Microscopic Theory of Intersubband Thermophotovoltaics Mauro F. Pereira
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Microscopic Theory of Intersubband
Thermophotovoltaics
Mauro F. Pereira
Theory of Semiconductor Materials and Optics
Materials and Engineering Research Institute
Sheffield Hallam UniversityS1 1WB Sheffield, United Kingdom
Department of PhysicsJazan University, Jazan, Saudi Arabia
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Outline
• The Solar Paradox• Challenges for next generation solar cells• Nonequilibrium Green's Functions
approach to absorption and gain• ISB Thermophotovoltaics• Summary
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Solar Potential
Average power > 100 W/m2 in populated areas
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The Solar Paradox• Infinitely abundant energy
– Fusion reactor– Solar constant: 1360 W/m2 (CN@6000K)– Surface incidence: ~ 1000 times the need
of primary energy– Sub products at the origin of > 90% of
commercial energy
• A resource negligibly exploited for energy production
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Conventional PVs - Problems to be Solved
• Light with Energy below Eg will not be absorbed
• Excess photon energy above Eg is lost in form of heat
• Possible solutions :– multi-junction– Intermediate bands– hot carrier solar cells– TPVs
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Third Generation PV Challenges
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III-V Multi-Junction Solar Cell
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Challenges - Multi-juntcion and IB
Further microscopic analysis is required with full quantum transport and optics - NGF method is ideal!
Slide courtesy of S. Tomić
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Challenges - Hot Carrier PVs
Energy Loss Mechanisms
• Heat transfer to lattice (LO Phonon emission)
• Heat leakage to contacts as they are extracted from the absorber
• The NGF method used for complex QCL structures is ideal to address those difficulties
Solutions sought• Nanostructures to
reduce cooling rate due to phonon emission
• Energy selective contacts allowing carrier transmission at a single energy level - however difficult to achieve good selectivity and high current densities
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Thermophotovoltaics Convert IR radiation (heat) into electricity• technology very closely related to MJPV
Many potential applications• Portable, low emission generators for military and civilian
use• Generation from ‘waste’ industrial heat• Domestic boilers• Automotive industry
Market size (2000 estimate) $85 – 265 million possible for non-auto
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TPV Structures
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Calculated Photocurrent
• photon flux at 1 sun and 1.5 am
• photocarrier generation at depth z
• photocurrent
1
423 1)exp(105.3)(
sourceB Tk
hcF
))(()()(1)(),( nzzn eFRzG
nn
n
dz
z
nn dzzGqJ ),()(
Ref: V. Aroutiounian et , J. Appl. Phys. 89, 2268–2271 (2001).
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Dyson equation solvers for realistic structures Many Body + Nonequilibrium +
Bandstructure engineering
= +G 0G 0G G
)2()1()12( iG
Theoretical Approach to obtain the microscopic optical response - Nonequlibrium Keldysh Greens
Functions (NGF)
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• Both coherent transport and scattering described on the same microscopic footing with Green's functions.
• Relation to the (single particle) density matrix
• The GF's contain more information than conventional semiconductor Bloch equations derived directly from
tikk etdtikG
)0()(),( *
dkGik ),()(
)(k
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• Other GF's complete the picture
• Spectral function
• Lehman representation for the retarded GF
tikk etdtikG
)()0(),( *
),(),(),(ˆ kGkGikG
i
kGdkG ret
'
)',(ˆ
2
'),(
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• electron-electron selfenergy GWapproximation
• impurity scattering selfenergy second born approximation
• interface roughness second born approximation
)','()','(
2
'
2
'),( 2
2
kGkkW
dkdk
),'()'()'(
22
'),( 2
2
EkGqkkWqkkWdqkd
nk zs
zsz
imp
),'()'()'(
2
'),( 2
2
EkGkkVkkVkd
k roughrough
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initial guess:
Gret(ω,k,α,β)G<(ω,k,α,β)
evaluate:
Σret(ω,k,α,β)Σ<(ω,k,α,β)
evaluate anew:
Gret(ω,k,α,β)G<(ω,k,α,β)
evaluatecurrent densities
populations
converged?
yes
no
new guess:
Gret(ω,k,α,β)G<(ω,k,α,β)
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Projected Greens Functions Equation - Intersubband
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Correlation Contribution
Dynamically screened, nondiagonal and frequency dependence dephasing mechanisms are described.
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Gain/Absorption Calculated through the Optical Susceptibility (Imaginary Part)
)'()'(),(
)()(),'()(),(),()(
1
''
''
knkMk
knkVkknkkike
kkkk
kkkk
)}({4
)( mcnb
),()(2
)(,
kkV k
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Conduction x Valence Subband Structures
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Summary of the Numerical Method
• Solve the 8 × 8 K∙P Hamiltonian for QWs• Solve the selfconsistent loop for the
selfenergy and G< (occupation functions).• Solve the integral equation for the
polarization by numerical matrix inversion• Calculate the absorption• Calculate the semiclassical photocurrent
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ISB Thermophotovoltaics
TE Mode Tsource = 1000 K• (a) 5 nm QW• (b) 10 nm QW• solid: many body effects• dashed: free carriers• bottow and top curves in each
panel: 1 and 3 × 1012
carriers/cm2
• extra features on absorption due to a combination of nonparabolicity and many body effects
M.F. Pereira, JOSAB 28, 2014 (2011)
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ISB Thermophotovoltaics
TM Mode - carriers at 300K Tsource = 1000 K
• (a) 5 nm QW• (b) 10 nm QW• solid: many body effects• dashed: free carriers• bottow and top curves in each
panel: 1 and 3 × 1012
carriers/cm2
• Strong redistribution of oscillator strength due to many body effects
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Interplay of Irradiance and QW ISB Absorption
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ISB Thermophotovoltaics
TE Mode - carriers at 300 K Doping: 3 × 1012 carriers/cm2
• (a) 5 nm QW• (b) 10 nm QW• solid: many body effects• dashed: free carriers• bottow and top curves in each
panel: Tsource= 500 and 1000 K
• If the peak flux overlaps with certain spectral regions, the many body effects are highlighted
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ISB Thermophotovoltaics
TM Mode - carriers at 300 K Doping: 3 × 1012 carriers/cm2
• (a) 5 nm QW• (b) 10 nm QW• solid: many body effects• dashed: free carriers• bottow and top curves in each
panel: Tsource= 500 and 1000 K
• If the peak flux overlaps with certain spectral regions, the many body effects are highlighted
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ISB Thermophotovoltaics
Doping: 3 × 1012 carriers/cm2
• (a,c) 5 nm QW• (b,d) 10 nm QW• solid: many body effects• dashed: free carriers
• (a,b) Tsource= 500
• (c,d) Tsource= 1000 K
• There is a region in the far infrared where TE > TM even without considering projection losses on TM, which are unavoidable.
TE vs TM (max) Mode in the far infrared - carriers at 300 K
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ISB Thermophotovoltaics
• There is a region in far infrared where the TE mode that does not require prisms and couplers dominates even though the MIr dipole is much larger for TM.
• Many-body corrections are important if high densities are reached - hot carrier devices???
• Full nonequilibrium required for hot carriers - forthcoming.