ELCOREL Workshop, Oud Poelgeest Castle, Oegstgeest

An Introduction to Semiconductor Electrochemistry

Laurie Peter University of Bath Basic Ideas: Band Diagrams

energy energy

CB CB

DOS Egap Egap

VB VB

Egap

energy D(E)

density of states parabolic approximation simplified band diagram Fermi-Dirac Function Determines Concentrations of Electrons and Holes in Electronic Energy Levels

1 f() E = FD E− E  1+ exp F  kB T 

EF – the Fermi level 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

∞ E− E  n= D()() EfEdE = N exp − C F  ∫ C C k T EC B  EV E− E  =()()() −  = V F p∫ DEV 1 fE  dEN V exp   −∞ kB T 

E  In thermal equilibrium = − gap np NC N V exp   kB T  CB

EF

Integrate the green curve to obtain the concentration of electrons

VB Intrinsic and Doped Semiconductors

CB

EF

VB

intrinsic n-type p-type

→→→ + →→→ - D D + eCB A A + hVB Energy diagram for an n-type semiconductor

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. The Fermi level, Free Energy and the Electrochemical Potential of Electrons and Holes

_  Electrochemical potential or Partial Molar _ ∂ G µ =   Gibbs Free Energy of Charged Species i ∂  ni   ϕ chemical potential T, P , n j , potential dependent term _ ϕµ µ = + i iz i F µ µ µ =+0 n =+0 p nnBkTln ppB kT ln NC N V _ _ µ µ = = − nEF p E F Nc and N V – effective density of states

thermodynamic standard states Equilibration of Fermi Levels in Solid State Junction Formation of a Schottky Barrier

- e vac

ϕϕϕ -q -qϕϕϕ µµµ n,sc E µµµ C EC n,m EF EF

EV EV

semiconductor metal semiconductor metal Vacuum Energy Scale and Electrode Potential Scale

e 0 vac

≈ ≈ ≈ ≈ ≈ V vs. SHE eV -3.5 -1.0

Cr 3+ /Cr 2+

-4.5 H+/ 0

H2

Fe 3+ /Fe 2+ -5.5 +1.0 0≈ − − 0 EF, redox4.5 eV qU redox Outer Sphere Redox Systems – Marcus Theory

2 ()E− E 0  ()=1 −   W E 1/2 exp ()πλ 4λk T  4 kB T B 

− − − o − o N Redox =−= µµ µ  −  + R EF, redox RO R O  kB T ln Fermi level   NO Equilibration of Fermi Levels in Electrolyte Junctions

CB

VB

metal/redox semiconductor/redox not to scale! 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 Band Bending as a Function of Applied Potential

E F EF

EF,redox EF EF,redox EF,redox EF,redox

EF

equilibrium reverse bias flat band accumulation In the dark the electrode behaves as a diode

depletion region

accumulation region

n-type electrode Mott Schottky Plots of the Space Charge Capacitance

  1= 2 ∆φ − kB T 2 ε ε sc  CqNsc d SC 0  q  1/C 2

p-type n-type

=( ) − Ufb U fb pH0 0.059 pH

U EF p-type

EF n-type CB VB Band Edge Positions at pH = pH pzc Nanostructured Electrodes Space Charge in Spherical Particles

qN 2r  φ ()r=() r − r 2 1 + W εε W   6 0 r 

Band bending for a spherical 17 -3 anatase particle ( Nd = 10 cm , 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 . Influence of Doping Density on Space Charge in Spherical Particles

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 . Space Charge in Nanorod Electrodes

qN 1 r   ∆=−φ ()R2 −+ r 2 R 2 ln W SCεε  W    20  2 R  

rW

R Fe 2O3 nanorod arrays

, Non-Ideal Systems: Surface States

E

E C empty

EF

filled

EV

gss (E) Surface States and Fermi Level Pinning Equivalent Circuit Including Surface State Capacitance

Rss CSs

CH

CSC

RF

What happens when we illuminate a semiconductor electrode?

e- CB We create electron-hole pairs

Thermal equilibrium no longer applies

→ ∆ neq neq + n VB + → ∆ h peq peq + p

∆n = ∆p Quasi-Fermi Levels and Fermi Level Splitting

EC E F nEF

ννν h pEF

EV E− E E+ E +∆= −C n F +∆= V p F n n NC exp p p N V exp  kTB  kTB majority carriers - small change minority carriers - large change µ µ = + = − stored free energy eh n p nFE pF E Quasi Fermi Level in Illuminated Semiconductor-Electrolyte Junction

photocurrent due to reaction of minority carriers (holes)

e- O

nEF

R pEF

R + h + → O

h+ Generation and Collection of Minority Carriers

Gärtner Equation j exp − (αW ) EQE =photo =1 − SC +α qI01 L min

k T L= D µ τ τ = B min min minq min min 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 Delayed Photocurrent Onset Due to Surface Recombination of Electrons and Holes

jphoto

no recombination

jG

jrec

recombination

jphoto U E - Efb End of the first lecture….