Semiconductor physics
Every topic in this subject left a piece ready for this one: the photon, tunneling, Fermi-Dirac statistics, the Pauli principle. Here they all come together in the band gap and the Fermi level, which is where a diode comes from.
01From atomic levels to bands
The unit on atomic models ended with a pending idea: when \( 10^{22} \) atoms come together in a crystal, Pauli forces the single atomic level of each one to split into as many levels as there are atoms, so close together that they form a continuous band of allowed energies. Between two allowed bands there can be a band gap, of width \( E_g \), where no possible state exists. All the electricity of a material depends on how this scheme is arranged.
| Material | Bands | \( E_g \) | Example |
|---|---|---|---|
| Conductor | The valence band is partially filled, or overlaps the conduction band | — | Copper, aluminum |
| Semiconductor | Full valence band, narrow band gap: thermal agitation is enough to cross it | 0.5 to 1.5 eV | Silicon 1.12 eV · germanium 0.66 eV · gallium arsenide 1.42 eV |
| Insulator | Full valence band, wide band gap: almost nobody crosses it | > 4 eV | Diamond, 5.5 eV |
The difference between a semiconductor and an insulator is not one of nature, it is one of the size of \( E_g \) compared with \( kT \): at room temperature, \( kT \approx 26 \) meV, an \( E_g \) of 1.1 eV still lets a fraction —small, but not zero— of electrons cross on their own.
02The effective mass
Inside a crystal, an electron feels not only the external force: it also feels the periodic potential of all the nuclei. The net result can be summarized, for most calculations, in a single number: the effective mass \( m^* \), which replaces \( m_e \) in \( F=m^*a \) and makes the electron in a crystal behave like a free particle, but lighter or heavier depending on the curvature of its energy band.
| Material | \( m^*_e / m_e \) (electrons) | \( m^*_h / m_e \) (holes) |
|---|---|---|
| Silicon | 0.26 | 0.39 |
| Germanium | 0.12 | 0.30 |
| Gallium arsenide | 0.067 | 0.45 |
The electron in gallium arsenide weighs, for practical purposes, fifteen times less than a free one. It responds faster to an electric field and reaches a higher velocity before colliding with the lattice: that is why GaAs is used in the highest-frequency transistors, even though it is more expensive and harder to work with than silicon.
03Intrinsic carriers and the law of mass action
In a pure crystal —with no added impurity—, thermal agitation moves some electrons from the valence band to the conduction band. Each electron that crosses leaves behind a hole: an empty state that behaves, for almost all purposes, like a mobile positive charge. The concentration of each one, in that pure material, is the intrinsic carrier concentration \( n_i \).
You choose the material, doping type and concentration; the band diagram places the Fermi level and the panel calculates \( n \), \( p \) and \( n_i \), verifying the law of mass action in each case.
04Doping and the Fermi level
Adding impurities —phosphorus to donate electrons, boron to accept them, in silicon— does not change \( n_i \), which depends only on the material and the temperature. What changes is how far \( n \) departs from \( p \), and with that, where the Fermi level falls relative to the midpoint of the band gap, \( E_i \).
The same calculation explains why a semiconductor device stops working if it is heated too much: \( n_i \) grows exponentially with \( T \), and a point is reached —the intrinsic temperature— at which it exceeds the doping concentration. From then on the material forgets that it was doped and behaves as intrinsic: germanium, with its narrow band gap, reaches that temperature well before silicon, and that is one of the reasons germanium practically disappeared from modern electronics.
05The end of the bridge
This unit closes the subject where its curriculum description promised: in silicon. Each earlier topic contributed a piece that this last one uses without explaining it again. The photon from the photoelectric effect unit is the light that a photodiode converts into current by crossing this very band gap. The tunneling of the barrier is the leakage through the gate oxide of a MOSFET. The Fermi-Dirac statistics of the previous unit is, quite literally, the formula \( n = N_c e^{-(E_c-E_F)/kT} \) that was just used in the lab on this page. And the Pauli exclusion principle, two units back, is the ultimate reason bands exist instead of isolated levels.
A diode is a junction between an n-doped region and a p-doped one, with their Fermi levels forced to equalize at equilibrium. A bipolar transistor is two such junctions, back to back. A MOSFET controls, with an electric field, how many carriers are available in a region of the crystal. None of those three sentences makes sense without the band gap, the Fermi level and the law of mass action on this page.
06In the lab
With the lab on this page, for n-type doped silicon at \( 10^{17} \) cm⁻³, note down \( n \) and \( p \). Verify by hand that \( n\,p = n_i^2 \) and calculate how many orders of magnitude separate the majority carriers from the minority ones.
Raise the temperature until \( n_i \) of germanium approaches \( 10^{16} \) cm⁻³. Repeat for silicon. Compare the two temperatures and explain why germanium diodes fail sooner with heat.
Calculate \( E_F - E_i \) for p-type doped silicon at \( 10^{15} \), \( 10^{17} \) and \( 10^{19} \) cm⁻³. Plot it against the logarithm of the concentration and verify that the relationship is linear.
07Common mistakes
- Thinking that doping changes \( n_i \). \( n_i \) depends only on the material and the temperature; doping changes \( n \) and \( p \) separately, not their product.
- Forgetting the law of mass action when calculating the minority carrier. Once the majority carrier is known, the minority one follows directly from \( p = n_i^2/n \).
- Confusing effective mass with the actual mass of the electron. It is a parameter that summarizes the effect of the crystal, and it can be larger, smaller or —in some bands— even negative.
- Believing that the Fermi level is an occupied energy level. In a semiconductor it typically falls right in the band gap, where there is no state at all.
- Ignoring the temperature dependence of \( n_i \) when designing a circuit. It is exponential, and it is the cause of a good share of thermal failures in power electronics.
- Thinking that germanium is inferior to silicon in everything. Its higher \( n_i \) is a thermal disadvantage, but its narrow band gap also gives a lower forward voltage drop, useful in some signal diodes.
08Self-assessment
What distinguishes a semiconductor from an insulator?
The size of the band gap \( E_g \): narrow in the semiconductor (thermal agitation crosses it), wide in the insulator.
What does the effective mass represent?
The net effect of the periodic potential of the crystal on the electron, summarized in a single number that replaces the actual mass in \( F=m^*a \).
What does the law of mass action say?
That the product \( n\,p = n_i^2 \) is constant in thermal equilibrium, for a given material and temperature, doped or not.
What happens to the Fermi level with n-type doping?
It rises from the midpoint of the band gap toward the conduction band.
Why can a semiconductor device stop working if it is heated too much?
Because \( n_i \) grows exponentially with temperature and, past the intrinsic temperature, exceeds the doping concentration: the material stops behaving as extrinsic.
Why is gallium arsenide used in high-frequency transistors?
Because its electron effective mass is much lower than that of silicon, so its electrons respond faster to an electric field.
09Further reading
- Simon M. Sze and Kwok K. Ng. Physics of Semiconductor Devices. 3rd ed., Wiley, 2007. The reference book for this whole unit: bands, effective mass, carrier statistics and the law of mass action, with the tables of constants for each material.
- Ben G. Streetman and Sanjay Banerjee. Solid State Electronic Devices. 7th ed., Pearson, 2014. A more gradual introduction to the same topics, designed for the first devices course that follows this subject.
- Charles Kittel. Introduction to Solid State Physics. 8th ed., Wiley, 2004. The origin of the band structure from the crystal, with the detail that this unit takes for granted.
- Robert F. Pierret. Semiconductor Device Fundamentals. Addison-Wesley, 1996. The classic treatment of the Fermi level and the intrinsic temperature, with the \( n_i(T) \) graphs for the most widely used materials.