Energy bands, step by step: from one atom to a band diagram
This chapter starts with a single silicon atom and builds up the ideas behind every semiconductor device: energy levels that split into bands, the band gap, electrons and holes, how they are counted, doping, and band diagrams drawn across a device. Every picture is live. It needs only chapter 1 (voltage and current). Then chapter 6 joins p- and n-type silicon into a diode, chapter 7 builds nMOS and pMOS transistors, chapter 8 wires them into CMOS logic, and chapter 9 derives all of it from the Schrödinger equation.
Twelve steps from a silicon atom to a logic gate: this chapter covers the first eight
How to read any band diagram
- Vertical axis: energy of one electron, in electron-volts. 1 eV is the energy an electron gains falling through 1 volt (1.602·10⁻¹⁹ J). Up means more energy for an electron.
- Shaded regions are allowed energies (bands). White between them is forbidden: the band gap.
- Ec: bottom edge of the conduction band. Ev: top edge of the valence band. EF (dashed orange): the Fermi level, where a state is filled with 50 % probability.
- Electrons roll down like marbles. Holes float up like bubbles.
- Horizontal axis: it depends. Atom spacing in section 3, position inside a device from section 8 on (and in chapters 6 to 8), and the wave vector k in chapter 9.
- A slope of the bands in space is an electric field. A flat EF means equilibrium: no current.
In an atom, an electron can only have certain energies
The nucleus holds its electrons by electric attraction. In classical physics an electron could circle at any distance with any energy. Quantum mechanics says no: the electron behaves like a wave, and only certain standing-wave patterns fit around the nucleus (chapter 9 shows why). Each pattern is an orbital, and each orbital has a fixed energy, its energy level.
The zero of energy is the vacuum level: an electron at rest, far away from the atom. Bound electrons have negative energy. That is the energy you must pay to pull them out. Each level offers a fixed number of states, think of them as seats: an s level has 2, a p level has 6 (3 orbitals × 2 spin directions). The Pauli exclusion principle allows at most one electron per state. Silicon has 14 electrons: 1s² 2s² 2p⁶ 3s² 3p². The 10 inner core electrons sit deep and tight around the nucleus. The 4 outer valence electrons (3s² 3p²) do all of chemistry and all of electronics.
Light proves that the levels are discrete. A photon of energy is only absorbed if it lifts an electron exactly into a free seat. Silicon atoms absorb and emit only at a set of sharp lines. One of the strongest, 3p → 4s at 4.93 eV, sits at λ = 1240 nm·eV / 4.93 eV = 251.6 nm. The demo shows only this line (real lines are even far sharper than here). Photons with energies in between pass straight through. Above 8.15 eV, the ionization energy, every photon can be absorbed: the electron leaves the atom, and the rest of the energy becomes its kinetic energy. Above the vacuum level, energies are not quantized any more.
The ladder is to scale above the break. The core levels (2p ≈ −100 eV, 2s ≈ −150 eV, 1s ≈ −1840 eV) are so deep that chemistry and electronics never touch them. The 3s value (≈ −15 eV) is a calculated orbital energy. The 3p value is minus the measured ionization energy.
Bring atoms together and every level splits. Many atoms make a band.
When two atoms come close, their outer orbitals overlap and an electron can tunnel from one atom to the other. The two equal levels couple, like two identical pendulums joined by a spring: instead of one shared frequency you get two. In the lower level the waves on both atoms are in phase (bonding: the electron likes the region between the nuclei, attracted by both). In the upper level they are out of phase (antibonding: a node between the atoms).
The number of states never changes: N atoms with one level each give exactly N levels. They spread over a width of about 4t. The hopping energy t measures how easily an electron jumps to a neighbour, and it grows quickly with overlap. So the band width depends on how close the atoms are, not on how many there are. More atoms only pack the levels more densely. A cubic centimetre of silicon holds 5·10²² atoms, so the levels are ~10⁻²² eV apart, far below anything measurable. This quasi-continuum of levels is an energy band. For a chain of N atoms (tight-binding model) the levels are exactly:
Click a level on the right to see its wave pattern on the atoms (red = +, blue = −). The lowest level never changes sign, the highest flips sign from atom to atom. Then fill in electrons, 2 per level (spin up and spin down). With 1 electron per atom the band is half full. Empty states sit right next to the highest filled one, so the smallest push can move electrons: a metal (sodium, copper). With 2 electrons per atom the band is completely full, and a full band cannot carry current (section 4).
How silicon gets its band gap
Each silicon atom brings two valence levels: 3s (2 states) and 3p (6 states). N atoms give 2N + 6N = 8N states for 4N valence electrons. Drag the atoms closer, from the right to the left of the diagram:
- Far apart: sharp atomic levels. The 2N 3s states hold 2N electrons (full). The 6N 3p states hold the other 2N (one third full).
- Closer: both levels broaden into bands, as in section 2.
- Overlap: the s and p bands merge into one band of 8N states holding 4N electrons, half full.
- Closer still: s and p mix into four sp³ hybrid orbitals that point to the 4 neighbours of each atom (the tetrahedral bonds). Bonding combinations form a lower band, antibonding ones an upper band, with exactly 4N states each. Between them a gap opens that contains no states.
- At the real spacing d₀ (0.235 nm between neighbours in Si) the lower band holds 4N states and 4N electrons: completely full. This is the valence band. The upper band holds 4N states and no electrons: the conduction band. The gap is Eg = 1.12 eV.
The curves are schematic. Their shape follows the classic calculation for diamond (Kimball, 1935). Only the gap at d₀ is a measured number for silicon. Drag the vertical line or use the slider. The inner-shell levels (1s, 2s, 2p) stay sharp at every spacing: their orbitals are too small to reach the neighbours.
| Group IV crystal (same structure) | bond length | band gap at 300 K | type |
|---|---|---|---|
| C (diamond) | 0.154 nm | 5.47 eV | insulator |
| Si | 0.235 nm | 1.12 eV | semiconductor |
| Ge | 0.245 nm | 0.66 eV | semiconductor |
| α-Sn (grey tin) | 0.281 nm | 0 eV | semimetal |
Same crystal structure, same 4 valence electrons per atom. The bigger the atoms, the longer the bonds, the weaker the bonding–antibonding splitting and the smaller the gap.
Only a partly filled band conducts
To carry a current, electrons have to pick up a little extra speed in one direction, which means moving into slightly different states. In a completely full band that is impossible: every state is taken, and for every electron moving right there is one moving left. Net current: zero. An empty band has nobody to carry anything. Only a partly filled band conducts.
The highest occupied band is only partly full, or two bands overlap. EF lies inside a band. Plenty of free electrons at any temperature: copper has 8.5·10²² per cm³.
Full valence band, empty conduction band, wide gap: diamond 5.47 eV, SiO₂ ≈ 9 eV. Heat practically never lifts an electron across.
At T = 0 exactly like an insulator, but the gap is small (Si 1.12 eV). At room temperature a few electrons get thermally lifted across. Each one leaves an empty seat behind, a hole. Both conduct.
kT is the typical thermal energy, with the Boltzmann constant k = 8.617·10⁻⁵ eV/K. At 300 K, kT = 0.0259 eV, 43 times smaller than the silicon gap. The electrons that make it across come from the far exponential tail of the thermal distribution. That is why the intrinsic carrier density ni reacts so violently to temperature and gap. Nc and Nv (~10¹⁹ cm⁻³) are the effective numbers of states at the band edges (section 6).
A cube of pure silicon 1 µm on a side contains on average only 0.01 free electrons at room temperature, one per 100 µm³. Pure silicon is a decent insulator: ρ ≈ 3·10⁵ Ω·cm. A diamond as big as the Earth would hold only a handful of thermally freed electrons. Semiconductors become useful only through doping (section 7).
A missing electron moves like a positive particle
Lift one electron from the valence band into the conduction band. Now there are two ways to carry current:
- The electron in the almost empty conduction band moves freely. In a field it drifts against E, because it is negative.
- The empty seat in the almost full valence band lets valence electrons move as well. A neighbour hops into the empty seat, the next one into the seat just vacated, and so on. These valence electrons all move against E, so the empty seat moves along E, exactly like a positive charge. This empty seat is a hole.
Both carriers make current in the same direction, along E. Following one hole is much easier than following 10²³ valence electrons, so we treat the hole as a particle of its own, with charge +e and its own effective mass (chapter 9). It is a quasi-particle, not a positron. In band diagrams, an electron above a deep hole can drop into it and release energy, which moves the hole up. So holes float to the top of the valence band like bubbles, while electrons sink to the bottom of the conduction band. When an electron falls back into a hole, both disappear: recombination, releasing the energy as heat or light (LEDs).
Density of states × Fermi–Dirac = carriers
To count the electrons in the conduction band you need two ingredients:
Multiply both and add up over the band. As long as EF stays more than about 3kT away from both bands, the integrals simplify (Boltzmann approximation):
Nc ≈ 2.8·10¹⁹ cm⁻³ and Nv ≈ 2.65·10¹⁹ cm⁻³ (Si, 300 K) are the effective densities of states: it is as if all states of the band sat right at its edge. Multiply n and p, and EF drops out:
Pushing EF up raises n and lowers p by the same factor. Drag EF and watch the four panels: band diagram, seats g(E), chance f(E), and their product, the electrons n(E) and holes p(E).
Where EF lies in the gap there are no states at all. The Fermi level is not "where the electrons are". It is a bookkeeping energy that sets the occupation of every state. In equilibrium it is the same everywhere in a connected system (section 8).
Swap one atom in a million and the conductivity jumps a millionfold
Replace one silicon atom by phosphorus, which has 5 valence electrons. Four of them form the four bonds. The fifth has no bond to join and is held only loosely by the one extra positive charge of the P nucleus. It behaves like the electron of a hydrogen atom, but inside silicon: the attraction is weakened by the dielectric constant εr = 11.7, and the electron moves with the effective mass m* ≈ 0.26 m₀ (chapter 9). Scaling the hydrogen atom accordingly:
The measured binding energy for P is 45 meV (the P core pulls a bit harder at short distance). The orbit spans hundreds of Si atoms, which is why the bulk values εr and m* are allowed here. With kT = 26 meV at room temperature practically every donor is ionized: its electron roams free in the conduction band. Phosphorus is a donor, and the silicon becomes n-type. Boron has only 3 valence electrons, one short for its four bonds. It grabs an electron from a neighbouring bond for about 45 meV and leaves a hole behind: an acceptor, giving p-type silicon.
In the band diagram, donors are localized states just below Ec (at Ed) and acceptors just above Ev (at Ea). Doping pulls EF towards the conduction band (n-type) or the valence band (p-type). Ei is the Fermi level of undoped silicon, practically mid-gap:
Click atoms in the lattice to turn them into P (donor) or B (acceptor). The lattice is a cartoon. The band diagram on the right is calculated from the sliders: charge neutrality p + ND⁺ = n + NA⁻ with Fermi–Dirac statistics and partly ionized dopants.
Typical doping levels are 10¹⁵ to 10²⁰ cm⁻³, while silicon has 5·10²² atoms per cm³. One phosphorus atom per million silicon atoms gives n = 5·10¹⁶ cm⁻³: five million times more electrons than in pure silicon. The minority carriers (here holes) drop by the same factor, because n·p = ni². The temperature plot shows three regimes. Cold (freeze-out): electrons fall back onto their donors. Middle (extrinsic): n = ND, flat, the range where devices work. Hot (intrinsic): ni overtakes ND and doping stops mattering. Long before that, leakage currents (proportional to ni or ni²) rise steeply, the main reason chips are rated only up to about 125–150 °C. Above ~10¹⁸ cm⁻³ the donor orbits overlap, the donor levels merge into a band, and the simple freeze-out model overestimates the bound fraction.
Voltage tilts the bands. Electrons roll down, holes float up.
From here on the horizontal axis is the position x inside a device. The band edges shift with the local electrostatic potential φ(x). A negative electron at a place with high potential has low energy:
Four rules cover every band diagram: (1) the slope of the bands is the electric field ℰ. (2) Electrons roll down the Ec slope, holes float up the Ev slope. (3) In equilibrium EF is flat: no net current. (4) When current flows, EF tilts, or splits into separate quasi-Fermi levels EFn and EFp for electrons and holes. Their slope drives the current: , which contains both drift and diffusion.
In a uniform resistor the bands and EF tilt together by qV across the length. Electrons drift towards the + contact, holes towards the − contact. Both carry current from + to −: Ohm's law in band language (compare the voltage page). The band gap itself never changes; the whole band structure is shifted up or down by −qφ(x).
From band diagrams to devices: diode, transistor, CMOS
Everything so far happened inside one piece of silicon. Devices appear when you put different pieces together. The band picture of section 8 then explains each device in one sentence:
The whole chain in one table
| Step | Key idea | Key formula |
|---|---|---|
| 1 Atom | only certain energies; at most one electron per state | |
| 2 Many atoms | N levels spread into a band of width ~4t | |
| 3 Silicon | sp³ bonding: 4N full states, 4N empty states, a gap in between | Eg = 1.12 eV |
| 4 Materials | only a partly filled band conducts | |
| 5 Holes | a missing electron acts like a +e particle | |
| 6 Counting | seats × chance | |
| 7 Doping | donors/acceptors ~45 meV from the bands | |
| 8 Bands in space | bands follow −qφ; slope = field; flat EF = equilibrium | |
| → ch. 6: pn junction | a built-in hill; bias changes its height | |
| → ch. 7: MOS | the gate bends the bands at the surface | |
| → ch. 7: MOSFET | gate-controlled barrier between source and drain | S ≥ 60 mV/decade |
| → ch. 8: CMOS | complementary pair: no static path to ground |
Six classic misunderstandings
- "kT = Eg" in some figures: wrong. Eg(Si) ≈ 43 kT at room temperature; only the exponential tail crosses.
- "EF is where the electrons are": no. In a semiconductor EF usually lies in the gap, where there are no states at all. It sets the occupation of all states.
- "Band bending changes the gap": no. The whole band structure shifts up or down by −qφ(x); Eg stays the same.
- "The depletion region is empty": it has almost no mobile carriers, but it is full of fixed ionized dopants. That charge creates the field.
- "Holes are positrons": holes are the collective motion of the many valence electrons, described as one quasi-particle.
- "Band diagram vs position = band structure E(k)": two different plots. E(x) diagrams show only the band edges Ec, Ev taken from E(k) (chapter 9), shifted by the local potential.
Next: chapter 6 puts p- and n-type silicon together into a diode. Later, chapter 9 starts from the Schrödinger equation and derives levels, bands, effective mass, density of states and the band-bending picture used here.
Every term on this page
| acceptor | dopant with one valence electron too few (B in Si); captures an electron and creates a hole |
| accumulation | MOS state with extra majority carriers at the surface |
| band, band gap | a quasi-continuous range of allowed energies; the forbidden range between two bands |
| bonding / antibonding | in-phase (lower energy) and out-of-phase (higher energy) combinations of atomic orbitals |
| built-in voltage Vbi | the potential step across a pn junction in equilibrium |
| CMOS | complementary MOS: logic built from nMOS and pMOS pairs |
| conduction band, Ec | lowest band that is (nearly) empty in a semiconductor; its bottom edge |
| core electrons | inner, tightly bound electrons that do not take part in bonding |
| C–V curve | capacitance of a MOS structure versus gate voltage |
| density of states g(E) | number of states per energy and volume |
| depletion region | zone emptied of mobile carriers, containing fixed ionized dopants |
| donor | dopant with one valence electron too many (P, As in Si); gives an electron to the conduction band |
| drift, diffusion | carrier motion driven by an electric field; motion from high to low concentration |
| effective mass m* | the mass with which an electron or hole responds to forces inside a crystal (chapter 9) |
| electron affinity χ | energy from Ec to the vacuum level |
| electron-volt (eV) | energy gained by an electron falling through 1 V: 1.602·10⁻¹⁹ J |
| Fermi–Dirac distribution f(E) | probability that a state at energy E is occupied in equilibrium |
| Fermi level EF | energy at which f = ½; constant throughout a system in equilibrium |
| flat-band voltage VFB | gate voltage at which the semiconductor bands are flat |
| hole | missing electron in an almost full band, acting like a particle with charge +e |
| hopping energy t | coupling between neighbouring atomic orbitals; sets the band width (~4t in a chain) |
| interface traps Dit | electronic states in the gap at the Si/SiO₂ interface |
| intrinsic, ni, Ei | undoped; its carrier density n = p = ni; its Fermi level, near mid-gap |
| inversion | MOS state in which the surface holds more minority than majority carriers |
| kT | thermal energy scale: 25.9 meV at 300 K |
| mass-action law | n·p = ni² in equilibrium, independent of doping |
| MOSFET | metal-oxide-semiconductor field-effect transistor |
| n-type / p-type | doped so that electrons / holes are the majority carriers |
| orbital, energy level, state | a standing-wave pattern of an electron; its energy; one seat for one electron (orbital × spin) |
| Pauli exclusion principle | at most one electron per quantum state |
| Poisson's equation | links charge density to the curvature of the potential: d²φ/dx² = −ρ/ε |
| quasi-Fermi levels EFn, EFp | separate Fermi levels for electrons and holes out of equilibrium |
| recombination | an electron falls into a hole; both disappear, releasing heat or light |
| saturation | MOSFET region where the drain current hardly depends on VDS |
| subthreshold swing S | gate voltage per decade of current below threshold; ≥ 60 mV/dec at 300 K |
| surface potential ψs | band bending at the semiconductor surface, in volts |
| threshold voltage VT | gate voltage at which strong inversion starts (ψs = 2φF) |
| vacuum level | energy of an electron at rest just outside the material |
| valence band, Ev | highest band that is (nearly) full in a semiconductor; its top edge |
| valence electrons | outer electrons that form bonds (4 in Si) |
| work function Φ | energy from EF to the vacuum level |
