Download Chapter 5: Optical Processes in Semiconductors

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5 Optical Processes in Semiconductors:
22/03/2010
Optisch induzierte Polarisation
Eines Moleküls
Polarized Atomic Dipole in Optical Field
Blue Silicon Carbide Light Emitting Diode LED
Representation of band-to-band transitions in Semiconductors
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Goals of the chapter:
Design of optical SC-devices (Laser, Photodetectors, optical amplifiers, etc.) depends critically on the calculation of the
carrier transition rates R, the rate-equations of carriers and photons and the optical gain g(ω,n)
• Nonclassical description of the motion of bound electrons in the static force fields of the atomic nucleus
or crystal lattice and a oscillating light wave
• The equation of motion and polarization has to be described in a quantized 2-energy level material system
by a quantum mechanical model based on the perturbed time dependent Schrödinger-equation
• The description of the interaction between light field and a 2-energy band Semiconductor needs an
extension of the discrete 2-level concepts to continuum-to-continuum, resp. band-to-band optical
transitions
Methods for the Solution:
• The classical model for electron motion r(t) of “electrons-on-springs” will be replaced by the time-dependent
Schrödinger-equation (TSE) for a harmonic oscillator providing a statistical expectation value 〈r(t)〉
• The quantum mechanical equation of motion of the expectation value 〈r(t)〉 of bound or quasi-free charges
in atoms and SC will be obtained for weak optical fields.
• Perturbation-theory leads to the Transition-Rates R and a realistic microscopic susceptibilities
χ(ω,n2,n1) depending on the of charge carriers densities n1, n2 in the lower and upper levels
• The quantum-mechanical analysis shows for carrier inversion n2>n1 optical gain g(ωopt) occurs
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5 Optical Processes in Semiconductors:
Introduction:
The classical dipole oscillator model „electrons on springs“ breaks down for
a) the quantized energy exchange between atom and light field,
b) the microscopic force constant and the relation to carrier state occupancy, that is:
„
the „spring force constant“ kDP and the damping γ D of the dipole motion are not physically defined
„
Realization of a positive χ’’>0, resp. γ<0 to obtain optical gain is not obvious?
¨ Carrier motion in a atomic system with discrete quantized energy levels E2 and E1 and
¨ Distribution of carriers n2, n1 occupying these levels influence the optical gain g ?
„
Separation of transparent and absorbing spectral regions of SC
¨ Goal: Semi-classical QM-Theory of the atomic oscillator and a light field with
- a classical optical field and
- a quantum mechanical (QM) description of the microscopic, induced dipole motion
by an optical field
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Step 1: Einstein-Theory of optical transitions
• Phenomenological semi-classical description of optical processes in a quantized 2-level medium by the so
called Einstein-Rate-equation formalism of optical transitions (Einstein 1905)
Step 2: Quantum mechanical model of optical transitions
• Quantum mechanical model for the dipole-oscillator (perturbation theory) (~1930)
Î
Relation between microscopic material parameters and the quantum mechanical susceptibility χ
Î Condition
for optical gain in atoms and SC and the concept of “pumping” and carrier inversion
Î Extension
of the 2-level-interaction to band-to-band transitions in SCs
Î calculation
of optical gain / absorption spectrum g(ω,n) in SC as a function of carrier density n, p.
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What are the concepts for a solution from a QM viewpoint ?:
A) Stationary Eigenstates:
Consider an electrons moving in the attractive and repulsive static potential field V(r) of an individual atom or a
crystal lattice without an optical field.
¨ Quantization of motion defines static discrete energy eigenstates ΨI(r,t), with energy eigenvalues Ei
(assumed dipole-free)
The matter wave functions ΨI(r,t) fulfill the
time-independent Schrödinger equation (SE):
⎛ p2
⎞
+ qV ( x ) ⎟ Ψ ( x,t ) = E Ψ ( x,t )
H Ψ ( x,t ) = ⎜
⎝ 2m
⎠
Solving the Schrödinger Eigenvalue problem → Ej (energy eigenvalue), Ψj (energy eigenfunction)
with ωj=Ej/ħ and E2>E1
V(x) is the potential of the atomic electric force field
p is the momentum operator
m is the mass of the particle
H is the Hamilton operator
Generic stationary Solution:
ψ i ( x,t ) = ui ( x ) e − jωit = ui ( x ) e − jEi /
t
General solution :
ψ ( x,t ) = ∑ Ci ψ i ( x,t )
i
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with the normalizations:
ψ ψ = ψi ψi = 1 →
∑ Ci = 1
2
i
ψ ψ i = Ci*
ψ i ψ = Ci
Interpretation of the Ci:
The coefficient Ci = Ci Ci i is proportional to the probability of finding the system in the state ψ i ( x,t ) with the energy
Ei.
2
*
B) Time dependence Eigenstate occupation
Perturbation of the static potential field V(r) by a superposed harmonic optical field (E, H, ω)
with its dynamic potential Vopt(r,t)
Definition: weak perturbation ¨ the optical field is much weaker than the static atomic field, Vatom >>Vopt
„
QM-interaction between N quantized 2-level system (E2,E1) (level E2 with N2 electrons, level E1 with N1 electrons) and a
classical, non-quantized optical harmonic EM-field (E, H) of spectral energy-density W(ω), W (ω ) =
„
∂ 2 EEM
∂ Vol ∂ω
The interaction changes of the occupation probabilities
(N2(t)/N , N1(t)/N) of the 2-level system by photon absorption or emission, resp. transitions between levels combined
with an energy exchange due to the EM-forces of the light field
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Quantum mechanical perturbation analysis: time-dependent SE
⎛ p2
⎞
∂
H Ψ ( x,t ) = ⎜
+ qVatom ( x ) + qVopt ( x,t ) ⎟ Ψ ( x,t ) = i
Ψ ( x,t )
m
∂
t
2
⎝
⎠
static potential
Materie
oscillating potential of the EM-field (ωopt)
E2
up- / down transitions
E1
ψ2
ψ1
EM-Feld
EM-field:
klassisch
E ,ω
0
Assumed dynamic solution Ψ ( x,t ) :
a superposition of static solutions Ψ1 ( x,t ) ; Ψ 2 ( x,t ) with time-dependent coefficients Ci(t)
¨ Ψ ( x,t ) = C1 ( t ) Ψ1 ( x,t ) + C2 ( t ) Ψ 2 ( x,t ) = C1 ( t ) u1 ( x ) e − jω1t + C2 ( t ) u2 ( x ) e − jω2t
C1(t), C2(t) are now time-functions
Ψ ( x,t ) describes the quantized system completely in its interaction with the optical field !
Calculation of the time dependent expectation values of the variables of the atomic system relevant for optical
properties:
¨
x ( t ) ; P ( t ) ; χ (ω ) ; N 2 ( t ) ; N1 ( t ) etc.
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5.1 Quantum mechanical model of the dipole-oscillator
5.1.1 Semi-classical model for absorption and gain
Optical Absorption and Amplification
„
The classical, mechanical model breaks down describing photon emission- and absorption properties and provides no clue
for amplification and energy quantization.
„
only the frequency dependence of the refractive index far away from the atomic resonance is well described
(with phenomenological parameters)
The classical, mechanical Lorentz-model is based on:
1. classical description of optical fields with continuous E- und H-field variables and a spectral field-energy density
W (ω ) =
∂ 2 Energy
∂ω ∂V
2. it neglects the particle aspect of the optical field, the quantization E=hv of the field
and the particle concept of the photon
3. the atomic harmonic oscillator, described by the coordinate x(t) of the charge separation¨
of the dipole and its system energy E (potential and kinetic) as continuous observables
of the Newton equation of motion.
The model does not describe:
1. Optical Gain and Absorption
2. Optical spontaneous emission
Extension to the semi-classical quantum mechanical-model:
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5.1.2 Interaction of a EM-field with a discrete 2-level system
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Introduction: (Lit. Loudon)
„
„
The optical field is represented by a classical, not quantized field H(r,t), E(r,t) → description by Maxwell’s equations
The medium response and its microscopic electrodynamics with respect to energy quantization E1, E2 under
interaction with the optical field E(t) (QM-description of the equation of motion for the electron position x(t))
→ description of expectation value of <x(t)> of the QM-dipole by the time-dependent Schrödinger equation
Atomic 2-level systems (E1, E2 , N1, N2) which interact with an optical EM-field (ω, E, H) exchange energy resulting in a
change of the occupation probability of the of the electrons, resp. in a change of the electron numbers N1, N2.
The goal is to calculate the time evolution of N1(t) and N2(t) for an idealized “2-level atom” and the formation of an atomic
dipole p(t) under the influence of the harmonic optical field E ( r,t ) , H ( r,t ) .
Strong interaction only close to the atomic transition frequency ω21 = ( E2 − E1 ) /
Quantum mechanical 2-Level-System
as a model: (quantitative description, QM >1930)
„
„
1-electron-atom-model with
(no collision processes → infinite state-lifetime)
2 discrete „sharp“ energy-levels E1 and E2 (E1<E2)
-eV(x)
Ψ3 ( x ) , E3
for the electron in the stationary, parabolic potential V(x)
of the nucleus:
(3. excited state)
Ψ 2 ( x ) ,N 2 ,E2 2.excited state
E2
The 2 energy levels are Eigenvalues E1 und E2 of the
1-D time independent Schrödinger-equation
Energy E
Transition 2↔1
E1
Ψ1 ( x ) , N1 ,E1 1.ground state
x
Energy-Levels of a harmonic Oscillator (electron in a parabolic potential V(x)= -kx2 , no light field)
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Stationary state without optical field: Time independent Schrödinger-Equation
The matter wave Ψ(x,t) representing the probabilistic motion of a particle of constant total Energy E=Etot (kinetic and potential) in a
stationary electrostatic potential V(x) is a solution (eigenfunction) with the energy eigenvalue E of:
⎛ p2
⎞
H Ψ ( x, t ) = ( H kin + H pot ) Ψ ( x, t ) = ⎜
+ qV ( x ) ⎟ Ψ ( x, t ) = E Ψ ( x, t )
⎝ 2m
⎠
H=Hamilton (Energy)-Operator
Stationary Schrödinger equation with time-invariant potential V(x)
⎛ p2
⎞
H =⎜
+ qV ( x ) ⎟ = Hamilton − Operator
⎝ 2m
⎠
∂
= impulse operator ; p 2 = −
p=−j
∂x
( of the time-independent potential V ( x ) of the positive nucleus )
2
∂2
∂ x2
E = total energy = potential and kinetic energy
V ( x ) = time-independent potential of the nucleus
q = charge of the particle (-e)
ψ ( x,t ) = Eigenfunction of the matter wave
with the time-space-dependent Eigenfunctions ψ ( x,t ) = u ( x ) e− jωt = u ( x ) e− jE / = t ,
resp. spatial term u ( x ) and Energy Eigenvalues E as solutions :
⎛ p2
⎞
+ qV ( x ) ⎟ ui ( x ) = Ei ui ( x,)
H ui ( x ) = ⎜
⎝ 2m
⎠
with ui ( x ) = spatial Eigenfunction and Eigenvalue Ei
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Eigenfunction are a complete orthonormal set μ i ( = orthogonal and normalized ) :
ui u j =
E1 :
+∞
∫
−∞
u*i u j dx = δ ij
Ψ 1 ( x,t ) = u1 ( x ) e − jE1 /
t
E2 : Ψ 2 ( x,t ) = u2 ( x ) e − jE2 /
¨ General solution:
(without proof)
with : δ ii = 1 , δ ij = 0 if i ≠ j
Standing wave with the oscillation frequency ω1=E1 /
Standing wave with the oscillation frequency ω2 = E2 /
t
ψ ( x, t ) = C1ψ 1 ( x, t ) + C2 ψ 2 ( x, t )
2
C1, C2 = constant
2
C1 and C1 are the probabilities of finding the system at E1 or E 2
;
2
2
C1 + C1 = 1
„
Ψ(x,t) describes the statistical behaviour of the electron in the potential field V(x) with constant total Energy E.
„
ΨΨ*=IΨI2 is interpreted as probability density of finding the electron at the position x, if normalized to: ∫ψψ * dx = 1.
„
the statistical expectation value <x> for the space variable x of the particle is (probability of finding the particle at x) :
x =
+∞
∫ Ψ ( x,t ) x Ψ ( x,t )
∗
dx =
−∞
we will use x
p = −e x
„
+∞
∗
∫ u ( x) x u ( x)
−∞
dx
=
N
Definition
of BRA− KET
notation
u x u
BRA KET
to describe the dipole
→ χ
etc.
the stationary solutions Ψ(x,t) are, in order to be stationary (E=const.), non-radiating, resp. free of a dipole moment.
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Definition of bra-ket notation:
expectation value <f> of an observable f(x) of a QM-system is defined as:
f =
+∞
∫ Ψ a ( x,t ) f ( x ) Ψ b ( x,t )
−∞
∗
dx
=
N
Notation
Ψa
f Ψb
Non-stationary states with a perturbing optical field: Time dependent Schrödinger-Equation
It is expected that the nonstationary state with an interacting optical field changes the occupation probabilities
ICj(t)I2 by exciting or de-exciting charge carriers by optically induced state transitions between energy states Ei, Ej.
¨ the coefficients Ci(t) become time functions
¨ State transitions will be described by transition probabilities r and transitions rates R of
charge carriers
G G
G G
Electron dynamics in an optical EM-field ( E ( r ,t ) , H ( r ,t ) ):
G G
The optical EM-field exerts forces F ( r ,t ) on the moving electron
G
G
Felektrisch
z
⎧
⎩
b) time-dependent magnetic force
v
e -
a) time-dependent electrostatic force Felektrisch = −eE ( x,t ) ⎨= − e
and
FLorentz
G
G G
FLorentz = eμ0 v × H
¨ producing energy exchange and state transitions
⎫
∂
Vopt ( x,t ) = ? ⎬
∂x
⎭
E
pi(t)=-eri(t)
+
x
y
G
G
F
>>
F
In most cases we can neglect the H-field because elektrisch
Lorentz
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Schematic representation of basic interactions of a 2-level system and an optical field:
Energy E
Photon Absorption Spontaneous Photon-Emission
e
excited
state
Stimulated Photon-Emission
e
e
Exciting photon
In
hv=E2-E1
out
out
ground
state
h
h
in
Stimulated Photon
h
4
(Electron-Hole-Pair Generation)
Photon-Annihilation
e-transition from: E1 → E2
(Electron-Hole-Pair Annihilation)
Photon-Generation
E2 → E1
E2 → E1
¨ EM-Radiation field changes occupation probabilities by level-transitions
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What are the transition rates R and how do they depend on the optical field ?
In historical order:
1) Einstein Theory: (A. Einstein, 1905 without QM)
The 2-level system exchanges energy with the EM-field of frequency ω only if the photon energy E=ħω=E2-E1 (resonance)
1) electrons in the ground state E1 absorb a photon and make an up-ward transition to the excited (upper) state E2.
2) electrons in the excited state E2 make a down-ward transition to the ground state E1 emitting an incoherent
spontaneous photon or emit a coherent stimulated photon, if a stimulating field is present.
3) Stimulated absorption and emission are proportional to the energy density of the optical field, the probabilities of
the occupation of the initial state and the final state being empty.
Definition: Transition-Pair (E2, E1) = initial and final state:
The initial and final state (E2 and E1) of an atom form a local transition pair. In isolated atoms the occupation of a level
means that the other level is necessarily empty and available for a transition.
Details see p.5-16.
2) Quantum Mechanics: (after ~1930)
The motion of the bound electron in an optical field (where the energy E of the 2-level system is not constant) is described by the matter
wave Ψ(x,t) of the electron in the time dependent potential V(x,t) = V(x) + Vopt(t).
„
„
The time-dependent EM-forces induce a microscopic dipole extracting (attenuation) or transferring
(amplification) energy from or to the field depending on the relative phase.
the energy-exchange excites electron transition from the ground to the excited state and vice versa if the
optical frequency ωopt is close to the transition frequency ω21.
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Time-dependent Schrödinger equation with the time-dependent Hamilton-Operator Hww(x,t) of the optical field:
∂
⎡⎣ H ( x ) + H ww ( x,t ) ⎤⎦ Ψ ( x,t ) = jh Ψ ( x,t )
∂t
⎛ p2
⎞
H ( x) = ⎜
+ qV ( x ) ⎟
⎝ 2m
⎠
HWW ( x,t ) = ?
time-dependent Schrödinger equation
Static Energy-Operator H ( x ) for the interaction with the positive charge of the nucleus V ( x )
time-dependent Energy-Operator for the interaction electron - optical field
( how does the EM energy of
the electron in the optical field formally enter H ww )
Task: find the time-dependent solution Ψ(x,t) for Hww(t) of the harmonic optical field E(t) of the form (weak perturbation assumption):
ψ ( x, t ) = C1 ( t )ψ 1 ( x, t ) + C2 ( t )ψ 2 ( x, t )
C1 ( t ) , C2 ( t ) = time dependent
C1 ( t ) and C1 ( t ) are the time-dependent probabilities of finding the system at E1 or E 2 at a time t ;
2
2
C1 ( t ) + C1 ( t ) = 1
2
2
The solution-“Ansatz” for “weak” perturbation is a time-dependent superposition of the stationary solutions.
¨ the occupation probabilities IC(t)I2 of the levels become time dependent due to the energy exchange with the field.
¨ Details in Chap.5.4.1.1 ,
first we consider the classical Einstein-Theory:
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5.1.3 Einstein-Theory of the optical transition in 2-level systems
(see lit: Loudon)
Concept: Phenomenological theory of A. Einstein (1905).
Light fields and matter interact by EM-field induced stimulated and spontaneous level-transitions in 2-level
atomic systems by photon emission and absorption at the transition frequency.
The transition modify the occupation probabilities pi of the energy levels Ei in the atomic system (rate
equations).
1) Transition Postulate:
„
Matter is built from quantized 2-level systems (transition pairs) representing atoms undergoing interaction with a classical
optical wave.
„
Interaction with the EM-field induces “upward or downward” transition between the levels (absorption, emission).
The ↑- or ↓-transitions between levels i and j are described by changes occupation probability ICi(t)I2 and ICj(t)I2:
a) Definition : Transition Probability rij (i → j)
change of level occupation probability pj per unit time in a single 2-level system
2
rij = ∂ p j / ∂ t = ∂ C j / ∂ t
i = int ial state
; j = final state
b) Transition Rates Rij (i→j)
change of the particle density Nj in state Ej per unit time and volume of a collection of atomic density N
Rij = ∂ 2 N j / ∂ t ∂V = rij N i = rij N Ci
2
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Describing the occupation probability of state i of the transition as
pi ( t ) = Ci ( t )
2
we get for the 2-level system E2 and E1 with
C2 ( t ) and C1 ( t )
2
rij =
∂ C j (t )
2
2
∂t
=−
fulfilling
∂ Ci ( t )
C2 ( t ) + C1 ( t ) = 1
2
2
2
i= initial state, j= final state
∂t
and obtain for the transition rate of Ni identical pairs
Rij = N i
∂ C j (t )
∂t
2
= − Ni
∂ Ci ( t )
∂t
2
Ni=density of transition pairs i→j, resp. atoms
2) Spontaneous emission Postulate:
For thermodynamic reasons or due to the additional quantization of the optical field (zero-point energy) spontaneous
emission transitions (downward transitions) are possible without an exciting optical field.
Spontaneous emission is spatially isotropic.
3) Stimulated emission Postulate:
The stimulated emitted photon induced by an external field has the same frequency ω, phase φ and propagation direction
(wave number) k as the stimulating photon.
¨
The two photons are indistinguishable.
¨
Stimulated emission results in a coherent amplification of the EM-field.
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5.1.3.1 Transition rates Rij between discrete energy levels in atoms
Goal:
We want to relate the “up” and “down” transition rate R12, R21 and R21,spont to the properties of the spectral energy density
W(ω), resp. W(E) or photon density sph of the optical field and the densities of the available transition pairs N1 (level E1 for
up-ward transitions) and N2 (level E2 for up-ward transitions).
Ni is the occupation density of level Ei.
Definition: spectral energy density W(ω) (SED)
Optical broad-band fields E(r,t), H(r,t) are characterized in the frequency domain by their cycle averaged spectral energy
density W(ω)
W (ω ) =
∂2 E
∂ω ∂ V
;
W (E = ω) =
∂2 E
∂E ∂ V
with the relation : W (ω ) = W ( E ) (SED)
{
For a monochromatic, harmonic optical fields (as an ideal case): E ( t ) = E0 cos (ωt ) = Re e
+ jωopt t
} = E / 2(e
¨ the SED W(ω) becomes a δ-function (1-sided spectrum):
1
ε 0 E02
W (ω ) = ED =
δ (ω − ωopt ) = s ph ωδ (ω − ωopt )
2
2
0
+ jωopt t
+e
− jωopt t
)
W(ω)
E0 ⇔ sph
using the concept of the photon as a energy quantum E=ħωopt and the photon density sph
Δω
ω0
ω
At Einstein’s time lasers producing quasi-ideal monochromatic fields (cos-field) did not exist. Only optical broadband (thermal)
sources characterized by W(ω) were available.
So it was natural to formulate the dependence of the transition probability rij related to W(ω).
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a) Graphical
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representation of the transition rates for a discrete 2-level system:
We consider N identical atomic 2-level systems with a atom density N and a occupation density n2 for the N upper level 2
(↓-pairs) and a occupational density n1 (↑-pairs) for the N lower level 1.
Therefore N=n1 + n2.
N states at E2 occupied with a density n2
R12 = R12,abs =
Rate of stimulated emission
R21 = R21,stim
Rate of spontaneous emission
R21,spont =
n2 zzz
E2
n2
R12
N
R21
R21, netto
E1
N states at E1 occupied with
a density n1
n1
zzzzzz
n1
∂
n1 = r12 n1
∂t
∂
=
n2 = r21n2
∂t
Rate of stimulated absorption
Net-Rate of stimulated Emission R21,net
∂
n2 = r21,spont n2
∂t
= R21 − R12 = r21n2 − r12 n1
(net down-ward transition, defines gain if >0)
N
Resulting in a total particle rate in (+) and out(-) flow for N levels at E2 occupied with n2 electrons:
Rate-Equation
∂ n2
∂n
= − R21,stim + R12,abs − R21,spont = − r21n2 + r12 n1 − r21,spont n2 = − 1 = rate of inflowing particles – rate of the outflowing particles
∂t
∂t
∂ n1
= + R21,stim − R12,abs + R21,spont = + r21n2 − r12 n1 + r21,spont n2
∂t
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Relations between transition rates R and W, n1 and n2:
We are looking for a relation of the rates Rij=rijNi between energy levels Ei → Ej and
1) the electron occupation density ni=Nfi of the levels Ei and the occupation probability function f(Ei)
(fi = f(Ei)= occupation probability function of level Ei , eg. Boltzmann- or Fermi-distribution)
2) the total transition pair (corresponding initial and final state) or atomic density N and
3) the SED W(ω) or W(E) of the exciting optical field
4) the values and microscopic dependence of transition probabilities rij
A. Einstein proposed in 1905 based on plausible arguments the following
assumptions for the transition probabilities rij and the transition rates Rij
(which will be proven quantum mechanically later):
1) the transition probability r for a 2-level transition pair (1,2) is postulated as:
r21=B21W(ω21)
stimulated Emission 2 € 1
r12=B12W(ω21)
„stimulated“ Absorption 1 € 2
rspont=A21
spontaneous Emission into all modes 2 € 1
A. Einstein postulates
ω21=(E2-E1)/ħ transition frequency
• the transition probabilities rij of the stimulated processes are related by the Einstein-coefficient Bij to
spectral energy density W E ji , resp.W ν ji assuming an optical field with a spectral density much broader than
( )
( )
the sharp atomic transitions.
• The spontaneous transition probability Aji is independent of the photon field.
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Remark:
W ( E ji ) is the energy of the light field per unit volume and energy, W (ω ji ) is the energy of the light field per unit volume and frequency !
¨
W ( E ji ) =W (ω ji ) and B(E) = B(ω)
2) the transition rate Rij (iŒj) for atomic 2-level systems:
- Rij is proportional to ni the carrier density of the occupied “Start”-levels Ei (resp. transition pair density)
- Rij is proportional to W(ω21) the spectral energy density
- for atoms all final states Ej of a transition pair with density nj are per definition unoccupied, nj=N-ni.
3) level occupation probability functions fi(Ei) for a transition-pair:
fi(Ei) is the occupation probability function that the level i is occupied by an electron.
Per definition: fj=(1-fi)= probability that state j is empty
Transition Rates:
1) Stimulated Emission 2 → 1:
R21,stim = W B21n2 = W B21 Nf 2
for isolated atoms
B21 is the Einstein-coefficient for stimulated Emission.
W(ω) B21 is the transition probability per excited energy state at E2
2) stimulated Absorption 1→2 :
R12,abs = W B12 n1 = W B12 N f1
for isolated atoms
B12 is the Einstein-coefficient for stimulated Absorption
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3) spontaneous Emission 2→1 per mode:
R21,spont = A21n2 = A21 N f 2
for isolated atoms
A21 is the Einstein-coefficient for spontaneous Emission
4) net stimulated Rate R21,netto 2 → 1:
R21,net = R21,stim − R12,abs = R21,net = W ( B21n2 − B12 n1 ) = WB21 Nf 2 − WB12 Nf1
for isolated atoms
n1 = density of the transition pairs with the electron in the ground state E1 (Valence band)
n2 = density of the transition pairs with the electron in the excited state E2 ( Conduction band)
N = density of all atomic transition pairs, resp. density of the atoms
f1 = occupation probability for an electron in the ground state E1 (eg. Boltzmann or Fermi-Function)
f2 = occupation probability for an electron in the excited state E2 (eg. Boltzmann orFermi-Function)
Condition for optical amplification and population inversion:
¨ The net-rate of stimulated emission (difference between stimulated Emission und Absorption) must be R21,net >0, in
order that the photon density in the optical field is increased ¨ optical gain g is possible.
For B12=B21 the occupation probability of the upper energy level 2 must be larger than the lower level 1:
Ö POPULATION INVERSION n2 =Nf2> n1=Nf1 , → f2(E2)>f1(E1)
Important Remark:
In thermal equilibrium with the thermal blackbody radiation at a finite temperature T the electrons are distributed according to a
Boltzmann-Distribution with n2<<n1.
This means that no gain is possible in thermally excited materials.
Electrons have to be “pumped” or transferred actively to the upper level to create population inversion by non-equilibrium
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5.1.3.2 Concept of thermodynamic equilibrium, optical pumping and population inversion
Problem:
Is it possible to establish a population inversion n2>n1, resp. amplification R21,net>0 by optical absorption in a 2level system to “pump” the upper level?
Introducing a 2-level system with an atom density N in an optical field of a resonant spectral energy density W(ω21) leads to a
change in the level population densities n1 and n2 until a stationary state is reached when the down- and up-ward transition
rates are equal and no change in n1 and n2 occurs any more
¨ no optical population inversion is possible in a 2-level system.
→
∂ n2 / ∂ t = ∂ n1 / ∂ t = 0 → equilibrium
→ R12 = R21 + Rspont equilibrium condition (rate equation)
by insertion of the expressions for the rates R gives:
W B12 n1 = W B21n2 + A21n2
→
with n1 + n2 = N
W B12 ( N − n2 ) = W B21n2 + A21n2
W B12 N = (W ( B21 + B12 ) + A21 ) n2
W B12 N
n2 =
(W ( B21 + B12 ) + A21 )
for
W →∞
ntransparency
→
W
; n1 = N − n2
B12
n2 =
N < N / 2 resp. R21 < R12
B12 + B21
Absorption N2<N/2
→
Transparency N2~N/2
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The rate of absorption is always larger than the rate of stimulated emission, therefore
„
a simple 2-level system “optically pumped” by a light field remains always absorbing or at most becomes
transparent (n2=n1).
„
No population inversion (n2>n1) or gain can be reached by optical pumping at the transition frequency !
The same formalism as above can be used for the
Thermal Boltzmann equilibrium with the blackbody radiation: W(ω) = ρbb(ω) (see chap.5.2.1)
In reality a 2-level system with no external light field W=0 must be in equilibrium with the always present „black body radiation“
with the spectral energy density ρbb(ω) and exhibit an electron distribution according to a Boltzmann-distribution, where
n1>>n2 , resp.
n2 / n1 = exp ( − ( E2 − E1 ) / kT )
The resulting relationship was used by A. Einstein to relate the constant A to B
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Optical population inversion in a 3-Niveau-System:
Reaching optical amplification g>0 in a 2-level system requires: R21,stim > R12 ,abs resp. R21,net > 0 resp. n2 > n2 ,transp > N / 2
To enforce a population inversion, resp. inverting the population in a 2 → 1 transition (ω21) by an optical „pump“
field Wpump(ωpump) such that n2 > n2,transp > n1 it is necessary to introduce a 3rd level for artificially injecting carriers in
level E2 without removing them by stimulated emission of the pump light again.
Establishment of such a non-equilibrium (Inversion) n2 > n2,transp is called Pumping or Inversion of the medium
Optical pumping of a 3-level system:
The simplest way to achieve population inversion between level E2 and E1 is the introduction of a third, lower level E0
ωpump=ω20=(E2-E0)/ ħ
Pump level
2: n2
A21
ω21=(E2-E1)/ ħ (active transition 1-2)
and ω20> ω21
n1 +n2 + no=N
(B12)
1: n1 → ~0
B20
B02
Wactive=0
A20
A10→∞
0: n0
Ideally the transition 1 → 0
occurs infinitively fast (r10 → ∞, n1=0)
meaning level 1 is always empty !
Assumptions:
- active optical transitions (eg. gain) occur only between 1 ↔ 2
- optical pumping take place between 0 ↔ 2
- the transition probability r10 →∞ (“empty” 1-level)
- A20 should be small
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Calculation of the equilibrium rates between the 3 levels leads to:
Lasing transition: 2-1 (without lasing W12 =0), pump transition: 0-2 (ρpump)
static :
∂
= 0 ; n0 + n1 + n2 + = N = density of atoms ; W12 = 0
∂t
level 2 : W pump B20 ( n0 − n2 ) − n2 ( A20 + A21 ) = 0
A21n2 − A10 n1 = 0 → condition for inversion n2 > n1 → 0
level 1 :
n2 A10
=
>1 →
n1 A21
replacing
0 < n2 =
τ 10 < τ 21
n0 and n1
→
W pump B20 N
⎛A
⎞
W pump B20 ⎜ 21 + 2 ⎟ + ( A20 + A21 )
⎝ A10
⎠
<
N
A21
+2
A10
<
N
2
and
n1 = n2 A21 / A10 < n2
Depending on the optical and electrical properties of the gain-medium the inversion of transition 2 → 1 is realized by:
a) absorption of pump-light of a shorter wavelength than the active transition
(many solid state lasers, Er-doped fiber amplifiers etc.)
b) Injection of carriers by an electrical current in forward biased pn-diodes (SC-Diode Lasers)
c) Ionization in gases (eg. HeNe-Gas Laser)
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5.1.4 Stimulated Emission Rate R21,net and Optical Amplification g
Goal:
• Relation between gain and optical transition rate
g ↔ R21,net
Stimulated emission generates temporal and spatial coherent photons (ω, k, φ) which amplify the stimulating light field.
• Relation between the net stimulated emission rate R21 or the carrier densities difference n2-n1 and the
gain constant g(R21,net) = g(n2,n1)=dI/dx/I.
• Relation between the susceptibility χ( n2-n1) for the wave-picture.
Wave representation as a photon stream: (necessary for coupling to transition rates)
„
„
The optical field is represented as coherent superposition of elementary wave packets with an energy quantum ω
In analogy to the electrical current a photon flux density (intensity) can be associated to a current of photons
with the photon density sph which propagate with the group velocity vgr (particle picture of light)
Relation between photon density sph and field amplitude E=E0cos(ωt) for a harmonic TE plane wave sph ⇔ E0
1
energy density : w = ε E0 E0* = ω s ph → E0 = 2 ω s ph / ε = E ( s ph )
2
or from the int ensity I :
ε ε 2 1
1
1
I = S = E × H = E0 E0
= E
= wvgr = ω s ph vgr
2
2
μ0 2 0 εμ0
with the spectral energy density : ( 1 − sided )
vgr =
1
ε μ
⎯⎯⎯⎯
→
E0 = 2 ω s ph / ε
; ε = ε 0ε r
1
W (ω ) = w δ (ω − ωopt ) = ε E 0 E0 * δ (ω − ωopt ) = ωopt s ph δ (ω − ωopt )
2
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Relation between the energy density w and the photon density sph of an optical field:
classical energy density of optical EM-fields
Graphical representation of photon-flux through a volume element:
1
DE
2
b) magnetic energy density 1 BH
2
a) electric energy density
E
c) cycle and spatially averaged energy density
∂W ε 2
of an harmonic optical field w =
= E0
∂V
hω
photon inflow
z
2
d) intensity (Pointing-Vector)
H
I = S = 1 / 2 E × H = wvgr = ω n ph vgr
associated photon density: n ph = s ph
x
ε E02
=
=
ω 2 ω
w
y
dx
x
sph
dVol
vgr
β
photon outflow
photon
generation / destruction
dx
x+dx
nph = Photon density = sph
E = electrical field strength
H = magnetic field strength
S = Pointing-vector
w=energy density
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Calculation of the relation g(Rstim(n2,n1)):
¨ the net-rate of stimulated emission R21,net changes the photon density sph in transit. Depending R21,net> or <0
optical amplification g or attenuation α will result.
The optical field with photon density sph(x,t), resp. intensity I(x,t)=sph(x,t)ħωvgr travels with vgr in the x-direction through a thin
layer Δx of amplifying medium in time Δt.
¨ The medium increasing its intensity by ΔI, resp Δsph.
Δsph
sph
z
Photon generation / annihilation
R21,netto
sph + Δsph
Photon
inflow
x
R21,netto
Photon
outflow
y
hω
I
sph
x + Δx
x
Δt=Δx/vgr
R21,net
From the definition of the attenuation α:
α =−
∂I 1
∂x I
I + ΔI
→
I ( x ) = I ( 0 ) e −α
x
x
dx
x+dx
using this definition of α: α>0 means absorption, α<0 gain !
because the intensity I=sphvgrħω is proportional sph, we write:
I ( x + Δx ) = I ( x ) + ΔI
I ( x ) − I ( x ) αΔx
s ph ( x + Δx ) = s ph ( x ) + Δs ph
I ∼s
ph
⎯⎯⎯→
s ph ( x ) − s ph ( x )αΔx
Δs ph
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To travel through the layer Δx and increase its intensity by Δ I, resp. Δs ph the wave needs the time Δt:
Δt = Δx / vgr
during the time Δt the wave increases its photon density by Δs ph = R21,netto Δt of stimulated photons:
Δs ph
Δs ph = R21,netto Δt = ( R21 − R12 ) Δt = − s ph ( x ) α Δx = − s ph ( x )α vgr Δt = s ph ( x ) g vgr Δt
α = − R21,netto / ( s ph ( x ) vgr ) = − g
Important relation ship to connect fields and rates: g ⇔ R21
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5.2 Spontaneous emission in 2-Level Systems
K5
(see lit. Coldren, Yariv)
Goal:
„
To determine 2 relations between the 3 Einstein-coefficients A, B21, B12 we can use the thermodynamic equilibrium
situation between a
2-level system and the
Blackbody-Radiation with a spectral power density W(E)=ρbb(E) (obtained from thermodynamics) and
a carrier distribution obeying the Boltzmann-distribution (see Physik II, 2.Sem.)
„
Finally there is still one unknown remaining, eg. B which can only be determined only from a quantum mechanical
description of atomic polarization or experimentally from an absorption measurement.
5.2.1 Spontaneous Emissions-Processes
R21,spont Spontaneous-Emission-Postulate
„
„
The spontaneous transition of an excited QM-system without external optical disturbance into the lower energy-level has no
classical equivalent.
This can only be understood by QM if also the optical field is quantized into a harmonic oscillator
(2.quantization, zero-point fluctuations of an optical mode)
Spontaneous emission is a necessary process to explain that an optical 2-level system perturbed by an external optical
field (resulting in non-equilibrium carrier densities n2+Δn2 and n1+Δn1 with Δn= carrier disturbance) can relax back to
equilibrium (Bolzmann-distribution) after switching off the disturbance.
¨ Spontaneous emission is a thermodynamic necessity to establish equilibrium
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5.2.2 Thermodynamic equilibrium:
K5
quantized atomic system ↔ blackbody radiation
Following A. Einstein we consider the rate equations for the electrons of a 2level system in thermodynamic equilibrium
(Boltzmann) with the known blackbody EM-radiation field with a spectral energy density ρBB(ω) inside a „2-level
blackbody“ of temperature T. The carrier distribution obeys Boltzmann distribution.
Idea: In thermodynamic equilibrium the spectral energy density is W = ρ BB and the electron densities n2 and n1 obey Boltzmann-statistics.
In equilibrium the transition rates between levels 2 → 1 and 1 → 2 must be equal:
ª
R21 ( ρ BB ) + R21,spont = R12 ( ρ BB )
©
ρbb = ?
Spectral mode ρmode and energy ρBB density of blackbody-radiation:
(without proof, see Physik II)
We obtain the spectral energy density ρ BB of blackbody radiation by counting the EM-modes inside a Volume V and multiplying the
spectral mode density ρmode with the average number of thermal photons per mode
n
and the photon energy E. n=refractive index.
ρ BB ( E ) = ρ mode ( E ) n ( E ) E
dn
8π n E
dE
ρ BB ( E ) = 3 3
h c ⎡⎣exp ( E / kT ) − 1⎤⎦
3
3
1 + ( E / n)
ds ph,BB hv
8π n3 E 3
1
=
dE
⎡⎣ exp ( E / kT ) − 1⎤⎦
h3c 3
no dispersion ρ
mode E
n
of n
≅
dn
ds
hv
8π n E
dE
ρ BB ( v ) = 2 3
= ph,BB
dv
h c ⎡⎣exp ( E / kT ) − 1⎤⎦
3
3
1 + ( E / n)
E=hv= energy of blackbody radiation photons
energy density ρ BB ( E )
volume V
energy density ρ BB ( v )
n= refractive index of blackbody
T= temperature of the blackbody
h= Plank constant
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For the derivation of the ρBB we made use of:
Spectral blackbody mode density:
8 π n3 E 2 ⎡
d n ⎤ 8 π n3 E 2 d 2 N mode
ρ mod e ( E ) =
⎢1 + ( E / n ) d E ⎥ ≅ h3c 3 = d E dVol
h 3c 3
⎣
⎦
blackbody-mode density
8 π n3v 2 d 2 N mode
=
ρ mod e ( v ) ≅
d v dVol
c3
I
Photon-occupation n ( E ) per mode in thermal equilibrium:
In thermal equilibrium each mode i carries according to the Bose-Einstein-Statistic for photons
ni photons.
Bose-Einstein-Statistic:
ni ( Ei ) =
1
⎡⎣exp ( Ei / kT ) − 1⎤⎦
Therefore we get
<< 1
occupation probability of mode i with photons ofenergy Ei = ωi
Ei >>kT
ρ BB ( E ) = ρmode ( E ) ni Ei
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5.2.3 Relation between the Einstein-coefficients A21, B21 and B12
Example: atomic 2-level transition pairs
In the thermodynamic equilibrium with the spectral energy density ρBB of the thermal radiation:
1) total emission rate = total absorption rate
ª A21 n2 + B21 n2 ρ BB = B12 n1 ρ BB ©
A21n2 = spont. photon emission rate into all vacuum modes with density ρbb
2) Boltzmann-equilibrium of the level occupation for n2, n1
n2
⎛ E ⎞
= exp ⎜ − 21 ⎟ ; E 21 = E 2 − E1
n1
⎝ kT ⎠
insertion of the ratio of n2/n1 leads to:
3
A21 / B21
8π n3 E21
1
=
= ρ mod e ( E21 ) n21 E21
ρ BB ( E21 ) =
3 3
⎛ E21 ⎞
⎡⎣exp ( E21 / kT ) − 1⎤⎦
hc
( B12 / B21 ) exp ⎜ ⎟ − 1 ρ E
mod e 21
⎝ kT ⎠
n21
Comparision of the coefficients on both sides of the equation results in:
B21 = B12 = B
A = B(E)
3
⎛ 8π n 3 E21
⎞
⎛ 8π n 3 h v 3 ⎞
⎜
⎟ = B (v) ⎜
⎟ = B ρ mode ( E ) E
3 3
3
h
c
c
⎝
⎠
⎝
⎠
for energy density
per energy E
Relation between Einstein-coefficients
B and A
for enery density
per frequency v
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Important remark:
The relations are valid at a particular energy difference E21=E2-E1 of a particular mode.
The coefficient A which is the inverse of a spontaneous carrier lifetime A = 1 / τ spont ( Ei ) is the Einstein coefficient for the
spontaneous emission into all modes of free space. (also not only lasing modes or resonator modes, see spont. emission
factor β in chap.6)
Considering the ratio between the net stimulated rate and the spontaneous rate:
(measure for the ease to invert the material)
R21,net = R21 − R12 = B ( n2 − n1 ) ρ opt
R21,spont
⎛ 8π n 3 h v 3 ⎞
= An2 = B ⎜
⎟ n2
3
c
⎝
⎠
In thermal equilibrium ρopt ( E21 ) = ρbb ( E21 ) = ρmod e ( E21 ) n21 E21 we have the following magnitude-relation of the net stimulated
and spontaneous rates:
R21,net
1
=
= n << 1
R21,spont exp ( E / kT ) − 1
R21,spont >> R21,stim , R12,abs
if E = hv > kT
and
n2 << n1
kT~26meV@300K, hv~1eV → E/kT~40
Thermal Radiation Source
Conclusion:
Systems in thermodynamic equilibrium (eg. lamps) do virtually not emit coherent, resp. stimulated radiation (Rstim << Rspont).
Only for LASERs, excited by pumping into a strong non-equilibrium state (inverted) with n2>n1 , optical gain and coherent
radiation becomes possible and dominant.
Optical gain is more difficult to achieve at short wavelengths (microwave Masers have been invented before optical Lasers).
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Relation between the Einstein coefficient B and the attenuation α:
A direct experimental measurement of A or B is difficult, however we will show that there is a simple relation ship between the
Einstein coefficients and the attenuation α.
The attenuation α, describing the decrease of the optical beam intensity I(L)=I(0)e
medium of known absorption state, is relatively easy to measure.
-αL
with propagation distance L in the
Assuming a weak monochromatic optical field (close to thermal equilibrium) with the spectral energy density:
W (ω ) = s ph hv δ (ω − ω0 ) →
R21,net = B ( n2 − n1 ) s ph ω
We have already shown that α and R21,netto are related by:
α ( E ) = − R21,net ( E ) / ( s phvgr ) = − s ph hv B ( n2 − n1 ) / ( s phvgr ) = − hv B ( n2 − n1 ) / vgr
h
Using a weak optical beam for attenuation measurements, which does not disturb the equilibrium Boltzmann population:
n2
⎛ E ⎞
= exp ⎜ − 21 ⎟
n1
⎝ kT ⎠
B ( E ) = −α ( E )
vgr
→
n2 − n1
n −n ⎡
⎛ E ⎞ ⎤
≅ 2 1 = ⎢exp ⎜ − 21 ⎟ − 1⎥
n1 n ~ N N
⎝ kT ⎠ ⎦
⎣
1
v
1
= −α ( E ) gr
hv ( n2 − n1 )
hv
1
⎡
⎛ E ⎞ ⎤
N ⎢exp ⎜ − 21 ⎟ − 1⎥
⎝ kT ⎠ ⎦
⎣
with known N=atoms / unite volume
; E2 − E1 = hv
From the measurement of α with known atom density N and group velocity vgr we obtain B and also A.
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Conclusions:
• The Einstein model postulates
1) the energy quantization of the medium and
2) transition rates between the states
• The Einstein coefficients B12, B21 and A21 have been introduced formally as postulates and can not be related
directly to the material properties (eg. electronic potentials, eff. mass) on a microscopic level
• The model allows the description of optical gain g by population inversion
• The spontaneous emission is a quantum-mechanical phenomenon and has no classical equivalent.
However it is a thermodynamic necessity for the classical system to relax to thermal equilibrium.
• The model is also capable of representing the black-body radiation, resp. the relation between A and B
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5.3 Pumping and Population Inversion
Optical amplification α<0 or g>0 requires a population inversion → R21,net>0 → n2>n1.
„
„
A population inversion can not be realized in thermodynamic equilibrium (eg. heating T→∞).
Non-equilibrium is enforced by an influx of carriers into the higher energy level E2 ¨ PUMPING
(pumping transitions in 3- or multi-level systems).
5.3.1 Optical and electrical pumping
Inverted transition pairs are realized by Absorption of „pump-light“ (optical pumping) in the isolating crystal matrix containing
the active atoms. The atoms must provide at least an additional 3rd energy level E3 (3-level-system), at λpump<λ21.
For so called solid-state lasers based on active metal atoms like Nd, Er, Cr, etc. embedded in an isolating crystal matrix of eg.
Erbium in Glass, Cr in Ruby, Nd in Glass etc. there is no possibility for pumping by an electrical current or impact-ionization.
a) Optical Pumping-Scheme of a 4-level system:
•
It is assumed that the active atoms (often metals in
isolators) provide the atomic levels
•
In addition we assume that there is a very fast
transition with a short life time r=1/τ between
levels 3 → 2 and 1 → 0.
¨ n3~0 , n1~0 , n2>0
level 3
3: n3
PumpTransition
0→3
2: n2
1: n1
0: n0
level 2
Laser-Transition
2→1
level 1
n0 + n1 + n2 + n3 = N
level 0
See also Er-doped-Fiber-amplifier in chap.6.
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b) Electrical Pumping: (mostly used with semiconductor based components, see chap.6)
„
Inversion of the excited states in the conduction and valence band of SC can be realized by optical pumping, impact
ionization, but in particular by current injection in pn-junctions.
„
Minority carrier-injection (electron-injection into the conduction band of a p-SC, hole-Injection into the valence band of nSC) is realized simply by current injection through a depletion layer of a forward biased pn-diode.
Schematic Carrier (electrons / holes) injection in forward biased PiN-Diode:
a) amplification in SC (n1<<n2):
Injection current
• Elektrons (n2) in Conduction band, E2
P
o Löcher (n1) in Valence band, E1
Output-wave
b) attenuation in SC (n1>>n2):
Input-wave
N
Carrier non-equilibrium by
carrier injection
the active i-layer:
n>>n0 , p>>p0
und np>>ni2
of a forward biased diode.
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Technical Realizations of optical and electrical Pumping:
1) Optical Pumping of Solid State Lasers:
Nd:YAG-Laser
3) Current-Injection-Pumping of pn-Diodes Lasers:
Band diagram-Representation
Excitation of Neodymium atoms (Nd) in an YAG-crystal-matrix
by Absorption of pump-light from a gas discharge lamp
Inversion by minority carrier injection (chap. 6)
ws
Nd doped
YAG-X-tal
ws = depletion layer width
VD=0 , ID=0
VD~Eg/e , ID>0
YAG: NdxY3-xAl2O12
Yttrium-Aluminium-Granat
electrons
2) Gas-Discharge-Pumping (electrical pumping)
of Gas-Lasers (He-Ne)
Excitation of gas atoms (Ne) by Impact ionization in gas-discharge
VD>Eg/e (High current-Injection)
holes
d: area if inversion
eh-
Non-equilibrium
(carrier inversion)
ID
VD
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5.3.2 Carrier Rate Equations and total Spontaneous Emission
Total Spontaneous Emission Rate Rspont:
(see lit. Coldren)
The Einstein coefficient A describes the spontaneous emission rate R21,spont into all modes of a particular frequency ω21, in a
spectral width Δωspont and all propagation directions in a “given volume” V.
Therefore an excited 2-level system can emit spontaneously
1) in all spatial directions resp. into all available optical modes ρmode in the a given device volume Vspont
2) at all frequencies Δωopt of the involved active transition pairs
¨ The total spontaneous emission rate R21,spont (appearing n the carrier rate equation) is obtained as the spontaneous emission
3 2
3
rate per mode R21,spon,mode (eg. the lasing mode) multiplied by the spatial mode-density ρ mode (ω ) ≅ 8 π n v / c , the
spectral band-width Δωspont of the total spontaneous transition and the “available” volume for spontaneous
emission Vspont :
n2 A )
od
e
(
R21,spont ,mod e = γ R21,spont
(
Vspont
un
gu
ide
dm
R21,spont = R21,spont ,mod e ρ mode (ω ) Δω spontVspont
fictive mode volume
)
with γ = β =spontaneous emissions factor=1/ Vspont ρ mode (ω ) Δω spont << 1
(one often assumes as a first order approximation Vspont=Vmode)
(10
−4
− 10
−5
for SC-Diode Lasers
spontaneous
dipole
)
wave guide
wave guide mode
R21,spon,mode=γAn2 is the spontaneous emission into a single mode (eg. lasing or amplified mode)
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Carrier-Rate Equations including Pumping
Concept of Rate Equations: (carrier continuity equation for particle currents per energy level)
Optical transitions produce particle currents (rates) R into and from a particular energy level Ei. Particles can also be stored
in the energy states of the energy level Ei in the volume Vactive (particle reservoir).
Rate equations are just the continuity equation for particle in an energy level (reservoir):
Temporal change of particle density dn/dt = Σ in-flowing particle currents - Σ out-flowing particle currents
The pump-mechanism (optical, electrical) can be accounted by the continuity equation for the carrier densities n2, n1 by the
addition of an inflowing carrier generation rate Rpump.
∂ n2
∂n
= − 1 = Rpump − R21,net − R21,spont
∂t
∂t
Rate equation of the carriers in the excited state E2 and E1
We express the optical field by the photon density sph and relate it by the gain g to
the transition rates:
R21,net = g ( n2 ,n1 ) s ph vgr
Rspont = n2 / τ spont
∂n
= R pump − g ( n2 ,n1 ) vgr s ph − n2 / τ spont
∂t
Rpump
Vmode
n
S
Rspont Rstim
Vaktive
L
Remark: the active volume for carriers Vactive and the mode volume for photons Vmode must not necessarily be equal !
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Amplification and Inversion:
Static : ∂ / ∂ t = 0
g ( E21 ,ntr ) = −
→ ntr
R21,net ( E21 ,ntr )
=0
s phvgr
Transparency condition g ≥ 0 defines
transparency density
Pump rate for Transparency n = ntr and R pump,tr ( n2,tr ) :
0 = R pump ,tr − 0 − ntr / τ spont
→ with R21,net = 0
R pump,tr = ntr / τ spont
Conclusions:
Materials without pumping are absorbing for the wavelength of interest.
Pumping reduces the attenuation until the material becomes transparent and finally amplifying.
For low pump rates for transparency Rpump,tr
• small spontaneous emission (“lost” photons) and small density of atoms (gives also only small gain)
• small Einstein coefficient A, eg. long carrier lifetime of the excited state
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5.3.3 Photon Rate Equation
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Photons propagate as photon streams of density sph or are contained as standing waves in resonators.
Continuity equations per unit volume can also be formulated for in- and out-flowing photons into the propagating field of a
single optical mode i. (note that here the spontaneous emission rate per mode R21,spont,mode=γR21,spont has to be used)
The photon density sph is changed by:
1) the stimulated emission (net) R21,net and
2) the spontaneous emission per mode R21,spont, mode=γ R21,spont
( γ = β = spontaneous emission factor)
3) Scattering losses from the waveguide, Rloss (residual absorption, scattering)
Photon number Sph=sphVmode of the mode volume Vmode:
∂ S ph
∂t
∂ s ph
∂t
(
)
(
)
= R21,net + γ R21,spont Vactive − RlossVmode
= R21,net + γ R21,spont
Vactive
Vmode
R21, netto
Mode i
→ :Vmode
− Rloss
Vmode
Vactive
confinement factor Γ
Rate equation for photon density per mode
Rloss
Rloss= losses per unit length due to scattering,
different absorption mechanisms, etc.
For a 2-level system we get:
∂ s ph
∂t
= Γg ( n2 ,n1 ) s phvgr + Γγ n2 / τ spont − Rloss
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Example:
Graphical representation of particle “reservoirs” and particle flow rates:
Particle transition rates between an active, pumped medium and an optical mode field
n2 , n1
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5.4 Quantum Mechanics of Optical Transitions
5.4.1 Quantum-Mechanical concepts of the dynamic of bound
electrons in an optical E/M-field
What do we need beyond the Einstein-Theory ?
(see lit: Loudon, Yariv)
¨ Perturbation Theory for the motion of electrons in the potential V(x) of atomic or crystal lattice forces and a superposed
dynamic optical field (optical potential Vopt(t))
¨ Correlation of the transition rates R or Einstein coefficients A, B to microscopic properties (eg. V(x)).
Quantum Mechanics (QM) allows the self-consistent calculation of B, resp A from first principles
Procedure of solution:
1) the classical EM light field superposes a time dependent, high frequency (f~200 THz) electrical potential Vopt(x,t) on the
static microscopic potential field V(x) of the atom or crystal lattice.
Assumption: weak perturbation Vopt(x,t)<<V(x).
2) Weak Perturbation Theory solves approximately the time-dependent Schrödinger-equation (TSE) for ψ(x,t)
The interaction of the E(M) forces of the optical field appears as time-dependent electrical (magnetic) „potential“ in the
time-dependent Energy (Hamilton)-Operator HWW(t): Vopt(x,t) ¨ HWW(x,t)= ?.
a) the solution ψ(x,t) of the TSE, allows the calculation of statistical QM expectation values of:
position :
C1 ( t ) ; C2 ( t )
x ( t ) = ∫ψ ( x,t ) x ψ ∗ ( x,t ) dx →
dipole :
p ( t ) = −e x ( t )
→ susceptibility :
χ (ω )
→ r12 ; r21
b) the transitions probabilities rij =
∂
Cj
∂t
2
between discrete atomic states
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5.4.1.1 Time dependent Schrödinger-Equation for a 2-level system in a
optical field (perturbation theory):
Questions:
What is the electron motion <x> in a one-electron atom exposed to the EM-forces of a harmonic optical field with
frequency ωopt and field strengths E and (H) ?
Schematic Potential of the harmonic dipole
Assumptions:
oscillator in an oscillation optical field E
The forces acting on the moving electron result from:
E(x)
1) time invariant potential field of nucleus or crystal field V(x)
2) time variant, oscillating (with
e
jωopt t
-eVKern
) EM-potential of the light field Vopt(t)
simplified optical potential (H neglected)
E(t,x)=E0 ejωopt t
using the definition
∂V
E = − opt ; Vopt = − E (t ) x
∂x
-
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
→ Hww(t,x) = -eVopt(t)
=+eE0x
(Note: we assume the positive time dependence e
+ jωopt t
-
ejωopt t
+
with ∂ • − −○ + jω ! )
∂t
We neglect the magnetic forces for simplicity and consider only the electrical force.
-
-eVopt(x,t')
Eopt(0,t')
x
-eVopt(x,t'')
Eopt(0,t'')
The TSE describes the probabilistic electron motion by the wave function ψ(r,t),
resp.by the spatial probability Iψ(r,t)I2 of finding the electron at position x.
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Time dependent Schrödinger-equation:
H ( t,x ) Ψ ( x,t ) = ⎡⎣ H pot ( x ) + HWW ( t,x ) ⎤⎦ Ψ ( x,t ) = i
∂
Ψ ( x,t )
∂t
H=Hamilton-Energy –Operator, Hpot >>HWW
Hww(t,x) is the Hamilton-Operator of the optical field. 2 is the upper excited level, 1 is the lower ground state with E2 > E1.
Stationary solutions (Hww=0): ( → discrete energy levels without the optical field)
without an optical field: HWW =0
∂
H Ψ ( x,t ) = i
Ψ ( x,t ) = E Ψ ( x,t ) time independent Schrödinger equation
∂t
→ H μ ( x) = E μ ( x)
with the two stationary solutions (eigenfunctions) and the energy-eigenvalues: E1 , E2
Eigenfunctions for H ww=0
E1 : Ψ 1 ( x,t ) = u1 ( x ) e -jE1 / h t = u1 ( x ) e -jω1 t
; ω1 = E1 /
E 2 : Ψ 2 ( x,t ) = u2 ( x ) e -jE2 / h t = u2 ( x ) e -jω2 t ; ω2 = E2 /
The solutions ui ( x ) fulfill the Orthonormalization Relations (orthogonal and normalized):
∫ uk ( x ) ul ( x ) dx = Δ kl
∗
,
∫ uk ( x ) uk ( x ) dx = Δ kk = 1
∗
,
∫
k ≠`l
uk ( x ) ul∗ ( x ) dx = Δ kl = 0
( without
prove )
The stationary total solution for the 2-level-system is a superposition:
Ψ ( x,t ) = C 1 u1 ( x ) e -jω1 t + C 2 u2 ( x ) e -jω2 t
2
2
with C1 + C2 = 1
,
ωi = Ei /
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Interpretation:
IC1I2, (IC2I2) is the probability of finding the electron in the state E1 (E2) resp. Ψ1 ( x,t ) ,
( Ψ2 ( x,t )) .
C1 and C2 are constants for the stationary state, Hww=0.
System in Ground state: C1=1, C2=0
System in Excited state: C2=1, C1=0
System in Mixed state: C2≠0, C1≠0
5.4.1.2 Optical Perturbation calculation of the transition probability rij or transition rate Rij
1) Non-stationary solution for a harmonic perturbation HWW≠0:
H Ψ ( x,t ) = ⎡⎣ H pot ( x ) + HWW ( t ) ⎤⎦ Ψ ( x,t ) = i
∂
Ψ ( x,t )
∂t
with the postulated solution as a superpostion of the stationary ground and excited states described by C1 ( t ) , C2 ( t ) as time functions
Ψ ( x,t ) = C1 ( t ) u1 ( x ) e− jω1 t + C2 ( t ) u2 ( x ) e − jω2 t
with
2
2
C1 ( t ) + C2 ( t ) = 1
„
The perturbation of the 2-level-system by the optical field (switched on at t=0 with eg. the system initially in the ground state
IC1(0)I2=1 and IC2(t)I2=0) causes an absorption Process E1 → E2 changing the occupation probabilities IC1(t)I2 and
IC2(t)I2.
„
For simplicity the system is at t=0 in a „pure“ state, either at E1 (leading to absorption) or at E2 (leading stimulated emission).
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Knowing C1(t), C2(t), we calculate the transition probabilities rij(t) as change of the occupation probability per unit time:
r21 =
2
2
∂
∂
C1 ( t ) = − C2 ( t )
∂t
∂t
Transition probability r21
Generic solution-procedure for Ci(t):
1) insertion of the assumed solution Ψ ( x ,t ) into the TSE, then
2) we first right multiply both sides of the TSE with u*k(x) and then
3) we left multiply the TSE by ul(x) and finally
+∞
4) we integrate both sides by
∫
dx using the orthonormality of the static solutions ui(x) and uk(x) and the abbreviation:
−∞
ui H uk =
∫
ui∗ H uk dx Matrixelement between the states i and k of Energy Operator H
volume
Considering only the time-dependent part of the TSE, we get ( after a simple but lengthy calculation, see appendix !) the fundamental
nonlinear and coupled 1.order differential equations of for C1(t) und C2(t):
∂
j
C1 = − ⎡⎣C1 u1 HWW u1
∂t
+ C2 u1 HWW u2 exp ( − j (ω2 − ω1 ) t ) ⎤⎦
∂
j
C2 = − ⎡⎣C1 u2 HWW u1 exp ( − j (ω1 − ω2 ) t ) + C2 u2 HWW u2 ⎤⎦
∂t
Coupled Differential-Equation for C1(t) and C2(t)
with the definition
H kl = uk HWW ul = ∫ uk∗ HWW ul dx
Perturbation - Matrix Element
With the definition ω 2 − ω1 = ( E2 − E1 ) / = ω = ω21 > 0
Transition frequency
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Observe that Matrix element <ukIHwwIul> is the key parameter containing the materials parameter and the optical
perturbation with the frequency ωopt because Hww=Hww(E0, ωopt).
<ukIHwwIul> contains only the spatial parts ui(x) of the solutions Ψ(x).
2) Calculation of the Interaction Hamilton Operator HWW(x,t) for monochromatic harmonic optical fields:
We calculate the interaction hamiltonian Hww(t,x) as the time dependent potential energy - eVopt(x,t) of a bound electron in
the time dependent potential of the electrical field strength E(t) of the harmonic optical field (magnetic energy is neglected).
The matrix element Hkl. describes the atomic forces acting on the electron inducing the transition k ↔ i resp. 2 ↔ 1.
Potential energy of electrons in the optical E-field (atomic nucleus at x=0):
On atomic dimension (few Å) the optical field is considered spatially constant E(x,t)~E(0,t).
Vopt ( x,t ) ≅ − x E ( 0,t ) →
HWW ( t,x ) = −eVopt ( t,x ) = e x E ( 0,t )
x
because
Felectrical ,opt ( x ) = −eE ( x ) and Vopt ( x ) = − ∫ E ( x ) dx = − E ( 0,t ) x
0
Considering only harmonic optical fields with frequency ωopt:
1
1
+ jω t
E ( t ) = E0 cos (ωopt t ) = E0 ⎡⎣exp ( jωopt t ) + exp ( − jωopt t ) ⎤⎦ = E0 ⎡e opt + cc ⎤
⎦
2
2 ⎣
¨
Remark: we just use the positive frequency e
+ jωopt t
-term
H ww ( t ) = ex E0 cos ωopt t Interaction-Hamiltonian
The atomic resonance occurs at the transition frequency: ω = ω12 =
E2 − E1
= ω2 − ω1 ≅ ωopt
Inserting Hww(t) into the differential equation for the coefficients C1(t) und C2(t) we obtain:
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using
H kl = uk HWW ul =
∗
∫ uk HWW ul dx = eE0 cos ωopt t
u k x uk
K5
Interaction Hamiltonian
Vol
and the definition :
uk x ul = ∫ uk∗ x ul dx
dipole matrixelement of the transition k ↔ l
leading to : H 11 = H 22 = 0 ( because x is an odd function )
Inserting Hww and ω21=ω in the diff. equation for Ci ¨
∂
j ⎡
C1 =
exp j (ωopt − ω ) t + exp − j ( ωopt + ω ) t ⎤ C2 u1 eE0 x u2
⎣
⎦
∂t
2
(
)
(
)
;
∂
j ⎡
C2 =
exp j ( ωopt + ω ) t + exp j ( −ωopt + ω ) t ⎤ C1 u2 eE0 x u1
⎣
⎦
∂t
2
(
)
(
)
Coupled differential equation for C1(t) and C2(t) for harmonic optical excitation
Remark: Ci(t) is determined by ω21, ωopt and u2 eE0 x u1
For the calculation of the transition (↑, ↓) rates we will solve this diff. eq. for 2 different situations (initial conditions):
1) Switching on the field at t=0 if the system is initially either
- in the ground state 1 (C1(0)=1) → absorption, upward transition or → r12
→ r21
- in the excited state 2 (C2(0)=1) → stimulated emission, downward transition
3) Calculation of the photon absorption (ground state transition 1 Π2):
we simplify the solution for C1(t), C2(t) for a weak perturbation (only small changes from the initial condition at t=0 !) of the
system, which is assumed to be initially in the ground state.
1) Initial condition for Absorption: C1 ( 0 ) = 1 , C2 ( 0 ) = 0
ground state
E2>E1
2) the changes of C1(t) and C2(t) are still small with respect to the initial state, C1(t)~1-ε , C2(t)~ε <<1
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t
With these simplification a simple integration
∫
dt' of the differential eq. for C2(t) leads to simple solution (appendix):
0
C2 ( t ) =
C2 ( t )
sin ⎡⎣(ω − ωopt ) t / 2⎤⎦
j
u2 eE0 x u1 exp j (ω − ωopt ) t / 2
2
(ω − ωopt ) / 2
(
2
1
= 2
4
u2 eE0 x u1
)
2
sin2 ⎡⎣(ω − ωopt ) t / 2⎤⎦
⎡(ω − ωopt ) / 2⎤
⎣
⎦
2
and
~ E02 ≈ Intensity I ;
C2 ( t ) << 1
2
Approximation for a weak perturbation
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Approximation of the evolution of the occupation probability IC2(t)I of the excited state at E2:
2
(excitation with both exactly defined optical frequency ωopt and transition frequency ω of the 2-level system, unphysical )
The occupation probability IC2(t)I2 of the excited state 2 increases resulting in absorption.
The evolution of IC2(t)I2 depends on time t and frequency detuning ω-ωopt between transition and monochromatic E-field.
• only close or at resonance ω≈ωopt there is a substantial absorption
• at resonance ω=ωopt the occupation probability of the excited state increases with t2 (experimental ~t !)
→ unphysical, in most experimental situations, because state energies E and photon energies hωopt need to be exactly
defined over long times without any perturbation.
~t2
δ(ω−ωopt) t
ICI2
~t2
IC2(t)I
2
IC2(t)I
2
1_
resonance
IC1I2
weak
perturbation
limit
t
~2π/ t
area
~2πt
ω-ωopt
IC2I2
t
0
ω-ωopt
2
e2 E0 2
• at resonance ω = ω opt C2 ( t ) =
u2 x u1 t 2 increases ~t2, the „width“ of the resonance decreases as Δω ~2π/t,
2
4
¨ the area under the curve C2 ( t ) 2 increases ~2πt ( constant rate at constant field).
2
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4) Fermi’s Golden Rule
To get out of the dilemma we use a procedure known as “Fermi’s Golden Rule”:
• The calculation contains the unphysical assumptions
1) the light field is ideal monochromatic, characterized by a δ-function for the SDF W(ω) (infinite duration of the light wave !)
2) the energy levels Ei are defined exactly and characterized by a δ-function for the density of state function
(assuming infinite duration of atomic states no collisions, dephasing processes, etc.).
In reality the density of state function of the electrons is broadened and Ψi shows an energy ΔEi uncertainty due to
the short (~0.1ps) lifetime of the state
Solution: Averaging the transition probabilities over the optical power density ρopt(ω)=W(ω)
We average the transition functions over the energy width of the optical field or the electronic states by an
integration over the frequency domain of C2(t,ω)
For averaging over the optical spectral width of the transition we replace the field amplitude term E02 by the spectral energy
(
density ρopt , defined with the relation ε E02 / 2 = ∫ ρopt (ωopt ) ... dωopt , and integrate the time-dependent probability C t,ωopt
)
2
weighted by ρopt(ωopt) over the finite spectral width of W(ω)=ρopt(ω) of the field:
C2 ( t )
2
e 2 2W (ωopt )
= ∫
u2 x u1
2
ε
−∞ 4
+∞
sin 2 ⎡⎣(ω − ωopt ) t / 2 ⎤⎦
2
⎡(ω − ωopt ) / 2 ⎤
⎣
⎦
2
lim
¨
⎯⎯⎯ ⎯⎯⎯→
ωopt →ω
dωopt = e 2
W (ωopt ) π
u2 x u1
ε
2 2
2
t
2π t δ ⎛⎜ω −ωopt ⎞⎟
⎝
⎠
the averaged occupation probability increases linear with t leading to a constant transition probability r12 which is
proportional to the spectral energy density W(ω)=ρopt(ω).
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a) the term dipole matrixelement u 2 x u1 contains the dynamic microscopic properties of the 2-level system
b) a transition with a matrix element u2 x u1 =0 is not possible and is called forbidden transition.
c) detuned transitions ωopt ≠ ω only show weak oscillations of IC(t)I2 → polarization mode
• Interpretation of the dipole matrix elements: uk x ul = ∫ uk∗ x ul dx
The dipole moment <p>=-e<x> is proportional to uk x ul . For strong polarization and absorption uk x ul has to be
large, meaning that the wave functions uk and ul should reach large values at large values of x → only weakly bound
electrons.
Stimulated emission proceeds in the same way but with changed initial conditions giving the same results:
C1 ( 0 ) = 0 , C2 ( 0 ) = 1
excited state as "start" state
Einsteins Coefficient with an optical field of finite spectral energy density W(ω):
From C2 ( t )
2
e2 1
= 2
u2 x u1
4 ε
2
ρ opt (ω ) 2π t averaged occupation probability (linear in time t) ¨
Per definition the transition rate r12 per transition pair 1 → 2:
2
∂ C2
e2 π
r12 =
= 2
u2 x u1
∂t
2 ε
2
e2 π
B (ω ) = 2
u2 x u1
ε
2
W (ω ) = BW (ω ) →
2
Einstein-coefficient B for stimulated transitions (Absorption)
⎛ n3 ω 3 ⎞
From A = ⎜ 2 3 ⎟ B (ω )
⎝ π c0 ⎠
→
e2ω 3
A=
2π c03ε
u2 x u1
2
= 1 / τ spont
Einstein-coefficient A for spontaneous emission
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Calculation of transition rates with a spectral energy density: (self-study)
In order to avoid the averaging by the Fermi-Golden Rule we consider that the optical field is not monochromatic but described by a
spectral power density ρopt(ω). For being able to work with Fourier-transforms we consider all signal to be time-limited (compare chap.7).
The optical field amplitude is described by a time-limited envelop-function A(t) and we consider a the situation of absorption:
time-limited field to the interval [0,T ] :
AT ( t ) jωopt t
− jω t
+ e opt
e
for 0 < t < T ; ET ( t ) = 0 for T < t and t < 0
2
Differential equation for occupation probability for absorbtion with C1 ( t ) ≈ 1: ( p.49 )
(
ET ( t ) =
)
(
jeE ( t )
jeA ( t )
∂
+ j (ω +ω )t
+ j (ω −ω )t
C2 ( t ) =
1 x 2 e + jω21t =
1 x 2 e 21 opt + e 21 opt
∂t
2
Initial condition : C1 ( 0 ) = 1 ; C2 ( 0 ) = 0
)
; ω21 = ( E2 − E1 ) /
T
∫ dt
Integrating both sides over t:
and considering time-limited functions:
0
T
⎛T
⎞
je
+ j (ω +ω )t
+ j (ω −ω )t
1 x 2 ⎜ ∫ A ( t ) e 21 opt dt + ∫ A ( t ) e 21 opt dt ⎟ =
2
0
⎝0
⎠
+∞
+∞
je
je
+ j (ω21 +ωopt )t
+ j (ω −ω )t
A
t
e
dt
1
x
2
AT ( t ) e 21 opt dt =
1x2
+
(
)
∫ T
∫
2
2
−∞
−∞
C 2 ( T ) − C2 ( 0 ) = C2 ( T ) =
C2 ( T ) =
je
1x2
2
(
Fourier −Transform AT ω21 +ωopt
C2 ( T ) =
2
)
(
Fourier −Transform AT ω21 −ωopt
(
AT (ω21 + ωopt ) + AT (ω21 − ωopt )
≈0
)
≠`0
)
⎯Mixed
⎯⎯⎯⎯
→
products =0
⎞
e2
2⎛
∗
∗
+
+
+
−
−
1
x
2
A
A
A
A
ω
ω
ω
ω
ω
ω
ω
ω
(
)
(
)
(
)
(
)
T
21
opt
T
21
opt
T
21
opt
T
21
opt
⎜
⎟
4 2
⎝
⎠
≈0
Using the relation between sectral power density and spectral amplitude density (chap.7):
ET2 =
2 +∞
ε 0∫
ST (ω )dω =
C2 ( T ) =
2
+∞
1
ET (ω )ET (ω ) dω
∫ Tlim
→∞ T 2π
∗
if T → ∞ ST (ω ) T 2π = ET (ω ) ET∗ (ω )
( one − sidedspectrum )
0
e 2π
e 2π
2
2
1
x
2
S
ω
T
=
1 x 2 ρopt (ω ) T
(
)
2
2
2 ε
2 ε
( same result as Fermi − Golden − Rule )
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Electronics Laboratory: Optoelectronics and Optical Communications
22.03.2010
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5.4.1.3 Expectation value
x(t )
and
p(t )
K5
of the dipole oscillator (field viewpoint)
Goals:
Transition rates R and their dependency on carrier density n do not describe the frequency dependent properties of
the QM dipole, in particular the expectation value of the expectation value of the susceptibility 〈χ(ω)〉.
〈χ(ω, N1, N2)〉 ?
we need to determine 〈x(ω)〉 first !
− jω t
− jω t
Knowing C1(t) and C2(t) gives the transient mixed state Ψ ( x,t ) = C1 ( t ) u1 ( x ) e 1 + C2 ( t ) u2 ( x ) e 2 and the time
dependent expectation value 〈x(t)〉 and 〈p(t)〉= - e〈x(t)〉 by the basic definition
x (t ) = Ψ x Ψ =
+∞
∗
∫ Ψ ( x,t ) x Ψ ( x,t ) dx .
−∞
C1(t) and C2(t) can be related to 〈x(t)〉 (Appendix 5A):
x(t ) =
C1 ( t ) C2∗ ( t )
depending on occupation,
resp.carrier densities n1 ,n2 and ωopt ,Eopt
u2 x u1
e j(ω2 −ω1 ) t
Oscillation at ω21
+ cc
cc = conjugate complex part,
ω21= transition frequency
Expectation value of position 〈x(C)〉 of the electron of the atomic DP (only in mixed state ≠0 during 1-2, 2-1transition)
Graphical interpretation of the Dipole-Formation during the level transition: (see also chap. 5.4.1.5)
„
〈x(t)〉 oscillates with transition frequency ω = ω 2 − ω1 = ( E2 − E1 ) / .
„
the functions C1 ( t ) C2∗ ( t ) of 〈x(t)〉 depend on the optical field by E02 and ωopt and are calculated from the coupled Dgl. for Ci(t)
„
stationary pure states have no dipoles because C1=0, C2=1 (excited state) or C1=1, C2=0 (ground state) → 〈x(t)〉 =0.
„
〈x(t)〉 has a well defined phase relation ship to the exciting field Eo(t)
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K5
Dipole oscillation for the transition between a symmetric ground state (u100)- and a
nonsymmetric excited state (u210) of the H-Atom:
Consider the total mixed wave function Ψ ( z,t ) and Ψ ( z,t ) , where we moved the phase factor out of the bracket expression:
2
symmetric
ground state
/
anti-symmetric wavefunction
Ψ ( x,t ) = Ca ( t ) u100 ( x ) e − jωa t + Cb ( t ) u210 ( x ) e − jωb t
⎧⎪
⎫⎪
− j ω −ω t
= e − jωb t ⎨Ca ( t ) u100 ( x ) e ( a b ) + Cb ( t ) u210 ( x ) ⎬ = e − jωb t Ψ ' ( x,t )
⎩⎪
⎭⎪
excited state
Phase term oscillating
with the transition
frequency ωa-ωb
Ψ'
where Ψ ( x,t ) = Ψ ' ( x ,t )
2
Ψ ' ( z,t ) and Ψ ( z,t ) at times tn
2
mixed state
2
(b)
tn = 2π n / (ωb − ωa ) = nTba ,
n = 0,1,2,3 ... →
phase term e− j 2π n = 1
(c)
tn ' = π ( 2n + 1) / (ωb − ωa ) = ( n + 1) Tba ,
n = 0,1,2,3...
→
+ -
- +
<p(tn)>
<p(tn’)>
phase term e
− jπ ( 2n +1)
= −1
Dipole-oscillation occur only during the transition between 100 ↔ 210.
The transient mixed state ψ(z,t) is asymmetric and forms a
oscillating charge dipole <p> with the transition frequency
ωba=ωb−ωa
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Electronics Laboratory: Optoelectronics and Optical Communications
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Remark:
the electron charge distribution ~ uiui* is symmetric to the nucleus for a pure state, therefore stationary pure states have no
dipole moment (für ρopt=0).
The phase of the dipole relative to the exciting field depends on the relative phase of Ci resp. carrier occupation n2, n1 and the
opt. Frequency ωopt
Depending on the phase the dipole absorbs or emits radiation.
5.4.1.4 Dipole-Matrix-Elements for allowed and forbidden transitions
Possible Dipole-Matrix-Elements of a 2-level system:
The time invariant potential V(x) has been assumed as symmetric around x=0 for our 1-dimensional case. Therefore the
solutions u(x) are either symmetric or anti-symmetric. However the probability density uu*(x) must be symmetric.
The dipole operator x in
Initial state:
u 2 x u1
is a asymmetric function, therefore we have 4 possibilities:
final state:
transition:
u1 symmetric
- u2 symmetric
→ forbidden, because
u2 x u1 =0
u1 asymmetric
- u2 asymmetric
→ forbidden, because
u2 x u1 =0
u1 symmetric
- u2 asymmetric
→ allowed,
because
u2 x u1 ≠0
u1 asymmetric
- u2 symmetric
→ allowed,
because
u2 x u1 ≠0
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K5
5.4.1.5 QM-calculation of the Susceptibility 〈χ(ω)〉
Procedure: we show that the quantum-mechanical expectation value of 〈x(t)〉 leads to the same differential
equation of motion as for the classical oscillator for x(t).
The important difference is that 〈x(t)〉 depends on the populations n1 and n2, and the initial occupation state
leading to negative absorption, resp. gain.
„
For the analysis we consider the stationary harmonic solutions of the differential equation for C1(t) and C2(t) with weak
harmonic optical excitation.
„
Calculation of the expectation value of the “dipole distance“ 〈x(t)〉, resp 〈X(ω)〉 determines the quantum mechanical
Susceptibility 〈χ(ω)〉.
Classical mechanics:
D = ε 0 E + P = ε 0 E + ε 0 χ E = ε 0 (1 + χ ) E = ε 0 ε r E
(electrons on springs)
D (ω ) = ε 0 E + P (ω ) = ε 0 E + eN X (ω ) = ε 0 E + ε 0 χ (ω ) E →
Quantum mechanics:
χ (ω ) = eN / ε 0 X (ω )
expectation values of physical observables
Solution-Procedure:
To calculate the quantum mechanical susceptibility 〈χ(ω)〉, we determine from the diff.eg. for C1 and C2 first an oscillator
equation of motion for 〈x(t)〉, resp. 〈X(ω)〉 having the same form as the classical diff.eq. for x(t):
x + ω02 x + γ D x = −e / me Ex cos (ω t )
classical driven oscillator with damping γD and resonance frequency ω0.
1) from the oscillator differential equation of 〈x(t)〉 we calculate 〈χ(ω)〉, resp. 〈n(ω)〉, 〈k(ω)〉 as for x(t).
Definitions and assumptions:
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Electronics Laboratory: Optoelectronics and Optical Communications
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K5
Definition of position expectation value : < x >= ∫ Ψ ( x,t ) x Ψ ( x,t ) dx
∗
(
)
E0 + jωopt t
− jω t
+jω t
e
+ e opt
assuming: harmonic optical field with positive frequency e opt
2
ε E
+ jω t
− jω t
P ( t ) = 0 0 χ (ωopt ) e opt + χ ( −ωopt ) e opt
harmonic polarization ( assumption )
2
E + jω t
− jω t
H ww ( t ) = ex E ( t ) = ex 0 e opt + e opt
2
− jω1 t
Ψ ( x,t ) = C1 ( t ) u1 e
+ C2 ( t ) u2 e − jω2 t assumed solution for harmonically perturbed wavefunction
E (t ) =
(
)
(
)
Therefore the differential equations for C1 , C2 (p. 5-42) transform to:
•
(
)
(
E0 e
u2 x u1 ⎡ exp j (ωopt − ω ) t + exp − j (ωopt + ω ) t
⎣
2
•
Ee
j C2 ≅ 0 u2 x u1 ⎡ exp − j (ωopt − ω ) t + exp j (ωopt + ω ) t
⎣
2
j C1 ≅
(
)
(
)⎤⎦ C
2
)⎤⎦ C
with C =
1
∂
C
∂t
By insertion of the of assumed solution for ψ(x,t) into <x> we obtain:
∗
< x >= C1 ( t ) C2∗ ( t ) u1 x u2 e + jω t + C1∗ ( t ) C2 ( t ) u1 x u2 e jω t
with transition frequency: ω = ω2 − ω1
(used p.5-49)
For the derivation of the QM-equation of motion of 〈x(t)〉 we evaluate the first and second derivatives < x >=
∂
<x>
∂t
∂2
and < x >= 2 < x > (see appendix Chap.5). Making use of the above differential equation for C1(t) and C2(t) gives finally an:
∂t
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Electronics Laboratory: Optoelectronics and Optical Communications
22.03.2010
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K5
QM Oscillator-Equation of the expectation value 〈x(t)〉:
< x > +ω 2 < x > = 2
E0 e
u2 x u1
2
ω cos (ωopt t )
×
{
C 2 ( t ) - C1 ( t )
2
2
}
(without damping force!)
determines the phase of the oscillator
relative to the driving optical field
fundamental driven oscillator equation for 〈x(t)〉 with the self-resonance ω=ω21
{
−1 ≤ C 2 ( t ) - C 1 ( t )
2
2
} ≤ 1 depends on the occupation n , n
1
2
of E1, E2 determines the phase of the DP to the driving E-field
2
}
F-transforming gives: x(t) → X(ω)=F(x(t))
→
X (ω ) =
ω
2
ω 2 − ωopt
⎛ 2E0 e ⎞
⎜
⎟ u2 x u1
⎝
⎠
2
{
2
C 2 ( t ) - C1 ( t )
(frequency domain with undamped resonance at ω)
Remark: Incomplete solution, because we do not know the time-dependence of C(t) !
The equation simplifies (C ~ constant and known) under the assumption of
1) a weak field, that does not change the initial condition for C1 and C2 substantially during time t and
2) that the system was initially eg. in the ground state E1 (absorption process):
C 1 (t ) ≅ 1 und
C 1 (t ) >> C 2 (t ) → medium is absorbing
{ C (t )
2
2
- C1 ( t )
2
}∼ −1
Quantum mechanic equation of oscillation for the charge position <x> for ABSORPTION (C2(t)~0, C1(t)~1)
< x > +ω 2 < x > = − 2
E0e
u2 x u1 ω cos (ωopt t )
2
undamped driven oscillator for absorption
Inphase or antiphase oscillating DP (classical situation)
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Electronics Laboratory: Optoelectronics and Optical Communications
22.03.2010
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→
X (ωopt ) = −
ω
⎛ 2E0 e ⎞
⎟ u2 x u1
2
2 ⎜
ω − ωopt ⎝
⎠
2
K5
(frequency domain solution for absorption)
The QM-equation for the harmonic oscillator <x(t)> for absorption is formally equivalent to the classical undamped dipole,
therefore all relations for the frequency dependence of the real part-(χ‘) and imaginary part (χ‘‘) of the QMsusceptibility <χ> will be equal (no damping γD=0).
Substitution / equivalence classical ↔ QM: (absorption)
classical:
Quantum mechanical dipole:
ω0
−
e
Ex
me
γD
→
ω = ( E2 − E1 ) /
→
−2
→
E0 e
γD =0
u2 x u1
2
( absorption, gain : + )
ω
( undamped →
no loss, gain )
h
Depending if the system is initially in the a) ground state (absorption) or b) excited state (emission), there is a
change in the sign of the driving field in the oscillator equation (dipole phase relative to the optical field E):
a) ground state:
C1 ( 0 ) = 1, C2 ( 0 ) = 0
→
b) excited state:
C1 ( 0 ) = 0, C2 ( 0 ) = 1
→
(
(
C2 ( 0 )
2
− C1 ( 0 )
2
C2 ( 0 )
2
− C1 ( 0 )
2
) ≅ −1
) ≅ +1
Absorption
(n1~N , n2~0)
Emission
(n2~N , n1~0)
• the susceptibility for absorption and stimulated emission have opposite signs.
• similar to the classical case χ‘‘=0 results in no optical gain at this level of QM-oscillator model. Only by introducing
damping γ we will be able to realize χ‘‘≠0.
• The necessary damping mechanism (deexcitation of excited state) by the spontaneous emission can not be introduced in
the semi-classical QM-model self-consistently ( Quantization of the optical field is required).
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Electronics Laboratory: Optoelectronics and Optical Communications
22.03.2010
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Relation of polarization density and transition pair densities n1, n2:
We have a transition pair density N (=atomic density) with an excited (E2) transition pair density n2 and a ground-level (E1)
density n1 , so that N=n1 + n2.
From <xi> we get the polarization density <Pi>= - ni e<xi> per type of transition pair density ni:
χi (ωopt ) = − ( + ) ni e X i (ωopt ) / ε 0 ; + gain ,
− absorption
For a medium containing a mixture of transition pair in the ground state (n1) for absorption and the excited state (n2) for
gain with the densities n1 and n2, we get for the total polarization density the superposition (using phase change for emission
2 → 1 transitions):
(
)
χ (ωopt ) = χ1 (ωopt ) + χ 2 (ωopt ) = − n1e X 1 (ωopt ) + n2 e X 2 (ωopt ) / ε 0 = e X 1 (ωopt ) ( n2 − n1 ) / ε 0
The susceptibility depends on the OCCUPATION DENSITY DIFFERENCE (n2-n1) !
From the QM-oscillator equation for <x> we obtain for an assumed solution 〈x(t)〉= 〈X(ω)〉e
P = n1 p1 + n2 p2 = − n1e X 1 (ωopt ,Eo ) + n2e X 2 (ωopt ,Eo ) = −
χ (ωopt ) = −
Absorption:
( n2 − n1 ) e2
u1 x u2
ε0
2
( n2 − n1 ) e2
u1 x u2
2
+jωt
2ω
E0 = ε 0 χ (ωopt ) E0
2
ω 2 − ωopt
2ω
ω 2 − ωo2pt
n2=0, n1=N
Stimulated emission: n1=N, n1=0
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Electronics Laboratory: Optoelectronics and Optical Communications
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QM: absorption, n1=N (stimulated emission, n2=N)
χ ' (ωopt ) = ( − )
Ne2 u1 x u2
ε0
2
2ω
2
ω 2 − ωopt
χ '' (ωopt ) = 0
Susceptibility of the undamped,
, lossless QM-Dipole with density
N in absorption
Characteristic of χ(ω): n1>n2 (absorption)
χ'(ω)
χ’’=0
with N = dipole density = n1 ; n2 = 0 absorption
χ'’(ω)=0
Klassisch :
ω
Ne2
2
P = − Nex ( Eo ) = +
E0 = ε 0 χ ' E0
2
2
m ω − ωopt
χ ' (ωopt ) = +
Ne2
2
2
mε 0 ω 2 − ωopt
, χ '' = 0
ωopt
stimulated
emission
Formal introduction of QM damping by spontaneous emission:
(Appendix 5A)
Goal:
The QM-model assumed unrealistically „sharp“ energy levels and exact optical frequency meaning infinite carrier lifetimes.
As can be seen from the differential equations for C1 and C2:
An excited system can not return to equilibrium after switching-off of the optical field (E0=0) because the is no damping term in
∂
Ci = 0 .
the diff.eq. for C1, C2 (same argument as in Einstein-model) →
∂t
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Electronics Laboratory: Optoelectronics and Optical Communications
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To formally introduce damping and optical gain (χ’’≠0) into the diff.eq. for C1, C2 we modify them by additional a damping term
γ (representing finite carrier lifetime τ~1/γ):
Only the introduction of damping als the QM representation of loss and gain ! (from first principles required the
quantization of the optical field)
∂
jE e
C2 = − 0
2
∂t
(
γ = A / 2 = 1 / ( 2τ spont )
)
(
)
⎡ exp j (ωopt + ω ) t + exp j (ω − ωopt ) t ⎤ C1 − γC 2
⎣
⎦
u2 x u1
carrier lifetime damping of excited level E2
damping constant
The damping γ relaxes C2 after a disturbance (E0 → 0) with the time constant τspont back to equilibrium →
∂
Ci = −γ Ci .
∂t
We repeat the previous calculation as for the undamped case to get the frequency dependence of the susceptibility 〈χ(ω)〉 of
the damped QM-oscillator:
χ = χ ground state ( n1 ) + χ excited state ( n2 )
e2
χ (ωopt ) = Re χ + j Im χ = χ ' + j χ '' =
ε0 2
e2
χ ' (ωopt ) =
ε0 2
e2
χ '' (ωopt ) = −
ε0 2
u2 x u1
u2 x u1
u2 x u1
(
2
2ω ω 2 + γ 2 − ωopt
2
(ω
2
+γ −ω
2
2
opt
2
(
(
)
) + ( 2γω )
2
2
+ γ −ω
2
)
2
− j2γωopt ⎤
2ω ⎡ ω 2 + γ 2 − ωopt
⎣
⎦ n −n
( 1 2)
2
2
2
+ ( 2γωopt )
ω 2 + γ 2 − ωopt
2
)
( n1 − n2 )
Positive frequencies have been
opt
assumed
2ωγ Dωopt
2
(ω
n1= transition pairs in ground state ; n2= transition pairs in excited state , n1 + n2 = N =atom density
) + ( 2γω )
2 2
opt
2
( n1 − n2 )
≠0
opt
e
+ jω opt t
Absorption χ’’<0
χ’’>0
Gain
using the attenuation relation from chap. α = 2 β0κ ≈ − χ ''/ 2 / 1 + χ ' from chap.2.
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Quantum mechanically susceptibility with damping
QM allows describing optical gain as a function of population inversion (n2 > n1):
n2>n1 → Gain (for population inversion), χ“>0
n2<n1 → Attenuation,
χ“<0
Real- and imaginary part of the QM-susceptibility with damping
χ', χ'’
• The frequency dependence of χ ' (ω ) , χ ' ' (ω ) is similar to the
classical damped dipole.
n1<n2
However for population inversion n2>n1 we get χ ' ' (ω ) ≤ 0 and
optical gain g(ω)= - ∼ χ '' (ω )
n1>n2
n1<n2
ω0
ωopt
• The detuned QM-resonance frequency is:
ωres ≅ ω 2 − γ 2
γ / ω <<1
⎯⎯⎯⎯
→ ω = ( E2 − E1 ) /
n1>n2
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Conclusions:
• QM showed us, that the emission of a stimulated, coherent photon becomes possible for n2>n1, because the 2level oscillator can be synchronized in such a way, that there is a dipole component orthogonal to the exciting
field , resp. that χ‘‘>0. In a classic description this was not possible (χ‘‘<0).
• In case of the population inversion n2>n1 , then χ‘‘>0 and the medium produces optical gain g>0. Thus we
succeeded to relate population (n2-n1) to gain and absorption.
• The stimulated global optical field E0 enforces a macroscopic synchronization of all atomic dipoles in a
medium with a phase difference according to the frequency response of χ(ωopt).
• The synchronized dipoles reemit (scatter) a macroscopic dipole field, that is synchronized with the exciting field
and superimposes to a 1) amplified or attenuated (χ‘’, α) and 2) phase modified (χ‘, n) total field.
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Linewidth functions and reduction of maximal gain:
(self-study)
For weak damping (γ>>ω) g(ω) simplifies to a Lorenzian-shaped frequency dependence L(ωopt):
e2
g (ωopt ) ≅ −α (ω opt ) = χ '' (ω opt ) =
ε0 2
u2 x u1
2ωγ Dω opt
2
(ω
2
+ γ −ω
2
) + ( 2γω )
2 2
opt
The width Δω of the linewidth function L(ωopt) is: L (ω opt ) =
opt
2
e2
( n1 − n2 ) =
ε0 2
u2 x u1
2
L (ω opt ) ( n1 − n2 )
1
1
Δω / 4
=
2γ 1 + (ω − ω )2 / γ 2 ( Δω / 2 )2 + (ω − ω )2
21
opt
21
opt
; Δω = 2γ
The width of linewidth function is proportional to the inverse of the spontaneous carrier lifetime τspont (typ.~1ns for SC).
Additional lifetime effects such as carrier collisions leading to a dephasing of the dipole (typ. ~100fs in SC) cause much
stronger broadening and reduction of the gain by the linewidth functions.
In the following we will neglect the effect of the linewidth functions for mathematical simplicity.
Remarks:
The damping γ of the level populations n1 and n2 is the origin for the appearance of the imaginary part χ’’ of the
susceptibility χ.
In the excited state the polarization is opposite to the “classical” polarization (1800 out of phase).
In the excited state the dipole acts as a source of radiation or as an active source of the field.
The role of the external stimulating field is to synchronize the phases of C1(t) und C2(t) in such a way, that there
results an active dipole, emitting radiation coherent to the simulating field.
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5.5 Optical band-to-band transitions in Semiconductors
In Optoelectronics active (amplifying) III-V semiconductors, such as GaAs, AlGaAs, InP, InGaAsP, GaN etc. play a key role
for optical component, therefore we will concentrate on optical processes in semiconductors (SC).
In the courses Physik II (3.Sem) and Bauelemente (4.Sem) it was demonstrated that the coupling of the atoms in the solid and
the condensation into a periodic lattice leads to:
„
energetic splitting of discrete energy levels of the electrons into energy-bands (Conduction-, Valenceband)
„
a band structure with a band gap Eg and quantized, quasi-free electron states (no localized states as in atoms)
„
distributed, non-localized matter waves ψ(x,t) and quasi-free motion of the electrons in the bands
To analyze the optical gain in SC we extend the notion of discrete transition pairs from atoms to a quasi-continuum of
state-pairs distributed over energy-bands (Conduction and Valence band) of the SC.
Similar to the averaging process for the transition rate over energy spectrum of the optical field W(ω), we assume here a
mono-chromatic light field E0cos(ωot) (typical for modern optoelectronic devices) and average the transition over the energy
state density distributions of the transition pairs ρcontinuum(E) or ρcontinuum(ω) (≠ρC or ρV).
Conduction band:
¨
density of energy states ρC(E2)
ψ C ( E2 ,x )
E 2,
ΔE 2
Δn2
ΔE2
EC
ΔR12
Valence band:
¨
density of energy states ρv(E1)
ΔR21
transition pair density
ρr(E21)= ρr(ρC(E2), ρv(E1))=?
Eg
EV
E 1,
ΔE1
Δn1
ψ V ( E1 ,x )
ΔE1
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Band-to-Band-Transition Rates between energy bands of semiconductors:
For SC the optical transitions occur between the densely distributed energy states Ec,i, ψ C , j ( x ) in the C-band and the
Ev,i , ψ V ,i ( x ) in the V-band forming transition pairs (defined by selection rules).
„
For the total energy interval ΔE= ΔE2+ΔE1 we will prove later that a reduced transition-pair density (RTPD) ρr(E2-E1) can
be defined as the number of transition pairs pro unit volume and unit energy ΔE.
„
In SC the electrons are quasi-free to move and the final state is not necessary empty as in atoms. The final state can be
occupied by an electron according to the Quasi-Fermi-occupation probability function fQ,c(E2), resp fQ,v(E1).
We consider incremental sub-populations Δn2, Δn1 around the energy levels at E2 and E1 and distributed in energy intervals
ΔE2 and ΔE1 with the individual density of state functions ρC(E2) and ρV(E1).
Generalization of Einstein-Formalism:
Conduction band
E 2,
ΔE 2
ΔR12
E 1,
ΔE1
Valence band
Transition probability of a single transition pair
Δn2
Stimulated Emission:
r21 = W B21
ΔR21
Δn1
fC ( E2 )
(1 − fV ( E1 ) )
ìnitial , occupied
final , empty
Transition rate in the energy interval ΔE with the
postulated pair density ρr(E12)
Stimulated Emission:
ΔR21 = W B21 ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) ΔE
Stimulated Absorption:
Stimulated Absorption:
r12 = W B12 fV ( E1 ) ( 1 − fC ( E2 ) )
ΔR12 = W B12 ρ r ( E12 ) f v ( E1 ) ( 1 − f c ( E2 ) ) ΔE
Spontaneous Emission:
Spontaneous Emission:
r21,spont = A21 fC ( E2 ) ( 1 − fV ( E1 ) )
ΔR21,spont = A21 ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) ΔE
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Transition rates rij ~ to the density of transition pairs ρr × occupation probability fi of initial state × probability
of final to be empty 1-fj
Generalization of transition rates for non-monochromatic light fields and distributed levels:
In the general case when optical field and transition pairs are described by spectral density functions, W(E) and ρr(E) we
generalize the transition rates as for the example for the stimulated transition rate:
dR21 ( E21 ) = B21W ( E21 ) ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) dE
R21 ( E21 ) =
+∞
∫
−∞
B21W ( E21 ) ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) dE
→
∫ .... dE
total transition rate C → V
Considered the two extreme cases:
a) optical field described by a broadband spectral density function W(Ε) and a “sharp” discrete transition-pair density N
δ(E-E21) (eg. atomic states with no collision processes)
W ( E ) ; ρ r ( E ) = N δ ( E − E21 ) ; N = atom density
R21 ( E21 ) = B21 ∫ W ( E ) N δ ( E − E21 ) fC ( E ) ( 1 − fV ( E ) ) dE = B21 W ( E21 ) N fC ( E2 ) ( 1 − fV ( E1 ) ) Einstein-Ansatz (broad band sources)
b) monochromatic optical field E(t)=E0cos(ωoptt) described by a spectral density function W(E)=Nphћωopt δ(E- ћωopt) and
a distributed transition-pair density ρr(E) (eg. bands-to-band transition in semiconductors)
W ( E ) = N ph ωoptδ ( E − ωopt ) =
ε 0 E02
2
δ ( E − ωopt ) ; ρ r ( E ) ; N ph = photon density
R21 ( E21 ) = B21 ∫ N ph ωoptδ ( E − ωopt ) ρ r ( E ) fC ( E ) ( 1 − fV ( E ) ) dE = N ph ωopt B21 ρ r ( E21 = ωopt ) fC ( E2 ) ( 1 − fV ( E1 ) )
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c) monochromatic optical field E(t)=E0cos(ωoptt) with W(E)=Nphћωopt δ(E- ћωopt) and a “sharp” discrete transition-pair
density N δ(E-E21)
Here the transition rate diverges because the situation is unphysical.
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5.5.1 Dipole-Matrix-Element of the transitions between energy bands
For the calculation of the transition rates between valence-continuum and conduction-continuum we have to determine
the dipole-matrix element uC x' uV between a delocalized discrete valence and a delocalized discrete conduction-band
state ψ C , j ( x,t ) and ψ V ,i ( x,t ) of the continuum:
→ continuum-to-continuum transition
∑ ψ ( x,t )
v, j
j
⎯⎯⎯⎯⎯→
E0 cos(ωopt t )
∑
ψ c ,i ( x,t )
i
In a SC the wave functions ψ C , j ( x,t ) and ψ V ,i ( x,t ) are extended and periodic with the lattice (lattice-modulated plane waves)
The calculation of total transition rates in SC is further complicated because the transition pairs are distributed over two
continuums in the conduction and valence bands
→ Summation, resp. integration over allowed band-to-band (interband) transitions
r
∑
ij − pairs
ij
Assumptions:
1) The light field is monochromatic with an amplitude E0 and a frequency ωopt resulting in a δ spectral energy density:
ρopt(ω)=1/2εEo2δ(ω − ωopt)
(1-sided)
2) Wave functions in SC are non-localized Bloch-functions uj(x), having the periodicity a of the lattice.
Ψ j ( x,t ) = F j ( x ) u j ( x ) exp ( − jE j t /
) resp. = ψ 0 exp ( jk j x ) u j ( x ) exp ( jk j x ) exp ( − jE j t / )
Fj(x) slow spatial envelope function, determined by the macroscopic crystal-potential (macroscopic crystal dimension L)
uj(x) fast varying, lattice-periodic Bloch-function, determined by the microscopic lattice potential (a lattice period a ~5A)
kj(ω) crystal wave vector
ψ 0 normalization constant
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Schematic representation of the wave function and e-h-dipoles in SC:
extended, crystal periodic Bloch wave functions
Ek
j↔k
Ej
Lattice-periodic potential
a = lattice period
L = macroscopic crystal dimension
3) For transitions in SC the final states are not necessarily empty, but are occupied by electrons according to the Fermi-
statistics F(E, EFQ) (in atoms the final state of an atomic transition-pair was per definition empty).
„
Dipole-transitions are represented as a superposition of wavefunctions ψC , ψV of the conduction- and valence-band.
lattice cell
a
Crystal-potential (schematic, period a)
Wavefunctions of conduction (e) and valence band (h)
.
- Dipol +
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5.6 Basic Properties of Electrons in Semiconductors
(Recapitulation, 4.Sem. Hablbleiterbauelemente)
For the calculation of the rates of stimulated R21.netto and spontaneous R21,spont emission, as well as optical amplification g in
SC and their relation to the electron- and hole-densities n and p we review the following basic relations on
(
wavefunctions in conduction (valence) band ψ c( v ) r ,kc( v ) ,t
(
dispersion relation, Ec( v ) kc( v )
)
Quasi-Fermi-functions
effective mass meff
density of states ρ c( v ) ( E ) and
)
density of states
E
E
EQF,c
carrier densities in SC: n, p
Dispersion relation
ρC
EC(kC)
E
EC2
EC
fQ,C
kC2
EV
EQF,v
fQ,v
Ev(kv)
ρV
fQ
kV1
k
EV1
ρ
„
Density of states ρc(ECi), ρv(EVi) and carrier densities n, p in conduction- and valence-band at energy Eci and Evi
„
Reduced density of state-pairs ρr(Eij), describing the density of electrons and holes of transition pairs having the
same crystal momentum p , resp. the same propagation vector k (conservation of k of the transition) for an energy
difference Eij, resp. EV1C2 : kv(Ev1)=kc(Ec2)
„
The dispersion relation E(ki) for electron and holes, expressing the dependence of the quantized particle energy Ei
on its crystal momentum pi = ħ ki , resp. wave vector ki.
„
The carrier occupation probability in valence- and conduction-band described by assumed Quasi-Fermidistribution functions fC(V)(E, EFQ) of the individual bands C, V. EFQ is the Quasi-Fermi-level of the particular band.
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5.6.1 Density of states and dispersion-relation of electrons in an ideal SC
Electrons of a precise total energy E=Ekinetic + Epotential moving in the periodic potential Vlattice(r) of a crystal can be represented
quantum mechanically by a planar Bloch-wave Ψ ( r ,t ) as solution of the stationary Schrödinger equation.
Planar matter wave for a particle of energy E=hv, of mass m and a crystal momentum p=ħk:
p ⎞
⎛ E
Ψ ( r ,t ) = u ( r ) exp ⎜ j t − j r ⎟ = u ( r ) e xp j ω t − j k r
⎝
⎠
(
)
u(r) has the periodicity of the crystal lattice (a)
To first order the motion of the electron in an energy band can be described as a quasi-free “classical” particle with
1) an effective mass meff,
2) a kinetic energy Ekin=E-Ec (for the electrons in the conduction band and Ekin=Ev-E for holes in the valence band) and
3) a crystal momentum p= ħk
Dispersion-Relation E(k) and density of states ρ:
The kinetic energy Ekin of the electron as a quasi-free particle in the SC-crystal is related to its momentum p in the same way
as for a free particle:
Ekin
2 2
p2
k
=
=
2meff 2meff
Bild / Formel
In analogy to classical particles there is a square law relation
between the quantized crystal momentum p, resp. k
and the quantized Ekin:
p= k
k= wave vector of matter wave
ω=E/
ω= frequency of the matter wave
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the kinetic energy Ekin=E-EC of the electron is positive from the conduction band edge Ec into the band
(similar for holes in the valence band)
Dispersion-Relation Ekin(k) for quantized matter waves in (1D) semiconductors:
Ekin ( p ) =
Ekin ( k ) =
C-band: Ψc,i, Ec,i , kc,i
1
p2
2mn. p
2
2mn,p
k
Ekin,n = E − Ec
Ekin,p = − E + Ev
2
V-band: Ψv,i , Ev,i , kv,i
( electrons )
( holes )
Ec(k)
E
ΔE
EC
Conduction band
k
Energy gap
EV
E
mn( v ) = effective electron (hole) mass
wave vector (k) and energy (E(k)) of
the quantized conduction band state
Valence band
Δk=2π/L
EV(k)
momentum quantization
Unlike a classical particle the crystal momentum p and wave vector k are quantized ( by the boundaries of the macroscopic crystal
volume Lx):
kx =
2π
n = nΔk
Lx
; n = 0 ,1,2 ,3, ....
the momentum p=ħk is quantized equidistantly by Δk=2π/Lx
Δk = kn − kn−1 = 2π / Lx
Therefore also the energy levels E are quantized :
Ekin ( kn ) =
2
2mn ,p
2π 2 2 2
kn =
n
Lx mn ,p
2
n=0,1,2,3,4 ....the energy level quantization is not equidistant !
The allowed energy-levels and momentums on the dispersions-relation E(k) are quasi-continuous, because Δk and ΔE very small.
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Occupation of states in the k-diagram:
Temperature T=0
(Valence band completely occupied, conduction band empty)
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(undoped SC → in thermal equilibrium Fermi-Energy EF lies in the band gap)
Temperature T>0
(Valence band and conduction band partially occupied)
parabolic
approximation
EF
EF
f(E,EF)
f(E,EF)
−π/a
π/a
−π/a
kn
π/a
k
a=lattice period
Bild
For each wave vector kn there are two associated energy-levels EC(kn) and EV(kn) in the conduction and valence
band forming a discrete transition pair (EC(kn) ↔ EV(kn))at kn.
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Repetition: carrier densities in C- and V-bands:
In the course „Halbleiterbauelemente“ (4.Sem) the number N of solution of the Schrödinger equation in k-space has been
calculated per unit volume and unit energy interval – this density is called
1. Density of states ρ(E) (DOS), resp. ρ(k) for electrons and holes as a function of Energy
ρc(E), ρv(E) using a constant effective masse for parabolic bands
⎛ 2m ⎞
ρC ( E ) = 4π ⎜ 2n ⎟
⎝ h ⎠
3/ 2
⎛ 2m p ⎞
, ρV ( E ) = 4π ⎜ 2 ⎟
⎝ h ⎠
E − Ec
3/ 2
EV − E
ρ describes the state density in a band at a particular energy, it does not describe the transition pair density ρr directly
2. Fermi-Occupation Probability of the energy level E by an electron
1
f E,EQF =
, EQF=EF (Quasi-) Fermi-Energy of the corresponding band for equilibrium
E − EQF ) / kT
(
1+ e
(
)
resp. Quasi-Fermi-Distribution for electrons and holes in non-equilibrium EQF,n ≠ EQF,p
fQ ,n ( E ) =
1
1+ e
( E − EQF ,n ) / kT
;
fQ,p ( E ) =
1
1+ e
( E − EQF ,p ) / kT
3. Electrons- and Hole-Density n, p for parabolic Bands as a function of Fermi-energies
From the DOS ρ(E) and from f(E,EQF) the carrier densities n, p are determined by integration over the corresponding band
n=
EC ,Top
∫ ρc ( E ) f c ( E )
EC
p=
EV ,Bottom
4 ⎛ 2π mn,eff kT ⎞
dE =
⎜
⎟
h2
π⎝
⎠
∫ ρv ( E ) fv ( E )
Ev
3/ 2
4 ⎛ 2π m p ,eff kT ⎞
dE =
⎜
⎟
h2
π⎝
⎠
with the Fermi − Integral :
⎛ E − EF
F1 / 2 ⎜ V
⎝ kT
3/ 2
⎞
⎛ EF − EC ⎞
⎟ = N C ,eff F1 / 2 ⎜
⎟
⎠
⎝ kT ⎠
⎛ E − EV
F1 / 2 ⎜ F
⎝ kT
⎞
⎛ EV − EF ⎞
⎟ = NV ,eff F1 / 2 ⎜
⎟
⎠
⎝ kT ⎠
Neff= effective density of states
∞
x1 / 2
F1 / 2 ( x, xF ) = ∫
dx
x − xF
0 1+ e
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4. Carrier Transport by Drift and Diffusion
∂n
∂p
,
j p = epμ p E − eDn
∂x
∂x
Drift current Diffusion current
jn = enμn E + eDn
μ = mobility, D=diffusion constant
Important for carrier transport in photodiodes and current-injection in Laser diodes and LEDs
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Graphical representation of the density of states ρC, ρv and electron and hole occupation n, p in the
density of states diagram:
(at different temperatures T and for different positions of the Quasi-Fermi-levels EQF,C ---, EQF,V ---)
ρn
n electrons
ΔE
EQF,C
EQF,C
The carriers are mainly distributed over an energy-interval ΔΕ ≅ EQF,C-EC or Ev-EQF,V
typ. ~ a few kT determined by the
1) position of Fermi-levels EQF,C , EQF,V (band filling) relative to the bandedge EC, EV
EF
2) temperature dependence of the Fermi-functions ( ΔE~kT temperature-“smear-out”)
EQF,v
EQF,v
p holes
ρp
T=0
Ev < EQF,c=EQF,V < Ec
Equilibrium
T=0
EQF,C > Ec
EQF,V < Ev
T>0
EQF,C > Ec
EQF,V < Ev
non-equilibrrium
For equilibrium:
n p=ni2
EQF,n=EQF,p=EF
Quasi-Neutrality for undoped SC:
n=p
np≠ ni2 , EQF,n≠ EQF,p (nonequilibrium)
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5.7 Optical Transition Rates and Amplification in SC
Band-to-band Transitions in SC are continuum-to-continuum transitions and the total transition rates are obtained by an
integration over all involved allowed state-pairs (state-pair = initial + final state):
- allowed pair transition are (in most cases) energy (E) and momentum (k)-conserving
- the matrix element is determined by the Bloch-wave function for the C- and V-band
- the transition-pair density function ρr(E21) is given by the bandstructure (dispersion relation k(ω)) and the density of state
functions of the C- and V-band, ρn(E2) and ρp(E1).
Goal:
z
Extension of the transition rate R(ω,n) for Band-to-Band-transitions
• Calculation of the reduced transition pair density ρr(ω) for parabolic conduction- and valence-bands
• Determination of the spectral dependence of the transitions R(ω)
• Relation between transition rate R(n) and the carrier densities n and p ,
resp. the quasi-Fermi-levels and pump current Ipump
5.7.1 Optical Interaction between C- and V-band electrons in direct SC
Schematic Band-to-Band-Transitions with momentum (k)-conservation
We will show that the matrix element Hik between 2 discrete states out of the continuum is only non-zero (allowed) if the
transition is between 2 states with the same momentum p, resp. k
¨ Vertical or direct transitions (in the dispersion diagram)
The common k-vectors k=kc,i= kv,i defines with the dispersion relations of the bands kc(ωc,i) and kv(ωv,j) the corresponding
transition energy Ecv,i,j(k)= Ec,i(k) - Ev,,j(k)
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The k-conserving optical Band-to-Band-Transitions E2(k) ⇔ E1(k) in the dispersion diagram E(k):
Individual dipole-transitions can only take place between wave functions of the same k-vector.
Postulated momentum conservation: ¨ k of start- and final state are equal (vertical transitions)
EC(k)
e
E(k)=ωħ=EC(k)-EV(k)
k
h
EV(k)
no momentum kv ≠ kc
conservation
momentum (k) kv = kc (kintial = kfinal)
conservatuion
forbidden
allowed
Dispersions-Diagram represents energy- and momentum-conservation:
From a k the corresponding energy difference of the pair are:
E ( k ) = ω = Eg + EC ( k ) + EV ( k ) = Eg +
2
2meff ,c
2
k +
2
2meff ,v
k2
→ k (ω ) or ω ( k )
The reduced pair state density ρr has to be determined separately from the densities of states ρc , ρv.
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5.7.2 Direct Semiconductors
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The transition rates between C- and V-band levels j and k depend essentially of the matrix-elements u j x
if u j x uk = 0
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uk
.
then this optical transition is forbidden.
Transition energy Ψ j ( E j ) → Ψ k ( Ek ) ; E jk = E j − Ek = ω conserved by converting potential/kinetic energy into a radiation quantum:
ω = E j − Ek
Energy-conservation
During band-to-band transitions in SC the momentum of the electron-hole pair must be identical
(„vertical“ transitions, other transitions are forbidden):
k j = kk
momentum conservation
5.7.2.1 Momentum conservation in optical transitions in undoped SC
Simple calculations will show that in ideal SC:
1) the transition matrix element is large for k-conservation and small
(forbidden) for non-k-conserving transitions.
2) indirect transitions need a 3rd particle (eg. phonon) for k-conservation
→ low interaction probability for 3 particle interaction.
„
a transition is direct if the energy-minimum of the C-band and the
energy –maximum of the V-band have the same k-vector
„
a SC with a direct optical transition (usually at k~0) is called a direct
Semiconductor (eg. AlGaAs, InGaAsP, GaN, etc. but not Si, Ge)
electrons occupy the energy minimum of the conduction band and holes occupy the maximum of the valence band with
high probability.
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Allowed and forbidden transitions in SCs:
Direct, momentum conserving transitions
To proof the k-conservation we calculate the dipole matrix-element for the electron and hole wave function in the C- , resp.
V-band :
Bloch-matter wave for e and h with k − vectors kc , resp. kv :
Ψ c ,k ( x ) = uc ,k ( x ) e − jkc x
matter wave of e in conduction-band ; EC = ωc
Ψ v ,k ( x ) = uv ,k ( x ) e − jkv x
matter wave of h in valence-band ;
E ( x ) = e− jβ x
optical wave with propagation vector β = 2π / λ ; ω
(
Not all transitions between ψ C ,kc , j x,kc , j ,t
)
EV = ωv
( ωc − ωv )
→ ψ V ,kv ,i ( x,kv ,i ,t ) are allowed. Allowed transitions (
uc , j x uv , j ≠ 0 ) conserve:
1. Energy conservation: ħωopt = Ec,j – Ev,i
2. Momentum conservation: kc,j = kv,i
The dipole-matrix-element of the valence-conduction band transition Ψv → Ψc becomes for an extended polarized ( e , β )
optical plane EM-wave:
(
{
)
eE0 cos ωt − β r = eE0 Re e
(
− j ωt − β r
)
}
=
eE0 j β r − jωt
e e
+ cc
2
E (r )
Electrical plane wave : e β = 0 ; r = ( x, y,z )
E ( r ) = E0 e e− jβ r
and
E ( r ,t ) = E ( r ) e+ jωt
; r = spatial coordinate
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Electrical potential :
V ( r ,r ' ) = − E0 e− jβ r e r '
; r ' = dipole displacment coordinate at position r
Including the distributed wave and EM-wave functions in the dipole matrix elements gives integrated over the total
crystal volume:
H 21 = ψ c∗ ( r ) H ww ( r ,r ',t ) ψ v ( r ) = ψ c∗ ( r ) eE0 e− jβ r e r ' ψ v ( r ,t )
Matrixelement
≡
∫∫∫
(
Fc = Fv =1
)
uc∗ ( r ) e+ jkc r eE0 e− jβ r e r ' uv ( r ) e− jkv r dr = eE0
crystal
volume
= eE0 uc∗ x' uv
∫∫∫
uc∗ ( r ) uv ( r ) ( e r ' ) e
crystal
volume
(
) dr
− j kv − kc + β r
=
(
)
eE0 uc∗ ( e r ' ) uv
falls kv −kc + β =0
The function u(x) is periodic with the lattice-constant a and the exp-term containing [kC-kV-kopt]x oscillates with unity absolute
value in the integral and thus the volume integral over L3 averages to <uCIxIuV>=0 if [kC-kV-kopt] ≠0 ¨.
k v − k c + β opt = 0
momentum conservation in optical transitions
momentum of a photon:
β opt = 2π / λ
; → β opt ≈ 10 4 cm −1
momentum of a electron with the thermal energy Ek=kT:
2m e kT
pc = k c
kc ≈
≅ 10 9 cm −1
Because the photon wave vector kopt is much smaller than the electron-momentum k c , k v >> β opt we get:
kc ≅ kv
Direct Optical Transition requirement
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Remarks:
• The matrix element in a crystal is often a function of the optical propagation direction β , resp. of the polarization
optical field, therefore in SC the optical properties are often non-isotropic.
e of the
• The lattice constant (a~5Å) is much smaller than the wavelength λ (~1μm) of the light, a<<λ,. The optical field is
homogenous for atomic dimensions, resp. over the lattice cell.
5.7.3 Stimulated and spontaneous transition rates in direct SC
We extended from Chap.5.5 Einstein’s-transition rates to the transition probabilities and differential transition rates in ΔE as:
Conduction band
E 2,
ΔE 2
ΔR12
Transition rate in the energy interval ΔE with the
pair density ρr(E12)
Stimulated Emission:
Transition probability of a transition pair
Δn2
Stimulated Emission:
ΔR21
E 1,
ΔE1
Valence band
ΔR21 = W B21 ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) ΔE
r21 = W B21 fC ( E2 ) ( 1 − fV ( E1 ) )
Stimulated Absorption:
Stimulated Absorption:
Δn1
r12 = W B12 fV ( E1 ) ( 1 − fC ( E2 ) )
ΔR12 = W B12 ρ r ( E12 ) f v ( E1 ) ( 1 − f c ( E2 ) ) ΔE
Spontaneous Emission:
Spontaneous Emission:
ΔR21,spont = A21 ρ r ( E21 ) f c ( E2 ) ( 1 − f v ( E1 ) ) ΔE
r21,spont = A21 fC ( E2 ) ( 1 − fV ( E1 ) )
B=
π e2
2
2
uc x uv
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Wave functions of electrons in bands of SC are not localized and the final state may already be occupied by a particle
according to the Fermi-statistic thus blocking the transition (because of Pauli exclusion principle):
• the transition rate r12 (eg. absorption 1→2)
is ~ to the probability f(E1), that the start-level E1 is occupied
• the transition rate r12 (eg. absorption 1→2)
is ~ to the probability [1-f(E2)], that the final-level E2 is empty
• rij ∼ f ( Ei ) ( 1 − f ( E j ) )
occupied state:
empty state:
fx = Fermi-Function of band x
z
{
eh-pair
Concept of the reduced density ρ r (E ) for k-conserving pairs and the reduced
mass mr :
Concept:
For optical transitions rates with k-conservation we are interested in the number of energy-pairs (in the 2 bands) with the
same k-vector in the interval Δk, resp. per energy interval ΔEs at E2(k) and E1(k) in C- and V-band.
(the density of states ρn and ρp describe only the number of energy-levels (not pairs !) per energy interval ΔE in one band)
Definition of the Reduced Density ρr for transition-pairs in conduction and valence bands:
the reduced density ρ r ( E ) is defined as number of energy level-pairs with the energy-difference E per energy
interval ΔE where the initial and final states in the conduction and valence band have the same k-vector
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¨ The continuum-continuum transition rates R are proportional to the so defined reduced pair density
occupation probabilities of the initial and final state-pair.
ρr
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weighted by the
Graphical representation of continuum → continuum transitions 2 (C) € 1 (V)
E
Ε2
All ΔN transition pairs of the momentum
interval Δk are distributed in the total energy
interval ΔE21:
ΔΕ2
ρC ΔΕ
ΔΝ Uebergangspaare
Δk
ρr
?
k
ΔE21= ΔE2+ ΔE1
ΔΕ1
Ε1
δk (equidistant k-quantization) → in ΔE1 und in ΔE2 we have ΔN=Δk/δk
Dispersion relation E(k):
ΔN-states in Δk
ΔE2
E2 − Ec =
2 2
k2
/ 2mn
conduction band
Ev − E1 =
2 2
k1
/ 2m p
valence band
ΔE21
ΔE1
ΔN-states in Δk
Δk
momentum conservation: k1 = k2
Remark:
k determines (by the dispersion relation E(k)) the energy levels E2(k) and E1(k), E(k)=E2(k) – E1(k) of an individual pair
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Procedure for the calculation of the reduced density ρr:
„
calculate the reduced density ρr for direct transition (k2=k1) between E2 and E1 as a function of the known densities of
states of conduction and valence band ρc(E2), ρv(E1) and as a function of k, resp E21:
ρr{E21}= ρr{ρc(E2), ρv(E1)} = ?
„
The calculation of the number of transition pairs ΔN(Δk) in the momentum interval Δk is simple, because all states are
separated by the equal momentum quantization δk.
We then transform the interval Δk into the associated energy interval ΔE21(Δk) with the help of the dispersion relation of
the C- and V-bands.
In addition we also use the fact that ΔΝ can also be expressed by both energy state densities
ΔN=ρv(E)ΔE1=ρc(E)ΔE2 with ΔE1(Δk) ≠ ΔE1(Δk).
a) Calculation of the number of transition pairs ΔN in the interval Δk and ΔE:
Due to the quantization δk of kn there are ΔN pairs in the momentum interval Δk:
k n = nδk with δk = h / L (1-D case) , resp. δ 3 k = h / L3 (3-D-case)
 ΔN = Δk / δ k
transition pairs  distributed over the total energy interval ΔE21= ΔE2+ ΔE1
¨ ΔN transition pairs are distributed in each energy interval ΔE2 and ΔE1 at E2 and E1.
b) Calculation of the relation between ΔE2 , ΔE1 , (E2-Ec) and (Ev-E1) with k and Δk:
Δk, ΔE21, ΔE2, ΔE1 are related by the dispersion relation E(k)
2
( E2 ( k ) − Ec ) = 2m
k
2
and ( − E1 ( k ) + Ev ) =
n
ΔE2 =
2
mn
k Δk
and
ΔE1 =
2
mp
2
2m p
k2
; →
differentiation by k:
^
k Δk
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⎛ k
k⎞
k ⎛ mn + m p ⎞
→ ΔE21 ( k ) = ΔE2 + ΔE1 = ⎜
+
Δk =
⎟
⎜⎜
⎟⎟ Δk
⎜ mn m p ⎟
m
m
n
p
⎝
⎠
⎝
⎠
2
2
2 ⎞
⎛ 2
2
2
→ E21 ( k ) = E2 ( k ) − E1 ( k ) = Eg +
+
k +
k = Eg + ⎜
k2
⎟
⎜ 2mn 2m p ⎟
2mn
2m p
⎝
⎠
2
E21 ( k ) = Eg +
with
mr =
2
2mr
k2
2
2
⇒
Dispersion relation for the transition pairs
mn m p
Reduced mass ( definition )
mn + m p
c) calculation of the relation between ΔΝ and the density of state functions ρn, ρp:
Because the number ΔN of electron and hole states are equal in Δk, ΔE21, ΔE2, ΔE1 it follows:
ΔN = ρc ( E2 ) ΔE2 = ρv ( E1 ) ΔE1
ΔE2 =
ρr
ρ c ( E2 )
ΔE21
und
=
definition of ρr
ΔE1 =
ρr
ρ v ( E1 )
ρ r ( E21 ) ΔE21 = ρ r ( E21 )( ΔE2 + ΔE1 )
ΔE21
→ ρ r ( E21 ) =
→
E
⎛ 1
⎞
1
ΔE21 = ( ΔE2 + ΔE1 ) = ρ r ⎜
+
⎟ ΔE21
E
E
ρ
ρ
(
)
(
)
c
2
v
1
⎝
⎠
from the equality
ρ r ( E21 ) =
→
ρC
=1
ρc ( E2 ) ρv ( E1 )
= ρ r ( E2 ,E1 )
ρc ( E2 ) + ρv ( E1 )
if ρv >> ρc then → ρ r ( E21 ) = ρ c ( E2 )
ΔN
ΔN
=
ΔE21
Δω21
Reduced Density ( Definition )
( is the case for many
III − V − SC )
ρ(E)
ρV
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we get with ρ r ( E ) =
3
transition with a monochromatic field W ( E21 ) =
R21,abs ( E21 ) =
(
π e2 E02
2
)
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∂ N
∂ N
for the Transition Rates R ( E ) =
by generalizing to continuum (C:2) – continuum (V:1)
∂Vol ∂E
∂Vol ∂E ∂t
2
2
uc, j x uv,0
B = π e2 / 2 2ε uc, j x uv,0
2
ε E2
∂2 E
= 0 δ ( E − ( E2 − E1 ) ) :
∂Vol ∂E
2
ρr ( E21 ) ( 1 − fc ( E2 ) ) fv ( E1 ) = ε E02 Bρ r ( E21 ) ( 1 − fc ( E2 ) ) fv ( E1 )
2
R21,stim ( E21 ) = ε E02 Bρ r ( E21 ) ( 1 − fV ( E1 ) ) fC ( E2 )
R21,net ( E21 ) = ε E02 B ρ r ( E21 ) ( fC ( E2 ) − fV ( E1 ) )
ΔR21,spont ( E21 ) = Aρ r ( E21 ) ( 1 − fV ( E1 ) ) fC ( E2 ) ΔE → Rspont = ∫ dR21,spont = ∫ Aρ r ( E21 ) ( 1 − fV ( E1 ) ) fC ( E2 ) dE
with ω = E2 − E1
fv, fc are the Quasi- Fermi-functions at E1 and E2
Remark:
if (E2-E1)=E21=ћω in a practical situation is specified, we get the corresponding k by solving a quadratic equation for E21(k).
From k we obtain E2, E1 and ρr(E2, E1) by using the corresponding dispersion relations of the C- and V-band.
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Transform Eo to photon density sph of the momochromatic field:
Modern optoelectronic devices often operate with almost monochromatic fields compared to the energy width of the band-toband transitions in SC. Therefore we use discrete field-amplitudes Eo or photon densities.
Using
2
w = ε Eo / 2 = s ph ωopt
1
2
R21,netto (ωopt ) = ε Eo B ρ r ( f 2 − f1 ) = s ph ωopt B ρ r ( f 2 − f1 )
2
1
2
R21 = w B ρ r f 2 (1 − f1 ) = ε Eo B ρ r f 2 (1 − f1 ) = s ph ω B ρ r f 2 ( 1 − f1 )
2
1
2
R12 = w B ρ r f1 (1 − f 2 ) = ε Eo B ρ r f1 (1 − f 2 ) = s ph ω B ρ r f1 (1 − f 2 )
2
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5.7.4 Optical Amplification and Attenuation in ideal SCs
Goals:
Combining the transition pair density ρr and the Fermi-functions f for the occupation probability of the energy-levels we can in
principle calculate all transition rates R (at least numerically) and their frequency dependence, determined mainly by
ρr(ω) and f(EF,ω):
• Emissions-/Absorption rates R21,net(ω), R21(ω), R12(ω) und R21,spont(ω) vers. ω resp λ, E21 (rates per unit volume at the optical
energy ħω21 or wavelength λ21) as a function of
1) the Quasi-Fermi-Levels EFQc and EFQv or
2) the carrier densities n and p (to combine with the carrier rate equation !) because n=n(EFQc) and p=p(EFQv)
• Amplification g(ω) versus carrier density n or finally as a function of pump current: g(ω,n), resp. g(ω,I)
• Wavelength λmax(n) and value of the maximal optical gain gmax(n, λmax)
These rates will be fundamental for the design of the characteristics of diode lasers and optical amplifiers in chap.6.
5.7.4.1 Gain spectrum g(ω,EQF,n, EQF,p) ; g(ω,n=p)
Procedure:
For the computation of the frequency dependence of g(ω,n=p), g(ω,EQF,n, EQF,p) the transition rates R21,net(E=ħω) , R12(E) ,
R21(E) , Rspont(E), R21,spont are evaluated at
a) given optical energy E=ħω, resp. a wavelength λ and
b) given quasi-Fermi-level EFQ,n, EFQ,p, resp. carrier density n(EFQ,n)=p(EFQ,p) assuming carrier neutrality
EFQ,n , EFQ,p are not independent because of the charge neutrality n=p condition (pumped undoped SC) !
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Relation between optical amplification g(ω,n) and netto-rate of stimulated emission R21,netto(ω,n):
From R21,netto(ω,n) we obtain immediately the important spectral gain function g(ω,n) often used in device design.
Assuming given quasi-Fermi-energies EFQc, EFQv (with the restriction n=p) and using the fundamental relation between the gain
g(ω, EFQc, EFQv ) and R21,netto
g (ω ) =
1 ∂ s ph
1 ∂ s ph
1
1
=
=
R21,net =
( R21 − R12 ) = g (ω ) →
s ph ∂ x vgr s ph ∂ t
vgr s ph
vgr s ph
g (ω ,EFQc ,EFQv ) =
=
s ph hv
vgr s ph
(B ρ (E
r
21
) ⎡⎣ fc ( E2 ,EFQ ,c ) (1 − fv ( E1 ,EFQ ,v ) )⎤⎦ − B ρr ( E21 ) ⎡⎣ fv ( E1 ,EFQ ,v ) ( 1 − fc ( E2 ,EFQ ,c ) )⎤⎦
hv
B ρ r ( E21 ) ⎡⎣ f c ( E2 ) − f v ( E1 ) ⎤⎦
vgr
(
π e2
with B ( E21 ) = 2 uc x uv
2 ε0
2
)
, ρ r ( E21 ) , ω = E21 = ( E2 − E1 ) and the relation between E and E2 , E1 we express g (ω ) as →
g (ω ) = g max ( E21 ) ⎡⎣ fc ( E2 ) − f v ( E1 ) ⎤⎦ ; g max ( E21 ) =
> ,<0
)=
ω
vgr
B ρ r ( E21 )
⎡⎣ f c ( E2 ) − f v ( E1 ) ⎤⎦ defines pumping condition and “parabolic or bell-shaped” frequency dependence of g(ω)
For the numerical evaluation of g(ω) the energies E1(ω), resp. E2(ω), assuming k-conservation, are determined first followed by
the computation of f(E2) and f(E1) using the charge neutrality n=p.
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From the functional dependence of g(ω,n) some basic conclusions about the gain-spectrum can be drawn:
¨ Condition for amplification in SC (Bernard-Duraffourgh condition)
a) SC in thermal equilibrium (no population inversion)
In thermal equilibrium in undoped SC the Fermi level is approx. in the middle of the bandgap: E FQ ,c = E FQ ,v = E F ≅ ( Ec + Ev ) / 2
therefore fc(E2) < fv(E1) (few electrons in the C-band)
As a result g is negative and the SC is for all frequencies absorbing (attenuating), so stimulated absorption is the
dominating rate.
g (ω ) = g max (ω ) ⎡⎣ f c ( E2 ) − f v ( E1 ) ⎤⎦
π e 2 2ω
with g max (ω ) ≅ .
u*c x uv
2 ε vgr
for ω ≈ E g /
( Absorption
∼ − g max (ω ) f v ( E1 )
2
ρ r ( E21 )
remark: here ρr is the density per energy interval !
at the Bandedge )
In general we can estimate g(ω) by considering:
[
]
• for the term [ f c (E c ,2 ) − f v (E v ,1 )] we have the following inequality: − 1 < f c (E c ,2 ) − f v (E v ,1 ) < 1
• the dependence of gmax(ω) is dominated by ρ r ( E ) ≅ ρ c ( E ) = 4π ( 2meff ,c / h 2 )
3/ 2
E − Eg ,
if ρc<< ρv as is the case for many III-V-SC
then
g max ( ω ) ≈
( E21 − Eg ) /
max. absorption for parabolic bands
(assuming that the matrix element is only weakly dependent on the transition energy E21)
• Optical amplification is only possible if fc(E2) >fv(E1), that means [fc(E2) -fv(E1)] >1
¨ POPULATION-INVERSION
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b) Properties of the spectral dependence of optical gain g(ω) and absorption α(ω):
For simplicity we assume T=0.
1) Intrinsic SC at T=0 and in thermal equilibrium: n=ni, p=ni
EFQ ,c = EFQ ,v = EF ≅ Ei
(~
g (ω ) = g max (ω ) ⎡⎣ f c ( E2 ) − f v ( E1 ) ⎤⎦ = g max (ω ) [0 − 1] = − g max (ω ) < 0
(no gain-situation)
E − Eg
)
fC(E2)-fV(E1)
E
+1
-1
EF
g(E)
g(ω)
gmax(ω)
transparency
Verstärkung
gain
f(E)
Ö the semiconductor absorbs at all frequencies
ω > Eg
Eg
transparency
Absorption
E
ω=(E-Eg)/ h
loss
Ö Absoption becomes stronger for shorter wavelengths.
Ö Above the bandgap wavelength the material becomes transparent.
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2) Intrinsic SC at T=0 and in thermal non-equilibrium (Inversion)
Assumption : EFQ ,c > EC , EFQ ,v < EV
→ EFQ ,c − EFQ ,v > E g
( degenerated
n>ni , p>ni but n=p (quasi-neutrality)
SC )
transparency
g (ω ) = g max (ω ) ⎡⎣ f c ( E2 ) − f v ( E1 )⎤⎦
[ +1] = gmax (ω )
g (ω ) = g max (ω ) [ −1] = − g max (ω )
g (ω ) = gmax (ω )
for E g < E21 < E FQ ,c − E FQ ,v
range of amplification
for E21 > EFQ ,c − EFQ ,v
range of absorption
fC(E2)-fV(E1)
+1
E
EFQ,c
EFQ,c
E2
fc(E)
EC
Eg
-1
(EFQ,c-EFQ,c)
g(E)
g(ω)
Verstärkung
gain
transparency
EV
E1
EFQ,v
Eg
EFQ,v
Absorption
E
ω=(E-Eg)/ h
fv(E)
Quasi-Neutrality: n=p
transparent
I gain I losses/absorption
Ö the SC amplifies in the frequency range Eg < ω < EFQ,c − EFQ,v
In the frequency range ω > EFQ ,c − EFQ ,v the SC is absorbing.
Ö Condition for amplification in SC by Bernard-Duraffourgh
the condition Eg < ω < EFQ,c − EFQ,v is called the amplification-condition of Bernard-Duraffourgh.
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3) Intrinsic SC at T>0 and thermal non-equilibrium (population inversion):
For T≠0 the expression for g(ω,n) is “smeared out” due to the continuous distribution of the Fermi-function at EQF.
The following figure gives a qualitative picture of the role of the factor [ f c (E 2 ) − f v (E1 )] in the expression for
g (ω ) = g max (ω )[ f c (E 2 ) − f v (E1 )] .
Thermal broadening of g(ω): graphical construction of the spectral dependence of the gain g(ω)
Band-filling and thermal broadening:
The spectral width Δω of g(ω,n) is
determined by:
Gain
Transparency
g(ω)
Absorption
•
Filling of the bands by the carrier
density n
•
Temperature dependence of the Fermifunction
The higher the carrier density n or the
higher the temperature T
- the broader the positive gain-spectrum
- the smaller the peak gain
ΔE ~kT
transparent I gain
I losses/absorption
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If the quasi-Fermi-levels are assumed to be known under the constraint of charge neutrality n=p, then the electron density n in
the conduction band can be determined and we will calculate the gain g(ω,n) vers. carrier density n.
This dependence is essential for device design and combining g with the rate-equations.
Numerical example of gain spectra g(λ,n) for GaAs and InGaAs at different carrier densities n=p:
n
gain
n
¨
gain
To achieve substantial gain in SC the
“transparency” carrier density ntr
must be in the range of 3-4 1018 cm-3
loss
Parameter: carrier density (electron density /1018cm3)
loss
¨ gain is narrowband: Δλ~0.1λ
Remark on densities of states and quasi-Fermi-levels:
For the self-consistent calculation of g(ω,n) we have to eliminate the quasi-Fermi-levels EFQc(n), EFQv(p).
As EFQc, EFQv determine n and p and together with the boundary condition of quasi-neutrality in the SC n=p, the elimination
can be carried out numerically as no analytical inverse Fermi-function F1/−12 exists .
n=
EC ,Top
∫ ρn ( E ) f n ( E ) dE = N C ,eff F1 / 2 ( ( EFQn − EC ) / kT )
EC
p=
EV ,Bottom
∫ ρ p ( E ) f p ( E ) dE = NV ,eff F1 / 2 ( ( EV − EFQv ) / kT )
∞
x1 / 2
dx
x − xF
0 1+ e
F1/2 are Fermi-integrals F1 / 2 ( x , xF ) = ∫
Ev
n(EQF,C)=NC,eff F1/2([EQF,C-EC]/kT) and with the inverse function [EQF,C-EC]/kT= F1/2-1(n/ NC,eff)
p(EQF,V)= NV,eff F1/2([EV-EQF,V]/kT) and with the inverse function [EQF,V+EV]/kT= F1/2-1(p/ NV,eff)
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5.7.4.2 Spontaneous Emission versus carrier density n and wavelength λ
a) incremental spontaneous emissions rate R21,spont into all modes
In the analysis of the thermal equilibrium with the blackbody radiation, we considered the radiation equilibrium with all
modes of the blackbody-radiation.
¨ so the Einstein-coefficient A for discrete energy states describes the spontaneous emission rate into a all modes.
The incremental spontaneous emission rate ΔR21,spont(ω) for a the frequency ω, resp. energy E =
⎛ ω3 ⎞
A(ω ) = B⎜⎜ 2 2 ⎟⎟ Einstein-Relation.
⎝ π c0 ⎠
ω can be calculated from
• the spontaneous emission rate results from downward-transitions and has to be weighted by the level-occupancy (Fermifunctions fC and fv:
(
ΔR21,spont ( ω ) = A ρr ( ω ) .⎡⎣ fc ( Ec,2 ) 1 − fv ( Ev,1 )
)
⎛ ω3 ⎞
⎤ ΔE = B⎜ 2 2 ⎟ ρr ( ω ) .⎡ fc ( Ec,2 ) 1 − fv ( Ev,1 ) ⎤ ΔE
⎦
⎣
⎦
⎝ π c0 ⎠
(
)
Spontaneous Emission in ΔE
b) Total spontaneous emission rate Rspont into all possible spatial and frequency modes:
„
The total spontaneous emission is generated isotropic in all directions by all transition pairs distributed over the whole
energy range ΔE=ħΔωspont. For the total spontaneous emission rate Rspont we have to integrate over all filled states in the
bands resp. to the carrier density n under assumption f charge neutrality n=p.
The total spontaneous emission rate Rspont enters the carrier rate-equation for n and p.
„
Rspont is emitted into all modes keeping in mind that Rspont >> R21,spont ,mode.
∞
(
)
Rspont = ∫ A (ω ) ρr (ω ) . ⎡⎣ fc ( Ec,2 ) 1 − fv ( Ev,1 ) ⎤⎦ dω
0
=
1
γ
DEFINITION of γ
R 21,spont,mode
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For lasers also need the fraction γ, β of spontaneous emission into a single mode R21,spont, mode:
Definition of the spontaneous emissions factor γ =β :
γ = 1/
∫
Δωspont
ρmode (ω )Vspont dω ≅ 1/ (Vspont ρmode (ω ) Δωspont )
ρmode = mode density of free space
If A and B are approximately frequency independent, we can use for simplicity the following
life-time approximation for Rspont:
∞
(
)
Rspont = A ∫ ρ r ( ω ) ⎡⎣ f c ( Ec ,2 ) 1 − f v ( Ev,1 ) ⎤⎦ dω
0
≈
Definition of τ spont
n / τ spont
→
Numerical Simulation of Gain- and
spontaneous emission-spectra for GaAs:
at high carriere densities τ spont is not really constant but depends also on n
∞
τ spont = n / ∫ Aρ r ( ω ) ⎣⎡ f c ( Ec ,2 ) ( 1 − f v ( Ev,1 ) ) ⎦⎤ ≈ 1 / n
( not
constant ! )
0
→
Rspont = n / τ spont ≈ BBM n 2
( Bimolecular
Recombination )
Numerical calculation of Rspont(n) confirms a square-law dependence
~n2 (– n3) of the carrier density n (Bimolecular Recombination).
Δωspont
Δωstim
•
the spectral width Δωspont of the spontaneous emission R21,spont(ω) is in general larger than the width Δωstim of stimulated
emission Rstim(ω)
•
for mathematical simplicity spontaneous emission rates are often approximated by a constant carrier lifetime τspont,tot.
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5.7.4.3
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Numerical simulations of optical gain-spectra g(ω,n):
The numerical simulation of g(ω,n) using the appropriate material parameters gives for the practically important SC like
InP (λ=1.3μm und 1.5μm) und GaAs (λ=0.85μm) demonstrated the following generic properties:
Amplification and absorption spectrum g(ω,n) for GaAs: (Simulation)
With increasing carrier density n the bands are filled with carriers to higher energy levels resulting in
1) Increase of maximum gain
2) decrease of the wavelength of the maximum gain
3) increase of the width of the gain-spectrum
Gain-Spectrum vers. carrier density n
Maximum Gain gmax vers. carrier density n:
gmax(n)
Band filling € λmax-shift
Transparency density ntr
Parabolic spectral approximation of g(ω,n):
g (ω ,n ) ≅ g max ( n )
1
1 + ( (ω − ω max ( n ) ) / Δω ( n ) )
2
,
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Linear Gain dependence for bulk semiconductors:
Simulation show that the maximum gain gmax is approximately linear dependent on the carrier density n > ntr.
gmax(ωmax ,n) ≅ a (n − ntr )
Linear approximation of max. gain in function of n
Definition: ntr = Transparency-carrier density ( g max (ω max ,ntr ) ≅ 0 ) and a = differential gain a ( n ) = ∂g max / ∂ n
•
•
•
at n=0 the material is absorbing with f1~1 and f2 ~0 , g(0)~ -gmax(ω)
at n=ntr (transparency density) the material is transparent ,
g(ωmax,n)
g(ntr)=0
at n>ntr the material is amplifying with f1~0 and f2 ~1 , g(n)~ +gmax(ω)
Nonlinear gain-carrier density relation for Quantum Wells
(QW) in SCs:
gmax(n) relations show in general the nonlinear behavior (logarithmic for QWs)
of the figure below and a parabolic frequency dependence:
n
ntr
gmax (ωmax ,n) ≅ g0 ln( n/ ntr )
g
Parabolic approximation of the spectral dependence of g(ω,n):
⎧⎪ ⎛ ω − ω max ⎞ 2 ⎫⎪
⎧⎪ ⎛ ω − ω max ⎞ 2 ⎫⎪
g (ω ,n ) = g max (ω max ,n ) / ⎨1 + ⎜
⎟ ⎬ = a n − ntr / ⎨1 + ⎜
⎟ ⎬
Δ
ω
⎝
⎠ ⎭⎪
⎪⎩ ⎝ Δω ⎠ ⎭⎪
⎩⎪
(
)
Valid approximation because often only the gain close to the maximum is relevant !
gmax
Δω
ωmax
ω
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Temperature dependence of the optical gain g(ω,n,T) in SC:
Thermal gain broadening and gain reduction
Because of the temperature T dependence of the Fermi-functions and subsequent broadening of the carrier distribution:
• the spectral width Δω of g(ω,n) increases with increasing temperature
• the maximum gain gmax(ωmax,n) decreases with increasing temperature (RT-operation of diode lasers was a challenge)
T=297K
T=77K
Bild Temp-abhängigkeit
The broadening and reduction of the optical gain with temperature has a very detrimental effect on devices such as diode
lasers and amplifiers. It is also responsible for the high temperature sensitivity of these devices.
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5.7.4.4 Rate equation for electrons and photons, electrical pumping
Concept of carrier rate equations in SCs:
Rate equations are continuity equations for particle (electrons, n , holes, p , and photons sph) currents flowing in and out
of a particle reservoir (energy bands: energy states / volume, optical resonator for photons) per unit volume.
„
Because each optical transition results in the generation or annihilation of a electron-hole pair in the conduction- and
valence band, we formulate the continuity or rate equation for the carrier densities n and p.
We postulate that there is an external pump mechanism (eg. carrier injection in pn-junctions) available, which pumps
electrons from the valence band up to the conduction band with a rate Rpump:
Rpump = ∂ n pump / ∂ t .
The continuity equation for n and p becomes for charge neutrality n=p:
∂
∂
n = + Rpump − g (ω ,n ) vgr s ph − Rspont ( n ) = − p
∂t
∂t
Rspont = Rate of the total spont. emission in all modes in Volspont
;
Carrier (electron)-Rate equation (assuming Vactive=Vmode)
R21,spont,mode = Rate of spont. emission in a single lasing mode
(
)
The stimulated emission R21,net can be related to the gain g(w,n): R21,net ω ,n,s ph = g (ω ,n ) vgr s ph
Photon-Rate-Equation:
Because each optical transition creates (stimulated or spontaneous emission) or destroys (absorption) one photon in the
optical field (mode) we formulate the continuity equation for the photon density sph:
∂
s ph = + R21,net + R21,spont,mode = + Γg (ω ,n ) vgr s ph + β ΓRspont
∂t
Photon-Rate Equation for 1 optical mode
β=γ is the spontaneous emission factor (β<<1)
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Conclusions SC:
• At low electron densities n< ntr the SC is absorbing for all frequencies ω with ω > E g . This means that the
simulated absorption is dominating for g(ω,n)<0
• At the carrier density ntr (transparency density) g(ω,ntr)=0 for a particular frequency ω. Stimulated emission almost
equals absorption – therefore the SC is transparent at ω.
• At carrier densities n>ntr there is optical gain g(ω,n)>0 over a limited optical frequency range ~ ( EQF ,n − EQF ,n ) > ω > E g ,
The optical bandwidth increases with increasing n. The stimulated emission dominates.
• The frequency ωmax , where g(ω) reaches a maximum gmax(ωmax,n), increases with increasing n (filling the
conduction- and valence band from the bottom)
• For bulk-SC gmax(ωmax,n) increases approximately linear with (n-ntr), for Quantum-Wells ~ln(n-ntr)
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Conclusions:
• For the description of the interaction of the optical field with the quantized medium near resonance (amplification)
needs an quantum-mechanical treatment
• The influence of the material properties on the dynamic of the dipole is described quantum-mechanically by the
dipole-matrix-element
• Population inversion creates a non-classical net dipole with an out-of-phase component in anti-phase to the
exciting field and which delivers energy to the field
• Populations-Inversion in quantized media allow a) optical amplification and b) generation of coherent
optical wave
• The energy conservation in the interaction of optical field – carrier population can be described by rate
equations for charge carriers and photons.
• The polarization and the e-h-pair generation can be described by the susceptibility and the transition
probability r, resp. by the quantum-mechanical transition rate R of the carriers
• QM shows that the “electron-on-springs”-model is an adequate description for loss-less media
• Without the quantization of the optical field (photons) the spontaneous emission can not be calculated from
first principles – it remains just a thermodynamic necessity for equilibrium
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5.8 Population inversion by electrical pumping of pn-diodes
(optional, see Coldren)
To satisfy the condition of Bernard-Duraffourgh in SC a carrier pump mechanism is required for achieving population
inversion between electrons and holes n> ntr.
A simple, fast and efficient realization of electrical pumping is carrier injection through the depletion layer of
a forward biased pn-diode or into the I-layer of a PIN-diode.
A key benefit electrical current pumping of a pn-diode is current modulation of the light generation up to high frequencies and
with high efficiency.
5.8.1 Carrier injection and population inversion in homojunction pn-diodes
Considering a strongly forward biased GaAs-pn-diode with VD ≅ Eg/e:
•
•
electrons from the n-area (majority carriers in n+) are injected as minority carriers through the depletion layer into the p-area
in the p-area the injected electrons diffuse for a distance of a diffusion length LD,n (~2-3μm) before they have mostly
recombined by spontaneous emission and reach the electron equilibrium value no,p of the p-area.
„
Within a diffusion length LD in the p-area adjacent to the depletion layer we have np(x)>>no,p,. There is a local carrier nonequilibrium (np>>ni2).
„
Because the electrons in the conduction band and the holes in the valence band are not in equilibrium, the distribution of
electrons and holes is described by 2 separate Quasi-Fermi-distributions characterized by 2 different Quasi-Fermi-levels
EFQ,n and EFQ,p.
with:
EFQ,n − EFQ ,p ≅ eVD
Same arguments hold for hole-injection from the p-area into the n-region.
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The external diode voltage VD separates the quasi-Fermi-levels in the depletion-layer
Carrier injection in forward biased pn-diodes:
Equilibrium VD=0
p
Sperrschicht
Region of gain
n
EQF,n
EFn
EQF,p
EFp
EF=EFQn=EFQv
eUD
High injection VD~Eg/e (forward polarized) EFQn-EFQv~eVD
Area of non-equilibrium carrier injection
e-Injektion
Ln
h-Injektion
Lp
Considering the inversion condition by Bernard-Duraffourg, VD must fulfill:
eVD ≅ EQF ,n − EQF ,p > E2 − E1 = E21 = ω > E g
•
Under this condition the material on both sides of the depletion layer is inverted within about one diffusion length L
and can be optically amplifying g>0 resp. (n>ntr).
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As long as there is no strong light field, which is amplified by stimulated emission, the injected minority carriers recombine
only by spontaneous emission with a carrier-lifetime τ spont (approximation):
∂n
n
≅
= Rspont
∂ t τ spont
¨ it is obvious that for a given diode current I the diffusion length L must be a minimum in order to achieve a minority carrier
density n (or p) or an optical gain g(n,p) as heigh as possible.
For simple homojunction-pn-diodes the diffusion length LD is as a material parameter in the order of a few μm leading to very
high current densities for population inversion (~ several 10kA/cm2), resp. for achieving a certain optical gain g(n).
Quasi-Neutrality Condition in SCs:
We will saw that in the electrical pumping of SCs by carrier injection in pn-diodes:
•
The application of a forward diode voltage determines the separation of the quasi-Fermi-levels EQF,n and EQF,p
eVD= (EQF,c - EQF,v)
•
VD also causes a very strong carrier-injection and current flow through the diode
It is assumed that at the high carrier injection levels, the minority carrier density is comparable to the majority carrier density.
The carrier density is so high, so that there can be 1) no internal E-field or as an equivalent statement 2) the active area must
be electrically neutral requiring:
Quasi-Neutrality Condition for undoped SC:
n= p
¨ With the help of this equation it is possible to determine the positions of the quasi-Fermi-levels
for a given VD.
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5.8.2 Materials properties of direct Semiconductors (optional)
The crystal structure and composition of compound SC determines the bandstructure and thus the optical and electronic
properties. Thus the composition of binary, ternary, quaternary crystals is used to control:
1) Bandgap Eg → Emission-wavelength λg (direct Q indirect SC)
2) Bandgap Eg , Electron Affinity χ → Heterostructure-barriers, ΔEC, ΔEV
3) Refractive index n → Waveguides
AlxGa1-xAs-compound crystal:
(group III-element: Ga, Al group V-element: As)
Bandgap Eg vers. lattice constant a for ternary compound crystals:
o binary
---- ternary
In ternary AlGaAs-crystal
a fraction of Ga-atom is
replaced by Al-atoms.
Refractive index n of AlxGa1-xAs vers. Al-concentration:
The Al-concentration x is used to control the refractive index of AlxGa1-xAs.
Because the lattice constant of AlxGa1-xAs does not change with composition x the layers are
automatically
→ crystallographically lattice matched
InGaAs and InGaAsP are only for a precise composition lattice-matched to InP-substrates
(difficult growth) !
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