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Semiconductor Gain Media- Structure, Spectrum

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In this lecture, we will look at the basics of some LED and laser diode structures.
Double Hetero-structure design

The basic structure and the gain medium of a laser diode are also the same as of LED. In laser diodes, we have some additional mirrors that get fabricated at the end facets, but the
material used for the generation of photons and the mechanism of generation of photons are similar for LEDs and Laser diodes. Typically, it is a forward-biased pn junction which is shown in the figure above.
The pn junction diode is typically formed by doping the semiconductor substrate with p-type and
n-type material as shown, and the depletion region is formed in the junction across the structure as shown. There is a metallic contact through which the injection current is allowed to flow through the diode. The recombinations typically happen in the depletion region, with typical
widths of 1-10 μm in most of the semiconductor materials. The width of the depletion region is decided by the injection current and dopant concentration. Now, this kind of structure is called as
homo-junction because the material that constitutes the p-type and the n-type doping has the same bandgap. Since the bandgap on either side of the junction is the same, there is no possibility of confinement of the carriers. It means that in the presence of an injection current I, the carriers are generated anywhere within this width of the depletion region and so, the
recombinations can actually happen in a widely distributed fashion over this entire depletion region. It is a very inefficient way of generating light. It results in very low internal quantum efficiency. LEDs were initially made with this kind of arrangement. However, a modification
that is most popular is a hetero-junction, fabricated in a semiconductor structure. Hetero junction consists of a double heterostructure, described below.

Double heterostructure consists of p-type and n-type doping as before. However, this structure is different from homo-junction in that, a thin active layer is grown in between the two p and n type doped material. The bandgap of the p and n-type material is chosen to be identical (Eg1). The
thickness of this active layer is about 0.1 μm, which are several orders of magnitude smaller than the width of the depletion region itself. The material for the thin layer is chosen such that the bandgap of that material is smaller than Eg1. This results in two benefits - carrier confinement
and optical confinement. The band structure is shown in the right, which shows the lowest energy level of the conduction band and the highest energy level of the valence band.
In the presence of a forward-biased current, holes are injected from the p-side and the electrons
are injected from the n-side towards the junction. However, there is a large potential well at the junction due to the difference in band gaps, because of which all the carriers get confined in this active region. This results in carrier confinement. Since the width of the active layer is much smaller, the carrier density becomes larger. The larger carrier density increases the possibility of
radiative recombinations and thus, the process of growing a thin active layer between the p-type and n-type helps in increasing the radiative recombination. Now, the second benefit is optical
confinement. In semiconductors, it is found that materials with a larger bandgap have smaller refractive indices. It means that the light emitted in the lower bandgap active region actually
experiences the higher refractive index. The plot in red shows the refractive index profile across the band for a double-heterostructure. The generated photons in the active region are trapped in a region of higher refractive index. Thus, the structure acts as a waveguide. In fact, the refractive index profile of a step-index fiber looks exactly similar to the profile shown here. The boundary condition imposed by this refractive index profile allows only specific transverse field patterns –also referred to as the transverse modes of the waveguide. Depending on the refractive index contrast and the width of the active layer, the structure can support single-mode operation for a
given wavelength, which is the most preferred mode of operation in optical communication systems. The transverse profile of this single mode is expected to follow a Gaussian profile. We will see the details while we study the modes of an optical fiber.

One example of a double-heterostructure design with Gallium Arsenide substrate is shown in the figure above. On one side, it is doped with a p-type material and on the other side, with an n-type
material, thus forming the pn junction. The molar concentration of Gallium and Aluminium in the active layer is changed such that its bandgap Eg is now smaller than Eg1. This results in a higher refractive index in the active region and as we discussed earlier, this leads to carrier
confinement and optical confinement.

Band-gap engineering
The emission wavelength itself can be controlled through band-gap engineering. For example,
emission at a wavelength in the range 0.81 – 0.87 μm can be designed by engineering the
proportions of Ga, Al and As in GaAlAs. For the emission at 1 – 1.65 μm, which includes the
most commonly used communication band of 1.5 μm, the material used is Indium Gallium
Arsenide Phosphide (InGaAsP). These are compound semiconductors alloys made with different
molar weights of the constituent elements. One can basically engineer the value of Eg to get a
desired range of wavelength, by adjusting the molar concentrations. For example, take a
compound semiconductor - InGaAsP with x molar concentration of Gallium and y of Arsenic
with Indium of concentration 1-x and Phosphorus of concentration 1- y. Typically, x/y = 0.45 is
required for lattice matching otherwise the strain due to lattice-mismatch would result in
reduction inefficiency. The empirical relation connecting the bandgap and molar concentration
is shown here. Now, by choosing the appropriate molar concentrations, one can arrive at material
compositions that would emit at the desired wavelength ranges. Note that, the bandgap Eg of the
active region decides these properties and hence, the bandgap of p and n regions do not play any
role in deciding the emission wavelength. We now proceed to understand how the nature of
spectrum from a semiconductor source can be intuitively derived.
Spectral width

E-k diagram is useful to understand the wavelengths of emission and the spectral width of
emission. Silicon is not used as a source because it is an indirect bandgap material. The minimum frequency (νmin) emitted by a material corresponds to the smallest energy difference between the valence and the conduction band – which is in fact the bandgap energy, Eg. Thus,
νmin= Eg/h is the minimum frequency that can be emitted by a semiconductor with band gap Eg.
This can be represented in terms of the corresponding minimum wavelength as
λ௠௜௡(μm)
ଵ.ଶସ
ா೒(௜௡ ௘௏)
.

Now, we will find out the largest frequency or the shortest wavelength that can be
emitted from the structure. As we know, the band structure in a semiconductor can have a
continuum of energy levels. Thus, the shortest wavelength will get decided by the highest energy
level difference between which, the transitions can occur. This highest occupied energy level
separation is decided by the (a) joint density of states and (b) occupation probability of these
energy states. By definition, the joint density of states is the number of states per unit volume
with energies between  and  that can interact with (emit or absorb) the photons of energy
, satisfying the energy and momentum conservation conditions. So, the joint density of states
can be derived from the E-k diagram as follows.
Electron density of states :
The density of states, (k) is defined as the number of states occupied by electrons with wavenumbers (momentum) between k and k+dk. We use the wave model for the electron to derive the
density of states in the valence and conduction bands. In this model, the electron is described as a
wave function with wave vector k, energy E and spin. Electron near conduction band can be
visualised as a particle of effective mass mc, enclosed in a cube of volume, V in the
semiconductor. Standing wave solutions imposed by the boundary conditions of the volume
require that the x,y and z components of k assume discrete values, for the electron wave to
survive in the volume through constructive interference. Following the approach in resonator
optics (to be discussed later in the course), the electron waves can now be thought of enclosed in
a three-dimensional resonator, allowed to have wave vectors with its components assuming
discrete values ൫k௫, k௬, k௭൯ = ቀ
௤ೣగ

,
௤೤గ

,
௤೥గ

ቁ , with q௫, q௬, q௭

representing positive integers.

Thus, the allowed values of k can be thought to form a k-space, with,
The volume of the unit cell in the allowed k-space = ቀ





The number of points per unit volume in the allowed k-space = ቀ





No. of states with wave-number k =

௏௢௟௨௠௘ ௢௙ ை௖௧௔௡௧ ௢௙ ௔ ௦௣௛௘௥௘ ௢௙ ௥௔ௗ௜௨௦ ௞
௏௢௟௨௠௘ ௢௙ ௨௡௜௧ ௖௘௟௟

=


×


గ௥య×ଶ




య =


ଷగమ d


Note that, an octant of sphere is considered since k௫, k௬, k௭

can assume only positive values, and
the factor of 2 in the above expression is to account for the fact that the same state can be
occupied by two electrons with opposite spins.
Number of states with wave number k, per unit volume = ௞

ଷగమ
.

NPTEL-Fiber Optic Communication Technology –Lecture 10 Page 6
Therefore, the number of states with wave number between k and k+dk, per unit volume (density
of states), ρ(k)dk =

ௗ௞ ቀ


ଷగమቁ dk.

Therefore, density of states of electron with momentum k is given as,

ρ(k) =


గమ
(1)

To find the density of states in the conduction band :
Let c represent the number of energy levels in the conduction band per unit volume, with energy
between E and E+dE. Since there is a one-to-one correspondence between E and k,
ρ௖(E)dE = ρ(k)dk (2)

Therefore,

ρ௖(E) =
ఘ(௞)
ௗ௞
ௗா ൗ
(3)

Now, the energy, E in of the electron in the conduction band corresponding to a momentum, k,
can be written as,
E = E௖ +
ħ
మ௞

ଶ௠೎

(4)

where mc is the effective mass of the electron in the conduction band, Ec is the lowest energy of
the conduction band.
From Eqn (3),
ௗா
ௗ௞
=
ħ
మ௞
௠೎
(5)

Therefore, the density of states in the conduction band can be obtained by substituting Eqn (5)
and Eqn (1) in Eqn (2) as,
ρ௖(E) =


గమ

ħమೖ
೘೎
=
௠೎
గమħ
మ k (6)

Since the state corresponds to that in the conduction band, substituting for k from Eqn (4) in Eqn (6),

ρ௖(E) =
௠೎
గమħ
మ ට(E − E௖)
ଶ௠೎
ħ
మ௞మ =
(ଶ௠೎
)



ଶగమħ
య ඥ(E − E௖); E ≥ E௖

(7)
Similarly, the density of states in the valence band, ρ௩(E) can be derived as,

ρ௩(E) =
(௠ೡ
)



ଶగమħ
య ඥ(E௩ − E); E ≤ E௩ (8)

where E௩ represents the highest energy level in the valence band.

NPTEL-Fiber Optic Communication Technology –Lecture 10 Page 7
To find the joint density of states :
In order to achieve emission, it is not just enough to have density of states in the valence and
conduction bands, there should be a “joint density of states”, in both these bands, which can be
derived as follows. Intuitively, the joint density of states would mean the presence of states in the
the valence band corresponding to momentum k, and the corresponding states for the holes in the
conduction band for the same momentum k.
Consider a transition between an energy level E2 in the conduction band to E1 in the valence
the band is shown in the diagram.
Eଶ = E௖ +
ħ
మ௞

ଶ௠೎
(10)

Eଵ = E௩ −
ħ
మ௞

ଶ௠ೡ
(11)

Eଶ − Eଵ = E௖ − E௩ +
ħ
మ௞

ଶ௠೎
+
ħ
మ௞

ଶ௠ೡ
= E௚ +
ħ
మ௞

ଶ௠ೝ
(12)

where ଵ
௠ೡ
+

௠೎
=

௠ೝ
.

Since Eଶ − Eଵ = hυ, substituting in Eqn (9),
k
ଶ = ൫hυ − E௚൯
ଶ௠ೝ
ħ

(13)

Now, the joint density of states, ρ(ν)dν, is defined as the number of states per unit
volume with energies between  and  that can interact with the photons, satisfying the
energy and momentum conservation conditions. Since each frequency corresponds to a unique
energy state, ρ(ν)dν = ρ(Eଶ)dEଶ,
ρ(ν) = ρ(Eଶ)
ௗாమ
ௗఔ
(14)
Substituting from Eqn (7), (8) and (10), the joint density of states for transition involving a
photon of energy  is,
ρ(ν) = ρ(Eଶ)
௠ೝ
௠೎
=
(ଶ௠ೝ
)



గħ
మ ට൫hν − E௚൯ (15)

Occupation Probability :
In addition to having a non-zero joint density of states, it is also important that there
should be a nonzero probability for an electron to occupy an energy state E2 in the conduction
band and hole in the corresponding state, E1 in the valence band. The occupation probabilities
are decided by Fermi statistics as electrons are fermions. According to Fermi statistics, the
probability of occupation of an electron in energy state E is given by this function

NPTEL-Fiber Optic Communication Technology –Lecture 10 Page 8
f(E) =

ଵା௘

ಶషಶ೑
ೖ೅ ቇ
where the Ef is the Fermi energy. Fermi energy is the energy for which

this probability of occupation is1
2
ൗ . Now, the probability of emission can be calculated as the
joint occupation probability, pe of an electron in the conduction band with energy state E2 and
that of the absence of an electron in the valence band with energy E1.

p௘ = f(Eଶ)(1 − f(Eଵ) =

ଵା௘

ಶమషಶ೑೎
ೖ೅ ቇ
቎1 − ଵ
ଵା௘

ಶభషಶ೑ೡ
ೖ೅ ቇ
቏ ≈


ಶభషಶ೑ೡ
ೖ೅ ቇ


ಶమషಶ೑೎
ೖ೅ ቇ
(16)

where Efv and Efc represent the quasi-Fermi energy levels, under quasi-equilibrium. At
thermal equilibrium Efv =Efc =Ef, and hence, p௘ = e
ି൫௛ఔ
௞் ൗ ൯

The rate of spontaneous emission is thus proportional to the product of the joint density
of states and the occupation probability, and is given as,

r௦௣ ∞
(ଶ௠ೝ
)



గħ
మ ට൫hν − E௚൯e
ି൫௛ఔ
௞் ൗ ൯ = R଴ට൫hν − E௚൯e
ି


൫௛ఔିா೒൯
௞்



(17)

where R଴ is a constant.
Equation (17) describes the spectral behaviour of emission from a semiconductor source.
The occupation probability decreases as the energy (E) increases as given by this expression.
When energy E=hν is smaller than Eg, the photon of energy E cannot be generated as only the
positive value of the quantity under the square root in Eqn 17 can be considered.

NPTEL-Fiber Optic Communication Technology –Lecture 10 Page 9
Let us plot the rate of spontaneous emission versus frequency. So, we will have emission only
when hν > Eg. For values of frequency upto Eg/h =ν0, there is no emission because the rate of
spontaneous emission is zero in this case. As frequency (ν) increases, ඥ(hν − E௚) starts
increasing but the exponential term decreases. Thus, as the frequency increases, the rate of
spontaneous emission increases because the joint probability density increases, but then it starts
decreasing because occupation probability starts decreasing. Thus, the spectrum of the rate of
spontaneous emission peaks at a certain frequency, and decreases for larger frequencies.
Now, the peak can be easily derived by maximizing Eqn (17). Note that, the shape of the
spectrum cannot be described as a Gaussian, but it changes with temperature. We can also find
the full-width at half maximum of the function represented by Eqn (17), and the surprising fact is
that this full width at half maximum is 11 THz (at room temperature), irrespective of the material
used. It just depends on the temperature.

Now, this frequency range of FWHM can be converted to wavelength in nm. The following can
be used for this conversion.
c=νλ, λ=c/ν, |∆λ|=(c/ν2
) |∆ν|.

The power generated by LED

NPTEL-Fiber Optic Communication Technology –Lecture 10 Page 10
The next thing is to estimate the optical power generated by the LED. If I is the injected current,
the rate of injection would be I/q, where q is the charge of an electron. Power (P) generated by the
LED can be written as hν×rate of the generation of photons.
Now, the rate of generation of photons is the rate of radiative recombination. The power is
expressed as,
P = hν×rate of radiative combinations=hν×ηint × (I/q)
The rate of generation of radiative recombination is thus related to the internal quantum
efficiency (ηint). Note that, a plot of light intensity (proportional to power) vs current is referred
to as the “L-I” curve and the power versus current is referred to as the “P-I” curve of the
semiconductor source. The LI/PI curve uniquely characterizes a given semiconductor source .
From the expression, the LI/PI curve would be a straight line with a slope given by (hν×ηint )/q.
When the current is zero, the optical power is zero so it is a line that passes through the origin.
As the current increases, the optical power generated increases, but beyond a certain current, the
power does not increase. As the current keeps on increasing, all the dopants are depleted up to a
certain current and there may not be enough carriers to undergo the recombination. So, after a
very large value of current power values saturate. Typically, LEDs are not operated in this saturation condition.
Now, this LI curve is important because it distinguishes a laser diode from a LED. Suppose, a two-pin device is given and you are asked to find out whether it is an LED or a laser diode. The only conclusive test is actually the LI curve. The LI curve for laser diodes will show threshold behaviour