, differing from the free electron mass $m_{e}$ .

Doping. The carrier concentration in a semiconductor can be dramatically altered by adding small amounts of impurities (dopants). An n-type semiconductor is created by doping with atoms that have more valence electrons than the host. Phosphorus (5 valence electrons) substituted into silicon (4 valence electrons) donates an extra electron to the conduction band — P is a donor. The donor level lies just below $E_{c}$ , and at room temperature nearly all donors are ionised.

A p-type semiconductor is created by doping with atoms that have fewer valence electrons. Boron (3 valence electrons) in silicon creates an acceptor level just above $E_{v}$ . An electron from the valence band fills the acceptor, leaving behind a hole.

The relationship between carrier concentrations is:

n p = n_{i}^{2}

In n-type material, $n ≫ p$ ; in p-type material, $p ≫ n$ . The Fermi level shifts accordingly: upward toward $E_{c}$ for n-type, downward toward $E_{v}$ for p-type.

The p-n junction. When p-type and n-type semiconductors are brought into contact, electrons diffuse from the n-side to the p-side and holes diffuse from the p-side to the n-side. This creates a depletion region devoid of free carriers and an internal electric field that opposes further diffusion. The built-in potential $V_{0}$ is:

e V_{0} = k_{B} T ln \frac{N _{a} N _{d}}{n _{i}^{2}}

where $N_{a}$ and $N_{d}$ are the acceptor and donor concentrations. Applying a forward bias (p-side positive) narrows the depletion region and allows current to flow. Applying a reverse bias widens it and blocks current. This asymmetric current-voltage relationship is the basis of the diode.

Band gaps of common semiconductors:

Material	Structure	$E_{g}$ (eV)	Type
Si	Diamond	1.12	Indirect
Ge	Diamond	0.67	Indirect
GaAs	Zinc blende	1.42	Direct
InP	Zinc blende	1.34	Direct
GaN	Wurtzite	3.40	Direct
SiC (6H)	Wurtzite	3.02	Indirect
ZnO	Wurtzite	3.37	Direct
CdS	Wurtzite	2.42	Direct

Counterexamples to common slips

Metals do not have a band gap. A common error is to describe metals as having "a band gap of zero." Metals have no band gap — their valence and conduction bands overlap or a single band is partially filled. Describing this as "zero gap" conflates the distinct categories.
Semiconductor conductivity increases with temperature; metal conductivity decreases. In a semiconductor, more carriers are thermally excited across the gap as temperature rises, so conductivity increases. In a metal, the number of carriers is essentially fixed, but lattice vibrations (phonons) scatter electrons more at higher temperatures, so conductivity decreases. These opposite temperature dependences are a diagnostic test.
The band gap is not the energy of the highest occupied molecular orbital (HOMO). The band gap is the energy difference between two bands, not the ionisation energy of a single atom or molecule. Molecular orbital concepts (HOMO, LUMO) are useful analogies for isolated molecules but break down in the solid state where bands contain $\sim 1 0^{23}$ states.

Key theorem with proof Intermediate+

Proposition (Intrinsic carrier concentration). In an intrinsic semiconductor with band gap $E_{g}$ , the equilibrium electron and hole concentrations at temperature $T$ are $n = p = N_{c} N_{v} exp (- E_{g} /2 k_{B} T)$ , where $N_{c}$ and $N_{v}$ are the effective density of states of the conduction and valence bands.

Proof. The occupation probability for a state at energy $E$ is given by the Fermi-Dirac distribution:

f (E) = \frac{1}{1 + exp ( \frac{E - E _{F}}{k _{B} T} )}

The electron concentration in the conduction band is obtained by integrating the density of states $g_{c} (E)$ weighted by $f (E)$ over all energies in the conduction band. Near the band edge, the density of states follows a parabolic dispersion:

g_{c} (E) = \frac{4 π ( 2 m _{e}^{*} ) ^{3/2}}{h ^{3}} E - E_{c}

When $E_{c} - E_{F} ≫ k_{B} T$ (the nondegenerate limit, valid for intrinsic and lightly doped semiconductors), the Fermi-Dirac distribution approximates to a Boltzmann distribution: $f (E) \approx exp (- (E - E_{F}) / k_{B} T)$ . The integral becomes:

n = \int_{E_{c}}^{\infty} g_{c} (E) f (E) d E = N_{c} exp (- \frac{E _{c} - E _{F}}{k _{B} T})

where $N_{c} = 2 (2 π m_{e}^{*} k_{B} T / h^{2})^{3/2}$ .

Similarly, the hole concentration (unoccupied states in the valence band) is:

p = N_{v} exp (- \frac{E _{F} - E _{v}}{k _{B} T})

Multiplying:

n p = N_{c} N_{v} exp (- \frac{E _{c} - E _{v}}{k _{B} T}) = N_{c} N_{v} exp (- \frac{E _{g}}{k _{B} T})

For an intrinsic semiconductor, $n = p = n_{i}$ , so $n_{i}^{2} = N_{c} N_{v} exp (- E_{g} / k_{B} T)$ . Therefore:

n_{i} = N_{c} N_{v} exp (- \frac{E _{g}}{2 k _{B} T}) ■

Bridge. The intrinsic carrier concentration is the fundamental quantity governing semiconductor behaviour. For silicon at 300 K: $N_{c} \approx 2.8 \times 1 0^{19}$ cm $^{- 3}$ , $N_{v} \approx 1.0 \times 1 0^{19}$ cm $^{- 3}$ , $E_{g} = 1.12$ eV, giving $n_{i} \approx 1.5 \times 1 0^{10}$ cm $^{- 3}$ . This is extraordinarily small compared to the atomic density ( $5 \times 1 0^{22}$ cm $^{- 3}$ ) — roughly one carrier per $1 0^{12}$ atoms. Doping with $\sim 1 0^{16}$ cm $^{- 3}$ donors shifts the majority carrier concentration by six orders of magnitude while preserving the product $n p = n_{i}^{2}$ . The p-n junction built from these doped regions exploits this asymmetry to create rectifying behaviour — the foundation of all semiconductor electronics.

Exercises Intermediate+

Exercise 1 (easy, short-answer).

Distinguish between a conductor, a semiconductor, and an insulator in terms of their band structure.

Hint

Consider two properties: (a) whether the valence band is fully or partially occupied, and (b) the size of the band gap (if any).

Answer

Conductor: The highest occupied band is partially filled, or two bands overlap. There is no band gap — electrons can move to adjacent energy states with no activation energy.

Semiconductor: The valence band is completely full and the conduction band is empty, with a small gap between them ( $E_{g} < 3.5$ eV). Thermal energy promotes a small number of electrons across the gap.

Insulator: Same band arrangement as a semiconductor, but with a large gap ( $E_{g} > 3.5$ eV). Thermal energy is insufficient to promote electrons, so conductivity is negligible.

Exercise 3 (medium, short-answer).

Explain how doping silicon with phosphorus creates an n-type semiconductor. Where does the donor level lie relative to the band edges, and why is it essentially fully ionised at room temperature?

Hint

Phosphorus has 5 valence electrons. Silicon has 4. What happens to the extra electron? How much energy does it need to enter the conduction band?

Answer

Phosphorus has five valence electrons while silicon has four. When a P atom substitutes for a Si atom in the lattice, four of its electrons form covalent bonds with neighbouring Si atoms. The fifth electron is loosely bound to the P $^{+}$ ion by the Coulomb attraction, but the screening by the dielectric constant of silicon ( $ε_{r} = 11.7$ ) makes this binding very weak.

The donor level lies approximately 0.045 eV below the conduction band edge $E_{c}$ . This is a hydrogenic model result: $E_{d} \approx 13.6/ (ε_{r}^{2} \times (m_{e}^{*} / m_{e})^{2})$ eV, heavily screened and reduced from the hydrogen ionisation energy. At room temperature ( $k_{B} T = 0.026$ eV), thermal energy ( $\sim 2$ –3 $k_{B} T$ ) readily ionises the donor, releasing the electron into the conduction band. The donor ionisation fraction exceeds 99% at room temperature.

The material is called n-type because the majority carriers are negative (electrons from donors). The electron concentration $n \approx N_{d}$ (donor concentration) is much larger than $n_{i}$ , while the hole concentration $p = n_{i}^{2} / n$ is correspondingly suppressed.

Exercise 4 (medium, numeric).

A silicon wafer is doped with $N_{d} = 1 0^{16}$ cm $^{- 3}$ phosphorus donors. Calculate the electron concentration, the hole concentration, and the Fermi level position relative to $E_{i}$ (the intrinsic Fermi level) at 300 K. Use $n_{i} = 1.5 \times 1 0^{10}$ cm $^{- 3}$ and $k_{B} T = 0.026$ eV.

Hint

For an n-type semiconductor, $n \approx N_{d}$ and $p = n_{i}^{2} / n$ . The Fermi level shift from $E_{i}$ is $E_{F} - E_{i} = k_{B} T ln (n / n_{i})$ .

Answer

$n \approx N_{d} = 1 0^{16}$ cm $^{- 3}$ .

$p = n_{i}^{2} / n = (1.5 \times 1 0^{10})^{2} /1 0^{16} = 2.25 \times 1 0^{20} /1 0^{16} = 2.25 \times 1 0^{4}$ cm $^{- 3}$ .

The electron-to-hole ratio is $\sim 1 0^{12}$ — the vast majority of current is carried by electrons.

$E_{F} - E_{i} = k_{B} T ln (n / n_{i}) = 0.026 \times ln (1 0^{16} /1.5 \times 1 0^{10}) = 0.026 \times ln (6.67 \times 1 0^{5}) = 0.026 \times 13.4 = 0.35$ eV.

The Fermi level is 0.35 eV above the intrinsic level, placing it about 0.21 eV below $E_{c}$ (since $E_{i}$ is near mid-gap at 0.56 eV below $E_{c}$ for Si).

Exercise 5 (hard, short-answer).

Derive the built-in potential of a p-n junction from the requirement that the Fermi level is constant across the junction at equilibrium. Show that $e V_{0} = k_{B} T ln (N_{a} N_{d} / n_{i}^{2})$ .

Hint

On the n-side, $n_{n} \approx N_{d}$ and $E_{F} - E_{i} = k_{B} T ln (N_{d} / n_{i})$ . On the p-side, $p_{p} \approx N_{a}$ and $E_{i} - E_{F} = k_{B} T ln (N_{a} / n_{i})$ . The built-in potential is the difference in $E_{i}$ across the junction.

Answer

At equilibrium, the Fermi level $E_{F}$ must be constant across the entire junction. On the n-side, the electron concentration is $n_{n} \approx N_{d}$ , so:

E_{F} - E_{i, n} = k_{B} T ln \frac{N _{d}}{n _{i}}

where $E_{i, n}$ is the intrinsic Fermi level on the n-side. On the p-side, the hole concentration is $p_{p} \approx N_{a}$ , so:

E_{i, p} - E_{F} = k_{B} T ln \frac{N _{a}}{n _{i}}

where $E_{i, p}$ is the intrinsic Fermi level on the p-side. Since $E_{F}$ is constant:

E_{i, p} - E_{i, n} = k_{B} T ln \frac{N _{a}}{n _{i}} + k_{B} T ln \frac{N _{d}}{n _{i}} = k_{B} T ln \frac{N _{a} N _{d}}{n _{i}^{2}}

The built-in potential $V_{0}$ is defined by $e V_{0} = E_{i, p} - E_{i, n}$ (the electrostatic potential difference corresponds to the shift in the band edges across the depletion region):

e V_{0} = k_{B} T ln \frac{N _{a} N _{d}}{n _{i}^{2}}

For a Si junction with $N_{a} = 1 0^{17}$ cm $^{- 3}$ and $N_{d} = 1 0^{16}$ cm $^{- 3}$ : $e V_{0} = 0.026 \times ln (1 0^{17} \times 1 0^{16} / (1.5 \times 1 0^{10})^{2}) = 0.026 \times ln (4.44 \times 1 0^{12}) = 0.026 \times 28.8 = 0.75$ V.

Exercise 6 (hard, short-answer).

Explain why GaAs is used for light-emitting diodes while Si is not, even though both are semiconductors. Relate your answer to the distinction between direct and indirect band gaps.

Hint

An LED emits light when electrons in the conduction band recombine with holes in the valence band, releasing a photon. Consider whether this process conserves both energy and momentum. What does the band structure ( $E$ vs $k$ diagram) look like for direct vs indirect gap materials?

Answer

In a direct band gap semiconductor, the conduction band minimum and the valence band maximum occur at the same crystal momentum ( $k$ -vector, usually at the $Γ$ point). When an electron recombines with a hole, the energy is released as a photon with energy $h ν \approx E_{g}$ . Momentum is conserved because $Δ k = 0$ — the photon carries negligible momentum. The radiative efficiency is high (up to $\sim 30%$ for direct-gap LEDs).

In an indirect band gap semiconductor, the conduction band minimum and valence band maximum occur at different $k$ -values. Recombination requires a change in crystal momentum, which must be provided by a phonon (lattice vibration). This three-body process (electron + hole + phonon) is much less probable than the direct two-body process. The radiative efficiency is extremely low ( $< 1 0^{- 3}$ ), and most recombination occurs nonradiatively via defect states.

GaAs has a direct band gap of 1.42 eV at the $Γ$ point, making it an efficient light emitter (infrared, $λ \approx 870$ nm). Si has an indirect gap of 1.12 eV — the valence band maximum is at the $Γ$ point but the conduction band minimum is near the X point. The momentum mismatch makes radiative recombination so improbable that silicon is essentially useless as an LED material. This is the fundamental reason why GaAs, InP, GaN, and other III-V compounds dominate optoelectronics while Si dominates digital electronics.

Effective mass, carrier transport, and semiconductor devices Master

Effective mass. Near a band extremum, the $E (k)$ dispersion relation can be approximated as parabolic: $E (k) = E_{0} + ℏ^{2} k^{2} /2 m^{*}$ . The effective mass $m^{*}$ encapsulates how the periodic lattice potential modifies the response of an electron to an applied electric field. For isotropic bands:

\frac{1}{m ^{*}} = \frac{1}{ℏ ^{2}} \frac{d ^{2} E}{d k ^{2}}

In real materials, $m^{*}$ is a tensor. For silicon, the conduction band has six equivalent minima along the $⟨ 100 ⟩$ directions, giving longitudinal and transverse effective masses: $m_{l}^{*} = 0.98 m_{e}$ , $m_{t}^{*} = 0.19 m_{e}$ . The density-of-states effective mass is $m_{d os}^{*} = (m_{l}^{*} m_{t}^{* 2})^{1/3} \times 6^{2/3} = 1.08 m_{e}$ . The valence band has heavy and light holes: $m_{hh} = 0.49 m_{e}$ , $m_{l h} = 0.16 m_{e}$ .

The effective mass directly controls the density of states, carrier concentration, and the density-of-states effective mass enters $N_{c}$ and $N_{v}$ . Light effective masses give high carrier mobilities but also high intrinsic carrier concentrations (and thus higher leakage currents). This trade-off is a central consideration in semiconductor device design.

Direct vs indirect band gaps. A direct band gap occurs when the conduction band minimum and valence band maximum share the same $k$ -vector. Optical transitions are allowed without phonon participation — the absorption coefficient rises sharply at $E_{g}$ , reaching $α > 1 0^{4}$ cm $^{- 1}$ within 0.1 eV of the edge. Direct-gap materials (GaAs, InP, GaN, ZnO, CdS) are the basis of LEDs, laser diodes, and efficient solar cells.

An indirect band gap requires a change in crystal momentum, supplied by a phonon. The joint density of states at the threshold is much smaller, so the absorption coefficient rises gradually: $α$ near the edge scales as $(h ν - E_{g} + E_{p h o n o n})^{2}$ rather than the $(h ν - E_{g})^{1/2}$ dependence of direct transitions. Silicon's absorption coefficient at 1.2 eV (just above the gap) is only $\sim 10$ cm $^{- 1}$ , requiring $\sim 1$ mm of material for significant absorption — hence the thickness of silicon solar cells.

Carrier mobility and scattering. The drift velocity of carriers in an electric field $E$ is $v_{d} = μ E$ , where $μ$ is the carrier mobility. The conductivity is $σ = n e μ_{e} + p e μ_{h}$ . Mobility is limited by scattering from phonons (lattice vibrations), ionised impurities, neutral impurities, and crystal defects.

The temperature dependence of mobility distinguishes the dominant scattering mechanism. Acoustic phonon scattering gives $μ \propto T^{- 3/2}$ . Ionised impurity scattering gives $μ \propto T^{3/2} / N_{I}$ . At high temperatures, phonon scattering dominates (mobility decreases with T). At low temperatures in heavily doped material, impurity scattering dominates (mobility increases with T).

Typical room-temperature mobilities in pure materials: Si electrons 1500 cm $^{2}$ /Vs, Si holes 450 cm $^{2}$ /Vs, GaAs electrons 8500 cm $^{2}$ /Vs. The much higher electron mobility in GaAs makes it preferred for high-frequency devices.

The Hall effect. When a current-carrying semiconductor is placed in a perpendicular magnetic field $B$ , the Lorentz force deflects carriers to one side of the sample, building up a transverse voltage (the Hall voltage $V_{H}$ ). The Hall coefficient $R_{H} = V_{H} d / (I B)$ (where $d$ is the sample thickness) gives the carrier concentration:

n = \frac{1}{e ∣ R _{H} ∣}

and the sign of $R_{H}$ identifies whether the majority carriers are electrons (negative $R_{H}$ ) or holes (positive $R_{H}$ ). The Hall mobility is $μ_{H} = ∣ R_{H} ∣ σ$ . Combined resistivity and Hall measurements on the same sample yield both $n$ and $μ$ — the primary characterisation technique for semiconductors.

Semiconductor devices. The p-n junction diode is the simplest semiconductor device. Under forward bias $V$ , the current follows the Shockley diode equation:

I = I_{s} [exp (\frac{e V}{k _{B} T}) - 1]

where $I_{s}$ is the reverse saturation current, limited by the minority carrier generation rate in the depletion region.

The bipolar junction transistor (BJT) consists of two p-n junctions in close proximity (emitter-base-collector). A small base current modulates a large collector current, providing current amplification. The field-effect transistor (FET) controls current through a channel by varying the gate voltage, which modulates the channel conductivity via the field effect. The MOSFET (metal-oxide-semiconductor FET) is the building block of modern integrated circuits.

LEDs and solar cells exploit the properties of direct-gap semiconductors. In an LED, forward-biased current through a p-n junction produces radiative recombination at energy $h ν \approx E_{g}$ . Band-gap engineering (alloying GaAs with AlAs or InP to tune $E_{g}$ ) produces LEDs and laser diodes across the visible and infrared spectrum. GaN-based LEDs ( $E_{g} = 3.4$ eV, blue/UV) completed the triad of primary colours, enabling white LED lighting — recognised by the 2014 Nobel Prize in Physics (Akasaki, Amano, Nakamura).

A solar cell is an LED in reverse: photons with $h ν > E_{g}$ are absorbed, creating electron-hole pairs that are separated by the built-in field of a p-n junction and collected as photocurrent. The theoretical maximum efficiency (the Shockley-Queisser limit) for a single-junction cell is $\sim 33%$ , reached near $E_{g} \approx 1.34$ eV. Silicon ( $E_{g} = 1.12$ eV) achieves $\sim 26%$ in the lab and $\sim 22%$ commercially.

Transparent conducting oxides (TCOs). Indium tin oxide (ITO) and fluorine-doped tin oxide combine high electrical conductivity ( $\sim 1 0^{4}$ S/cm) with optical transparency ( $> 80%$ in the visible). This is possible because the carrier concentration ( $\sim 1 0^{21}$ cm $^{- 3}$ , degenerate semiconductor) is high enough for metallic conductivity but the band gap is large enough ( $> 3.5$ eV) that visible photons cannot excite interband transitions. TCOs are essential for flat-panel displays, touch screens, and thin-film solar cells.

Topological insulators. A relatively recent discovery (2000s) is the class of materials that are insulators in the bulk but have conducting surface states protected by time-reversal symmetry. In Bi $_{2}$ Se $_{3}$ , the bulk band gap is 0.3 eV, but the surface has a Dirac cone dispersion where electrons behave as massless relativistic particles. The surface states are topologically protected — they cannot be destroyed by nonmagnetic perturbations. This field connects solid-state physics to topology and has potential applications in spintronics and quantum computing.

Connections Master

Crystal structures and close-packing 16.07.02 pending. The band structure of a solid depends on the crystal structure because the periodicity of the lattice determines the allowed $k$ -vectors in the Brillouin zone. The diamond structure of Si and Ge, the zinc-blende structure of GaAs, and the rock-salt structure of PbS all produce different band structures despite having the same FCC lattice. The coordination geometry and bond lengths control orbital overlap, which controls bandwidth and hence whether the material is a metal or insulator.
Defects in solids 16.07.03 pending. Point defects in semiconductors are both the basis of doping (intentional impurities) and the source of nonradiative recombination centres (unintentional defects). A vacancy in silicon introduces deep levels near mid-gap that act as traps, reducing carrier lifetime and degrading device performance. The Kroger-Vink notation for defect equilibria in oxides directly connects to the n-type or p-type character of the semiconductor: oxygen vacancies donate electrons (n-type), while cation vacancies create holes (p-type).
Molecular orbital theory 16.03.01. Band theory is the extension of molecular orbital theory to the thermodynamic limit ( $N \to \infty$ ). The bonding-antibonding splitting of a diatomic molecule generalises to the valence-conduction band separation in a solid. The tight-binding method for calculating band structures is literally a molecular orbital calculation on a periodic lattice. Understanding why O $_{2}$ is paramagnetic (two electrons in degenerate $π^{*}$ orbitals) prepares the student for understanding why a half-filled band gives metallic conductivity.
Coordination chemistry 16.04.01. Transition-metal oxides show a rich interplay between crystal field effects and band formation. In NiO, the crystal field splitting and Hubbard $U$ (on-site Coulomb repulsion) conspire to create a charge-transfer insulator — the band gap is not between bonding and antibonding bands but between the O 2p valence band and the Ni 3d upper Hubbard band. This Mott-Hubbard physics goes beyond simple band theory and requires a many-body treatment.
Thermodynamics 16.01.01. The temperature dependence of semiconductor conductivity is fundamentally thermodynamic: the carrier concentration $n_{i} \propto exp (- E_{g} /2 k_{B} T)$ is a Boltzmann factor. The Fermi-Dirac distribution that governs electron occupation is the grand canonical ensemble result for fermions. The requirement that $E_{F}$ is constant across a p-n junction at equilibrium is a statement about thermodynamic equilibrium — the electrochemical potential must be uniform.

Historical notes Master

The distinction between conductors and insulators predates band theory. By the mid-nineteenth century, metals were known to be good conductors and nonmetals poor ones, but there was no microscopic explanation. The key theoretical development was Felix Bloch's 1928 doctoral thesis at Leipzig, supervised by Werner Heisenberg. Bloch showed that electrons in a periodic potential are described by wavefunctions of the form $ψ_{k} (r) = u_{k} (r) exp (ik \cdot r)$ , where $u_{k}$ has the periodicity of the lattice — the Bloch theorem. This established that electron states in a crystal are labelled by a wavevector $k$ within the first Brillouin zone and form continuous energy bands.

Alan Herries Wilson, also working in Heisenberg's group, applied Bloch's theorem to explain the difference between metals and insulators in two seminal papers in 1931. Wilson proposed that insulators have completely filled bands separated from the next empty band by a gap, while metals have partially filled bands. This is the band theory of solids in its essential form.

The concept of the semiconductor emerged gradually. In the 1830s, Michael Faraday observed that the conductivity of silver sulfide increases with temperature (opposite to metals), but this was not understood theoretically until band theory. The term "semiconductor" (Halbleiter in German) was introduced by Josef Weiss in 1911 but did not gain wide usage until the 1940s.

The transistor era began at Bell Labs. Russell Ohl discovered the p-n junction in silicon in 1940 while investigating why a cracked silicon rectifier produced a photovoltaic effect. The pure region on one side of the crack was p-type (boron-doped) and the other was n-type (phosphorus-doped). In 1947, John Bardeen and Walter Brattain demonstrated the point-contact transistor, and William Shockley developed the junction transistor in 1948. The three shared the 1956 Nobel Prize in Physics.

The development of band-structure calculation methods transformed the field. The tight-binding method (Bloch, 1928), the nearly-free electron model, the augmented plane wave method (APW, Slater 1937), the Korringa-Kohn-Rostoker method (KKR, 1947–1954), and the pseudopotential method (Phillips and Kleinman, 1959) progressively improved the accuracy of calculated band structures. The empirical pseudopotential method of Cohen and Bergstresser (1966) produced band structures for all zinc-blende and diamond-structure semiconductors that agreed with optical measurements to within 0.1 eV.

The Hall effect was discovered by Edwin Hall in 1879, during his doctoral work at Johns Hopkins University. Hall found that a magnetic field applied perpendicular to a current-carrying gold leaf produced a transverse voltage. The quantitative theory was developed by Ludwig Boltzmann and later by the Soviet physicist Abram Ioffe, who recognised in the 1930s that the Hall effect could distinguish n-type from p-type semiconductors — essential for the development of semiconductor technology.

The theoretical understanding of p-n junctions was developed by William Shockley in 1949 in his paper "The Theory of p-n Junctions in Semiconductors and p-n Junction Transistors" (Bell System Technical Journal, 1949). The Shockley diode equation and the depletion approximation remain the starting point for semiconductor device physics. The complete theory, including generation-recombination in the depletion region, was developed by Clifford Sah, Robert Noyce, and William Shockley in 1957.

The concept of effective mass was introduced by Bardeen in 1937 and developed systematically by Charles Kittel and Arthur Slater in the 1940s–1950s. Cyclotron resonance experiments by Dresselhaus, Kip, and Kittel at Berkeley (1955) measured the effective mass tensor directly by observing the resonant absorption of microwave radiation by carriers executing circular orbits in a magnetic field.

The discovery of topological insulators began with the theoretical work of Kane and Mele (2005), who predicted that graphene with spin-orbit coupling would be a quantum spin Hall insulator. The first experimental three-dimensional topological insulator was Bi $_{2}$ Se $_{3}$ , characterised by Yulin Chen and colleagues at Stanford in 2009 using angle-resolved photoemission spectroscopy (ARPES). The 2016 Nobel Prize in Physics was awarded to David Thouless, Duncan Haldane, and Michael Kosterlitz for theoretical discoveries of topological phase transitions and topological phases of matter.

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