27 Physics of Semiconductors and semiconductor laser
Devendra Mohan
Learning Outcomes
From this module students may get to know about the following:
• Semiconductor Diode
• Semiconductor Laser
Physics of Semiconductors and semiconductor laser:
Introduction
In semiconductor laser theory, the optical amplification is achieved in a semiconductor material. The choice of material depends on the desired wavelength and properties such as modulation speed. It may be a bulk semiconductor, but more often a quantum hetero-structure. For creating the population inversion, pumping may be done electrically or optically.
Light is generated in a semiconductor laser by radiative recombination of electrons and holes. In order to generate laser different models are assessed wherein many –particles interactions are considered which otherwise in simple models are neglected. In Free carrier model carrier plasma is simply seen as a reservoir that relaxes the carrier distributions. Simple models for the gain coefficient are used to obtain a set of rate equations that enable us to dynamically calculate the time-dependent laser response.
The semiconductors materials can be broadly classified into two categories namely, Direct band gap and Indirect band gap materials .A general picture of the same is explained as:
(Ref: No. 6)
Crystal momentum p or the wave vector k for electron in a crystal is represented by the relation
P=2πhk,h being the Plank’s constant.
When electron makes a transition downwards from conduction band to valence band and recombines with hole, a photon is emitted with corresponding energy difference of E2-E1
So that
ℎf− ?2 − ?1 = Eg
However, in case of indirect band gap, the lowest energy conduction band electrons have a non zero momentum.
(Ref: No. 6)
Thus radiative recombination in this case requires momentum transfer due to photon participation .so frequency of emitted radiation is given by the equation
ℎf= ?2 − ?1; ℎfp ≈ Eg
Here fp is photon frequency. Thus recombination probability of an electron hole pair leading to the emission of radiation is much smaller in indirect band gap materials than in direct band gap materials .hence the direct band gap materials like gallium arsenide(GaAs) are used in the fabrication of semiconductor lasers.
The p-n junction
The commonly used semiconductor material for fabricating laser diodes is gallium arsenide which emits light at a wavelength ~ 8400 Ao in the near infrared of optical Spectrum. The Wavelength of emission can be varied from 6300 A (visible) for the III-V mixed semiconductor gallium arsenide phosphide to 8.5 μm (infrared) for the IV-VI semiconductor lead selenide.
In intrinsic semiconductor materials, the ambient temperature raises some electrons from the valence band into the conduction band by thermal excitation, leaving behind an equal number of holes in the valence band. Photon emission due to the recombination of electron and holes in intrinsic semiconductors is too small. Therefore, to increase the photon emission, we need to increase the charge carrier density by adding dopnts .Thus the energy level diagram modifies accordingly, resulting in an n-type semiconductor material when donor materials having excess electrons are added to it . On the other hand, when acceptor material rich in holes are added to it, a p-type semiconductor material is formed.
((Ref: No. 6)
When these two materials are joined to form a p-n junction, the electrons and holes tend to move in opposite directions across the physical boundary by a diffusion process, setting up a depletion layer sandwiched between the p-type and n-type materials .across the depletion layer, a barrier potential is set up in such a way that it inhibits further flow of charge carriers, thus establishing equilibrium condition. The electrons and holes which manage to find themselves in the depletion region may recombine with the emission of a photon corresponding to the energy gap Eg. In a normal p-n junction in electronic devices, these photons never see the light of the day since their numbers are small and efficiency of the production is small in Si and Ge materials typically used in electronic devices.
(Ref: No. 6)
Degenerate Semiconductor Diode
When heavy doping is done then the semiconductor material almost acts like a conductor and is referred to as degenerate semiconductor.
Degenerate p-type and n-type diodes of band gap semiconductor materials whose energy level diagram is.
(Ref: No. 6)
- In ordinary semiconductor junction, electrons and holes will tend to move towards the centre under the influence of diffusion and will be balanced by the barrier potential set up across the junction.
- Charge carriers in this case are expected to recombine in the depletion region and generate photons, this region is now referred as active region.
- When p-n junction is forward biased, charge carriers are injected into the active region from either side producing photons as shown
(Ref: No. 6)
- When degenerate semiconductor p-n junction is forward biased, holes are injected into active region from n-side. So these devices are simple referred as ‘injection laser diodes’ or ILDs or called ‘laser diodes’.
- In case of non radiative recombination of charge carrier, energy released is dissipated in the form of lattice vibrations. but in case of radiative recombination , energy released in the form of photon whose frequency is
E? = ℎf = hc/?
Where ג is optical wavelength.
By putting values of h and c we get ?= 1.24/E?
Units of ג and Eg are µm and eV.
Estimation of forward bias voltage:
When the degenerate semiconductor diode is forward biased, potential hill is decreased
(Ref: No. 6)
- Under the influence of external voltage, the Fermi level on the n-side is different from that of the p-side and is shown as
(ф?−ф?)=??
Where ф? Fermi is level in conduction band of n-side and ф?is Fermi level in valence band of p-side.
- The distribution of charge carriers in semiconductor materials obeys Fermi statistics. The probability that a state in the conduction band at energy Ec is occupied by an electron is given by the Fermi function:
??(??)=[1+???(??−ф?/??)]−1
Similarly the probability that an electron will occupy a state in the valence band on the p-side will be
?? (?? ) = [1 + ??? (??−ф?/?? )]−1
Now assuming that stimulated emission process is overwhelming, the contribution due to spontaneous emission can be ignored. If stimulated emission coefficient Bcv is defined as the probability that a transition per unit time will occur from a state in the conduction band to one in the valence band, the inverse process which is stimulated absorption, will have
a probability Bvc. The photon cloud density in the active region is ρ(f), where f is frequency corresponding to the energy gap Eg.
Rate at which the photons are generated /absorbed due to transition between conduction band and valence band is
|??/??|???????? = ???? ?? (1 − ?? )?(?) ………………………………………1
and
|??/?? | ?????????? = ???? ?? (1 − ?? )?(?) …………………………………2
?ℎ??? ? ?? ?ℎ? ??????????????? Constant which contains a factor concerning the density of states in the conducting and valence bands. The term ?? (1 − ?? ) gives a composite probability that a state will be occupied by an electron in the conduction band and that there is a state which is vacant in the valence band (presence of a hole) so that a transition will occur causing emission. ?? ????? ?ℎ?? ?ℎ? ???????? ?????? ?? ? ??? ??????? ?? ?ℎ????? , the photon emission rate should exceed the absorption rate , that is
|??/??|???????? > | ??/ ??|??????????
Using eq. 1 and 2 in eq 3, we get
Threshold current density:
The photons that are generated in the active region lack direction because of spontaneous emission. To impart direction to the emitted light and therefore to achieve laser output rather the just light emission, it is necessary to provide a semiconductor junction in the vicinity of the active region. This can be done first cleaving the two sides of the crystal, making them parallel to each other, to a high degree of accuracy. Then by polishing these two faces of the crystal, the crystal air boundary provides a natural reflecting surface. in order to channel the photons along the length of the device, the other faces along the breadth are roughly ground the length of the device, the other faces along the breadth are roughly ground or deliberately etched or roughened, to eliminate reflections from these two sides. By further mirroring the back side of device by coating it with a totally reflecting material, the laser beam can be made to emerge from the front surface of the device as shown in fig.
Because a large number of charge crriers move into the active region when the current flows across the junction due to external bias voltage, the net area of the active region is about 1mm2 and its width is 5 to 10μm. these junction diode lasers are more efficient than the conventional lasers because almost every hole-electron pair injected into active region generates a useful photon, however part of electrical energy is lost in the form of heat across the electrical resistance internal and external to the diode. But not all generated photons will contribute to the laser radiation emerging from the junction.
In order to sustain laser action in the junction, the emitted photon rate must overcome following losses:
(a). the absorption coefficient α0 of the device for the photon generated.
(b). the losses due to the transmission at the partially reflecting surface at the emitting end of the laser diode.[if R is the optical power reflection coefficient of this surface, then the transmission loss per unit length is given by 1/L log(1/R), where l is the active layer length].
(c). the diffraction losses in the active region, which are approximately given by an empirical relation given below
??=0.42(?/??2)
Here ? is the refractive index of the semiconductor material and w is the width of the active region.
If the total current paasing through the active region from positive terminal (p-side to the negative terminal (n-side) when forward biased is I, then the current density J is expressed as
?=?(?? ???????)/???? ?????? ?ℎ??ℎ ?? ?? ??????? ?? ??.??????
Thus the number of charge carriers injected into active region per unit area per second is J/e. if the quantum efficiency ? is the ration of number of photons produced per unit hole –electron pair, then number of photons produced per unit area per second is given by ?J/ew, where w is the width of the active region. If the effective line width of photon emission due to spontaneous is given as Δf, then rate of spontaneous emission per unit frequency interval per unit volume is given by J?/ewΔf.
As the active region of volume V acts like a resonator cavity, as per the theory of a resonator cavity, it can sustain a number of modes Nf per unit frequency interval per unit volume neglecting dispersion
??=8??3/?2?
Thus rate of spontaneous emission per mode is
?=??/??Δ???
=????Δ?∗?2?8??3
Hence gain β per unit length of the active medium is
?=??/?
=???2/8???Δ??2
The point at which the gain in the medium is able to overcome the net loss in the medium is usually referred to as the threshold condition, which is represented in terms of the current density as Jth.
??ℎ?2 / 8????Δ?=?0+??+????
If the two ends of the optical resonator cavity on the either side of the active region have reflections coefficients r1 and r2, then the transmission losses through them is given by the eq.
??=1/????1/?1?2
Therefore
??ℎ=8????Δ?/?2[?0+1/????(1?1?2⁄)+????]
If one end is mirrored with 100% reflecting material, and r1 =1 and r2 =R then
??=1/????1/?
Since the partially reflecting side of the optical cavity is nothing but the semiconductor –air interface, the reflection coefficient can be calculated from the relationship
?=√??+1/√??−1
Where εr is the ratio of the relative permittivity of the semiconductor material to the permittivity of free space.