2. Diodes and transistors

A Quantum Mechanical View

Abb. ##: Bohr's atomic model and ribbon model

The above model of conductivity in semiconductors will now be considered in a little more depth. In the Bohr atomic model (Abbildung ##, 1), it is assumed that the electrons in the atom move in certain circular orbits around the nucleus - similar to the planets in the planetary system. Here, more strongly bound electrons are in closer orbits and weaker ones are in orbits further out. This also behaves similarly to satellites in the gravitational field, which, when farther from the center, are more weakly attracted. Bohr postulated 3 axioms to make the model and measurement results fit together plausibly:

1. Circular orbits are discrete. There are only certain paths on which the electrons may move
   (and thus discrete energies for the electrons).
2. Each jump of an electron from one orbit to another is accompanied by an energy absorption or release.
3. The exact energy amounts of the orbits result from quantum physics.

Unfortunately, this representation produces quite a few physical contradictions - but the model is sufficient for explaining conductivity in semiconductors¹⁾. The Bohr atomic model and the Octet rule (tendency of higher orbits to be saturated with 8 electrons) are enough to gain a deeper insight into semiconductor physics.

Abbildung ## 1a shows the electrons in the discrete circular orbits, i.e., in a $x$-$y$ coordinate system. More strongly bound electrons are shown in black on inner orbits; on the outermost noncompletely occupied green orbit, electrons are shown in blue. In addition to the occupied orbits, other, outer, nonoccupied orbits are also present (blue in Abbildung ## 1a).
The same electrons can also be sorted into an $x$-$W$ coordinate system (see Abbildung ## 1b). Here $W$ is the binding energy, or work released when an unbound electron jumps into the orbit under consideration. The origin of the binding energy (i.e., the binding energy of an unbound electron: $W=0$) is above the unoccupied levels. Thus, as expected, the magnitude of the binding energy of the fully occupied level is the highest. The discrete orbits also result in discrete energy levels on the energy axis.

If we consider a section of a solid instead of a single atom, the electron configuration changes. In Abbildung ## 2a, the situation is again shown in the $x$-$y$ coordinate system. Here, the inner electrons and the nucleus are now reduced to a single, yellow circle with the resulting charges. The electrons from the (in the example atom) partially occupied levels now satisfy the octet rule.
However, depending on the element, there are different properties of the electrons here. In metals the electrons are freely movable - thus a good conductivity is measurable, but in semiconductors initially not. This statement cannot be explained by the Bohr atomic model, but by the band model and some quantum physics very well. As already for the atom, the electrons of the solid are now entered into a $x$-$W$-coordinate system. Here are now many electrons from the same atomic levels close to each other. The laws of quantum physics forbid that electrons occupy exactly the same energy level at the same location. This results in a broadening of the discrete levels into energy bands (Abbildung ## 2b). In the example, a semiconductor is drawn. In the semiconductor, the energetically highest-lying band is completely occupied. The energetically highest-lying and occupied band is called the valence band, and the next highest non-occupied (or not fully occupied) band is called the conduction band. The energetic gap between the conduction and valence bands is called the band gap. The conduction band of the semiconductor just corresponds to the electrons strongly bound in the $x$-$y$ coordinate system. Thus, there are initially no mobile electrons in the semiconductor (the conduction band is unoccupied, and the valence band is fully occupied). The band gap in semiconductors is approximately in the range of $0.1 ... 4eV$ ²⁾
Electrons can be released from bonds with addition of energy. An electron can get the energy it needs in two ways: Either by an excitation of the electromagnetic field, i.e. a quantum of light, or by an excitation of the elastic field, i.e. lattice vibrations of the crystal. Light quanta are also called photons, quantized lattice vibrations are also called phonons. In Abbildung ## 2a, top left, a photon is absorbed by an electron, thus breaking the bond. The electron absorbs the energy of the photon. It is excited and raised by that amount on the $W$ axis. It also follows that only quanta of energy can be absorbed that allow it to be lifted to an existing and free level. The energy absorption results in an electron in the conduction band that is mobile in the crystal. In addition, the electron leaves a positively charged hole in the valence band. This process is called generation of electron-hole pairs. Both electron and hole conduction contribute to conductivity in the undoped semiconductor. The reverse process - the recombination of electrons with holes, occurs in silicon after a few tens of microseconds, or a few tens of micrometers. In this process, the amount of energy in the bandgap is released again.
Abb. ##: band model and doping

Since the crystal lattice already contains thermal energy at room temperature (the atomic trunks move), phonons are also present in the crystal. The phonons have a broad, energetic distribution. At room temperatures, the average energy of a phonon is $k_B\cdot T = 26 meV$ ($k_B$ is the Boltzmann constant). In silicon, about 0.000 000 01% (about one in $10^{13}$) of phonons have sufficient energy to lift an electron from the valence band to the conduction band. However, this is sufficient to provide about 10 billion charge carriers ($10^{10}$) to pure silicon at room temperature and a volume of $1 cm^3$ (about $5\cdot 10^{22}$ atoms). These charge carriers enable the intrinsic conduction described above.

The previous subchapter also described another way of increasing the number of charge carriers: doping with impurity atoms. This requires that the semiconductor material used is very pure and crystalline. Impurities and crystalline impurities can also produce conductive charge carriers. The semiconductor material should have less than one defect per $10^{10}$ atoms (equivalent to about one person to humanity). In this case, intrinsic conduction would predominate in it. For doping, one impurity atom is added to $10^5...10^{10}$ semiconductor atoms. In the band model, n-doping results in additional electrons in the conduction band and additional positively charged fixed recombination centers due to the fixed positive atomic hulls, so-called (electron) donors (Abbildung ##: red marking for n-doping in b,c,d). A p-doping creates additional holes in the valence band and fixed negatively charged recombination centers, so-called (electron) acceptors.

2.2 PN junction and operating principle of a diode

Abb. ##: circuit symbol of a diode, with the designations of the doping and electrodes

For the circuit symbol there are the following mnemonics: Viewed from the cathode side, the circuit symbol resembles a „K“. From the anode side, the circuit symbol resembles a horizontal „A“. Mnemonic for sorting: Kathode - Negative - Anode - Positive.

In the simulation shown below, three examples of diodes in circuits are considered.
In the first example on the left, the voltage source is polarized so that the diode is forward biased. The light bulb is on.
In the first example on the right, the diode is reverse biased. The light bulb remains dark.
In the second example (middle), an ideal diode - i.e. a directional current valve - can be seen. Next to it is the transfer characteristic or current-voltage characteristic (in this case also called diode characteristic). The voltage at the diode is plotted on the x-axis, the current through the diode on the y-axis. The diode is non-conducting at all voltages below 0V, and conducts current at all voltages above 0V.
In the last example (right) a real diode is connected. The real diode differs from the ideal diode in the following ways:

1. The real diode does not have such a steep slope.
2. The real diode has a non-linear resistance; it is not an ohmic resistor.
3. The real diode seems to require a minimum voltage to allow a current to flow.

The details of the real diode are described below.

Abb. ##: Evolution of the p-n junction

In a diode, two differently doped layers of silicon collide: p-doped silicon („p-crystal“) on one side and n-doped silicon („n-crystal“) on the other.

The situation without external voltage will be considered first (compare Abbildung ##). On the n-doped side, many free-moving electrons will dissolve at room temperature, leaving acceptors stationary. The same can be seen on the p-doped side: the free-moving holes leave behind donors. in the middle, at the pn-junction, both moving charge carriers, electrons and holes, meet. When they meet directly, the two charge carriers will cancel each other out, they recombine. This creates a photon (electromagnetic vibration) and/or a phonon (lattice vibration). The recombination forms a layer, the barrier layer, which is largely free of free moving charge carriers. The barrier layer initially acts as an insulator.

With external voltage $U_D$ on the diode, two cases are to be distinguished (Abbildung ##):

1. Applying a positive voltage from p-doped side to n-doped side
   (diode voltage = forward voltage $U_D = U_F$, $U_F>0$).
2. Applying a negative voltage from p-doped side to n-doped side
   (diode voltage = reverse voltage $U_D = -U_R$, $U_R>0$).

Abb. ##: Functionality of a semiconductor diode

A triangular or sawtooth signal can be applied to create the diode characteristic (see Falstad simulations).

Forward voltage $U_F>0$

If a positive potential is applied to the p-doped side, the freely moving holes there are driven towards the pn-junction. Negative potential is then applied to the n-doped side, which also drives the freely moving electrons towards the pn-junction. At the pn-junction, holes and electrons can neutralize each other. Thus, holes from the positive terminal and electrons from the negative terminal can continue to move in, and an electric current flows through the diode. The diode is connected in the conducting direction. In common diodes, about $0.7 V$ is dropped in the forward direction. This means, of course, that the current does not pass the diode completely without resistance, but that the forward voltage $U_S$ of about $0.7 V$ must be applied from the outside.³⁾. This voltage results from the energy difference of the band gap related to one electron, which is about $1.1eV$ for silicon, but is reduced by thermal energy (phonons). On closer inspection, the curve resembles an exponential function. This can be described by the Shockley equation:

Several consequences can be derived from the exponential function:

1. The forward voltage $U_S$ of about $0.7 V$ depends on which current (/voltage) range is considered. $0.6...0.7V$ is a suitable value for currents in the range of $5...100mA$. This range is used in most circuits. For smaller currents, the forward voltage $U_S$ also decreases (e.g., for $5...100mA \rightarrow$ about $0.4V$, $0.1...1mA \rightarrow$ about $0.2V$, see the following Falstad simulation).
2. The forward voltage and the voltage response are temperature dependent. The higher the temperature, the more current flows for the same voltage. So if a diode is connected directly to a voltage source, at currents higher than about $50mA$ the current would increase directly via self-heating ⁴⁾ up to / above the maximum current.