Physics · Atomic and quantum physics · Year 1

Bands in solids

Why does copper conduct current but diamond does not: allowed and forbidden bands

The situation

The electric wiring in a house is made of copper, a phone processor is made from a silicon wafer, and the diamond in a jeweller's shop hardly conducts current at all. All three substances are made of atoms, and every atom has electrons. Copper wire conducts well at any temperature; silicon, when heated or lit, suddenly begins to conduct much better; diamond stays a transparent insulator. What energies can an electron in a crystal have, and why can it not carry current in some substances?

Topic

Band theory of solids. Why does copper conduct current but diamond does not: allowed and forbidden bands

Key ideas

Bloch theorem
Band gap
Fermi level

Formulas

ψ_{k} (x) = u_{k} (x) e^{i k x}, u_{k} (x + d) = u_{k} (x)

Bloch's theorem: an electron in a periodic potential is a plane wave multiplied by a function that repeats with the lattice period $d$ .

\cos k d = \cos α a \cosh β b + \frac{β^{2} - α^{2}}{2 α β} \sin α a \sinh β b

The Kronig–Penney equation ( $ℏ α = \sqrt{2 m E}$ , $ℏ β = \sqrt{2 m (V_{0} - E)}$ , $d = a + b$ ): energies at which the right-hand side has a modulus greater than 1 are forbidden bands.

v = \frac{1}{ℏ} \frac{d E}{d k}, \frac{1}{m^{*}} = \frac{1}{ℏ^{2}} \frac{d^{2} E}{d k^{2}}, ℏ \frac{d k}{d t} = - e ℰ

Group velocity, effective mass and the equation of motion in an external electric field $ℰ$ .

n_{i} = \sqrt{N_{c} N_{v}} e^{- E_{g} / 2 k T}, E_{c} - E_{F} = k T \ln \frac{N_{c}}{n}

The concentration of intrinsic charge carriers and the position of the Fermi level in an n-type semiconductor.

E_{F} = \frac{ℏ^{2}}{2 m} {(3 π^{2} n)}^{2 / 3}, C_{e l} = \frac{π^{2}}{2} n k \frac{k T}{E_{F}} = γ T

The Fermi energy of a free electron gas and the electronic heat capacity (Sommerfeld model).

Theory

From levels to bands

In an isolated atom the energy of an electron consists of definite levels. When two atoms come close, an electron can tunnel from one atom to the other and each level splits in two; $N$ atoms produce $N$ closely spaced levels from each level (each holding up to 2 electrons by the Pauli principle). In a crystal $N \sim 10^{23}$ — the levels merge into an almost continuous allowed band, and between the bands remains a forbidden band (band gap) in which an electron cannot exist. The width of a band depends on the overlap of the wave functions of neighbouring atoms: inner levels spread into narrow bands and outer levels into wide ones — this is the tight-binding approximation. The “Levels” mode is a simple model: $E_{j} = E_{0} + 2 t \cos \frac{j π}{N + 1}$ , $t \propto e^{- a / λ}$ ; the shift of the centre of the levels is not taken into account.

Periodic potential and Bloch's theorem

The second approach is weakly bound (nearly free) electrons: a weak periodic potential is added to a free electron ( $E = ℏ^{2} k^{2} / 2 m$ ). By Bloch's theorem the wave function in a periodic potential is $ψ = u_{k} (x) e^{i k x}$ — the probability repeats in every unit cell. At $k = n π / d$ (the edge of the Brillouin zone) the electron wave undergoes Bragg reflection from the lattice: two standing waves form and there is a jump in energy — a forbidden band opens. In the Kronig–Penney model, if $V_{0}$ is small, $E (k)$ is close to the free parabola (wide bands, narrow gaps); if $V_{0}$ or the barrier width $b$ is large, the bands narrow and the picture approaches the tight-binding one.

Effective mass, holes and motion in a field

The speed of an electron in a crystal is $v = \frac{1}{ℏ} \frac{d E}{d k}$ , and its response to an external force is determined by the effective mass $m^{*}$ : at the bottom of a band $m^{*} > 0$ , at the top $m^{*} < 0$ — there the electron accelerates against the force. A vacancy in an almost full band is conveniently regarded as a particle with positive charge and positive mass — a hole. In a field $ℏ d k / d t = - e ℰ$ : $k$ moves along the band, passes the edge and re-emerges at the other side of the band, while the velocity changes sign — a Bloch oscillation. In an ordinary crystal the electron is scattered after about $10^{- 14}$ s and the oscillation cannot be seen; it can be observed in artificial semiconductor superlattices. In the model the electron stays within one band (interband Zener tunnelling is not taken into account).

Metal, insulator, semiconductor

Everything depends on how the bands are filled. In copper the upper band is partly filled: the Fermi level lies inside the band, and even a very small energy can raise an electron into an empty state — a metal. If there is a gap $E_{g}$ between a full valence band and an empty conduction band: $E_{g}$ of several eV — an insulator (dielectric) (diamond ≈ 5.5 eV), $E_{g} \sim 1$ eV — a semiconductor (Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV). In a semiconductor heat lifts some electrons into the upper band — intrinsic conduction; if phosphorus (donor) or boron (acceptor) is added — extrinsic n-type or p-type conduction. A photon with energy $h ν > E_{g}$ creates an electron–hole pair — photoconductivity; this is why diamond does not absorb visible light (it is transparent) while silicon does. When p- and n-regions touch, a p–n junction forms — the basis of the diode, the transistor, the solar cell and the LED (in detail in the “Semiconductor physics” experiment).

Statistics of charge carriers and the Fermi level

Electrons occupy states according to the Fermi–Dirac distribution $f (E) = 1 / (e^{(E - E_{F}) / k T} + 1)$ : the Fermi level $E_{F}$ is the energy at which the probability of occupation is ½. The distribution of electrons in a band = density of states $g (E) \propto \sqrt{E - E_{c}}$ × $f (E)$ : on the graph, narrow “tails” at the band edges (the shape is normalised). In a pure semiconductor $E_{F}$ is near the middle of the band gap and $n = p = n_{i}$ ; donors move $E_{F}$ towards $E_{c}$ and acceptors towards $E_{v}$ . On heating, $n_{i}$ grows exponentially and $E_{F}$ returns to the middle. Model: all impurities are ionised (the “freeze-out” at low temperature is not taken into account), $E_{g} (T)$ follows the Varshni formula, the mobility is $μ \propto T^{- 3 / 2}$ (approximately), and the multimeter measures a 1 cm × 1 mm × 1 mm sample.

Free electron gas: Drude and Sommerfeld

Drude treated the electrons in a metal as a classical gas (drift and Ohm's law — the “Electric current in metals” experiment). Sommerfeld treated it as a quantum gas: by the Pauli principle, even at $T = 0$ the electrons fill a Fermi sphere of radius $k_{F}$ in $k$ -space; its surface is the Fermi surface, with energy $E_{F}$ (in copper 7.0 eV, that is $T_{F} \approx 82 000$ K) — the gas is degenerate. On heating, only a fraction $\sim k T / E_{F}$ of the electrons near the surface are excited, so the electronic heat capacity is small: $C = γ T$ . The quanta of lattice vibrations, phonons (Debye density of states $\propto ω^{2}$ ), give $β T^{3}$ at low temperature; by plotting $C / T$ against $T^{2}$ one finds $γ$ and $θ_{D}$ . Spins give Pauli paramagnetism $χ_{P} = μ_{0} μ_{B}^{2} g (E_{F})$ , and orbital motion gives Landau diamagnetism $χ_{L} = - χ_{P} / 3$ . In white dwarf stars the electron gas is relativistically degenerate: its pressure holds the star up to about 1.4 solar masses (the Chandrasekhar limit).

Did you know?

Felix Bloch proved his theorem about the electron in a crystal in 1928, in the doctoral dissertation he wrote at the University of Leipzig under the supervision of Werner Heisenberg.
Silicon hardly absorbs infrared light of wavelength longer than ≈ 1.1 µm, so lenses and windows for infrared cameras in the 3–5 µm range are made from it — even though to the eye it looks like a shiny grey opaque wafer.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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