A Technical Introduction to Laser Diodes

Chapter 1

405nm diode laser

Laser Diode Physics

Author: Dr. Matthias Pospiech and Sha Liu

University of Hannover, Germany

Page Updated: October 22, 2019

1.1 Band Stucture of a Semiconductor Laser

In a crystal, the discrete energy levels of the individual atom broaden into energy bands. Each quantum state of the individual atom gives rise to a certain energy band. The bonding combinations of states become the valence bands (VB) of the crystal, and the anti-bonding combinations of these states become the conduction band (CB). The energy difference between VB and CB is called energy gap. If the valence bands are partly filled, this material is p-type, if the conductive bands are partly filled, this material is n-type. Here Fermi level is used to label the occupation conditions of electrons in the semiconductor, it is the energy level to which electrons occupy. Fermi level (EFP) on p-type is near the valence band and EFN on the n-type is near the conductive band.

When two semiconductors with different band structures are combined, a heterojunction is formed, a p-n heterojunction is called a diode. Electrons and holes transfer to the other side, because of different Fermi levels respectively. They recombine with each other, leaving the p-side with negative charge and n-side with positive charge, this region is called space charge layer (SCL). A built-in voltage V0 appears because of the charge transfer and recombination. When there is no applied voltage, the Fermi level is continuous across the diode EFP = EFN, as indicated in Figure 1(a). The built-in voltage prevents electrons in conduction band on n-side from diffusing into conduction band on p-side, it is similar for holes in valence band, so the majority carriers can not flow into the space charge layer. An applied voltage V can separate EFP and EFN by eV, and the Fermi level is not continuous across the diode. The seperate Fermi level in each side is called quasi Fermi level, as indicated in Figure 1(b).

semiconductor-laser-heterojunction

laser diode heterojunction before and after current bias

Due to the applied voltage, the recombination process takes place and the diode current establishes. When applied voltage is greater than built-in voltage , the space charge layer is no longer depleted. Now at the junction, more electrons are injected into the conductive band at energies near Ec than electrons in valence band at energies near Ev. This is the population inversion, and the inversion region as indicated in Figure 1(b) is called active region. In Figure 2, the carrier concentration-x is shown in two dimensional coordinate in the space charge layer (SCL). [2]

laser diode p-n junction

carrier concentration under current bias

1.2 Recombination

There are three kinds of transitions that are important in laser diodes, which occur between the conduction and valence bands of the material. They are stimulated absorption, spontaneous emission and stimulated emission in Figure 3.

stimulated emission of a laser diode

laser diode stimulated emission

After defining R(abs), R(spon), R(stim) as the rate of absorption, spontaneous emission and stimulated emission respectively, the relationship between the three processes can be described by the following equation.

spontaneous emission mathematical model

And the rates can be expressed by the Einstein coefficients which are defined in the following way:

  • B(12) transition probability of induced absorption
  • A(21) transition probability of spontaneous emission
  • B(21) transition probability of induced emission

Here we only cite the the ratio of spontaneous emission to stimulated emission, the deduction details can be read in website [3]

transition probability of induced absorption laser diode physics

In the stimulated emission, a photon is strongly coupled with the electron, the photon can cause the electron to decay to a lower energy level, releasing a photon of the same energy. The emitted photon has the same direction and phase as the incident photon. When the stimulated emission is dominant, the light is amplified, and laser occurs. From this equation, we can see that stimulated emission is dominant when hw << kbT. From Fermi-Drac statistic law, under this condition, the probability of finding an electron in the conduction band has to be greater than the probability of finding an electron in the valence band, so there must be a population inversion. As mentioned before, in a laser diode, population inversion is achieved when EFN − EFP > Eg, where Eg is the bandgap energy and EFc and EFv are the Fermi levels of the conduction band and valence band, respectively. These Fermi levels can only be separated by pumping energy in the form of electrical current into the semiconductor laser. Electrons and holes are injected into the active region from n- and p-doped semiconductor cladding layers. The injection current required to achieve lasing is known as the threshold current, details will be given in section 2.2.

1.3 Laser Diode State Density

When we calculate various optical properties, such as the rate of absorption or emission and how electrons and holes distribute themselves within a solid, we need to know the state density. The density of state is described in the number of available states per unit volume per unit energy. So in conductive band and valence band, the state density of energy is:

   laser diode state density mathematical formula

According to Fermi-Dirac statistic law, electrons in the band obey the following equations:

Fermi-Dirac statistic law

so in conductive band and valence band, the electrons and holes distributions are:

laser valence band equation

The above equations are mainly cited from website [3]

Figure 4 is the schematic graph of the distributions described by the functions above. (a) is at 0K, and the charges first fill the lowest energy state. (b) is at a certain temperature above 0, so some charges are excited to higher energy states.

engery state equation for laser diodes

laser diode energy states

1.4 Direct Band Bap and Indirect Band Gap

The recombination process mainly depends on the band structure. Generally, there are two kinds of band structure, direct band gap and indirect bandgap. Direct band gap means that in the E-k diagram, electrons at the minimum of the conduction band have the same momentum as electrons at the maximum of the valence band, and for an indirect band gap, the electrons do not have the same momentum, as indicated in Figure 5. The recombination of an electron near the bottom of the conduction band with a hole near the top of the valence band requires the exchange of energy and momentum. For indirect band gap recombination, the energy may be carried off by a photon, but one or more phonons are required to conserve momentum. This multiparticle interaction is improbable and the recombination efficiency in the indirect band gap material is lower than in the direct band gap material. [6]

The majority part of semiconductors are indirect band gap material, compared with them, direct bandgap materials are preferred for laser diodes. Direct bandgap structures semiconductor-laser-band-gap maximize the tendency of electrons and holes to recombine by stimulated emission, thus increasing the laser effciency. For example, the direct band gap crystal aluminium gallium arsenide (AlGaAs) is often used for laser diodes with wavelengths between 750nm and 880nm. AlxGa1-xAs, through changing the x, the ratio of the aluminium to gallium can be adjusted to vary the band gap and thereby control the wavelength. [7]

1.5 Optical Feedback Mechanism

In a high efficiency laser, a resonator must be formed which has the ability not only to amplify the electromagnetic wave, but also to feedback to it. A laser resonator generally consists of two parallel mirrors perpendicular to the optical axis. Space between the mirrors is partially occupied by the amplifying material. This structure, called a Fabry-Perot Resonator, is obtained in a laser diode by cleaving the ends of the crystal. Because the refractive index has a jump at the interface of the crystal and other material, the mirror facet functions as a reflective surface. In some cases, special coatings are used to enhance either the reflectivity r, or transmissivity t, of the facet. When the resonator is brought to a state of population inversion, photons produced by spontaneous emission are amplified and repeatedly reflected by the front and rear facets. In homojunction LD, there is no optical confinement in the direction perpendicular to the optical axis, so the elelctromagetic waves in any direction not parallel to the optical axis of the resonator will pass through the sides of the resonator. In heterostructure LD, the waveguide will confirm the wave in the active region (see section 2). The component of the spontaneously emitted photons, which travels parallel to the optical axis, will be repeatedly reflected by the mirror facets. As the electromagetic wave travels through the semiconductor material, it is amplified by stimulated emission. At each reflection, the wave is partially transmitted through the reflective facets. Laser oscillation begins when the amount of amplification becomes equal to the total amount lost through the sides of the resonator, through the mirror facets and through absorption by the crystal. The details of Fabry-Perot can be found in the ‘Protokoll laser diodes, theory part’. [8]

fabry-perot-laser-resonator