Ennaoui cours rabat part ii

Photovoltaic Solar Energy Conversion (PVSEC)
‫إﻧﺘﺎج اﻟﻜﻬﺮﺑﺎء ﻣﻦ اﻟﻄﺎﻗﺔ اﻟﺸﻤﺴﻴﺔ‬
‫ﻴ‬ ‫ا‬ ‫إ ج ا ﻬﺮﺑ ء ﻦ ا‬

Courses on photovoltaic for Moroccan academic staff; 23-27 April, ENIM / Rabat
23 27

Ingot
PVSEC-Part
PVSEC P t II crystal
Fundamental and application of Photovoltaic solar cells
and system
Ahmed Ennaoui Wafer
Helmholtz-Zentrum Berlin für Materialien und Energie
ennaoui@helmholtz-berlin.de

Solar cell

Highlight:Photovoltaic Solar Energy Conversion (PVSEC)
Highlights
Basic of solar cells and Modules
Light absorption and band to band transition
g p
Quantum efficiency and absorption coefficient
Generation and recombination processes
Shockley-Read Hall Recombination (SRH)
Continuity equation and Transport process
Silicon to binary and ternary compounds
From silicon solar cell as example of PN j ti
F ili l ll l f junction
Performance of solar cells
Equivalent Circuit model: series (Rs) and shunt resistance (Rsh)
Change in cell performance with Rs and Rsh
Change in short circuit current and open-circuit with solar radiation
Change in short circuit current and open-circuit with the temperature
open circuit
Performance measurement standard conditions
Prof. Ahmed Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Basic of solar cells and Modules

Sun has roughly T = 5800 K Solar cell has roughly T = 300 K

Two basic functions of a solar cell
1.
1 Light absorption: generation of free excess charge carriers
photocurrent, I Power
2. Charge separation: separate/extraction of excess electrons and holes IxV
p
photovoltage, V
g ,
Conversion of the Sun light in the „Black Box“
• To absorb the solar spectrum as efficient as possible
• To collect photogenerated charge carriers
• Charge transport must be possible
• To make electron go to one side and holes to another current flow
Ahmed Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Basic: Task of Photovoltaic
Key aim is to g
y generate electricity from solar spectrum
y p
Power = Voltage x Current J [A/cm ] 2
. J xV
. (J ,V )
m m
[Watt/cm2] [Volt ] [A/cm2] Jm m m

Two challenges
Generating a large current. V [Volt ]
Generating a large voltage.
Vm

High current. High voltage
But low voltage But low current
E
Excess energy l t t h t
lost to heat Sub-band gap light is lost


Basic: Absorption-Separation-Collection
Photons absorbed Electron flow Electrical current
Photon flux gives number of photons/unit time/unit area/wavelength
Φ(λ) = Φ 0 .exp(−α λ x) ⎯R(λ )→
⎯⎯ Φ(λ) = Φ 0 (λ).(1 − R λ ).exp(−α λ x)

Electrons collected
dΦ Load
G(x) = − = αΦe−αx
dx dp
J = σE Dp
dx

ceptor
P = Voltage x Current
Voltage Δ = μe – μh μe
V lt Δμ

Acc
μ = chemical potential
Rec Voc
or

μh
Dono

0 W Ln La= 1/α


Basic: Quantum efficiency
• Photoccurrent = how much light converted? This ratio can be measured
Maximum short circuit current Electrons collected / Photons absorbed
• Limited information on the electronic properties
• Information on the optical properties of the device
electrons
N out =
J A/cm 2 [ ]
e[Coulomb]
hc 1239
hν = ⇒ EG (eV ) =
λ λ (nm) N photons
=
[
Φ Watt/cm 2 ]
hν [Joule ]
in

Load
L d
External Quantum Efficiency, EQE

cceptor
dp
J = σE D p N electron 1 J ( λ ) hc
dx EQE = photons =
Q out
N in Φ (λ ) e λ

Ac
→
→
∇p μe
E Internal Quantum Efficiency
EQE
Rec
R Voc IQE =
1 − R (λ )
μh Origine of the photovoltage
or

Chemical potential
Dono

x=0 La= 1/α
x=W x = Ln
EF,n = μe EF,h = μh


Basic: Quantum efficiency measurements
current/charge of 1 electron
External Quantum Efficiency" EQE" =
EQE
Total power of photon/energy of 1 photon

Beam splitter

Monochromator
Chopper equipped with more gratings*

EQE vs. λ

EG

*Gratings should have line density as high as possible for achieving high resolution and high
power throughput. (600 – 3000 lines/mm).

1 - Reference measurement 2 – Cell Measurement
3 – Final Result
J sc = q.EQE REF .Φ1
REF
J sc = q.EQECELL .Φ2
CELL

CELL MON,1
J sc J sc
EQE CELL = MON,2 . REF EQE REF
J MON,1
sc = q.EQE
q Q MON .aΦ 1 J MON,2
sc = q.EQE
q Q MON .aΦ 2 J sc J sc
MON,1 CELL
J sc J sc
EQE MON .a = EQECELL .a = MON,2 .EQEMON .a
qΦ 1 J sc

Basic: EQE and and absorption coefficient
( )
E(k) p
Photon absorption in a
direct band-gap
e.g. GaAs Conduction
Band
semiconductor
B
α= (h ν − E G ) 2
EC hν
1
Direct Bandgap Eg
EV
Photon
(α .h ν )2 vs. (hν − E G ) → E G

Valence
Band
Cut-off λ vs. EG
1.24
-k +k λ G [μm] =
E G [eV]
J sc = q ∫ Φ (λ ) EQE (λ ) dλ
E(k) λ

e.g. Si Conduction Photon absorption in an
Band indirect band-gap
semiconductor
1
EG+Ep
Phonon EC A
Eg α = (hν − EG ) 2
Photon
EV hν
Valence
Ep
(α .h ν )2 vs. (h ν − E G ) → E G
Band

-k +k Ahmed Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Basic: absorption coefficient and absorption length
Light absorption
E G (T) = E g (0) − A.f(T)
Si Ge GaAs
EG (eV) 1 12
1.12 0.66
0 66 1.42
1 42
Temperature changes:
EG ↑ as T ↓, Changing the absorption edge

Absorption ↔ Generation
hν Φ = Φ0 .(1 − R λ ).e−αx
1 Φ
Φ0(E) α λ = − . ln
d Φ0 .(1 − R λ )
ΦA(E)
Φt(x)
dΦ
G(E, x) = − = Φ o (E).(1 - R).α). -αx
dx
Φ0 = ΦR + ΦA + ΦT
Depth x
Φ
Surface x =1/α
Φr(E) 100% = R λ + A λ + Tλ

Basic: absorption coefficient and absorption length

100 nm


Basic: improuvement, Light traping

Influence of the layer thickness on the photocurrent of Si

Realization:
• Etching and texturing of semiconductors.
• Implementation of particles for scattering
deposition on rough or structured surfaces
d ii h d f


Basic: Challenging parameters
Important cost factor
All device parameters Reflection Loss
Material parameter €

J sc = q . ∫ [η(λ) . (1 − R λ ).Φ 0 (λ ) . exp - α λ .d ]dλ
η( ) ( ) p
EG Decisive Material parameter Light trapping
η(λ) =↑ % EQE or η or IPCE - incident photon to electron conversion efficiency)
p y)
1 J ph hc
EQE = .
Φ(λ) q λ

Resistive loss

Reflection loss Top contact
“loss”
loss
Recombination
loss

Back contact
„Loss“


Basic: Close look to EQE

1 J ph (λ ) hc
η(λ) = EQE = .
Φ(λ) q λ

(2) Losses due to reflection (3) Losses due to rear surface
and low diffusion length passivation and reduced
absorption at long wavelengths
and low diffusion length

(4) Complete loss due to
missing absorption

(1) Losses due to front surface
recombination and absorption Wavelength at 1.24
λ G [μm] =
in passivation and antirection the band gap E G [eV]
coating layers


Basic: PN junction Loss in Jph
Good surface passivation. Texturing in the form of pyramids so that
Antireflection coatings. light is trapped at the surface (<60nm)
Low metal coverage of the top surface.
Light trapping or thick material
(but not thicker than diffusion length).
High diffusion length in the material.
Junction depth optimized for absorption
in emitter and base.
Low reflection by texturing

Resistive loss

Reflection loss Top contact
“loss”
“l ”
Recombination
loss

Back contact
„Loss“

Generation vs. recombination processes
Generation (g) requires an input of energy given to an electron:
gy g
- Phonons - vibrational energy of the lattice
- Photons - Light, or electromagnetic waves
- Kinetic energy from another carrier (Impact ionization )
Electron
El t
EC thermalizes
Generation
to band edge
Ekin K.E. = E − EC
EV EC
Generation energy > EG
Ekin energy = EG
energy < EG

EV
- Impact ionization
The electron hits an atom, and break a covalent bond to generate an
electron-hole pair, if the kinetic energy is larger than the energy needed
to
t generate th pair. Th process continues with th newly generated
t the i The ti ith the l t d
electrons, leading to avalanche generation (e-h).


Generation vs. recombination processes
Recombination (r) is the opposite of generation, leading to voltage and current loss.
Non-radiative recombination phonons, lattice vibrations.
Radiative recombination photons (dominating in a direct bandgap materials )
Auger recombination charge carrier may give its energy to the other carrier.
Recombination processes are characterized b th minority carrier lif ti τ.
R bi ti h t i d by the i it i lifetime
Equilibrium: charge distributions np = ni2
Out of equilibrium: The system tries to restore itself towards equilibrium through R-G
Steady-state rates: deviation from equilibrium
y q
r = B .pn ⎫
(
⎬ R = r − g = B pn − ni
2
) B(Si) = 2 × 10 −15 cm 3 /s
g = B.p0 n0 = Bni2 ⎭
Electron thermalizes
to band edge

bination
xcess energy given
rrier in
EC EC
y

th same band d
EC Auger recomb

to another car
Radiative
Non-radiative
E(eV)

recombination
recombination
Phonon he
Ex
o
A

EV EV EV


Shockley-Read Hall Recombination (SRH)
The impurities create deep-level-traps (ET) within the bad gap
The electron in transition between bands passes through ET EC
(1)+(3): one electron reduced from Conduction band (1) (2)
and one‐hole reduced from valence‐band and ET
(2)+(4): one hole created in valence band and (3) (4)
one electron created in conduction band EV
Steady state
Steady-state rates: R = A (np-ni2) = deviation from equilibrium:
(np n
n, p and NT inside Δx are held constant by the balancing effect of distinct different process
np − n i2
R=
⎛ 1 ⎞ ⎛ ⎞
⎜ ⎟(n + n1) + ⎜ 1 ⎟(p + p1) cp,n: capture coefficient of the recombination process
⎜c N ⎟ ⎜c N ⎟
1⎝ p T ⎠ ⎝ n T⎠ NT: density of the recombination levels.
τp = ↑ 1
↑ τn = σn,p: capture cross sections for e and h.
σ p v t ,h N T σ n v t ,n NT ET: energy levels inside the energy gap.
( ET −Ei ) − ( E T −Ei )
vth: average thermal velocity of e and h.
pT, nT: number of empty states available
n1 = n i e k BT
p1 = n i e k BT
n, p: number of electrons or holes
n1 , p1: number of electrons and holes at ET
Low level i j ti
L l l injection
Δn Δp
n - type material R SRH = p − type material R SRH =
τn τp

Summary: Generation & Recombination
Auger recombination
(dominant effect at high carrier concentration) Ekin= -qELsc
q
EC
Direct Loss to thermal
Shockley-Read Hall recombination vibrations Ekin
recombination direct band EV

Impact ionization is a
generation mechanism.
⎛ 1 1 1 ⎞ When the electron hits an
R = RSRH + RDirect + RAuger = Δn⎜ + + ⎟
atom, it may break a
⎜τ ⎟
⎝ SRH τ Direct τ Auger ⎠
covalent bond to generate
(
= Δn cn NT + BN D + cn,Auger .N D
2
) an electron-hole pair.

(
⇒ τ eff = cn NT + BN D + cn,Auger .N
N D )
2 −1 The process continues with the newly
generated electrons leading to avalanche
electrons,
generation of electrons and holes.
τ : average time it takes an excess minority carrier to recombine
(1 ns to 1 ms) in Si
τ τ : depends on the density of metallic impurities and the density
t/teff of crystalline defects.

What we have learned?

Photo absorption and photo generation, Direct and indirect band gap, EQE, IQE
absorption coefficient, absorption length, excess minority carrier , carrier lifetime
Recombination: Non Radiative, Radiative, Auger
Shockley-Read Hall Recombination (dominant process in Si)
There are wide variety of generation‐recombination events that allow restoration of
equilibrium once the stimulus is removed.
Direct recombination is photon‐assisted, indirect recombination phonon assisted.
Recombination lifetime in Si is controlled by Auger recombination at high carrier
concentration
Recombination life time in Si is controlled by SRH at low carrier densities
and depends on the amount of impurities and defects.

http://en.wikipedia.org/wiki/Main_Page Ahmed Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Basic: Continuity equation and Transport process
Continuity equation for minority carriers:
y q y
∂ ( A . Δx . n ) A . J n (x) - A . J n (x + dx)
= + A . g n . Δx − A . rn . Δx
∂t −q

Φ = Φ0 .(1 − Rλ ).exp(−αx)
Light flux

∂n J n (x) - J n (x + dx) 1
= + g n − rn = ∇.J n + g n − rn
∂t − q.Δx q
∂
∇ ⋅ (∇ × H ) = ∇ ⋅ J cond + ∇⋅D = 0
∂t
( )
∇ ⋅ Jn + J p +
∂ρ
∂t
= 0, ρ = q( p − n + N D − N A )

Maxwell ⎧ ∂n = 1 ∇ ⋅ J + G − R
⎪ ∂t q n n n
⇒ ⎨
∂p 1
⎪ = − ∇ ⋅ J p + Gp − Rp
⎩ ∂t q

rain Evaporation
p

In flow

Rate of Out flow
dn
increase of = (in flow – out flow) + Rain - Evaporation
water level
t l l dt 1
in lake = ∇ .J n + gn - rn
q

∂n 1 ∂p 1
= ∇ .J n + g n - rn = ∇ .J p + g p - r p
∂t q ∂t q
J n = qnμn E + qDn∇ n J p = qnμ p E + qD p ∇ p

Carriers are collected when they are:
Generated closer t th j ti
G t d l to the junction
Generated within a diffusion length of the junction ∂n
=0
Key parameters for high collection are: ∂t
Minority carrier diffusion
y
Surface recombination
Difficult to achieve high collection near front surface and also rear
Differential equation is simple only when G = constant.
d 2 Δn Δn G(λ( x) d 2 Δp Δp G(λ( x)
2
= 2
− 2
= 2 −
dx Ln Dn dx Lp Dp
−x +x
Δn(x) = Aexp + Bexp + Gτ n ← Bondary conditions
Ln Ln

Acceptor
p(-αx)
ΕF,n=μe
Donor

ΕF,p=μh
Φ= 0(1-R)exp
D

A
Acceptor

Rec Rec Voc
=Φ

or

ΕF,p=μh
A

ΕF,n=μe
Dono

La= 1/α Lp W 0 0 W Ln La= 1/α

Basic Diode J-V equation
dΔp n
Applying the same boundary conditions as in the ideal diode case
case. J p = qD p
dx
Differentiating to find the current
dΔn p
Equating the currents on the n-type and p-type sides, we get: J n = qD n
dx
⎛ Dn Dp ⎞
J= ⎜q n p,0 + q p n,0
V
⎟ (exp qV − 1) − qG(L + L + W)
⎜ L Lp ⎟ k BT n p
⎝ n ⎠
J0 Photocurre nt J L

- JL + JD ⎛ qV
⎞
J = J 0 ⎜ exp n.k B T
− 1⎟ − J L
⎜ ⎟
⎝
1 4 42 4 43 ⎠
Dark current, J D

J0 : saturation current
kB : Boltzmann`s constant, 1.381 10-23 J/Kelvin
n : ideality factor
⎡ D n n i2 D p n i2 ⎤
J0 = q⎢ + ⎥ ni: carrier concentration
⎢ Ln N A L p N D
⎣ ⎥
⎦ NA,ND. Doping concentration


Silicon (Diamond) to Chalcopyrite (Tetragonal)
Diamond structure IV nq N + mqM
Grimm Sommerfeld rulea
Grimm-Sommerfeld-rulea =4
Si n + m + ...
N,M elements, n,m atoms/unit cell and qN, qM valence electrons

sp3 hybrid bonds

III-V zincblende structure II-VI
Epitaxial film: Polycrystalline thin film:
GaAs , InP… CdTe, ZnS

II-IV-V2 I-III-VI2
Epitaxial film: Polycrystalline thin film:
ZnGeAs, … Cu(In,Ga)(Se,S)2
(Chalcopyrite and related compounds)
I-III-VI2 Alloy: Group I= Cu, Group
III= In and Ga, Group VI = Se and S

Basic: how to make a solar cell: The p-n junction
IB IIB IIIB IVB VB VIB
Si
Periodic Table
5 8
Ge
6 7

B C N O GaAs
13 14 15 16

Al Si P S CdTe
29 30 31 32 33 34

Cu Zn Ga Ge As Se I P
InP
48 49 50 51 52

Cd In Sn Sb Te AlSb

CIGS
CZTS
Metallurgical Junction
NA ND
- - - - - - + + + + +
- - - - - - + + + + +
P - - - - - - + + + + +
N

- - - - - - + + + + +
Space
ionized acceptors Charge Region ionized donors
E-Field
h+ diffusion = h+ drift e- diffusion = e- drift

Basic: PN junction at equilibrium
⎧p p0 ≈ N A ⎧ n no ≈ N D
⎪ ⎪
⎨ n
2
⎨ ni
2
⎪n p0 ≈ i
⎪ p n0 ≈
⎩
EC
NA
⎩ ND n = p = ni
qVbi
n i = BT 3 e −EG
2 kT
Ei
EF
EV 300K : n i ≅ 1.5 × 10 10 cm −3

ρ(x) W
+
qND Built-in voltage Vbi

-qNA -
N x qVb = ( Ei − E F ) p + ( E F − Ei )n
bi
V (x) nn0 = ni exp[( E F − Ei ) k BT ]
Vbi p p 0 = ni exp[Ei − E F k BT ]
p
E(x) x ⎛ ⎞
k BT ⎜ p p 0 nn0 ⎟ ⎛ N AND ⎞
Vbi = ln ≈ VT ln⎜ ⎟
− xp xn q ⎜ n2 ⎟ ⎜ 2 ⎟
⎝ i ⎠ ⎝ ni ⎠
Emax x
A. Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Basic: PN junction in the dark
Depletion region width:
Solve 1D Poisson equation using depletion charge approximation, subject to the
following boundary conditions:
V ( − x p ) = 0, V ( xn ) = Vbi , E (− xn ) = E ( x p ) = 0

p-side: V p ( x) =
qN A
2k s ε 0
(x + x p )2
qN D
n-side: Vn ( x) = − ( xn − x )2 + Vbi
2k s ε 0
Use the continuity of the two solutions at x=0, and charge neutrality, to obtain the
expression for the depletion region width W:

xn + x p = W ⎫
V p (0) = Vn (0) ⎪ → W = 2k s ε0 ( N A + N D )Vbi
⎬
⎪ qN A N D
N A x p = N D xn ⎭


Depletion layer capacitance:
Consider a p+n, or one-sided junction, for which:
2k s ε 0 (Vbi m V )
W=
qN D

The depletion layer capacitance is calculated using:
dQc qN D dW qN D k s ε 0 1 2(Vbi m V )
C= = = → 2=
1 C2 dV dV 2(Vbi m V ) C qN D k s ε0

1 Measurement setup:
slope ∝
ND
W
dW
Reverse
bias Forward bias vac ~
V
Vbi − V V


Ideal Current-Voltage Characteristics:
Assumptions:
• Abrupt depletion layer approximation
• Low level injection
Low-level injected minority carrier density much smaller than the
majority carrier density
• No generation-recombination within the space-charge region (SCR)
Depletion l
D l ti layer:

EC
W
n.p = n exp(V / VT )
2
i

qV p n (x n ) = n n0 exp(V/VT )
EF
Fn
E Fp
EV
n p ( −x p ) = n p0 exp(V/VT )
k BT
− xp VT =
xn q


Forward bias: Reverse bias:

EC
W
Ln qV
EC q(Vbi + V )
q(Vbi − V )
E Fp
qV EV
E Fn E Fn
E Fp
p
EV
Lp
W

Reverse saturation current is due to
minority carriers being collected
over a distance on the order of the
diffusion length.

Quantitative p-n Diode Solution / Little MATH
Q. neutral Region Q. neutral Region
Depletion Region
P-type N-type
E≠0
E=0 E=0

-∞ -xp Electrical field +xn +∞
∂Δn p d Δn p
2
Δn p ∂Δp n d 2 Δp n Δp n
= Dn − + G L existe in the depletion = Dp − + GL
∂t dx 2 τn region the minority ∂t dx 2
τp
d 2 Δn p Δn p carrier diffusion
i diff i
0 = Dn − Does not apply here d 2 Δp n Δp n
dx 2 τn 0 = Dp −
dx 2 τp

Boundary condition Boundary condition Boundary condition Boundary condition
Δn p ( x → −∞) = 0 Δn p ( x → − x p ) = ? Δp n ( x → x n ) = ? Δp n ( x → +∞) = 0

n i2 ⎛ qV ⎞ n i2 ⎛ qV ⎞
Δn p (x = − x p ) = ⎜ exp
⎜ − 1⎟
⎟ Δp n (x = x p ) = ⎜ exp
⎜ − 1⎟
⎟
NA ⎝ k BT ⎠ ND ⎝ k BT ⎠

Total current density:
y
• Total current equals the sum of the minority carrier diffusion currents defined at the
edges of the SCR:
I tot = I diff ( n ) + I diff ( −x p )
p (x n

⎛ D p p n0 D n n p0 ⎞ V/V
I D = qA ⎜
⎜ L
+
Ln ⎟
(
⎟ e T −1 )
⎝ p ⎠
• Reverse saturation current density: I 0 = J 0 . A
current
Current density area
V (volt)
2 2
ni
p n0 ≈
ND n p0 ≈ ni
NA
⎛ D p p n0 D n n p0 ⎞ ⎛ Dp Dn ⎞
I 0 = qA ⎜ + ⎟ = qAn i2 ⎜ + ⎟
⎜ L Ln ⎟ ⎜L N ⎟
⎝ p ⎠ ⎝ p D LnN A ⎠


Basic: How to make a solar cell: Dark current + Dark
current
Φ(x) N P
Photocurrent C
Ph t t Current:
t
2
Φe-αx • Diffusion courant (electron, region 1)
1 3
• Generation current in SCR (region 2)
Ohmic xp xn
Ohmic • Diffusion current (holes region3)
contact E contact
⎛ ⎞
1 3 qV
2
J = J 0 ⎜ exp − 1 ⎟ − J ph
p-type n-type k BT
⎜ ⎟
⎝ ⎠
W
EC
J ph = J p , diff ( x n ) + J G ( x n ) + J n , diff ( x p )
qV 1 2 3
E Fn
E Fp
EV

− xp
xn

Basic: PN junction under illumination (Space Charge Region, SCR)

Φ(x) N P Generation current i space charge region 2
G ti t in h i
2 J G ,n = J G ,p
Φe-αx
1 3 Continuity
C ti it equation (for electron)
ti (f l t )
∂n 1 dJ n Δp n
Ohmic xp xn = − + GL
∂t q dx τp
contact E
Ohmic
1
2 3 contact Steady-state
p-type n-type
xn
1 dJ n
0= + GL J G = q ∫ G(x)dx
q dx xp

W G(x) = Φαexp( αx)
Φαexp(-

J G ,n ( x n ) = qΦ e ( - αx p
)
− e − αx n = qΦe
− αx p
(1 − e )
− αW


Basic: PN junction under illumination (diffusion current)
Φ( )
(x) N P Neutral region 3 n type
n-type
Diffusion current: holes
2
dΔp
Φe-αx J p ,dff . = qD p
1 3 dx
−x / Lp + x / Lp Φατ p
Δp = Ae + Be +
xp xn 1 − α 2 L2p
Ohmic
Oh i E
Ohmic Boundary conditions
contact 1 3 contact
2
p-type n-type
Δp = 0 x = x c → +∞ L p << d n ; 1 / α ⇒ B = 0
Δp = 0 x = x n ( E = 0)
Φατ p −αx n + x n / L p
⇒A= e
1 - α Lp
2 2

Δp =
Φατ p
1− α L 2 2
(
e − αx n e − α ( x − x n ) − e
−( x − x n ) / L p
)
p

αL p
J p ,diff . = −qΦ e − αx n
1 − αL

Basic: PN junction under illumination
Neutral region 1 p type
p-type
Φ(x) N P
2 Diffusion current: electron
Φe-αx dΔ
dΔn
1 3 J n ,dff . = qD n
dx
x / Lp Φατ n
Ohmic xć xp xn xc Δn = Ae − x / L n + Be +
1 − α 2 L2n
contact E Ohmic
1 3
contact
2
p-type n-type Boundary conditions
x = x p Δn = 0 (electrical field E)
x = x ć → Δn(x ć )
Δn(x ć ) depends on surface recombination (S 0 )
( p (
Surface recombination
S0

⎛ A n − xp /L n B n xp /L n Φα 2 τ n −αx p ⎞
J n,diff. ( x n ) = qD n ⎜ −
⎜ L e + e − e ⎟
⎟
⎝ n Ln 1 − α Ln
2 2
⎠

Basic: PN junction under illumination

1
For an efficient P - N junction d p << ⇒ J n,diff. ≈ 0
α

(
⎧J G ( x n = W ) = −qΦ 1 − e − αW )
⎪
Origine at x p = 0 and x n = W ⎨ αL p
⎪ J p,diff. ( x n ) = −qΦ e − αW
⎩ 1 + α 2 L2p

⎛ 1 ⎞
J ph = −qΦ ⎜1 − e − αW ⎟
⎜ 1 + αL ⎟
⎝ p ⎠
1
Maximum J ph
M i f αW >> 1 (W >> ) → e.g. pin - j
for i junction
ti
α


PN junction under illumination / Efficient p-n diode
⎛ 1 ⎞ ⎛ 1 ⎞
J ph = − qΦ ⎜1 − e − αW ⎟ Φ = Φ 0 (1− R )
⎯⎯ ⎯ ⎯→ J ph = qΦ 0 (1 − R ) ⎜1 − e − αW ⎟
⎜ 1 + αL ⎟ ⎜ 1 + αL ⎟
⎝ p ⎠ ⎝ p ⎠
⎛ 1 ⎞
A = cell area → I ph = J ph . A = AqΦ 0 (1 − R )⎜1 − e − αW ⎟
⎜ ⎟
⎝ 1 + αL p ⎠
Φ0 = Number of photon per unit area, per unit time, per wavelength increment
incident power: Pinput = hν . Φ0 . A λ(nm)
. EQE
1239
I ph AqΦ 0 (1 − R ) ⎛
⎜1 − 1 ⎞ q ⎛ 1 ⎞
= e − αW ⎟ = . (1 − R ) ⎜1 − e − αW ⎟
Pinput hνΦ 0 .A ⎜ 1 + αL p
⎝
⎟ hν
⎠
⎜ 1 + αL
⎝ p
⎟
⎠
Multimeter
I ph
geometry
q ⎛ 1 ⎞
EQE = = (1 − R ) ⎜1 − e − αW ⎟
Piinput ⎜ 1 + αL ⎟
⎝ p ⎠
Pyranometer
hν
reflexion Absorption Minority carrier
1 1 λ (nm) g
diffusion length
= = coefficient
hν hc 1239
q qλ

Basic: PN junction Loss in Jph

⎛ 1 ⎞
η = (1 − R) ⎜ 1 − e −αW ⎟
⎜ 1 + αL ⎟
⎝ p ⎠
SEM image

Basic: Open circuit Voltage, VOC
⎡ q
qV ⎤ nk B T ⎛ J L ⎞
J = J D + J ph = J 0 ⎢ exp − 1 ⎥ − J L = 0 ⇒ V OC = l ⎜
ln ⎜ + 1⎟
⎟
⎣ nk B T ⎦ q ⎝ J0 ⎠
J0 : saturation current , n : ideality factor, kB : Boltzmann ´s constant,
VOC: open circuit voltage JL or Jph photocurrent
voltage, h

- JL + JD
Open circuit voltage
O i it lt
nk B T ⎛ J L ⎞
V OC = ⎜
ln ⎜ + 1⎟
⎟
q ⎝ J0 ⎠


Open circuit Voltage, VOC
For a given band gap EG, we need trade-offs
trade offs
nk B T ⎛ J L ⎞ ⎛ D n n i2 D p n i2 ⎞
V OC = ⎜
ln ⎜ + 1⎟ J0 = q ⎜ + ⎟
q J0 ⎟ ⎜L N Lp N D ⎟
⎝ ⎠ ⎝ n A ⎠
Dp 1 Dn 1 qn i w E
J 0 = [qN C N V ( + )+ ]exp - G Diffusion length
τp ND τn NA τnτp k BT
Doping
Dn Dp kT
= = VT = (VT = 300K = 25mV )
μn μp q

VOC


Power output characteristics
mpp = Maximum Power Point
JSC.VOC.FF
V FF Jmpp.Vmpp J Rmpp
EFF.=
PSun JSC.VOC P=I.V

Fill Factor
J mpp x Vmpp
Pmpp= Impp x Vmpp
J SC .VOC V

Vmpp Jmpp . Vmpp
EFF=
PSun
Inverse of slope Vmpp/Impp
is characteristic resistance
is characteristic resistance
Jmpp mmp Jsc VOC Pmax


Importance of mobility μ and Diffusion length, Lp,n
The higher mobility μ, the better is the carrier extraction
g y
L : Mean free length of path (L2 = D.τ) gives how long charge carrier (Lp or Ln) can
travel in a volume of a crystall lattice before recombination takes place

dn
J n = qμ n nE + qD
dx

velocity v
Mobility μ = =
Field E


Dark current and photocurrent

⎡ qV ⎤ ⎛ qV D ⎞
ID = I 0 ⎢exp − 1⎥ I = I 0 ⎜ exp − 1⎟ - I L
⎣ nkT ⎦ ⎝ nkT ⎠

V (volt) V (volt)


Limitation of VOC by I0 and JSC
At room temperature: VT = kBT/q = 26 mV
VOC increases by 0.06 V if I0 decreases by one order of magnitude
VOC increases by 0.06 V if ISC increases by one order of magnitude

V (volt) V (volt)
Diode saturation current density for nearly ideal Si solar cells
I 0 (Si) ≈ 10 −13 A.cm −2

Energy conversion efficiency of a solar cell
Maximum Power Point = PMPP = V MP . I MP
POWER IN MPP VOC . I SC
EFFICIENCY = EFFICIENCY =
POWER OF SUN LIGHT PSun
V MP
Optimal load resistance in the MPP R MPP =
L Importance of the
I MP
solar cell efficiency

EFFECIENCY ↑
W/cm2)

↓
I(A/cm2) x V(W

MATERIAL + AREA
m

COST FOR PV ↓
€/Wp↓
V (volt)

One diode model / Equivalent Circuit
Ideal diode (dark current , ID) JD J. RS (Voltage drop)
(Shockley diode equation)
J
⎡ qV D ⎤
J D = J 0 ⎢ exp − 1⎥
⎣ nkT ⎦
P
VD V RLoad
add a serie resistance RS jsh . Rsh
JL N Current
V = V D + J .R S loss

Solar cell in the dark
⎡ q (V − J .R S ) ⎤
J D = J 0 ⎢ exp − 1⎥ Solar cell under illumination J D = J 0 ⎡ exp q (V − J .R S ) − 1⎤ − J L
⎢ ⎥
⎣ nkT ⎦ ⎣ nk
nkT ⎦
⎡ Dn ni2 D p ni2 ⎤ Dark characteristics being shifted down by photocurrent
J0 = q⎢ + ⎥ which depend on light intensity.
⎢
⎣ Ln N A L p N D ⎥
⎦ J = I/A
add a shunt resistance
Photogenerated carriers can also flow through the crystal J0 Forward
VOC
0
surfaces or grain boundaries in polycrystalline devices
Reverse V
i sh .R sh = V + J .R S 4TH Q d t
Quadrant
JSC
⎡ q ( V − J.R S ) ⎤ V + J.R S - JL
J = J 0 ⎢ exp − 1⎥ - J L +
⎣ nkT ⎦ R Sh

Two diodes model / Equivalent Circuit
4th Quadrant
⎡ q (V − J .R S ) ⎤ ⎡ q (V − J .R S ) ⎤ V + J.R S
J = J 01 ⎢ exp − 1⎥ + J 02 ⎢ exp − 1⎥ + - JL
⎣ n1 kT ⎦ ⎣ n 2 kT ⎦ R Sh

J
RS +
J
JL Rsh V RLoad
J01,n1 J02,n2 1st Quadrant

- 4th Quadrant
V

1st Quadrant
⎡ q (V − J .R S ) ⎤ ⎡ q (V − J .R S ) ⎤ V + J.R S
J = J L − J 01 ⎢ exp − 1⎥ − J 02 ⎢ exp − 1⎥ −
⎣ n1 kT ⎦ ⎣ n 2 kT ⎦ R Sh


Role of Rsh (Rp) for I-V-characteristics of solar cells
During operation, the efficiency of solar cells is reduced by the dissipation of power across
internal resistances which can be modeled as a parallel shunt resistance (RSH) and series
resistance (RS). For an ideal cell, RSH would be infinite and would not provide an alternate path
for current to flow, while RS would be zero, resulting in no further voltage drop before the load.

Voltage (Volt)
Voltage (Volt)
V lt (V lt)
Using LabVIEW analysis capabilities you can assess the main performance parameters for PV
cells and modules.


Role of Iph for the influence of RS

FF↓ and η↓ with increasing ISC

Voltage (Volt)
g ( )


Basic: solar cell is a sensor for solar radiation
The efficiency increases with increasing light intensity.
y g g y
We Compare the efficiency at two light intensities PSun and PSun > PSun
0 0

No additional heating I 0 (T0 ) = I 0 (T) = I 0
0 Linearity I Sun ∝ PSun
VOC . I SC
η = FF
PSun
0 0
VOC . I SC
η 0 = FF 0 0
PSun

nk B T ⎛ I ph ⎞ I
ln ⎜
⎜ I + 1 ⎟ SC
⎟P
η FF q ⎝ 0 ⎠ Sun
⇒ =
η 0 FF 0 nk B T0 ⎛ I ph0
⎞ I SC
0
ln ⎜ 0 + 1 ⎟ 0
q ⎜ ⎟
⎝ I0 ⎠ PSun
⎛ I ph ⎞ I SC
ln ⎜ ⎟
η FF T ⎜ I 0
⎝
⎟P
⎠ Sun PSun I ph
≈ = 0 =X
η 0 FF 0 T0 ⎛ I ph0
⎞ I SC
0 0
PSun I ph
ln ⎜ 0 ⎟
⎜ I ⎟ P0
⎝ 0 ⎠ Sun

The efficiency increases with increasing light intensity.
y g g y
Two light intensities
PS0 and PS > PS0
Sun Sun Sun

⎛ I ph ⎞0 0
I ph
ln ⎜X ⎟ ln X + ln
η FF ⎜ I0 ⎟ I0
= ⎝ ⎠ = FF
η 0 FF 0 ⎛ I ph ⎞
0
FF 0 ⎛ I ph ⎞
0

ln ⎜ ⎟ ln ⎜ ⎟
⎜ I0 ⎟ ⎜ I0 ⎟
⎝ ⎠ ⎝ ⎠

0
I ph ⎛ ⎞
⎜ ⎟
lnX + ln
η FF I0 FF ⎜ lnX ⎟
= = ⎜1 + ⎟>1
η 0 FF 0 ⎛ I ph ⎞
0
FF 0 ⎜ ⎛ I ph ⎞ ⎟
0

ln ⎜ ⎟ ⎜ ln ⎜ ⎟
⎜ I0 ⎟ ⎜ ⎜ I0 ⎟ ⎟ ⎟
⎝ ⎠ ⎝ ⎝ ⎠⎠


As light intensity changes
g y g
Φ: Photon flux photons/sec/cm²
dΦ
G(x) = − = αΦe − αx
dx
• JSC change much greater than VOC.
• Low light intensity still produces voltage.
• JSC increases proportionally with irradiance.
p p y
• MPP indicates Rload to achieve maximum power use.

⎡ qV ⎤ V + I.R S
1 sun I = I 0 . ⎢exp − 1⎥ + - IL
MPP ⎣ nkT ⎦ R Sh
JSC
0.8 sun ⎡ q(V − I.R S ) ⎤
I = I 0 . ⎢exp − 1⎥ - I L
0.6
0 6 sun ⎣ n.k T
n k B .T ⎦
nkT ⎛ J L ⎞
VOC = ln ⎜
⎜J + 1⎟
⎟
q ⎝ 0 ⎠
⎡ D n n i2 D p n i2 ⎤
J L = qG(L n + L p + W) J 0 = q⎢ + ⎥
⎢
⎣ LnN A LpND ⎥ ⎦
VOC

Basic: Temperature Effects
Solar cell operate best at lower temperature.
As the temperature decreases, the output voltage and efficiency increase.

The output voltage Voc, when Voc >> nkBT/q,
nkT ⎛ J L ⎞ nkT J
V OC = ln ⎜
⎜J + 1⎟ ≈
⎟ ln L
q ⎝ 0 ⎠ q J0

JL increase proportionally with irradiance
JL = K . I
nkT ⎛ KI ⎞
kT eV oc
V ⎛ KI ⎞
V oc = ln ⎜
⎜ J ⎟ ⇒ nkT = ln ⎜ J
⎟ ⎜ ⎟
⎟
e ⎝ 0⎠ ⎝ 0 ⎠
J0 is reverse saturation current and strongly
gy
depend on temperature:
⎡ Dn ni2 D p ni2 ⎤ −
EG
J0 = q⎢ + ⎥ n. p = n ≈ T e
2
i
3 kT

⎢
⎣ Ln N A L p N D ⎥
⎦


Basic: Temperature Effects
nkT ⎛ J L ⎞ nkT J
V OC = ln ⎜
⎜J + 1⎟ ≈
⎟ ln L
q ⎝ 0 ⎠ q J0

Assuming n = 1, at two different temperatures T1 and
g , p
T2 and the same illumination:
eVoc 2 eVoc1 ⎛ KI ⎞ ⎛ KI ⎞ ⎛ J 01 ⎞ ⎛ ni2 ⎞
− = ln ⎜
⎜J ⎟ - ln ⎜
⎟ ⎜J ⎟ ⎟ = ln ⎜
⎜J ⎟ ⎟ ≈ ln ⎜ 21 ⎟
⎜n ⎟
kT2 kT1 ⎝ 02 ⎠ ⎝ 01 ⎠ ⎝ 02 ⎠ ⎝ i2 ⎠
Eg ⎛ 1 1 ⎞
ni2 = N c N v exp(− E g kT ) ⇒ oc 2 − oc1 =
eV eV
⎜ − ⎟
kT2 kT1 k ⎜ T2 T1 ⎟
⎝ ⎠
⎛ T2 ⎞ E g ⎛ T2 ⎞
Voc 2 = Voc1 ⎜ ⎟ +
⎜ T ⎟ e ⎜1 − T ⎟
⎜ ⎟
⎝ 1⎠ ⎝ 1 ⎠
0.493V

Example, Si solar cell has Voc1 = 0.55 V at 20 oC (T1 = 293 K), at 50 oC (T2 = 323 K),

⎛ 323 ⎞ ⎛ 323 ⎞
Voc 2 = (0.55 V )⎜ ⎟ + (1.1 V )⎜1 − ⎟ = 0.493 V
⎝ 293 ⎠ ⎝ 293 ⎠

Basic: Examples
Consider a p–n junction diode at 25 °C
with a reverse saturation current of 10−9 A. Find the voltage drop across the diode when it is carrying the
following: (a) no current (open-circuit voltage), (b) 1 A, (c) 10 A.
q =1.602 × 10−19 C, k =1.381 × 10−23 J/K), n = 1 and T=25°C
(a) In the open-circuit condition ID = 0 and VD = 0
open circuit condition, 0, 0.
(b) With ID = 1 A, we can find VD by rearranging the Shockley diode equation
⎡ ⎤
⎡
J D = J 0 ⎢ exp
qV D
nkT
⎤
− 1⎥ = J 0 ⎢ exp
1 .602 x 10 −19 V D
1 .381 x 10 − 23 T
⎡ V ⎤
[ ]
− 1⎥ = J 0 ⎢ exp 11 .600 D − 1⎥ at T = 25 °C J D = J 0 e 38 .9V D − 1
T (K ) ⎦
⎣ ⎦ ⎣ ⎦ ⎣
1 ⎛J ⎞ 1 ⎛ 1 ⎞
(b) V D = ln ⎜ D + 1 ⎟ =
⎜J ln ⎜ − 9 + 1 ⎟ = 0 .532
⎟ 38 .9 ⎝ 10
38 .9 ⎝ 0 ⎠ ⎠
1 ⎛ 10 ⎞
( ) VD =
(c) ln ⎜ − 9 + 1 ⎟ = 0 .592
38 .9 ⎝ 10 ⎠
Consider a 100 cm2 PV cell
photovoltaic cell with reverse saturation current I0 = 10−12 A/cm2. In full sun, it produces a short-circuit
current of 40 mA/cm2 at 25°C Find th open-circuit voltage at full sun and again f 50% sunlight. Pl t
t f A/ t 25°C. Fi d the i it lt t f ll d i for li ht Plot
the results.
The reverse saturation current J0 is 10−12 A/cm2 × 100 cm2 = 1 × 10−10 A.
At full sun JSC is 0.040 A/cm2 × 100 cm2 = 4.0 A. The open-circuit voltage is
p g
⎛J ⎞
[ ] ⎛ 4 ⎞
J = J L − J 0 e 38 .9V − 1 = 0 ⇒ V OC = 0 .0257 ln ⎜ Sc + 1 ⎟ = 0 .0257 ln ⎜ −10 + 1 ⎟ = 0 .627 V
D
⎜ J ⎟
⎝ 0 ⎠ ⎝ 10 ⎠

Basic: One diode model / Equivalent Circuit
Since short-circuit current is proportional to solar intensity, at half sun ISC = 2 A and the open-circuit
short circuit open circuit
voltage is ⎛ 2 ⎞
V OC = 0 .0257 ln ⎜ −10 + 1 ⎟ = 0 .610 V
⎝ 10 ⎠
Plotting the relation belo gi es us the follo ing
below gives s following:


Lab. work
Varied d
V i d and measured parameters
d t
current
voltage
temperature
light intensity
wavelength of the light

⎡ q(V − I . R S ) ⎤ V + I.R S
I = I 0 . ⎢exp − 1⎥ + - I ph
⎣ n.k.T ⎦ R sh
Solar cell parameters: diode saturation
current density
ideality factor
series resistance
parallel resistance
short circuit current density
Derived parameters: fill factor FF energy , conversion efficiency thermal
factor,
activation energy, Ea
Sources: FU-Berlin A. Ennaoui / Helmholtz-Zentrum Berlin für Materialien und Energie

Lab. Work: ISC-VOC measurements
Very simple measurement no need for a load resistance with one multimeter only
y p y
Light intensity variation: ideality, I0 and Rp from ISC-VOC – characteristics
Temperature variation: thermal activation energy of I0

⎡ qV1 ⎤ decade
I D1 = I 0 ⎢ exp − 1⎥ qV1
⎣ nkT ⎦ exp
I D1 1 nkT
= ≈
I D2 10 qV 2
⎡ qV 2 ⎤ exp
I D 2 = I 0 ⎢ exp − 1⎥ nkT
⎣ nkT ⎦

k BT
ln10 = 2.3 = 26mV
q
ΔU/decade = U 2 − U 1
k BT
=n .ln10 → n.60mV
q
Room T: ΔU/decade = n.60mV
(Si: n = 1.1 – 1.3)


Lab. Work: ISC-VOC measurements


Lab-Work: Activation energy
Determination of EA from the slope in Arrhenius plots

EA = 0.5 eV corresponds to
about 2 orders of magnitude
for T1 = 300 K and T2 = 400 K


Lab-Work: Activation energy
Consequence of EA: Temperature dependence of VOC
q p p


Lab-Work: Measurements with loads
What is RL ? RL is the power taken from the illuminated solar cell.

V
RL = and P = V.I
I

Each RL corresponds to one point on I-V curve.
Simplest way: RL known, V measured.
(high accuracy for low cost)
Set-up: just using a voltmeter variation of known R
Good for ranges of RL between 1 Ω and 100 kΩ
(Si solar cells with small area, thi fil mini-modules)
l ll ith ll thin film i i d l )

sources of errors: accuracy of RL: Voltage (Volt)
g ( )
resistances of wires and contacts
i t f i d t t
internal resistance of the voltmeter


Choice of load resistance (RL) for simplest I-V measurements
IV
1. VOC and ISC are measured with a multimeter
2. RL* is calculated RL* = VOC / ISC (RL* is close to Maximum Power Point, MPP)
3. RL is changed towards ISC
RL is decreased by taking about 10 values up to RL ≤ RL*/10
4. RL is changed towards VOC
RL is increased by taking about 10 values up to RL ≥ 10 RL*

Determination of Rp
ΔV
Rp = −
ΔI V→0

Voltage (Volt)


Determination of RS
Measurement at two light intensities
Rp large enough
determination of the potentials
U1 at currents I1 = ISC1 - ΔI
U2 at currents I2 = ISC2 - ΔI

⎡ ⎛ q. (U 1 − I 1 .R S ) ⎞ ⎤
⎜
ΔI = I 0. ⎢exp ⎜ ⎟ − 1⎥
⎟
⎣ ⎝ k BT ⎠ ⎦
⎡ ⎛ q. (U 2 − I 2 .R S ) ⎞ ⎤
ΔI = I 0. ⎢exp ⎜⎜ ⎟ − 1⎥
⎟
⎣ ⎝ k BT ⎠ ⎦ Voltage (Volt)

U1 − U 2 Works well for conventional solar cells
RS = FF is relatively l
i l ti l large
I 2 − I1

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