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University Of Illinois University Of Illinois University Of Illinois University Of Illinois
at Chicagoat Chicagoat Chicagoat Chicago
College Of EngineeringCollege Of EngineeringCollege Of EngineeringCollege Of EngineeringNNNNNNNNano Engineering
Research Group
Design of a Novel Heterostructure Photodetectors with Dramatically Enhanced Signal-to-Noise based on
Resonant Interface-Phonon-Assisted Transitions and Engineering of Energy States to Enhance Transition Engineering of Energy States to Enhance Transition
Rates
Yi Lana, Nanzhu Zhanga, Lucy Shia, Chenjie Tanga, Mitra Duttaa,b, and Michael A. Stroscioa,b,c
Optics-2014, 8-10 Sept. 2014
aElectrical and Computer Engineering Department, U. of Illinois at Chicago (UIC), 851 S. Morgan Street, Chicago, Illinois 60607
bPhysics Department, U. of Illinois at Chicago, 851 S. Morgan Street, Chicago, IllinoiscBioengineering Department, U. of Illinois at Chicago, 851 S. Morgan St., Chicago, IL
University Of Illinois University Of Illinois University Of Illinois University Of Illinois
at Chicagoat Chicagoat Chicagoat Chicago
College Of EngineeringCollege Of EngineeringCollege Of EngineeringCollege Of EngineeringNNNNNNNNano Engineering
Research Group
A novel heterostructure photodetector design is presented that facilitates dramatic enhancements of signal-to-noise. The structure incorporates a single quantum well coupled to a symmetric double quantum well that makes it possible to engineer energy states with energy state separations equal to an interface phonon energy. In addition, quantum level energy degeneracy between states in the single-well and double-well
systems makes it possible to enhance the rate of interface-phonon-assisted transitions. systems makes it possible to enhance the rate of interface-phonon-assisted transitions. The techniques underlying this approach have been discussed previously by Stroscio and Dutta in Phonons in Nanostructures (Cambridge University Press, 2001). Together, these effects make it possible to greatly enhance signal-to-noise ratios in these heterostructure-based photodetectors. These designs are optimized based on Schrödinger equation calculations of the energy states and the determination of
interface phonon potentials and dispersion modes by applying boundary conditions for which the phonon potential has corresponding continuous normal components of the
displacement field and tangential components of electric fields. Novel photodetector designs with dramatically enhanced signal-to-noise will be presented for a number of
different heterostructure devices.
This energy-level structure facilitates the absorption of a photon, emission
of a phonon, and the absorption of a photon with the same wavelength as
the original photon. E1 is the first energy level of the single well, and E3 is
the second energy level. In addition, E2, E2’, E4, and E4’ represent the first,
second, third, and forth energy levels for the double quantum well.
With reference to Fig. 1, it is straightforward to see that there will be a
dramatic signal-to-noise enhancement in the current, Isn,E1, from the deepest
state E1, relative to Isn,E2, from the deepest state E2 (without phonon-assisted
transition and second photon absorption), as given by the Richardson sn,E2
transition and second photon absorption), as given by the Richardson
formula:
In this equation, E3 - E1 = E4’- E2 = Ephoton and E2’ - E2 = Ephonon.
For example if,
a dramatic 1/3,000 reduction can be realized.
u
Transverse Optical (TO) Phonon
Transverse Optical (TO) Phonon
u – displacementq
u – displacement
q - direction
Transverse Acoustic (TA) Phonon
LO and LA Phonons have
displacements along the
direction of q
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringPhonons: Some Basic Characteristics
Boundary Conditions:
Optical modes --- continuity of the tangential component of
the electric field and the z component of the displacement vector
must be continuous at the interfaces
Acoustic modes --- displacement and normal component of stress
tensor are continuous at interfaces
jj zz
j
zj
zz
j
zjz
z
z
z
=
+
+
=∂
∂=
∂
∂ )()( 1
,1,
φε
φε
tensor are continuous at interfaces
For example
I II III Quantized Confined Phonons
University Of Illinois
At Chicago
College Of Engineering
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Normalization and Applicability of Elastic
Continuum Theory --- One Monolayer Thick?
Normalization:
Mode amplitude normalized so that the energy in each model
is the quantized phonon energy – example 2D graphene
( )1
* *2 LOs
mn
u u v v dxdyS Mω
⋅ + ⋅ =∫h
McEwen & Park et al., Nature
September 2000 Elastic Continuum Results
Mode Energy (meV)
a0 62
a1 74
Elastic Continuum Theory Gives Correct Energy for a
Dominant Mode of the Nanomechanical Modes in
Carbon Nanotubes and in C60
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University Of Illinois
At Chicago
College Of Engineering
35 meV mode observed experimentally
a1 74
a2 111
b2 32
b3 38
Goal: Theoretical Description of Nanoscale Mechanical Structures for Nanodevice
and Sensor Applications including Nanocantilevers
Matches our theoretical results to 10%
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringPhonons: Some Basic Characteristics
Precision and Nature of
Optical Phonon Confinement
H. Sakaki et al.
Anharmonic Effects:
Klemen’s Channel with Keating Model ---Bhatt, Kim and Stroscio
q
m x
ZZ
mq q
L
π= =
• R. Fuchs and K. L. Kliewer, “Optical Modes of Vibration in an Ionic Slab,”
Physical Review, 140, A2076-A2088 (1965).
• J. J. Licari and R. Evrard, “Electron-Phonon Interaction in a Dielectric Slab:
Effect of Electronic Polarizability,” Physical Review, B15, 2254-2264 (1977).
• L. Wendler, “Electron-Phonon Interaction in Dielectric Bilayer System: Effects of
Electronic Polarizability,” Physics Status Solidi B, 129, 513-530 (1985).
• C. Trallero-Giner, F. Garcia-Moliner, V. R. Velasco, and M. Cardona, “Analysis of
the Phenomenological Models for Long-Wavelength Polar Optical Modes in
Selection of Major Theoretical Papers: Optical Modes
the Phenomenological Models for Long-Wavelength Polar Optical Modes in
Semiconductor Layered Systems,” Physical Review, B45, 11,944-11,948 (1992).
• K. J. Nash, “Electron-Phonon Interactions and Lattice Dynamics of Optic Phonons
in Semiconductor Heterostructures,” Physical Review, B46, 7723-7744 (1992). ---
For slab modes, reformulated slab vibrations, and guided modes, “intrasubband and
intersubband electron-phonon scattering rates are independent of the basis set used to
describe the modes, as lond as this set is orthogonal and complete.”
• F. Comas, C. Trallero-Giner, and M. Cardona, “Continuum Treatment of Phonon
Polaritons in Semiconductor Heterostructures,” Physical Review, B56, 4115-4127
(1997). --- Seven coupled partial differential equations; solutions for isotropic
materials; the non-dispersive case “leads to the the Fuchs-Kliewer slab modes.”
More on
Confined,
Interface, and
Half-Space
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University Of Illinois
At Chicago
College Of EngineeringVibrational Modes in Nanostructures
Half-Space
Phonon Modes
in
Phonons and
Nanostructures
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Research Group
University Of Illinois
At Chicago
College Of EngineeringMore on Interface Modes
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringMore on Interface Modes
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringMore on Interface Modes
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringMore on Interface Modes
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringPhonon “Bands”
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringPhonon “Bands”
Improved SemiconductorImproved SemiconductorImproved SemiconductorImproved SemiconductorLasers via PhononLasers via PhononLasers via PhononLasers via Phonon----AssistedAssistedAssistedAssisted
TransitionsTransitionsTransitionsTransitionsKey Point Key Point Key Point Key Point -------- Optical Devices not Electronic Devices!Optical Devices not Electronic Devices!Optical Devices not Electronic Devices!Optical Devices not Electronic Devices!
Why? Why? Why? Why? ENERGY SELECTIVITYENERGY SELECTIVITYENERGY SELECTIVITYENERGY SELECTIVITYA single engineered phonon mode may be selected A single engineered phonon mode may be selected A single engineered phonon mode may be selected A single engineered phonon mode may be selected
to modify a selected interactionto modify a selected interactionto modify a selected interactionto modify a selected interaction
photon emission
Interface Optical Phonons: Applications to Phonon-Assisted
Transitions in Heterojunction Lasers
phonon emission
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of Engineering
Interface-Phonon-Assisted Processes:
Double Resonance Process
A2
A1
phonon
assisted
depopulation
photon
emission
D2
D1
D1
A1
A2
D2
Gain enhancementsgreater than twoorders of magnitude
A
D
Interface phononmodes dominateover bulk modes
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of Engineering
a2 = 2 nm, a3 = 10 nm
ηtot = nA2 / nA1 = τ21/τout
= 6 for a1 = 8.5 nm
ηloc = (nA2 / nA1)k = 0
= ηtot (1 + τ11/ τout) EA2-A1/Ephonon
≥ 50 - 100
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes
Double Resonance Scheme
Michael A. Stroscio, Mikhail V. Kisin, Gregory Belenky,
and Serge Luryi, Phonon Enhanced Inverse Population in
Asymmetric Double Quantum Wells, Applied Physics
Letters, 75, 3258 (1999).
ps transition rates
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes
References 4 and 5:
M. Kisin, M. Stroscio, V. Gorfinkel, G.
Belenky and S. Luryi, Influence of
Complex Phonon Spectrum of
Heterostructure on Gain Lineshape in
Quantun Cascade Laser Quantun Cascade Laser
(QCL), Optical Society of America,
Technical Digest Series, Volume 11,
425 (1997).
Mikhail V. Kisin, Vera B. Gorfinkel,
Michael A. Stroscio, Gregory Belenky,
and Serge Luryi, Influence of Complex
Phonon Spectra on Intersubband Optical
Gain, J. Appl. Phys., 82, 2031 (1997).
----- 10 nm
____ 6 nm
AlGaAs-GaAs-AlGaAs
x = 0.3
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes, Con’t
6 nm, RT
6 nm, RT
6 nm, RT, 60 meV
6 nm, RT, 10 meV
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes, Con’t
GaAs
QW
AlGaAs
GaAs
QW
τout = 0.4 ps
τout = 0.6 ps
τout = 0.55 ps
--- all modes
__ w/o barrier
modes
A - 0.4 ps,
B - 0.5 ps,
C - 0.6 ps
τ1- 2 = 0.56 ps
• Dimensional Confinement and Boundary Effects Cause Plane Wave Phonons (Bulk Phonons) to by Replaced by Set of Modes --- Same as Putting Electromagnetic Wave in a Waveguide
• Bulk modes → Confined modes, plus interface modes, plus half-space modes with new energies, and spatial profiles.
Phonon Engineering: Some Key Techniques
half-space modes with new energies, and spatial profiles.
SINCE CARRIER INTERACTIONS MUST CONSERVE ENERGY AND MOMENTUM HAVING NEW PHONON ENERGIES LEADS TO WAYS TO MODIFY CARRIER
SCATTERING AND TRANSPORT…
EXPLOITING THE FACT THAT NEW ENERGIES LEADS TO WAYS TO MODIFY CARRIER
SCATTERING AND TRANSPORT ---
•Phonon assisted transitions → Example: use to enhance population inversions in Quantum
Cascade Lasers, Type-II Lasers, etc.
•Change phase space to modify interactions → In devices based on quantum wells, quantum
Phonon Engineering: Some Key Techniques
•Change phase space to modify interactions → In devices based on quantum wells, quantum
wires, and quantum dots reduces the set of phonon momenta and energies allowed in
transitions --- Example: Phase-space reductions in CNTs lead to enhanced carrier mobilities
• Modify materials to change phonons and thus interactions → Examples: (a) Form metal-
semiconductor inteface to eliminate selected interface modes; (b) Reduce carrier-phonon
interactions through the design of InxGa-xN-based structures exhibiting one mode behavior
•Modify phonon lifetimes (by arranging for different anharmonic terms) and phonon speeds
(by modifying dispersion relations)→ Reduce bottleneck effects; modify thermal transport
• Generate coherent phonons using Cerenkov effect (as an example) to amplify phonon effects
Some areas where phonon engineering has clear payoff:
improved gain in semiconductor lasers (especially lasers with narrow quantum wells like quantum cascade lasers),
enhance gain in Sb-lased lasers,
coherent phonon sources for non-charge-based binary switches and devices,
increasing carrier mobilities in CNTs, increasing carrier mobilities in CNTs,
improving CNT-based IR detectors based on understand phonon-assisted non-radiative recombination,
improving III-nitride-based device performance,
phonon engineering to modify thermal conductivity.
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of Engineering
Interface-Phonon-Assisted Processes:
Enhanced Population Inversion
United States Patent 6,819,696 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 6,819,696 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 6,819,696 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 6,819,696 Belenky, Dutta, Kisin, Luryi, and Stroscio. November 16, 2004November 16, 2004November 16, 2004November 16, 2004United States Patent 7,310,361 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 7,310,361 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 7,310,361 Belenky, Dutta, Kisin, Luryi, and Stroscio.United States Patent 7,310,361 Belenky, Dutta, Kisin, Luryi, and Stroscio. December 18, 2007December 18, 2007December 18, 2007December 18, 2007
Intersubband semiconductor lasers with enhanced subband depopulation rate
AbstractAbstractAbstractAbstractIntersubband semiconductor lasers (ISLs) are of great interest for mid-infrared (2-20 micron) device AbstractAbstractAbstractAbstractIntersubband semiconductor lasers (ISLs) are of great interest for mid-infrared (2-20 micron) device applications. These semiconductor devices have a wide range of applications from pollution detectionand industrial monitoring to military functions. ISLs have generally encountered several problems which include slow intrawell intersubband relaxation times due to the large momentum transfer and small wave-function overlap of the initial and final electron states in interwell transitions. Overall, the ISL's of the prior art are subject to weak intersubband population inversion. The semiconductor device of the present invention provides optimal intersubband population inversion by providing a double quantum well active region in the semiconductor device. This region allows for small momentum transfer in the intersubband electron-phonon resonance with the substantial wave-function overlap characteristic of the intersubband scattering. Inventors: Belenky; GregoryBelenky; GregoryBelenky; GregoryBelenky; Gregory (Port Jefferson, NY); Dutta; MitraDutta; MitraDutta; MitraDutta; Mitra (Wilmette, IL); Kisin; MikhailKisin; MikhailKisin; MikhailKisin; Mikhail (Lake Grove, NY); Luryi; SergeLuryi; SergeLuryi; SergeLuryi; Serge (Setanket, NY); Stroscio; MichaelStroscio; MichaelStroscio; MichaelStroscio; Michael (Wilmette, IL) Assignee: The United StatesThe United StatesThe United StatesThe United Statesof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Army (Washington, DC) Appl. No.: 957531 957531 957531 957531 Filed: September 21, 2001September 21, 2001September 21, 2001September 21, 2001
A2
A1
photon
emissionD1
A1
A2
D2
Phonon-Assisted Transitions in Heterostructure Lasers
A1
phonon
assisted
depopulation
D2
D1
A D
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of Engineering
Interface-Phonon-Assisted Processes:
Double Resonance Process
A2
A1
phonon
assisted
depopulation
photon
emission
D2
D1
D1
A1
A2
D2
A
D
References 1-9
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes
1. M. Dutta, H. L. Grubin, G. J. Iafrate, K. W. Kim and M. A. Stroscio, Metal-Encapsulated Quantum Wire for Enhanced
Charge Transport, CECOM Docket Number 4734, disclosed September 1991; filed September 15, 1992 (Serial No.
07/945040); Patent No. 5,264,711 issued November 23, 1993.
2. Michael A. Stroscio, Interface-Phonon--Assisted Transitions in Quantum Well Lasers, JAP, 80, 6864 (1996). ---
Strong interaction of interface modes pointed out.
3. SeGi Yu, K. W. Kim, Michael A. Stroscio, G. J. Iafrate, J.-P. Sun, and G. I. Haddad, Transfer Matrix Technique for
Interface Optical Phonon Modes in Multiple Quantum Well Systems, JAP, 82 3363 (1997).
4. M. Kisin, M. Stroscio, V. Gorfinkel, G. Belenky and S. Luryi, Influence of Complex Phonon Spectrum of
Heterostructure on Gain Lineshape in Quantun Cascade Laser (QCL), Optical Society of America, Heterostructure on Gain Lineshape in Quantun Cascade Laser (QCL), Optical Society of America,
Technical Digest Series, Volume 11, 425 (1997).
5. Mikhail V. Kisin, Vera B. Gorfinkel, Michael A. Stroscio, Gregory Belenky, and Serge Luryi, Influence of Complex
Phonon Spectra on Intersubband Optical Gain, J. Appl. Phys., 82, 2031 (1997).
6. Mitra Dutta and Michael A. Stroscio, Comment on Energy Level Schemes for Far-Infrared Quantum Well Lasers,
Appl. Phys. Lett., 74, 2555 (1999).
7. Michael A. Stroscio, Mikhail V. Kisin, Gregory Belenky, and Serge Luryi, Phonon Enhanced Inverse Population in
Asymmetric Double Quantum Wells, Applied Physics Letters, 75, 3258 (1999).
8. J. P. Sun, G. I. Haddad, Mitra Dutta, and Michael A. Stroscio, Quantum Well Intersubband Lasers, International
Journal of High Speed Electronic Systems, 9, 281 (1998).
9. Gregory Belenky, Mitra Dutta, Mikhail Kisin, Serge Luryi, and Michael Stroscio, Intersubband Semiconductor Lasers
with Enhanced Subband Depopulation Rate, invention disclosure filed November 2001;
U.S. Patent No. 6,819,696, November 16, 2004.
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of EngineeringInterface-Phonon-Assisted Processes: Applications
1. M. Dutta, H. L. Grubin, G. J. Iafrate, K. W. Kim and M. A. Stroscio, Metal-Encapsulated Quantum Wire for Enhanced Charge
Transport, CECOM Docket Number 4734, disclosed September 1991; filed September 15, 1992 (Serial No. 07/945040); Patent
No. 5,264,711 issued November 23, 1993.
2. Michael A. Stroscio, Interface-Phonon--Assisted Transitions in Quantum Well Lasers, JAP, 80, 6864 (1996).
5. Mikhail V. Kisin, Vera B. Gorfinkel, Michael A. Stroscio, Gregory Belenky, and Serge Luryi, Influence of Complex Phonon
Spectra on Intersubband Optical Gain, J. Appl. Phys., 82, 2031 (1997).
6. Mitra Dutta and Michael A. Stroscio, Comment on Energy Level Schemes for Far-Infrared Quantum Well Lasers, Appl. Phys.
Lett., 74, 2555 (1999).
9. Gregory Belenky, Mitra Dutta, Mikhail Kisin, Serge Luryi, and Michael Stroscio, Intersubband Semiconductor Lasers with
Enhanced Subband Depopulation Rate, invention disclosure filed November 2001;
U.S. Patent No. 6,819,696, November 16, 2004.
B. S. Williams, B. Xu, Q. Hu, Narrow-linewidth Terahertz Emission from Three-level Systems, APL, 75, 2927 (1999); Refs. 2 and 5.
V. M. Menon, L. R. Ram-Mohan, W. D. Goodhue, A. J. Gatesman, A. S. Karakashian, “Role of Interface Phonons in Quantum Cascade Terahertz Emitters,”
Physica B, 316-317, 212-215 (2002); Ref. 6 and Yu, Kim, Stroscio, Iafrate, Sun, and Haddad, JAP, 82, 3363 (1997).
V. Spagnolo, G. Scamarcio, M. Troccoli, F, Capasso, C. Gmachl, A. M. Sergent, A. L. Hutcheson, D. L. Sivco, and A. Y. Cho, Nonequilibrium Optical
Phonon Generation by Steady State Electron Transport in Quantum-Cascade Lasers, APL, 80, 4303-4305 (2002); Ref. 2 and Komirenko papers on
nonequilibrium phonons (APL, 77, 4178 (2000) and PRB, 63, 165308, (2000).
Mariano Troccoli, Alexey Belyanin, Federico Capasso,
Ertugrul Cubukcu, Deborah L. Sivco, and Alfred Y. Cho,
Raman Injection Laser, Nature, 433, 845-848 (2005).
Implemented in QCL-like Heterostructure
1
2
3
} ∆
NNNNNNNNano Engineering
Research Group
University Of Illinois
At Chicago
College Of Engineering
Interface-Phonon-Assisted Processes:
Enhanced Population Inversion
United States Patent 6,819,696 Belenky, Dutta, Kisin, Luryi, and Stroscio. November 16, 2004United States Patent 7,310,361 Belenky, Dutta, Kisin, Luryi, and Stroscio. December 18, 2007
Intersubband semiconductor lasers with enhanced subband depopulation rate
AbstractAbstractAbstractAbstractAbstractAbstractAbstractAbstractIntersubband semiconductor lasers (ISLs) are of great interest for mid-infrared (2-20 micron) device applications. These semiconductor devices have a wide range of applications from pollution detectionand industrial monitoring to military functions. ISLs have generally encountered several problems which include slow intrawell intersubband relaxation times due to the large momentum transfer and small wave-function overlap of the initial and final electron states in interwell transitions. Overall, the ISL's of the prior art are subject to weak intersubband population inversion. The semiconductor device of the present invention provides optimal intersubband population inversion by providing a double quantum well active region in the semiconductor device. This region allows for small momentum transfer in the intersubband electron-phonon resonance with the substantial wave-function overlap characteristic of the intersubband scattering. Inventors: Belenky; GregoryBelenky; GregoryBelenky; GregoryBelenky; Gregory (Port Jefferson, NY); Dutta; MitraDutta; MitraDutta; MitraDutta; Mitra (Wilmette, IL); Kisin; MikhailKisin; MikhailKisin; MikhailKisin; Mikhail (Lake Grove, NY); Luryi; SergeLuryi; SergeLuryi; SergeLuryi; Serge (Setanket, NY); Stroscio; MichaelStroscio; MichaelStroscio; MichaelStroscio; Michael (Wilmette, IL) Assignee: The United StatesThe United StatesThe United StatesThe United Statesof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Armyof America as represented by the Secretary of the Army (Washington, DC) Appl. No.: 957531 957531 957531 957531 Filed: September 21, 2001September 21, 2001September 21, 2001September 21, 2001
Interface Phonon-assisted Transitions in Interface Phonon-assisted Transitions in
Reduced Noise Single-Well--Double-Well
Photodetectors
E3=E2’
E3-E1=E4’-E2=Ephoton
E2’-E2=E phonon
Design
E2’-E2=E phonon
E1 is the first energy level of the single well, and E3 is the second
energy level of it. At the meanwhile, E2, E2’, E4, and E4’ represent
the first, second, third, and forth energy level for the double
quantum well
Phonon Potential
Let the phonon potentials (Φ) for the given structure be
defined as follow:
1 1
2 2
3 3
( ) ( )
( ) ( )
( ) ( )
qz
qz qz
q z d q z d
q z d q z d
q z d q z d
Ae
Be Ce
De Ee
Fe Ge
He Ie
−
− − −
− − −
− − −
Φ =Φ = +Φ = +Φ = +Φ = +
when z< 0
When 0 ≤ z < d1
when d1≤ z < d2
when d2≤ z < d3
when d3≤ z < d4
1 2( ) ( )Z ZΦ = Φ
4 4
5
( ) ( )
( )
q z d q z d
q z d
He Ie
Je Ke
e
− − −
− −
Φ = +Φ = +Φ =
At the heterointerface of region 1 and region 2, the dielectric
function of the semiconductor in the structure under study is ε, then
the following two condition have to be satisfied:
1 21 2
z zε ε
∂Φ ∂Φ=
∂ ∂
A, B, C, D, E, F, G H, I, J and K are constants in the potential equations.
when d4≤z< d5
when z≥d5(1)
(2)
From the previous equations we can get the relationship between
the constants:
1 1
1 1
1
2
1
2
1 1
2 2
1 1
(1 )2
(1 )2
1((1 ) (1 ) )
2
1((1 ) (1 ) )
qd qd
qd qd
AB
AC
D Be Ce
E Be Ce
ε
ε
ε
ε
ε ε
ε ε
ε ε
ε ε
−
−
= −
= +
= + + − = − + +
3 2 3 2
3 2 3 2
4 3 4 3
( ) ( )3 3
1 1
( ) ( )3 3
1 1
( ) ( )1 1
3 3
1((1 ) (1 ) )
2
1((1 ) (1 ) )
2
1((1 ) (1 ) )
2
q d d q d d
q d d q d d
q d d q d d
H Fe Ge
I Fe Ge
J He Ie
ε ε
ε ε
ε ε
ε ε
ε ε
ε ε
− − −
− − −
− − −
= + + −
= − + + = + + −
Phonon Potential
(3)
2 1 2 1
2 1 2 1
2 2
( ) ( )1 1
3 3
( ) ( )1 1
3 3
((1 ) (1 ) )2
1((1 ) (1 ) )
2
1((1 ) (1 ) )
2
q d d q d d
q d d q d d
E Be Ce
F De Ee
G De Ee
ε ε
ε ε
ε ε
ε ε
ε ε
− − −
− − −
= − + + = + + − = − + +
4 3 4 3
3 3
( ) ( )1 1
3 3
2
1((1 ) (1 ) )
2
q d d q d dK He Ie
ε ε
ε ε
ε ε− − −
= − + +
And we can also get the secular equation of this system
5 4 5 4
5 4 5 4
( ) ( )
3
( ) ( )
1
q d d q d d
q d d q d d
Je Ke
Je Ke
ε
ε
− − −
− − −
+= −
−
Plug the relationship between these constants into the secular
equation we can then solve it to get the interface phonon modes of
this system
(4)
Phonon Potential
222
2
( ) ( , )1 1( ( , ) )
2 4 2i
i i
i
R
q zdz q q z
L z
ε ω
ω π ω ω
∂ ∂Φ= Φ +
∂ ∂∑ ∫
h
For cubic material, the normalization condition is given by:
In order to calculate the potential of this system, we need to figure out
the constants in the potential equations. So here we will normalize the
potential of this system to get these constants.
(5)
iR
Then the normalization condition becomes:
1 2 1 2 1
3 2 3 2 4 3 4 3
2 2 ( ) 2 ( )2 2 2 2 2 21 2 1
2 ( ) 2 ( ) 2 ( ) 2 ( )2 2 2 23 1
223
( ) ( ) ( )( ( 1) (1 )) ( ( 1) (1 ))
( ) ( )( ( 1) (1 )) ( ( 1) (1 ))
( )( (
qd q d d q d dqd
q d d q d d q d d q d d
q
qA q B e C e q D e E e
q F e G e q H e I e
q J e
ε ω ε ω ε ω
ω ω ωε ω ε ω
ω ωε ω
ω
− − −−
− − − − − −
∂ ∂ ∂+ − + − + − + −
∂ ∂ ∂
∂ ∂− + − + − + −
∂ ∂
∂+
∂5 4 5 4( ) 2 ( )2 1
2
( ) 41) (1 ))
d d q d dK e q
L
ε ω π
ω− − − ∂
− + − − =∂
h
Plug the relationship between these constants into the normalization
condition we can get a equation with one unknown A, then we can solve
it to get constant A. As long as we know A we can calculate the rest
constants.
(6)
Results
SeGi Yu, K. W. Kim, Michael A. Stroscio, G. J. Lafrate, J,-P. Sun et al, JAP, 82, 3363 (1997)
GaAlAs/GaAs material system
We calculate the parameters we need
Phonon modes
(meV)
Ga0.452Al0.548As
(AlAs-like)
GaAs Ga0.741Al0.259As
(GaAs-like)
LO 48.44 36.25 34.67
TO 44.83 33.29 33.046
SeGi Yu, K. W. Kim, Michael A. Stroscio, G. J. Lafrate, J,-P. Sun et al, JAP, 82, 3363 (1997)
Results
Interfaces phonon modes
at q=1e8 (wavevector)
IF Phonon modes (meV)
33.38808 44.3023
33.8125 44.9045
34.193 46.278
34.6304 47.1212
35.57657 48.038603
Dispersion curve
Phonon Potential
Phonon Potential
33.8125 meV34.193 meV
33.38808 meV
34.6304 meV
Phonon Potential
44.9045 meV
47.1212 meV
44.9045 meV
46.278 meV
35.57657 meV
For InxGa1-xAs
InAs-like GaAs-like
Results
2
2
242.99 32.54 4.545
253.97 67.91 51.94
x x
x x
− +
− +
TO
LO
2
2
268 62 220
291 59.167 152.7789
x x
x x
− +
− +
TO
LO
InGaAs/InAs material system
Then, we calculate the parameters we need
Phonon modes
(meV)
In0.248Ga0.752As In0.59Ga0.41As InAs
LO 35.32 28.746 29.74
TO 32.89 27.93 27.01
ε∞
11.526 11.287 11.7
Results
Interfaces phonon modes
at q=1e8 (wavevector)
IF Phonon modes (meV)
29.542 33.6199006
30.351 34.72185
32.7285 35.1522
Results
29.542 meV
34.72185 meV 35.1522 meV
29.542 meV
30.351 meV32.7285 meV33.6199006
meV
For InxAl1-xAs
AlAs-like InAs-like
Results
2
2
361.5 24 9.5
401.5 55 20
x x
x x
− −
− −
TO
LO
2
2
229 22 13
229 22 9
x x
x x
+ −
+ −
LO
TO
InAlAs/InP material system
Then, we calculate the parameters we need
Phonon modes
(meV)
In0.36Al0.64As InP In0.61Al0.39As
LO 46.977 42.75 29.16
TO 43.57 37.63 29.23
ε∞
9.4344 9.61 10.32
Results
IF Phonon modes (meV)
Interfaces phonon modes
at q=1e8 (wavevector)
29.18687 44.314
38.383165 45.15
40.99825 45.8385
43.679
GaAlAs Design
GaAs/Ga1-xAlxAs
•Band Gap, Eg=(1.426+1.247x) eV
•Band alignment: 33% of total discontinuity in valence band, i.e.
∆VVB=0.33; ∆VCB=0.67
•Electron effective mass, m*=(0.067+0.083x)m0
From Quantum Wells, Wires and Dots (Paul Harrison)From Quantum Wells, Wires and Dots (Paul Harrison)
InGaAs Design
�
�
InAlAs/InP Design
From Appl. Phys. Lett. Vol. 58, No. 18, 22 April 1991 (Mark S.
Hybertsen)
�
early
early
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