1

2 Photons In, Electrons Out : Basic Principles of PV

2.1. Introduction

Efficiency : band gap, incident spectrum

Another way to look at it → Energy loss

2.2. The Solar Resouce

Black body radiation @ 5760K peaking in blue-green visible 400-800 nm

Spectral photon flux ,,, sEs : number of photons per unit energy, area, time

2m

W

dEdde

E

chdEddE

sBTkEs SSs

1

2,,,

2

23 (2.1)

Sd : the element of surface area around s

d : unit solid angle around the direction of emission of light ,

Ref. Planck’s Irradiation Formula (Statistical Mechanics on Thermal Photons)

Energy density of photons within frequency d~ from blackbody is d)(

d

ecd

TkB 1

3

32

Therefore, the density of photons within frequency d~ is

ddn

)()(

d

ecdn

TkB 1

1 2

32

Converting this into energy density of photons within energy dEEE ~ using E ,

2

de

E

cdn

TkB 1

1 2

322

dEe

E

cdEE

TkE B 1

1 2

323

--------End Planck’s Irradiation---------------------------------------------------------------------

Total flux issued normal to the surface ,, sEbs

dEde

E

ch

FdEddEEb

sBTkE

s

ss SSss

1

2cos,,,,

2

23 (2.2)

sF : geometrical factor based on solid angle

sF : at surface

ssF 2sin : at distant (2.3)

For object whose temperature is the same all over

1

2 2

23 sBTkE

ss

e

E

ch

FEb (2.2a)

Emitted energy flux density (irradiance) EL

EbEEL s (2.4)

Integrating (2.4) over E gives total emitted power density ssT where

428

32

45

/1067.515

2KmW

hc

kBs

(2A.1)

is called Stefan-Boltzmann constant.

------------------------------------------------------------------

[Question] Derive that the total energy flux density can be given as

4

ssTj (2A.2)

6.95x108 [m]

1.49x1011 [m]

Earth Sun 0.26°

3

(Statistical Mechanics)

-----------------------------------------------------------------

Spectrum of light arriving to the earth from the sun

Modified black body radiation due to absorption by gas and scattering

Air Mass factor : Attenuation by atomosphere

H2O : 900, 1100, 1400, 1900 nm

CO2 : 1800, 2600 nm

s

sAirMass ecoverheaddirectlySuniflengthpathoptical

Suntolengthpathopticaln

sin

1cos (2.5)

s : angle of elevation of the sun (Fig.2.2)

AM0 : in space

AM1.0 : Sun overhead

AM1.5 : Sun @42°

The lower the sun, the higher the AM

4

AM1.5 : spectrum normalized → integrated irradiance 1000 W/m2 .

Annual average in Fig.2.3

2.3. Types of Solar Energy Converter

1. Photovoltaic eg. semiconductor photo carrier, separation

2. Solar thermal thermal radiation

3. Photochemical eg. photocatalysts, photosynthesis

2.4. Detailed Balance

2.4.1. In equilibrium

Cell in dark: Assuming that the ambient is a black body with temperature aT ,

the photon flux spectrum provided to the cell at point s on the surface is from eq. (2.1) is

dEdde

E

chdEddE

aBTkEa SSs

1

2,,,

2

23 (2.1a)

5

Integrating over all directions ( Sdd ), following eq. (2.2), the incidence flux of thermal photons

normal to the surface of a flat plate solar cell is

1

2 2

23 aBTkE

a

ae

E

ch

FEb (2.6)

where aF assuming the ambient radiation is received over a hemisphere.

(and also following the argument for eq. (2.3))

Equivalent current density abosorbed from ambient (per unit area)

EbEaERqEj aabs 1 (2.7)

Ea : absorbance of photon at energy E

ER : reflectance of photon at energy E

Ejabs : photon current density at energy E

If rear surface contact the air, double (2.7), for area A

EbEaERqAEj aabs 12 (2.7a)

If rear surface contact material with refractive index sn

EbEaERqAnEj asabs 11 2 (2.7b)

If rear interface is a perfect reflector,

EbEaERqAEj aabs 1 (2.7b)

Spontaneous Emission

Cell at equilibrium temp aT emits thermal photons

Emissivity : probability of emission of a photon with energy E

6

Equivalent current density of photon emission through surface of cell (per unit area)

EbEERqEj arad 1 (2.8)

To reach a steady state, (2.7) and (2.8) must balance. Therefore,

EaE (2.9)

This is called Kirchhoff ’s law of thermal radiation.

Detailed balance

2.4.2. Under illumination

Cell under illumination by a photon flux Ebs absorbs photons of energy E at rate

EbEaER s1

Note that

(Total photon flux absorbed) = (Solar flux in) + (Thermal flux in) - (Thermal flux out)

Eb

F

FEbEaERqEj a

e

ssabs 11 (2.10)

Illumination → rise in chemical potential 0 → increase in spontaneous emission

(rise in chemical potential → rise in Fermi level 0 )

If the medium (cell) has a refractive index of sn and photons are emitted into ambient 0n

1

2,,,

2

23

aBTkE

e

E

chE

s 0nns (2.11)

By integrating over the angle in which the photons can be emitted is c 0

1

2,

2

23

2

aBTkE

see

e

E

ch

nFEb

(2.12)

where 2

2

02sins

cen

nF (2.13)

and

s

cn

n01sin (Snell’s law).

7

If the ambient is air, 10 n ∴ ase FnF 2 Therefore,

1

2,

2

23

aBTkE

ae

e

E

ch

FEb

(2.14)

For photon emission, the equivalent current density following eq. (2.8) is

,1 EbEERqEj erad (2.15)

Unlike equilibrium (steady) state

EaE (2.9)

is not necessarily valid, but is true when is constant throughout the device.

Net equivalent current density

Eb

F

FEbEaERqEj a

e

ssabs 11 (2.10)

,1 EbEERqEj erad (2.15)

(2.10) – (2.15) (the difference)

,11 EbEb

F

FEbEaERqEjEj ea

e

ssradabs (2.16)

Separate contributions from absorption and emission.

a) Net absorption (in excess to that of equilibrium)

Eb

F

FEbEaERqEj a

e

ssnetabs 1 (2.17)

b) Net emission / radiative recombination current density

0,,1 EbEbEaERqEj eenetrad (2.18)

where 0,Ebb ea

ns

n0

s

0

ns > n0

8

Radiative recombination results in loss of SCs. (ref. Chap. 4)

2.5. Work Available from a Photovoltaic Device

Photons

gEE

can contribute (Fig.2.6)

2.5.1. Photocurrent

In eq. (2.17)

Eb

F

FEbEaERqEj a

e

ssnetabs 1 (2.17)

2nd term can be neglected since angular range of sun is small

If each electron has a probability of Ec of being collected (at an electrode), photocurrent

density is

0

1 dEEbEaEREqJ scSC (2.19)

Compare with

dEEQEEbqJ SSC (1.1)

For perfectly absorbing, non-reflecting cells, all photons gEE are absorbed to generate one

electron to the upper band, and also no recombination takes place, i.e., 1Ec , then

9

g

g

EE

EEEaEQE

0

1 (2.20)

gE

sSC dEEbqJ (2.21)

2.5.2. Dark current

Current that flows through a PV device when a bias is applied in the dark.

・neglect loss through non-radiative recombination

・the only loss : spontaneous emission

Equivalent current density of photon emission through surface of cell (per unit area)

EbEERqEj arad 1 (2.8)

where Eba is the incidence flux of thermal photons normal to the surface of a flat plate SC

1

2 2

23 aBTkE

a

ae

E

ch

FEb (2.6)

For considering the net equivalent current density, it has been discussed in §2.4.2 that

0,,1 EbEbEaERqEj eenetrad (2.18)

( : chemical potential built up in the cell upon illumination)

10

[Ref. Sze, “Semiconductor Devices, 2nd ed., Wiley, 2002, Chap. 4. If the SC consists of a p-n

junction, which often is the case. Forward bias is defined as positive in Nelson.]

Integrating (2.18) over photon energy for a flat cell with perfect rear reflector

dEEbEbEERqdEEjEJ eenetradrad 0,,1 (2.22)

Once again, Kirchhoff ’s law of thermal radiation is taken into account.

EaE (2.9)

NB. The textbook does not really state to use (2.18) instead of (2.8)

In ideal material, one can assume that

qV

is constant throughout the material.

The overall net current is the sum of photocurrent and dark current. Therefore,

VJJVJ darkSC

Hence, the net cell current density is

11

dEEbqVEbEbEaERqVJ ees 0,,1 (2.23)

If the absorption occurs as a step-like function (the case of semiconductor),

g

g

EE

EEEaEQE

0

1 (2.20)

Then,

gE

ees dEEbqVEbEbqVJ 0,, (2.24)

Since qVEbEb ee ,, is expressed as

1

2

1

2,

2

23

22

23

2

aBaB TkqVE

seTkE

see

e

E

ch

nF

e

E

ch

nFEb

(2.12’)

(2.24) actually turns approximately into

10 TkqV

SCBeJJVJ 0J : constant (T dependent)

(Net current) = (Absorbed flux) – (Emitted flux)

gEE gE~

When V increases, the emitted flux increases and the net current decreases

When OCVV , LED.

Note that q

EV

g

OC

12

2.5.3. Limiting efficiency

Calculating the power conversion efficiency

Incident / extracted power from photon fluxes

Incident power : from eq. (2.4), emitted energy flux density (irradiance) EL

EbEEL s (2.4)

Integrating over photon energy

0

dEEbEP ss (2.25)

Output power : in ideal photoconverter, all electrons should have potential qV

13

Together with JVP (1.7)

VVJP

where VJ is given by (2.24)

gE

ees dEEbqVEbEbqVJ 0,,

Therefore, the power conversion efficiency is

ss P

VVJ

P

P (2.26)

Maximum efficiency achieved when

0VVJdV

d (2.27)

The maximum power bias mV introduced in §1.4.3 is the solution.

(Ref. Fig.2.7(b))

2.5.4. Effect of band gap

Using a single semiconductor, the efficiency is determined by the band gap.

1) Only photons with gEE contribute

14

2) Regardless of photon energy gEE , only will be delivered, i.e.,

E

: efficiency

2.5.5. Effect of spectrum on efficiency

Let’s consider how to increase the efficiency.

Assume a blackbody source KTs 5760

At AM0, eVEg 3.1 gives maximum conversion efficiency of 31%.

In the simplest model, the influence of the cell temperature aT

0 as TT

max 0aT

Here, no radiative current, and hence the optimum operating voltage is q

EV

g

Then,

0

max

dEEEb

dEEbE

s

Esg

g

Which has a maximum of ca. 44% at a band gap of 2.2 eV for KTs 6000

An alternative: increase the angular width of the sun → Details in Chap. 9

%37max for eVEg 1.1 predicted

2.6. Requirements for the Ideal Photoconverter

Energy gap

GaAs 1.42 eV InP 1.35 eV Si 1.1 eV

CdTe 1.44 eV CuInGaSe2 1.04～1.67 eV

Light absorption

Optical depth (usually O.K.)

Charge separation

Electric field or gradient in electron density

Lossless transport

Defects, impurities

No resistive loss, current leakage

15

p-n junction in Chap. 6.

Optimum load resistance

Circuit consideration

Present Challenges

1) Incomplete absorption of incident light

2) Non-radiative recombination – trap sites (defects)

3) Voltage drop due to series resistance, i.e., qV

2.7. Summary