Basics of Solar PV Cells •  Last Time: Review of Key Semiconductor Concepts –  Photon Energy Spectrum –  Conduction Band & Valence Band Electrons –  Energy Band-Gap –  Charge Transport in Solar Cell

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Basic Equations of Semiconductor Device Physics Poisson’s Equation:

dE = q( p − n + N D+ − N A− ) dx

Donor/Acceptor Ion Density N Current Transport

S.S. Charge Conservation:

+ D

≈ ND

N A− ≈ N A

dn J e = qµ e nE + qDe dx dp J h = qµ he pE − qDe dx 1 dJ e =U −G q dx 1 dJ h = −(U − G ) q dx

U~source G~Sink

Basic Equations of Semiconductors •  We Need Expressions for U and for G –  G ~ Generation Rate/Unit Volume of e/hole pairs via photon adsorption –  U ~ Loss Rate/Unit Volume of e/hole pairs via relevant mechanisms

•  è Must Examine Photon Adsorption Process & e/hole Loss Processes…

Solar Photon Spectrum •  Sun is ~blackbody at ~5800 deg K •  Emits photon spectrum •  Photon Energy related to wavelength: –  E=hc/l H=Planck’s Constant (h~6.6x10-34 J-sec) Ref: Ristinen & Kraushaar, “Energy and the Environment”, Wiley 1998

Typical Energy of a Visible-light Photon: λ = 5000 Angstroms = 5 × 10 −7 m ∴ E = hc / λ = 6.6 × 10 ≈ 10

−19

−34

J − sec3 × 10 m / sec/ 5 ×

J ⋅ 1eV / 1.6 × 10

8

−19

J ≈ 0.7eV

⇒ λ ~ 5000 Angstroms ⇔ E ~ 0.7eV NOW… Remember the Concept of the BAND-GAP ENERGY…

Band Theory of Solids: Semiconductors •  Some crystalline materials have smaller band-gap energy •  At low temperatures behave like insulators –  Ebg~1eV >> Temperature

•  With an electric field –  Electrons gain energy –  Can move into upper (conduction) band

Effect of III-IV Impurities on Energy Level of Silicon •  N-type impurities introduce new allowed energy bands just below bulk-Si conduction band •  P-type impurities introduce new allowed energy bands just above bulk-Si valence band •  Thus electrons can move into Si-conduction band and p-type bands and thus conduct !

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Model of Solar PV Cell Photon Flux with E>Egap Examine This p-n Junction… P-type

n-type

Simplified Model of PV Cell: No Light

Q: What Happens When Photon Flux with E>Egap Is Incident on Device ?

NEED TO BUILD MATHEMATICAL MODEL OF IDEALIZED SYSTEM…

Conduction Band & Valence Band Energy Levels in SEPARATE p-type and n-type materials

EF EF

Conduction Band & Valence Band Energy Levels in p-n junction •  THERE CAN ONLY BE ONE FERMI ENERGY IN A SYSTEM AT EQUIL (!) •  Result:

Can find carrier density distribution Also Known As “DEPLETION REGION”

P-type

QUASINEUTRAL

N-type

QUASINEUTRAL

EXAMINE TRANSITION REGION IN MORE DETAIL…

Idealized Model of Depletion Region: Can Find Electric Potential in Depletion Region: Charge Distribution From Semiconductor Material Theory E-field from Poisson’s Equation Potential Profile by Integrating E-field

Basic Equations of Semiconductor Device Physics Poisson’s Equation:

dE = q( p − n + N D+ − N A− ) dx

Donor/Acceptor Ion Density N

+ D

≈ ND

N A− ≈ N A

p, n distribution given by step functions seen in previous viewgraph

Integrate Poisson’s Equation to Find E(x); Integrate Again to find y(x) & equate to earlier result

Idealized Model of Depletion Region: RESULT: ⎡ 2q E max = − ⎢ (ψ 0 − Vext ) ⎣ε

⎛ 1 1 ⎞⎤ ⎜⎜ ⎟⎟⎥ + ⎝ N A N D ⎠⎦

⎡ 2ε w = l p + l n = − ⎢ (ψ 0 − Vext ) ⎣q

ND lp =W NA + ND NA ln = W N A + ND

1/ 2

⎛ 1 1 ⎞⎤ ⎜⎜ ⎟⎟⎥ + ⎝ N A N D ⎠⎦

1/ 2

Carrier concentration at edges of biased junction With NO Ext. Voltage: p nb

ni2 − qψo = p n0 = p p0 e( )≈ kT ND

nba = n p0

2 ⎛ − qψo ⎞ ni = nn0 e⎜ ⎟≈ ⎝ kT ⎠ N A

Forward bias injects minority carriers into quasineutral n and p regions Minority Carrier Density at edge of Quasineutral region Increases EXPONENTIALLY With forward bias voltage

n.b. Forward bias is defined so as to act to reduce or reverse the natural or inherent potential drop across the p-n junction

Distribution of Minority Charge Carriers at Interface Between Depletion and Quasineutral Regions: Concentration Of Carriers in the Minority (I.e. MINORITY CARRIERS) at Edge of Depletion Region INCREASES EXPONENTIALLY W/ Ext. Bias è KNOWN AS MINORITY è CARRIER INJECTION

p nb = p n0 e

n p a = n p0 e

qVapp

qVapp

kT

kT

ni2 qVapp kT = e nD ni2 qVapp kT = e NA

Charge Carrier Transport in Quasineutral Region •  Depletion Region Is Surrounded by two QUASINEUTRAL Regions •  Current Transport Occurs Via Diffusion in This Region

dn J e ~ qDe dx dp J h ~ −qDh dx

Summary so far: •  Can divide p-n junction diode into two regions –  Quasineutral & Depletion Regions

•  Minority Carrier Concentration at Depletion Edge Depends Exponentially on Ext. Voltage •  In Quasi-neutral Region Charge Carriers Flow by Diffusion

Consider n-side of diode: Current Diffusion:

dp J h ~ −qDh dx

Hole Conservation Law (similar to Fluid continuity eqn)

1 dJ h = −(U − G ) q dx

HOW TO TREAT U AND G TERMS?

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Charge Carrier Loss Mechanisms •  Radiative Recombination •  “Auger” Recombination •  Recombination at Traps –  Bulk Defects & Impurities –  Crystal Surfaces/Boundaries

I: Radiative Recombination REVERSE of Photon Adsorption Process:

I: Radiative Recombination What is the Rate, UR, of this process (#/unit volume/unit time) ? A.  In Thermal Equilibrium UR=0 (by definition) B.  Rate Proportional to e-density (n) and to Hole density (p) C.  Infer That Away from Equilibrium, Rate Is

U R = B (np − n ) 2 i

Where B is a rate constant that depends upon the material (for Silicon B~2x10-15 cm3/sec)

Carrier Lifetime Concept Carrier lifetimes, :

τe & τh

n − no Δn τe = = ; no ~ U U

equilibrium value of

p − po Δp τh = = ; po ~ equilibrium value of U U For Radiation Recombination with Δn = Δp

no p o τ= 2 Bni (no + po )

n

p

3 cm B ~ 2 ×10 −15 sec

Carrier Lifetime Concept Carrier lifetimes, :

τe & τh

Non-equilibrium Carrier Density Can Then Decay:

Δn; Δp

τ e or τ h 1/e time

II: Auger Recombination •  Energetic Electron Recombines with Hole •  MUST Get Rid of Excess Energy

Energy

e-

e-

hole

–  Transfers to Second Electron Conduction Band Valence Band

•  This Second Electron “Cascades” Back to Lower Energy •  Energy is Transferred to Material (as HEAT)

II: Auger Recombination •  Electron Auger Lifetime: •  Hole Auger Lifetime

1

τ

1

τ

= Cnp + Dn 2

= Cnp + Dp 2

•  U=Dn/te =Dp/th •  Usually An Important Loss Process

III. Defect & Crystalline Surface Recombination •  Defects Can Induce e/hole recombination •  Defects Consist of –  Unwanted Impurity Atoms –  Bulk Crystal Defects (I.e. missing atoms, missing rows, etc…)

•  Surfaces Can Also Induce e/hole recombination –  Adjacent microcrystal surfaces –  Solid-Air Interface

III. Defect & Crystalline Surface Recombination •  Implication –  è WANT TO KEEP UNWANTED IMPURITIES OUT OF PV CELL –  è WANT TO MINIMIZE NUMBER OF MICROCRYSTALS (I.E. MAKE OUT OF A SINGLE CRYSTAL) BOTH EFFECTS IMPACT MANUFACTURING TECHNIQUES AND COSTS

Consider n-side of diode: Current Diffusion:

dp J h ~ −qDh dx

Hole Conservation Law (similar to Fluid continuity eqn)

1 dJ h = −(U − G ) q dx

But we had U=(pn –p0 )/th=Dp/th è FIND A DIFFUSION EQUATION W/ SOURCE TERM:

d 2 Δp Δp G = 2 − 2 dx Lh Dh

Lh = D hτ h

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Un-illuminated (“DARK”) p-n Diode Response Set G=0; Use

∂ 2 pn0 ∂x

2

Find the Equation Gen. Soln:

Δp = Ae

=0 d 2 Δp Δp = 2; 2 dx Lh x

Ln

+ Be

−x

L2h ≡ Dhτ h Lh

B.C.’s: At Junction/n-type border (“x=0”) p nb = p n0 e

qV

kT

pn > 0 for x → ∞ ⇒ A = 0

Un-illuminated (“DARK”) p-n Diode Response Particular Solutions of Minority Carrier Densities in Quasineutral Regions

p n ( x) = p n0 + p n0 [e

qV

n p ( x' ) = n p0 + n p) [e

kT

qV

− 1]e

kT

−x

− 1]e

Where x, x’ are defined in figure…

Lh

−x

Le

Distribution of charge carriers under forward bias Densities Known Here

Exponential Decay Here

Charge Carrier Distributions Known… Can Find Currents Now: For Minority Carrier Currents in Quasineutral Region E.g. on n-type side

dp J h ( x) = −qDh dx

Thus find minority carrier currents:

J h ( x) =

Je =

qDh p n0 Lh

qDe n p0 Le

(e

qV

(e

kT

qV

− 1) e

kT

−x

− 1) e

Lh

− x'

Le

Known Current distribution across p-n diode (so far…) J h ( x) =

qDh p n0

J e ( x' ) =

Lh

(e

qDe n p0 Le

qV

kT

(e

− 1) e

qV

kT

−x

Lh

− 1) e

− x'

Le

Need Current Flow in Depletion Region (aka Transition Region or Junction) Current continuity equation gives

1 dJ h 1 dJ e − = (U − G) = q dx q dx Integrate across junction to find change in current: 0

δJ e = δJ h = q ∫ (U − G )dx −W

Usually W Egap Photons •  Assume e-h generation rate G = constant –  Corresponds to Ephoton ~ Egap Flux

•  Earlier Assumptions Still Valid –  Quasineutral Region, Depletion Region –  Drift Current Density ~ Diffusive Density in Depletion Region –  Small minority carrier population –  Diffusive minority carrier transport in quasineutral region

Photon absorption in semiconductor For Photon Energy, En > Egap Photon Is Adsorbed & Creates e/hole pair At Adsorption Site:

Photon absorption in semiconductor For Photon Energy, En < Egap Photon May be Adsorbed BUT DOES NOT CREATE e/hole pair… Or (more likely) it passes through material:

Approximate e/hole Generation Rate •  Assume All Photons Have Enough Energy to Create e/hole Pairs… Then G Is

G = (1 − R)αN exp(− xα )

Photon Flux vs Distance into PV Cell N(x)

Number of Photons/Unit Area/Unit Time

x

PV Cell

Approximate e/hole Generation Rate •  For Weak Adsorption Approximate Generation Rate, G (units: e/hole pairs/unit volume/unit time)

G ~ (1 − R)αN •  R = fraction of light reflected from surface (R~0.05 with Anti-reflection coating) N = number photons/unit area/unit time •  a = photon intensity e-folding attenuation length –  Typically few microns…

Typical Adsorption Coefficient •  Data for Ga-As •  Typical Penentration Depth for E>Egap,L: –  L=1/a~1 micron

•  Data for Silicon Not Too Different

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source

Effect of Illumination on diode MINORITY CHARGE CARRIER DISTRIBUTION: Diffusion Eqn for hole population distribution

General Solution

d 2 Δp Δp G = 2 − 2 dx Lh Dh

Δp = Gτ h + Ce

x

Lh

+ De

−x

Lh

Use Same Boundary Conditions to find Particular Solution

p n ( x) = p n0 + Gτ h + [ p n0 (e Similar Result for Electrons Holds

qV

kT

− 1] e

−x

Lh

Effect of Illumination on diode MINORITY CHARGE CARRIER CURRENT: Current is Diffusive, I.e. Jh ~ dp/dx, Je~dn/dx è

J h ( x) = J e ( x' ) =

qDh p n0 Lh qDe n p0 Le

(e

qV

(e

qV

kT

kT

− 1) e

−x

− 1) e

Next, need current across junction…

Lh

− x'

Le

− qGLh e

−x

− qGLe e

Lh

−x'

Le

Need Current Flow in Depletion Region (aka Transition Region or Junction) Current continuity equation gives

1 dJ h 1 dJ e − = (U − G) = q dx q dx Neglect Losses in Junction & Integrate across junction to find change in current: 0

δJ e = δJ h = −q ∫ Gdx ~ qGW −W

(since G ~ constant)

Distribution of carriers with Illumination

Diode Response w/ Illumination: Current Density vs Voltage Across Diode:

J total = J e + J h − dJ = ⎛ qDe n p0 qDh pn0 ⎜⎜ + Lh ⎝ Le

⎞ ⎟⎟(exp(qV / kT ) − 1) + qGW ⎠

Total Current, I (Amps), Is Just Jtotal * Area of Diode…

I = I 0 (e

qV

kT

− 1) − I L

I L = qAG ( Le + W + Lh )

I-V Characteristics of p-n diode

Characterizing Solar PV Cell Performance IL kT •  Open Circuit Voltage, Voc Voc = q ln( I + 1) 0

•  Short-circuit Current, I SC = I L = qAG( Le + W + Lh ) •  Maximum Power Point (Vmp,Imp) •  Fill Factor, VMP I MP FF ≡ Typically FF~0.5-0.7 VOC I SC •  Efficiency

VMP I MP VOC I SC η= = ~ 0.1 − 0.15 Pin Pin

Solar Photon Flux Spectrum & Maximum Short-Circuit Current

Solar Cell Efficiency Limits One Photon => One e-hole pair + Thermal Energy Recombination Losses (Surfaces, Defects, Impurities) Reflections & Scattering

Basics of Solar PV Cells •  Key Concepts –  Photon Energy Spectrum –  Charge Carrier Generation Via Photon Absorption –  Charge Carrier Loss Mechanisms –  Un-illuminated p-n junction diode –  Illuminated p-n junction diode: The Solar PV Cell –  Solar PV Cell’s as an Electricity Source