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Appendix 5
BEHAVIOUR MODEL FOR SEMICONDUCTOR LASER DIODES
1. WHY DO WE NEED THE BEHAVIOUR MODEL? ................................................... 1
2. SOME PHYSICAL CONCEPTS ................................................................................... 4
Recombination ................................................................................................................ 6
A) Spontaneous recombination.............................................................................. 6
B) Non-radiative recombination ............................................................................ 7
Photon absorption (stimulated carrier generation) .......................................................... 7
Stimulated emission. ....................................................................................................... 8
3. WORKING PRINCIPLES OF THE LASER ............................................................... 10
4. BALANCE EQUATIONS ............................................................................................ 11
Carrier Density Balance Equation ................................................................................ 12
Photon Number Balanced Equation: ............................................................................. 16
Photon Phase Rate Equation ......................................................................................... 18
5. STEADY STATE ANALYSIS..................................................................................... 23
1. WHY DO WE NEED THE BEHAVIOUR MODEL?
The laser transforms electrical signals into optical signals.
The behaviour model tells us
1) How the output optical power depends on DC input electrical current (the socalled STEADY-STATE ANALYSIS, Section 7)
1
a) The electrical signal is constant Iin; b) The optical signal is a sine wave with
optical frequency v (“nu”), but its amplitude, and therefore its power Pout, is
constant. The steady state analysis tells us how Pout depends on Iin.
2
2) How the output optical power depends on AC input electrical current.
(SMALL SIGNAL ANALYSIS)
a) The electrical signal changes slightly around some DC value with some
comparatively small frequency f; b) The optical signal is a sine wave with
optical frequency v, modulated by signal frequency f << v. Small signal
analysis tells us how the output modulation P depends on the input
modulation I and frequency f.
Note
The behaviour model depicts the laser as a black box with an input and
an output. It does NOT tell us about the physics of laser operation, but
we need physics to derive and understand the model.
In order to complete the course, you only need to know how the
behaviour model works. You do not need to understand all the physics
behind it. However, we will briefly introduce some physical concepts
that are necessary to build the simplest behaviour model. If you want to
learn more about them please refer to the special literature on
semiconductor devices.
3
2. SOME PHYSICAL CONCEPTS
The carriers of electrical current in semiconductor material are free electrons
and holes.
Free (conduction) electrons are the electrons that are not bound to any
atom. They are very mobile, move fast and therefore have high energy.
Electron density N is the number of free electrons per unit volume.
Bound (valence) electrons are electrons that are bound to an atom. They
cannot move away and we can conclude that they have lower energy than the
free electrons (in the same way that the potential energy of a body near Earth
is less than the potential energy of a body flying freely in space).
Note: Only 4 of 33 electrons are shown near each atom.
Holes are associated with the vacancies of bound electrons. For example, if a
nucleus has charge +33q (q = 1.6 x 10-19 is the charge of the electron) (i.e.,
Arsenic), then in a normal state it should have 33 electrons around it. If, for
some reason, one of the electrons was taken away, so that only 32 are left, the
total charge of the ion is +33q [nucleus] – 32q [remaining electrons] = +1q.
An electron from a neighbouring atom can occupy the vacancy, leaving
behind a hole. In this way, positively charged holes can travel from atom to
atom and therefore they are current carriers.
4
Hole density P is the number of holes (i.e., atoms with such vacancies) per unit
volume.
In a regular crystal structure, holes are mainly created by making some
electrons free (and letting them move away). Therefore, in a regular
(undoped) crystal structure the electron density should be approximately
equal to the hole density:
NP
This is NOT the only way to create holes (and free electrons).
N-doping. It is possible to blend two materials in such a way that the resulting
material has a significant number of free electrons and a very small number
of holes. Such semiconductor materials are called n-doped (from the word
“negative”, since the electron has a negative charge).
P-doping. Similarly, it is possible to blend two materials in such a way that the
resulting material has a significant number of holes and a very small number
of free electrons. Such semiconductor materials are called p-doped (from the
word “positive”, since the hole has a positive charge).
Note: Only 4 of 33 electrons are shown near each atom
5
PN junction. If we bring n- and p-doped material into contact, the electrons
driven by diffusion will go to the p-region and holes will go to the n-region.
Recombination. When they meet each other at the p-n junction, the free electron
can occupy the vacant places represented by holes. In this case, the electron
ceases to exist as a free electron and so does the hole. The electron density N
and the hole density P are each reduced by 1.
As mentioned above, the energy of a bound electron is less than the energy of
a free electron. According to the energy conservation law, when an electron
goes from a higher energy state to a lower energy state it should liberate some
energy in some form. We have two possibilities:
A) Spontaneous recombination is the process whereby such an electron releases
energy in the form of LIGHT. According to quantum mechanics principles,
the frequency of this light is
ν 
E state1  E state2
,
h
where h is the Plank constant h = 6.6x10-34 J.s. The explanation of these
principles is beyond the scope of this course.
I would like to point out that, although it may seem strange, WE DO NOT
HAVE TO KNOW THE AMPLITUDE OF THIS LIGHT. We know from the
energy conservation law that the TOTAL energy of this light is
E  E state1  E state2 . Therefore, if a certain number R of such electron-hole
recombinations occurs per unit time, the energy of the light released per unit
time is
R( Estate1  Estate2 )  Rhv
6
The amount of energy released per unit time is by definition the power of the
generated light. Since the power of the generated light is what we are looking
for, all we need to know is R and v.
Photon. A chunk of light generated by such recombination is called a photon. A
photon has two characteristics: frequency and phase (no amplitude).
The number of photons equals the number of recombinations, so we can say
that R is the photon generation rate. R and v are the variables that totally
determine the output power.
In simple words, spontaneous recombination of an electron-hole pair is the
process whereby the number of electrons and the number of holes are each
reduced by 1 and 1 photon is generated.
Balance: 1 EH pair  1 ph
B) Non-radiative recombination is when an electron recombining with a hole
releases energy in any other form. It is an undesirable phenomenon because
we reduce the number of carriers without gaining any light.
Balance: 1 EH pair  some useless energy
Photon
absorption
(stimulated
carrier
generation).
Is
spontaneous
recombination a reversible process? Yes. It is possible that a photon gives its
energy to a bound electron, making it a free (conduction) electron and leaving
behind a hole. This process reduces the number of photons by 1 and increases
the number of electrons and the number of holes each by 1.
Balance: 1 ph  1EH pair.
7
Stimulated emission. Now we would like to consider a very interesting
phenomenon that can be explained only by quantum mechanics. The
explanation of this phenomenon is beyond the scope of this course. In fact, we
are more interested in what happens rather than why it happens. The
phenomenon is the following: Consider a semiconductor material that has
free electrons and holes. As mentioned above, when an electron recombines
with a hole, it releases some energy. So we can assign a certain energy
difference E to each electron-hole pair (EH pair). This is the energy that
WOULD BE released in the case of recombination. Now consider a photon of
energy hv travelling through this material. Quantum mechanics says that
when this photon passes by an EH pair with E = hv it can trigger
recombination, so that the EH pair disappears and two photons travel
further: the original one and the one that was released. The energy
conservation law is satisfied:
Before “collision”: TOTAL ENERGY = hv + E
After “collision”: TOTAL ENERGY = hv + hv = hv + E.
A photon cannot trigger recombination for an EH pair with E hv. So we can
say that stimulated emission is the process whereby a photon makes an EH
pair disappear and generates another photon. The generated photon is
identical to the original one in frequency and phase.
Balance: 1 ph + 1EH pair  2 ph.
These 4 processes:
1)
Spontaneous recombination: 1 EH pair  1 ph;
2)
Photon absorption: 1 ph  1EH pair;
3)
Stimulated emission: 1 ph + 1EH pair  2 ph; and
4)
Non-radiative recombination: 1 EH pair  some useless energy
8
are the main processes that define the main work principles of the
laser. We will need them to build a simple behaviour model of the laser.
9
3. WORKING PRINCIPLES OF THE LASER
The typical structure of a semiconductor is shown in this figure.
The three main principles of how a laser works are listed and explained
below.
.
1. Power supply.
Holes from the p-doped region and free electrons from the n-doped
region are driven by the applied electrical field to the active region i.
This region is normally undoped (or weakly doped), so the number of free
electrons and the number of holes there are approximately the same (see
2. Some Physical Concepts, Hole Density, p. 3). In fact, we do want
them to be the same, because it is an EH pair rather than a single electron
or hole that generates light. Further, we will only consider the electron
density N, since the hole density is the same. We will call it carrier
density.
Another property of this region is that it is easy for electrons and holes to
get into it but difficult for them to get out (this phenomenon is explained
in the textbook).
To conclude, a high density of EH pairs that can release optical energy is
achieved in the active region by applying the forward electric bias
(“forward” means a positive terminal is applied to the p-doped region and
a negative terminal is applied to the n-doped region).
10
2. Power transformation is achieved by recombination processes (both
spontaneous and stimulated).
3. Feedback.
At the beginning of the operation of the laser, the main process that generates
light is spontaneous emission. The light generated in this way has a very
broad frequency band and therefore is not a good signal carrier. By applying
semitransparent mirrors at the front and rear facets of the laser, we make
some portion of the outgoing light go back into the active region. Such a
mirror structure filters out some frequency components, leaving some very
narrow-band peaks. When this filtered light travels through the active region
it stimulates the recombination of EH pairs and produces new photons of the
same frequency and the same phase (see 2. Some Physical Concepts,
Stimulated emission, p. 5).
Thus only filtered, narrow band light experiences significant gain.
4. BALANCE EQUATIONS
We would like to know the relation between the input current and the output optical
power.
As we said the optical power depends on the number of photons produced per unit
time (see 2. Some Physical Concepts, Photon, p. 4). The photons are produced
by
two
processes:
Spontaneous
Recombination
and
Stimulated
11
Recombination. In both cases, the rate of photon generation depends on the EH
pair density, or carrier density N. Stimulated recombination is triggered by the
photons, so its rate should depend on the number of photons S in the laser. The
carrier density for its part depends on the input current, and, due to Photon
Absorption (see 2. Some Physical Concepts, Photon Absorption, p. 5), on
the photon number.
If we assume the current to be a constant, we have two main variables defining
the work of the device: N and S. Let us write down two equations that show how,
respectively, carrier density N and the number of photons S change in time.
All the terms in our equations should be in one of three groups: (1) variables to be
determined; (2) variables that can be measured experimentally or determined from
the design; and (3) well known constants.
Carrier Density Balance Equation
dN
 GENERATION  RECOMBINATION
dt
GENERATION is the growth of the electron density due to the input current,
which, by definition, is equal to the incoming charge per unit time:
I
dQ
dt
12
The incoming charge is the number of incoming electrons multiplied by the
electron charge q. The number of electrons is the electron density multiplied by
the volume of the active region. Therefore
Q = nq = VNgen q
N gen 
Q
qV
and
GENERATION 
dN gen
dt

1 dQ
I

qV dt
qV
This is an ideal case, when all the electrons from the current go into the active
region. In real cases, only some portion of them does. For that reason, we
introduce a phenomenological variable called the current capture coefficient
(eta). So, finally
GENERATION 
ηI
qV
q is a known fundamental constant. I is the controlled input and  and V can be
measured experimentally.
THE RECOMBINATION RATE consists of three terms: stimulated emission,
spontaneous emission and non-radiative emission.
(1) RECOMBINATION . The STIMULATED EMISSION RATE is responsible for
the gain of light. Since gain can be measured experimentally, it is convenient to
express our stimulated emission rate in terms of gain.
Net gain is defined as the coefficient of the exponential growth of optical power,
or equivalently, the number of photons travelling along the active region. Let S
photons enter an active region of small length z. Through stimulated emission,
they experience gain gst, and through photon absorption they experience decay
ast. The net gain is
g = gst- ast. The number of photons leaving this small area will be
S out  S exp( gΔz )
.
13
Therefore, (Sout – S) photons are generated and the same number of carriers is
recombined in the time that it takes the light to travel through this area. This
time is
t 
Δz
vg
where vg is the light group velocity that can be determined experimentally.
Thus, the carrier stimulated recombination rate is
Rst 
v g ( S out  S ) v g S[exp( gΔz )  1]
S out  S


Vt
VΔz
VΔz
Note that we put the volume in the denominator because we need the carrier
density rate, not the carrier’s number. It can be derived using Taylor expansion
that if z is small enough then exp(gz)  gz + 1. Substituting this in the above
equation, we obtain
Rs t 
Sgvg
V
S/V is the photon density in this small region and Rst is the stimulated
recombination rate in this small region. To obtain an average stimulated
recombination rate, S should be the total number of photons and V should be the
total volume.
Note: Since g is the net gain, Rst takes into account both stimulated
emission and photon absorption. So strictly speaking, we should use the
term net stimulated emission. Since we can only measure the net
gain it is convenient to put these two phenomena into one term.
Again, the above formula is the ideal case when ALL photons are confined to the
active region. However, this is never the case in reality, so we have to introduce a
correction term called optical confinement factor . It is defined as a ratio of
14
the photons in the active region to the total number of photons and can be
estimated using the optical properties of the laser. Finally
Rst 
Sgv g
V
where S is the total number of photons in the laser. Variables g, vg,  can be
determined experimentally, and V is known from the design.
We can presume that the gain depends on the number of free electrons and holes,
while the absorption depends on the number of the valence (bound) electrons
(see 2. Some physical concepts. Photon absorption (stimulated carrier
generation)). When the valence (bound) electron density exceeds the carrier
density, gst< ast and g < 0. When the carrier density exceeds the valence (bound)
electron density, gst> ast and g > 0. The carrier density at which g = 0 is called the
transparency carrier density Ntr. It can be derived that, for bulk1 lasers, the
net gain
g = a(N-Ntr)
where a is a coefficient that can be calculated based on the properties of the
material.
(2) RECOMBINATION. The SPONTANEOUS EMISSION RATE depends only on
the electron and hole densities:
1
A bulk laser is a laser whose active region is one thick layer of semiconductor material.
There is another type of laser where the active region consists of several thin layers (on
the order of several nanometres) sandwiched between p and n regions. This type is
called a quantum well (QW) laser. It has better properties and is more widely used in
industry. For the QW laser
g = a ln(N-Ntr)
15
Rsp =Bsp NP ≈ Bsp N 2
since P ≈ N in a weakly doped active region. Bsp is some coefficient.
(3) RECOMBINATION. NON-RADIATIVE RECOMBINATION RATE depends
only on the carrier density. It is defined phenomenologically as
Rnr = AN + Bnr N2+CN3
For simplicity, the terms Bsp N 2 and Bnr N2 are combined into one:
BN2 = (Bsp + Bnr) N2, so that the combined spontaneous and non-radiative
recombination rate are expressed as
Rsp nr  AN  BN 2  CN 3 
N
c
where
 c (N ) 
1
A  BN  CN 2
τ c is called “carrier life time”. Coefficients A, B, C can be found
experimentally. Finally we can write
Sgvg
dN
I
N
η
Γ

dt
qV
V
τc
We considered all the terms that contribute to the carrier density change rate. We
took into account all 4 main processes and the injection current. We expressed
the carrier density change rate through N, S and several coefficients that can be
calculated theoretically, found experimentally, or determined from the design.
Photon Number Balanced Equation:
dS
 GENERATION  CONSUMPTION
dt
GENERATION: comes from the stimulated and spontaneous emission
(1) GENERATION. STIMULATED EMISSION
G s t  Γv g gS
16
This term is the same as that used for the carriers, but without V in the
denominator, because now we consider the total number of photons, not the
density.
(2) GENERATION. SPONTANEOUS EMISSION could be expressed in the same
way G s p  VRs p . Unfortunately, most spontaneous emission is noise. Only a
small part of it is coupled into the useful signal. So we have to introduce a
coefficient K < 1 called Petermann’s factor, and G s p  KG s p _ total .
Sometimes it is convenient to relate the spontaneous and the stimulated emission:
G sp _ total  n sp G st
where n s p
is the so-called population inversion factor. Thus, the
spontaneous emission term can be written as
G s p  KG s p _ total  Kn s pG s t  Kn s p Γv g gS
CONSUMPTION is defined by the losses inside the cavity αint (this is different
from photon absorption, described above, because it is already included the in
stimulation emission term) and the losses caused by some of the light leaving the
cavity αcav (this is nothing but our output light).
αtotal  αcav  αint
The formula for Gcons is similar to the one for Gst, except that it does not have
factor Γ and instead of gain g we have loss αtotal.
C  v g (α cav  α int ) S
Sometimes it is convenient to write the above formula as
C  v g ( cav   int ) S 
τp 
where
S
p
1
vg (αcav  αint )
is called photon life time.
Finally, we can write
dS
S
 Γv g gS  Kn sp Γv g gS 
dt
τp
17
Photon Phase Rate Equation
This is the third important equation. It shows how the phase of the photon
changes in time (which by definition equals light frequency).
Phasse is closely connected to frequency:
dφ
ω
dt
Therefore, let us see what frequency is produced by laser.
The field inside a laser is described by a traveling wave harmonic function:
z 

E( x , y, z , t )  E( x , y) exp  jω(t  )
v 

where v is a velociy of the light inside the laser:
v 
c
nr
c  3  108 m / s is the speed of light in vacuum, and nr  1 is the real part of
the refractive index2 of the active region.
n 

E( x , y, z , t )  E( x , y) exp  jω(t  z )
c 

2
The real part of the refractive index
the imaginary part
nr
is responsible for the change speed of light inside a medium and
n i is responsible for the loss or gain. For simplicity we ignored the imaginary part, but it
is easy to see what happens if we account for it:
n  jn i 

E( x , y, z , t )  E( x , y) exp  jω(t  r
z )
c


n
 ωn


 E( x , y) exp  jω(t  r z ) exp  i
c 

 c

z

The last term stands for a simple exponential growth or decay (depends on the sign of n i ).
18
As it is shown in the pictures above after making a round trip withing the cavity
the light can add with itself either in a constructive or destructive way.
Consider two waves at a time moment t 0 in a point z  z 0 : one of them has just
started travelling (blue) and the other has already made a round trip (red)
z 

E1 ( x , y, z , t )  E( x , y) exp  jω(t 0  0 )
v 

 
z  2 L   
E 2 ( x , y, z , t )  E( x , y) exp  jω t 0  0

v

 
where L is the length of the cavity.
In order to add constructively, the phase difference between them should be a
multiple of 2π :
2mπ  φ2  φ1 
2ωL
2ωnL

c
c
where m is any integer number. We can see that only those frequencies that
satisfy
ω  ωm 
mπc
nL
can add constructively and therefore can exist for a long period inside laser. All
the others add destructively and quickly disappear contributing very little in the
laser’s work.
19
Now let us assume that the carrier density N changed. It changes the ability of the
active region to produce light due to stimulated emission i.e. it changes gain.
Gain is defined as an exponential coefficient of the growth of the optical power:
P  P0 exp( gz )
We know that optical power is proportional to the square of the field amplitude:
P ~ E2
We know that
n
 ωn


E( x , y, z , t )  E( x , y) exp  jω(t  r z ) exp  i
c 

 c
and

z

n i stands for a simple exponential growth or decay of the field amlitude. Comparing
the three above formulas we can see that
g2
ωn i
c
Therefore, the change of the gain is automatically the change of the imaginary part of the
index
ni .
According to the complex analisys of analytical functions the real and imaginary part are
not independent and the change of
n i immediately causes the change of nr . Their
mutual dependence is described by so-called Kramers-Kronig relation which is beyong of
the scope of this course. Instead, we introduce a phenomenological coeffitient called
Linewidth Enhancement Factor that can be defined as
α LEF 
Note that in general
Δnr
Δni (only due to the change of N and T )
n i can change due to several factors: carrier density N, temperature
T, number of photons S. Mathematically it can be expressed as
Δni , general 
ni
n
n
ΔN  i ΔT  i ΔS
N
T
S
Linewidth Enhancement Factor DOES NOT take into account the last term. Sometimes it
is defined as
α LEF 
Δnr
n i
n i
ΔN 
ΔT
N
T
sometimes
20
α LEF 
Δn r
n i
ΔN
N
Since we do not consider the temperature change now the last definition is enough.
Let us go back to our chain of changes:
ΔN  Δg  Δni  Δnr .
From formula
ω  ωm 
we can see that the change of
mπc
nL
nr causes the change of ω and ω is a time derivative of the
phase. Thus our chain has two final links added
ΔN  Δg  Δni  Δnr  Δω  φ .
1st link
2nd link
3rd link
4th link
5th link
That was the purpose of this section: to see how the carrier density changes the
frequency of the output light.
Now we need to write this chain in the form of the equations.
Link # 5
dφ
ω
dt
Link # 4
ω  Δω 
mπc
(n r  Δnr ) L
ω  Δω(nr
 Δn r ) 
mπc
L
ωnr  ωΔn r  nr Δω  Δn r Δω 
mπc
L
Assuming Δnr  nr and Δω  ω and neglecting the last term of the LHS we
get
ωn r  ωΔn r  n r Δω 
mπc
L
Since
ωn r 
m πc
L
we can write
21
ωΔn r  n r Δω  0
Δω 
ω
Δn r
nr
Link # 3
Δnr  αLEF Δni
(remember:
Δn i is only due the carrier density change (and, in more advanced analysis,
temperature change), but NOT photon number change)
Link # 4
Δni 
Δgc
2ω
Finally
Δω 
α LEF c
α
Δg  LEF vg Δg
2 nr
2
If we are near the threshold where gain changes from zero to a certain value gnet
then
Δg  gnet  Γg  α
where
α  αint  αcav
Previously, we defined
1
 vg (αcav  αint )
τp
Thus, finally, we can write

dφ 1
1 
 α LEF  Γv g g 

dt
2
τ p 

where
α LEF 
Δnr
Δni (only due to the change of N and T )
22
5. STEADY STATE ANALYSIS
As we mentioned in the first part, in the steady state analysis we assume the DC
injection current that is converted in “DC” optical power. Of course, we should
remember that the optical signal is ALWAYS high frequency (on the order of 100
THz) signal, so when we say “DC” optical power we just mean this signal is not
modulated and the average power is constant.
This means that the number of photons is also constant:
dS
 0.
dt
We mentioned before that carrier density is a function of injection current I and the
number of photon S. Since neither of them changes, N should be the constant too.
The balanced equations become:
0η
Sgvg N
I
Γ

qV
V
τc
0  Γvg gS  Kns pΓvg gS 
S
τp
As mentioned above, the contribution of spontaneous emission in the useful signal
is very small and can be neglected: Kn s p Γv g gS  0 . Thus


 Γv g g  1  S  0

τ p 

23



This equation has two solutions: (1) S = 0; and (2)  Γv g g 
1 
 0.
τ p 
IN THE FIRST CASE:
0η
I
N

qV
τc

N 
ητ c
I
qV
N is directly proportional to the injection current and S and consequently the
output optical power are zero.
Can N grow indefinitely with growing I ? Considering the second case will answer
this question.
IN THE SECOND CASE:


 Γv g g  1   0

τ p 


g
1
 gth
Γv g τ p
No terms in RHS depend on the current I and, consequently, g does not depend on
the current either. We remember that in a bulk laser g = a(N – Ntr) or N 
g
 N tr .
a
Since g is a constant w.r.t. the current, N should be a constant too
N  N th  N tr 
g
1
 N tr 
a
aΓvg τ p
24
It means that N grows with growing I until N reaches the value
N th  N tr 
1
called threshold carrier density. After this it is clamped
aΓv g τ p
at its threshold value no matter how high the input injection current is. Let us
substitute Nth in the carrier density balance equation
0η
Sgth v g
N
I
Γ
 th
qV
V
τc

S 

V
Γgth v g
 I
N
 η
 th
τc
 qV




η 
qV
 I 
N th 
Γgv g q 
ητ c

Define threshold current as I th 
qV
1
N th . Substituting Ith and gth 
in the
ητ c
Γvg τ p
above equation, we obtain
S 
ητ p
q
I
 I th 
Now we have the opposite picture: N is constant and S is linearly dependent on the
input current.
As mentioned above, the output power is the energy of photons coming out of the
mirrors per unit time.
25
The number of photons leaving the cavity per unit time is expressed in the photon
number balance equation through the cavity loss term cav
C cav  α cavv g S
The output power is by definition the total energy of photons leaving the cavity per
unit time and , therefore, equal to the energy of one photon h miltiplied by Ccav
Pout  hνC cav  hναcavv g S
26