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

M.Pereira@shu.ac.uk

Outline

• The Solar Paradox• Challenges for next generation solar cells• Nonequilibrium Green's Functions

approach to absorption and gain• ISB Thermophotovoltaics• Summary

Solar Potential

Average power > 100 W/m2 in populated areas

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

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

Third Generation PV Challenges

III-V Multi-Junction Solar Cell

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

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

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

TPV Structures

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).

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)

• 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

• Other GF's complete the picture

• Spectral function

• Lehman representation for the retarded GF

tikk etdtikG

)()0(),( *

),(),(),(ˆ kGkGikG

i

kGdkG ret

'

)',(ˆ

2

'),(

• 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

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)

Projected Greens Functions Equation - Intersubband

Correlation Contribution

Dynamically screened, nondiagonal and frequency dependence dephasing mechanisms are described.

Gain/Absorption Calculated through the Optical Susceptibility (Imaginary Part)

)'()'(),(

)()(),'()(),(),()(

1

''

''

knkMk

knkVkknkkike

kkkk

kkkk

)}({4

)( mcnb

),()(2

)(,

kkV k

Conduction x Valence Subband Structures

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

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)

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

Interplay of Irradiance and QW ISB Absorption

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

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

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

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.