handout 29 optical transitions in solids, optical gain ...k q k cv k q k c k v k k q k c k v k r c k...
TRANSCRIPT
1
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 29
Optical Transitions in Solids, Optical Gain, and Semiconductor Lasers
In this lecture you will learn:
• Electron-photon Hamiltonian in solids• Optical transition matrix elements• Optical absorption coefficients • Stimulated absorption and stimulated emission• Optical gain in semiconductors• Semiconductor heterostructure lasers
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Interactions Between Light and Solids
The basic interactions between light and solids cover a wide variety of topics that can include:
• Interband electronic transitions in solids
• Intraband electronic transitions and intersubband electronic transitions
• Plasmons and plasmon-polaritons
• Surface plasmons
• Excitons and exciton-polaritons
• Phonon and phonon-polaritons
• Nonlinear optics
• Quantum optics
• Optical spintronics
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Fermi’s Golden Rule: A Review
Now suppose a time dependent externally applied potential is added to the Hamiltonian:
titio eVeVHH
ˆˆˆˆ
Consider a Hamiltonian with the following eigenstates and eigenenergies:
integerˆ mEH mmmo
Suppose at time t = 0 an electron was in some initial state k: pt 0
Fermi’s golden rule tells that the rate at which the electron absorbs energy from the time-dependent potential and makes a transition to some higher energy level is given by:
pmpm
mEEVpW
2ˆ2
The rate at which the electron gives away energy to the time-dependent potential and makes a transition to some lower energy level is given by:
pmpm
mEEVpW
2ˆ2
E
E
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Transitions in Solids: Energy and Momentum Conservation
k
For an electron to absorb energy from a photon energy conservation implies:
infm kEkE
Final energy
Initialenergy
Photonenergy
Momentum conservation implies:
qkk if
Initial momentum
Final momentum
Photonmomentum
Intraband photonic transitions are not possible:
For parabolic bands, it can be shown that intraband optical transitions cannot satisfy both energy and momentum conservation and are therefore not possible
Intraband
Intraband
Interband
Note that the momentum conservation principle is stated in terms of the crystal momentum of the electrons. This principle will be derived later.
E
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electromagnetic Wave BasicsConsider an electromagnetic wave passing through a solid with electric field given by:
trqEntrE o
.sinˆ,
The vector potential associated with the field is:
t
trAtrE
,
,
trqAn
trqE
ntrA
o
o
.cosˆ
.cosˆ,
The power per unit area or the Intensity of the field is given by the Poynting vector:
2
ˆ2
ˆ,,,222oo A
qE
qtrHtrEtrSI
noo
The photon flux per unit area is:
2
2oAI
F
E
H
q
The divergence of the field is zero:
0,.,. trAtrE
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Consider electrons in a solid. The eigenstates (Bloch functions) and eigenenergies satisfy:
knnkno kEH
,,ˆ
rVmPP
H latticeoˆ
2
ˆ.ˆˆ
Electron-Photon Hamiltonian in Solids
where:
In the presence of E&M fields the Hamiltonian is:
Pnm
eeAeH
PtrAme
H
PtrAme
trAPme
H
trAtrAm
ePtrA
me
trAPme
rVmPP
rVm
trAePH
tirqitirqio
o
o
o
lattice
lattice
ˆˆ2
ˆ
ˆ.,ˆˆ
ˆ.,ˆ2
,ˆ.ˆ2
ˆ
,ˆ.,ˆ2
ˆ.,ˆ2
,ˆ.ˆ2
ˆ2
ˆ.ˆ
ˆ2
,ˆˆˆ
ˆ.ˆ.
2
2
Assume small
Provided: 0,. trA
trqAntrA o
.cosˆ,
4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
k
Interband
E
Optical Interband Transitions in Solids
knnkno kEH
,,ˆ
Suppose at time t = 0 the electron was sitting in the valence band with crystal momentum :
ikvt
,0
nPm
eeAeHH
tirqitirqio
o ˆ.ˆ2
ˆˆ
ˆ.ˆ.
ik
The transition rate to states in the conduction band is given by the Fermi’s golden rule:
ivfckvkck
i kEkEVkWif
f
2
,,ˆ2
The summation is over all possible final states in the conduction band that have the same spin as the initial state. Energy conservation is enforced by the delta function.
ik
Comparison with:
nPmeAe
Vrqi
o ˆ.ˆ2
ˆˆ.
nPm
eAeV
rqio ˆ.ˆ2
ˆˆ.
titio eVeVHH
ˆˆˆˆgives:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Matrix Element
ivfckvrqi
kck
o
ivfckvkck
i
kEkEnPem
eA
kEkEVkW
iff
iff
2
,ˆ.
,
2
2
,,
ˆ.ˆ2
2
ˆ2
Now consider the matrix element:
rnPerrd
nPe
if
if
kvrqi
kc
kvrqi
kc
,.
,*3
,ˆ.
,
ˆ.ˆ
ˆ.ˆ
k
Interband
E
k
Interband
E
ruenPeruerdi
if
fkv
rkirqikc
rki
,..
,*.3 ˆ.ˆ
runPeeruerd
runkPeeruerd
ii
ff
ii
ff
kvrkirqi
kcrki
kvirkirqi
kcrki
,..
,*.3
,..
,*.3
ˆ.ˆ
ˆ.ˆ
0
Rapidly varying in spaceSlowly
varying in space
5
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Matrix Element
if kvrqi
kc nPe
,ˆ.
,ˆ.ˆ
k
Interband
E
k
Interband
E
jif
jfi
ii
ff
Rkvkc
j
Rkqki
kvrkirqi
kcrki
runPrurde
runPeeruerd
,,*
cell primitive th-
3.
,..
,*.3
ˆ.ˆ
ˆ.ˆ
nP
runPrurd
runPrurdN
runPrurde
cvkqk
kvkckqk
kvkckqk
Rkvkc
Rkqki
fi
iffi
iffi
jif
jfi
ˆ.
ˆ.ˆ
ˆ.ˆ
ˆ.ˆ
,
,,*
crystal entire
3,
,,*
cell primitive one any
3,
,,*
cell primitive one any
3.
N = number of primitive cells in the crystal
Interband momentum matrix element
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Crystal Momentum Selection Rule
iviccvo
ivfckqkcvk
o
ivfckvrqi
kck
oi
kEqkEnPm
eA
kEkEnPm
eA
kEkEnPem
eAkW
fif
iff
22
,
22
2
,ˆ.
,
2
ˆ.2
2
ˆ.2
2
ˆ.ˆ2
2
k
Interband
E
The wave vector of the photons is very small since the speed of light is very large
Therefore, one may assume that:
Crystal Momentum Selection Rule:
We have the crystal momentum selection rule:
qkk if
if kk
Optical transitions are vertical in k-space
6
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
iviccv
oi kEkEnP
meA
kW2
2ˆ.
22
Transition Rates per Unit Volume
Generally one is not interested in the transition rate for any one particular initial electron state but in the number of transitions happening per unit volume of the material per second
The upward transition rate per unit volume is obtained by summing over all the possible initial states per unit volume weighed by the occupation probability of the initial state and by the probability that the final state is empty:
R
ik
icivi kfkfkWV
R
1
2
If we assume, as in an intrinsic semiconductor, that the valence band is full and the conduction band is empty of electrons, then:
ik
ikWV
R
2
ikiviccv
o kEkEnPm
eAV
2
2ˆ.
222
k
Interband
E
k
Interband
E
ikW
ik
R
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transition Rates per Unit Volume
FBZ3
322
FBZ3
322
FBZ
22
3
3
22
22ˆ.
22
22ˆ.
22
ˆ.2
2
22
ˆ.2
22
kEkEkd
nPm
eA
kEkEkd
nPm
eA
kEkEnPm
eAkd
kEkEnPm
eAV
R
vccvo
ivici
cvo
iviccvoi
kiviccv
o
i
The integral in the expression above is similar to the density of states integral:
ce
cD EEm
EkEkd
Eg
23
22FBZ
3
3
32
2
1
22
e
cc mk
EkE2
22Suppose:
Then:
k
Interband
E
R
Joint density of states
7
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transition Rates per Unit Volume and Joint Density of States
FBZ3
322
22ˆ.
22
kEkE
kdnP
meA
R vccvo
e
cc mk
EkE2
22Suppose:
Then:
h
vv mk
EkE2
22
r
ghe
vcvc mk
Emm
kEEkEkE
211
2
2222
gr
cvo E
mnP
meA
R
23
22
22 2
2
1ˆ.
22
R
gE
Joint density of states
Reduced effective mass
k
Interband
E
R
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transition Rates per Unit Volume
gr
cvo E
mnP
meA
R
23
22
22 2
2
1ˆ.
22
2
22oA
I
The power per unit area or the Intensity of the E&M wave is:
gr
cv Em
nPI
me
R
23
22
2
2
2 2
2
1ˆ.
22
2
k
Interband
E
R
Interband Momentum Matrix Elements:
g
cv
her E
nP
mmmm
2
2
ˆ.4111
Recall the result from homework 7:
k
E
Eg
The above result assumed diagonal isotropic conduction and valence bands and also that no other bands are present, and is therefore oversimplified
8
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Because of transitions photons will be lost from the E&M wave traveling inside the solid. This loss will result in decay of the wave Intensity with distance travelled:
xIxxIxxI
xIdxxI
xIexI x
0
Loss coefficient or absorption coefficient
xI xxI
x xx
R
Transition Rates per Unit Volume and the Absorption or Loss Coefficient
Momentum Matrix ElementsIn bulk III-V semiconductors with cubic symmetry, the average value of the momentum matrix element is usually expressed in terms of the energy Ep as,
6ˆ.
2 pcv
mEnP
Parameters at 300K GaAs AlAs InAs InP GaP
Ep (eV) 25.7 21.1 22.2 20.7 22.2
Note that the momentum matrix elements are independent of the direction of light polarization!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transition Rates per Unit Volume and Loss Coefficient
xIxxIxxI
The wave power loss (per unit area) in small distance x is x I(x)
The wave power loss in small distance x must also equal:
xI xxI
x xx
R
I
R
IxxR
Therefore:
xR
gr
cv Em
nPme
23
22
22 2
2
1ˆ.
22
2
gr
cvo
Em
nPcnm
e
23
22
22 2
2
1ˆ.
Values of () for most semiconductors can range from a few hundred cm-1 to hundred thousand cm-1
k
Interband
E
R
9
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Loss Coefficient of Semiconductors
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Direct Bandgap and Indirect Bandgap Semiconductors
Direct bandgap (Direct optical transitions)
Direct bandgap(Indirect phonon-assisted transitions)
ehr mmm111
If me approaches -mh , then mr becomes very large
23
rmThen, also becomes very large
10
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Stimulated Absorption and Stimulated Emission
FBZ3
322
12
2ˆ.2
2
kEkEkfkf
kdnP
meA
R vccvcvo
Throwing back in the occupation factors one can write more generally:
Stimulated Absorption:The process of photon absorption is called stimulated absorption (because, quite obviously, the process is initiated by the incoming radiation or photons some of which eventually end up getting absorbed)
Stimulated absorption
k
EAn incoming photon can also cause the reverse process in which an electron makes a downward transition
This reverse process is initiated by the term in Hamiltonian that has the time dependence:tie
titio eVeVHH
ˆˆˆˆ
nPmeAe
Vrqi
o ˆ.ˆ2
ˆˆ.
nPm
eAeV
rqio ˆ.ˆ2
ˆˆ.
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Following the same procedure as for stimulated absorption, one can write the rate per unit volume for the downward transitions as:
FBZ3
322
12
2ˆ.2
2
kEkEkfkf
kdnP
meA
R vcvccvo
Stimulatedemission
k
E
Stimulated Emission:In the downward transition, the electron gives off its energy to the electromagnetic field, i.e. it emits a photon! The process of photon emission caused by incoming radiation (or by other photons) is called stimulated emission.
Spontaneous Emission: Electrons can also make downward transitions even in the absence of any incoming radiation (or photons). This process is called spontaneous emission.
Stimulated Emission and Spontaneous Emission
Spontaneousemission
k
E
11
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Stimulated Absorption and Stimulated Emission
The net stimulated electronic transition rate is the difference between the stimulated emission and stmulated absorption rates:
FBZ3
322
22ˆ.
22
kEkEkfkf
kdnP
meA
RR vccvcvo
And the more accurate expression for the loss coefficient is then:
I
RR
Stimulated absorption
FBZ3
322
22ˆ.
kEkEkfkfkd
nPcnm
evccvcv
o
Stimulatedemission
k
E
k
E
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Gain in Semiconductors
FBZ3
322
22ˆ.
kEkEkfkf
kdnP
cnme
vccvcvo
Note that the Intensity decays as: 0 xIexI x
What if were to become negative? Optical gain !!
Optical gain is possible if:
k
E
RR0
kEkEkfkf
kEkEkfkf
vcvc
vccv
for0
for0
A negative value of implies optical gain (as opposed to optical loss) and means that stimulated emission rate exceeds stimulated absorption rate
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Gain in SemiconductorsOptical gain is only possible in non-equilibrium situations when the electron and hole Fermi levels are not the same
Suppose:
KTEkEfhvv
KTEkEfecc
fhv
fec
eEkEfkf
eEkEfkf
1
11
1
fhfe
vcfhfe
fhvfec
EE
kEkEEE
EkEfEkEf 0
Then the condition for optical gain at frequency is:k
E
Efe
Efh
The above is the condition for population inversion (lots of electrons in the conduction band and lots of holes in the valence band)
kEkE vc
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Optical Gain in Semiconductors
The loss coefficient is a function of frequency:
FBZ3
322
22ˆ.
kEkEkfkf
kdnP
cnme
vccvcvo
gE
Optical gain for frequencies for which:
fhfeg EEE
Increasing electron-hole density (n = p)
Net Loss
Net Gain
0k
E
Efe
Efh
13
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor PN Heterojunctions2cE
2vE
fE2cE
2vEfE
1cE
1vE
P-dopedN-doped
2vE
fE fE
P-doped
N-doped
fE
fLEP-doped
N-dopedfRE
PN heterojunction in equilibrium
PN heterojunction in forward biaseVEE fRfL
PN heterojunction
Electron and hole Fermi level splitting
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
photons
stimulated andspontaneousemission
holes
electrons
A Heterostructure Laser (Band Diagram)
N-dopedP-doped
metal
A Ridge Waveguide Laser Structure
All lasers used in fiber optical communications are semiconductor lasers
Semiconductor Heterostructure Lasers
Efe
Efh
Non-equilibrium electron and hole populations can be sustained in a forward bias pn junction
eV
EE fhfe
14
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A commercial packagedsemiconductor laser
Semiconductor laser chip
actual laser(the long strip)
Wire bond
fiber
Semiconductor Heterostructure Lasers