An Introduction to Semiconductor

Electrochemistry

Laurie Peter

University of Bath

ELCOREL Workshop, Oud Poelgeest Castle, Oegstgeest

energy

DOS

energy

D(E)

Egap

VB

CB

Egap

energy

CB

VB

Egap

density of states parabolic approximation simplified band diagram

Basic Ideas: Band Diagrams

( ) 1

1 exp

= −+

FD

F

B

f EE E

k T

Fermi-Dirac Function Determines Concentrations of

Electrons and Holes in Electronic Energy Levels

EF – the Fermi level

( ) ( )

( ) ( )( )

exp

1 exp

C

V

C FC C

BE

E

V FV V

B

E En D E f E dE N

k T

E Ep D E f E dE N

k T

∞

−∞

−= = −

− = − =

∫

∫

To obtain the total concentrations of electrons and holes

we need to integrate the product of the occupation probability fFD(E) and the

density of electronic states in the conduction or valence bands, DC and DV

exp gapC V

B

Enp N N

k T

= −

In thermal equilibrium

CB

EF

VB

Integrate the green curve to obtain the concentration of electrons

CB

VB

EF

intrinsic n-type p-type

Intrinsic and Doped Semiconductors

D →→→→ D+ + eCB A →→→→ A- + hVB

Evac – vacuum energy level.

Ec conduction band edge energy.

Ev – valence band edge energy.

Egap – energy gap.

A – electron affinity.

I – ionization energy.

Φ - work function.

Φ EF Fermi energy.

Energy diagram for an n-type semiconductor

__

, , ,

_

0 0

_ _

ln ln

j

ii

T P n

i i i

n n B p p BC V

n pF F

G

n

z F

n pk T k T

N N

E E

ϕ

µ

µ µ ϕ

µ µ µ µ

µ µ

∂ = ∂

= +

= + = +

= = −

The Fermi level, Free Energy and the

Electrochemical Potential of Electrons and Holes

Electrochemical potential or Partial Molar

Gibbs Free Energy of Charged Species

potential dependent termchemical potential

Nc and NV – effective density of states

thermodynamic standard states

e-vac

-qϕϕϕϕ

EC

EF

EV

µµµµn,sc

µµµµn,m

semiconductor metal

EF

EC

EV

-qϕϕϕϕ

semiconductor metal

Equilibration of Fermi Levels in Solid State Junction

Formation of a Schottky Barrier

Cr3+/Cr2+

Fe3+/Fe2+

H+/

H2

-4.5

evac0

0

-1.0

+1.0

≈≈ ≈≈ ≈V vs. SHE

-3.5

-5.5

eV

Vacuum Energy Scale and Electrode Potential Scale

0 0, 4.5≈ − −F redox redoxE eV qU

Outer Sphere Redox Systems – Marcus Theory

Redox

Fermi level, ln

o oR

R O R OF redox BO

NE k T

Nµ µ µ µ− − − − = − = − +

( )( )

( )20

1/2

1exp

44 BB

E EW E

k Tk T λπλ

− = −

Equilibration of Fermi Levels in Electrolyte Junctions

metal/redox semiconductor/redox

CB

VB

Charge, Field and Potential Distribution

in the Case where a Depletion Layer or

Space Charge Region (SCR) is formed

The charge distribution corresponds to

a space charge capacitance

not to scale!

���� ��∆����

���

�/�

equilibrium flat band accumulationreverse bias

EF

EF,redoxEF

EF EF

EF,redox

EF,redox

EF,redox

Band Bending as a Function of Applied Potential

In the dark the electrode behaves as a diode

n-type electrode

depletion region

accumulation region

Mott Schottky Plots of the Space Charge Capacitance

1/C2

U

VBCB

p-type

n-typeEF

EF

p-type n-type

20

1 2 Bsc

sc d SC

k T

C qN qφ

ε ε

= ∆ −

( )0 0.059fb fbU U pH pH= −

Band Edge Positions at pH = pHpzc

( ) ( )2

0

21

6W

W

rqNr r r

rφ

εε = − +

Nanostructured ElectrodesSpace Charge in Spherical Particles

Band bending for a spherical

anatase particle (Nd = 1017 cm-3,

r = 25 nm, ε = 30) as a function

of rW .

rW defines the edge of the space

charge region.

The maximum band bending of

ca. 6 meV is reached when rW is

reduced to zero, i.e. the space

charge region extends to the

centre of the particle.

Band bending for complete depletion

in spherical anatase particles with

different doping densities (r = 25 nm,

ε = 30, doping density as shown).

In the case of the lower doping, band

bending is limited to only a few mV.

For the higher doping, saturation

occurs when the potential drop

across the depletion layer reaches ca.

120 mV. The effects of band banding

cannot be neglected in this case.

Influence of Doping Density on Space Charge in Spherical Particles

( )2 2 2

0

1ln

2 2φ

εε ∆ = − − +

WSC W

rqNR r R

R

Space Charge in Nanorod Electrodes

Fe2O3 nanorod arrays

rW

R

���,��� ���

����

��

E

gss(E)

EF

filled

emptyEC

EV

Non-Ideal Systems: Surface States

Surface States and Fermi Level Pinning

Rss CSs

CSC

CH

RF

Equivalent Circuit Including Surface State Capacitance

What happens when we illuminate a semiconductor electrode?

We create electron-hole pairs

Thermal equilibrium no longer applies

neq → neq +∆n

peq → peq +∆p

∆n = ∆p

e-

h+

CB

VB

Quasi-Fermi Levels and Fermi Level Splitting

EC

EV

EFnEF

pEFhνννν

µ µ µ= + = −eh n p n F p FE E

exp exp V p FC n FC V

B B

E EE En n N p p N

k T k T

+ −+ ∆ = − + ∆ =

majority carriers - small change minority carriers - large change

stored free energy

Quasi Fermi Level in Illuminated Semiconductor-Electrolyte Junction

O

R

nEF

pEF

e-

h+

R + h+ →→→→ O

photocurrent due to reaction of minority carriers (holes)

Generation and Collection of Minority Carriers

( )0 min

exp1

1photo SCj W

EQEqI L

αα

−= = −

+

Gärtner Equation

min min min min minBk T

L Dq

τ µ τ= =

Minority carrier diffusion length

Anodic Photocurrent for n-type Electrode

Photocurrent increases

as space charge region

gets wider

Simple theory predicts photocurrent onset at flat band potential

recombination

U

jphoto

jrec

jG

jphoto

no recombination

Delayed Photocurrent Onset Due to Surface Recombination of Electrons and Holes

E - Efb

End of the first lecture….