MENA 3200 Energy Materials

Materials for Electrochemical Energy Conversion

Part 2

Electrical conductivity – defects and transportDiffusion and stability

Truls Norby

Conductivity

Fundamentals of electrical conductivity

Conductivity requirements

Resistivity and resistance

Charged particles in an electric field E feel a force F The force sets up a net flux density and current density i

The ratio ρ (rho) = E/i is termed resistivity and is an intensive materials property

Resistivity has units (V/m)/(A/m2) = (V/A)m = ohm*m = Ωm

For an object we may instead express a current I and voltage U The ratio R = U/I (Ohm’s law) is termed resistance and is an extensive

property for the object Resistance has units V/A = ohm = Ω

The resistance of a current-carrying object is obtained from the resistivity ρ, length l, and cross-sectional area a: R = ρ*l/a

Conductivity and conductance

Conductivity σ (sigma) is the inverse of resistivity: σ = 1/ρ

Conductance G is the inverse of resistance: G = 1/R

The units for G and σ are S (siemens) and S/m, respectively.

(Other/older units for conductance comprise Ω-1, ohm-1, and mho)

G = σ*a/l

Exercises

A rectangular solid sample has a length of 2 cm and a cross-section with sides 5 x 5 mm2. Electrodes for merasurements are painted on its far faces. ◦ If its conductivity is 1000 S/cm, what is its conductance?

◦ And its resistance?

A circular disk has thickness 2 mm and diameter 2 cm. We paint electrodes on its two faces and measure the resistance. ◦ If the resistance is 10 Ω, what is the resistivity?

◦ If the conductance is 10 S, what is the conductivity?

Total conductivity, transport numbers The conductivity of a substance has contributions from all species,

mechanisms, and pathways of charge carriers:◦ Electronic and ionic

◦ Electronic: electrons and holes

◦ Ionic: cations and anions

◦ Or more detailed, for instance, protons, oxide ions, and metal cations

◦ Mechanisms: vacancies and interstitials

◦ Microstructural pathways: bulk, grain boundaries, surfaces…

The total conductivity is a sum of partial conductivities over all species, mechanisms, and pathways:

The fraction of the total conductivity (and ideally the fraction of any current going through the substance) is termed the transport number or transference number for s:

s

stot σσ

s

stot tt 1totss t tot

ss σ

σt

pnaceliontot

Exercise

Normally, only one or two charge carriers, defects, mechanisms, or pathways dominate to the extent that we need to take them into account. The others can be neglected.

What dominates the conductance in ◦ Si? As-doped Si?

◦ Pt?

◦ NaCl(s)? NaCl(aq)?

◦ H2O(l)? HCl(aq)?

◦ Y-doped ZrO2?

◦ La2NiO4+δ?

◦ Alumina single crystal? Dense alumina ceramic? Porous alumina ceramic?

Exercise

One can enhance or depress selected contributions for measurements or use

Discuss how you might affect the contributions below in the case of solid samples: ◦ Electronic conductivity vs oxide ion conductivity

◦ Proton conductivity

◦ Bulk conductivity

◦ Grain boundary conductivity

◦ Outer surface conductivity

◦ Inner surface (pore wall) conductivity

Series resistance contributions

Till now, we have looked at parallel possibilities that add to conductance and give more current

There are also many sources to series problems that add to resistance and give less current (more voltage):◦ Bulk resistance

◦ Traps

◦ Grain boundary resistance

◦ Electrode (contact) resistance

Note the difference between grain boundary conduction and grain boundary resistance

What is the source of each one? How can they be affected?

Conductivity; charge, concentration, and mobility

The conductivity of a species s is given by its charge zs, volume concentration cs, and charge mobility us.

The charge is an integer multiple zs of e or F, depending of whether the concentration is given in number of particles or moles of particles per unit volume:

The concentration cs may arise from different models comprising doping and thermodynamics for electrons and/or point defects.

Charge mobility us is the product of mechanical mobility Bs and charge zse:

ssss uecz ssss uFcz

sss eBzu

Charge mobility; itinerant carriers (metallic mobility)

In materials with metallic mobility (itinerant electrons or holes, broad bands) the mobility is determined by scattering, and the mobility is proportional to the mean free length between scattering events and inversely proportional to the electron or hole effective mass and the mean velocity at the mobile electrons’ energy level (Fermi level):

Scatterers are defects (e.g. impurities) or phonons (lattice vibrations) Both contribute to resistance in series:

Typical temperature dependencies:

Typically, impurities dominate at low T and lattice vibrations at high T.

Fe

me vm

eLu

latt,simp,s

s

uu

u11

1

230

/,imp,simp,s Tuu 23

0/

,latt,slatt,s Tuu

Charge mobility; diffusing carriers

For ions that move by defects in materials and for non-itinerant (trapped) electrons in semiconductors, the mobility of the ionic defect or electronic species is determined by diffusion; thermally activated jumps from site to site:

Note that usT (and thus σsT) is an exponential function of 1/T, and therefore the activation enthalpy may be extracted from the slope of a plot of ln(usT) or log(usT) versus 1/T (similar to an Arrhenius plot).

Such electronic charge carriers are called small polarons – the electron deeply trapped in the relaxation of the lattice around itself. Small polaron mobilities are orders of magnitude smaller than itinerant (metallic) mobilities.

Electronic charge carriers trapped in more shallow relaxations are called large polarons and have intermediate mobilities.

kT

Hexp

TuD

kT

ezeBzu s,m

,sss

sss

10

Concentrations cS of charge carriers - overview

Metals: Concentration of electrons approx. equal to the concentration of valence electrons

Electronic semiconductors: Concentration of electrons n or holes p fixed by donor or acceptor dopants

Solid ionic conductors: Concentration of defects (e.g. oxygen vacancies or protons) fixed by acceptors or structural disorder

Liquid ionic conductors: Concentration of ions…

ssss uecz

Conductivity of components and defects

For foreign species, like protons in an oxide, the conductivity of the defect is simply

e.g.

But for a component, like oxide ions in an oxide, conductivity can be expressed in terms of the component or the defect

Components need defects to move, and defects need components to move

dddd uecz

HHHHuecz

dddcccd ueczuecz

Exercise

Which is bigger? Cd or Cc?

Which is bigger: ud or uc?

Which is faster? The component atoms or the defects?

In order to understand, analyse, and affect the conductivity in crystalline solids, we need to

understand defect concentrations

Introductory on defect chemistry

Brief history of defects

Early chemistry had no concept of stoichiometry or structure.

The finding that compounds generally contained elements in ratios of small integer numbers was a great breakthrough!

H2O CO2 NaCl CaCl2 NiO

Understanding that external geometry often reflected atomic structure.

Perfectness ruled. Variable composition (non-stoichiometry) was out.

However, variable composition in some intermetallic compounds became indisputable and in the end forced re-acceptance of non-stoichiometry.

But real understanding of defect chemistry of compounds mainly came about from the 1930s and onwards, attributable to Frenkel, Schottky, Wagner, Kröger…, many of them physicists, and almost all German!

Frenkel Schottky Wagner

Defects in an elemental solid (e.g. Si or Ni metal)

Notice the distortions of the lattice around defects◦ The size of the defect may be taken to be bigger than the point defect itself

Adapted from A. Almar-Næss: Metalliske materialer, Tapir, Oslo, 1991.

Defects in an ionic solid compound

Bonding

Bonding: Decrease in energy when redistributing atoms’ valence electrons in new molecular orbitals.

Three extreme and simplified models:

Covalent bonds: Share electrons equally with neighbours!◦ Strong, directional pairwise bonds. Forms molecules. Bonding orbitals filled. ◦ Soft solids if van der Waals forces bond molecules. ◦Hard solids if bonds extend in 3 dimensions into macromolecules.

Examples: C (diamond), SiO2 (quartz), SiC, Si3N4

Metallic bonds: Electron deficiency: Share with everyone!◦ Atoms packed as spheres in sea of electrons. Soft. ◦Only partially filled valence orbital bands. Conductors.

Ionic bonds: Anions take electrons from the cations!◦ Small positive cations and large negative anions both happy with full outer shells.◦ Solid formed with electrostatic forces by packing + and – charges. Lattice energy.

Formal oxidation number

Bonds in compounds are not ionic in the sense that all valence electrons are not entirely shifted to the anion.

But if the bonding is broken – as when something, like a defect, moves – the electrons have to stay or go. Electrons can’t split in half.

And mostly they go with the anion - the most electronegative atom.

That is why the ionic model is useful in defect chemistry and transport

And it is why it is very useful to know and apply the rules of formal oxidation number, the number of charges an ion gets when the valence electrons have to make the choice

Bonding – some important things to note

Metallic bonding (share of electrons) and ionic bonding (packing of charged spheres) only have meaning in condensed phases.

In most solids, any one model is only an approximation:

◦Many covalent bonds are polar, and give some ionic character or hydrogen bonding.

◦ Both metallic and especially ionic compounds have covalent contributions

In defect chemistry, we will still use the ionic model extensively, even for compounds with little degree of ionicity.

It works!

…and we may understand why.

Formal oxidation number rules

Fluorine (F) has formal oxidation number -1 (fluoride) in all compounds.

Oxygen (O) has formal oxidation number -2 (oxide) , -1 (peroxide) or -1/2 (superoxide), except in a bond with F.

Hydrogen (H) has oxidation number +1 (proton) or -1 (hydride).

All other oxidation numbers follow based on magnitude of electronegativity (see chart) and preference for filling or emptying outer shell (given mostly by group of the periodic table).

Point defects

Kröger-Vink notation

CSA

We will now start to consider defects as chemical entities

We need a notation for defects. Many notations have been in use. In modern defect chemistry, we use Kröger-Vink notation (after Kröger and Vink). It describes any entity in a structure; defects and “perfects”. The notation tells us

What the entity is, as the main symbol (A)◦ Chemical symbol

◦ or v (for vacancy)

Where the entity is, as subscript (S)◦ Chemical symbol of the normal occupant of the site

◦ or i for interstitial (normally empty) position

Its charge, real or effective, as superscript (C)◦ +, -, or 0 for real charges

◦ or ., /, or x for effective positive, negative, or no charge

Note: The use of effective charge is preferred and one of the key points in defect chemistry. ◦ We will learn what it is in the following slides

Effective charge

-2iO

The effective charge is defined as

the charge an entity in a site has

relative to (i.e. minus)

the charge the same site would have had in the ideal structure.

Example: An oxide ion O2- in an interstitial site (i)

Real charge of defect: -2

Real charge of interstitial (empty) site in ideal structure: 0

Effective charge: -2 – 0 = -2

//iO

Effective charge – more examples

////Zrv

Example: An oxide ion vacancy

Real charge of defect (vacancy = nothing): 0

Real charge of oxide ion O2- in ideal structure: -2

Effective charge: 0 – (-2) = +2

Example: A zirconium ion vacancy, e.g. in ZrO2

Real charge of defect: 0

Real charge of zirconium ion Zr4+ in ideal structure: +4

Effective charge: 0 – 4 = -4

Ov

Kröger-Vink notation – more examples

/ZrY

Dopants and impurities

Y3+ substituting Zr4+ in ZrO2

Li+ substituting Ni2+ in NiO

Li+ interstitials in e.g. NiO

Electronic defects

Defect electrons in conduction band

Electron holes in valence band

/NiLi

iLi

/eh

Kröger-Vink notation – also for elements of the ideal structure (constituents)

xMgMgCations, e.g. Mg2+ on normal Mg2+ sites in MgO

Anions, e.g. O2- on normal site in any oxide

Empty interstitial site

xOO

xiv

Kröger-Vink notation of dopants in elemental semiconductors, e.g. Si

xSiSi Silicon atom in silicon

Boron atom (acceptor) in Si

Boron in Si ionised to B-

Phosphorous atom (donor) in Si

Phosphorous in Si ionised to P+

xSiB

/SiB

xSiP

SiP

Protonic defects

Hydrogen ions, protons H+ , are naked nuclei, so small that they can not escape entrapment inside the electron cloud of other atoms or ions

In oxidic environments, they will thus always be bonded to oxide ions –O-H

They can not substitute other cations

In oxides, they will be defects that are interstitial, but the interstitial position is not a normal one; it is inside an oxide ion.

With this understanding, the notation of interstitial proton and substitutional hydroxide ion are equivalent.

OOH

iH

A few tips: Defects and charges are done seemingly a little different in elemental

semiconductors and ionic solids◦ The donor and acceptor dopants are by tradition entered in doping reactions

neutral in the former and effectively charged (ionised to their preferred valency) in the latter. Don’t let it confuse or disencourage you.

◦ Physicists use + and – for effective and real charges alike, and actually don’t differentiate them much. Don’t let physicists confuse or disencourage you , and be kind with them .

Don’t mix real and effective charges in one reaction equation or electroneutrality consideration.◦ Use effective charges only in defect chemistry, which can only refer to one single

phase.

◦ Use real charges in all cases of exchange of charge between phases, like in electrochemistry.

I use v and i for vacancy and interstitial, while Kröger and Vink (and most of the rest still) use V and I.

Electroneutrality

Electroneutrality

One of the key points in defect chemistry is the ability to express electroneutrality in terms of the few defects and their effective charges and to skip the real charges of all the normal structural elements

positive charges = negative charges

can be replaced by

positive effective charges = negative effective charges

positive effective charges - negative effective charges = 0

Electroneutrality

0]z[ s

zs

The number of charges is counted over a volume element, and so we use the concentration of the defect species s multiplied with the number of charges z per defect

Example, oxide MO with oxygen vacancies, acceptor dopants, and defect electrons:

If electrons dominate over acceptors, we can simplify:

Note: These are not chemical reactions, they are mathematical relations and must be read as that. For instance, in the above: Are there two vacancies for each electron or vice versa?

][e][A]2[vor 0][e-][A]2[v MOMO////

][e]2[v /O

Examples of some important defect chemical reactions

Stoichiometric compounds – intrinsic disorders

Disorders that do not exchange mass with the surroundings, and thus do not affect the stoichiometry of the compound.

Schottky disorder in MO

xO

xMO

//M

xO

xM OMvvOM

O//M vv0or, equivalently:

new structural unit

M2+

O2-

Frenkel disorder in MO

M2+

O2-

i//M

xi

xM MvvM

Anion (anti-)Frenkel disorder in MO

M2+

O2-

//iO

xi

xO OvvO

Intrinsic electronic ionisation

/cv

xc

xv ehhe

hee /x

he0 /

Three equivalent reaction equations:

Consider charges, electrons and sites:

Simpler; skip sites:

Simplest; skip valence band electrons:

Valence defects – localised electrons and holes

M/M

xM MMM2

Fe/Fe

xFe FeFeFe2

/TiFe

xTi

xFe TiFeTiFe

Example: Ilmenite FeTiO3

Example: Fe2O3

Nonstoichiometric compounds – exchange of components with the surroundings

Disorders that exchange mass of one of the components with the surroundings, and thus change the stoichiometry of the compound.

We will take the first one – oxygen deficiency – in small steps, then the other ones more briefly.

Oxygen deficiency

)g(OvO 221x

OxO

)g(O)s(MO)s(MO 22y

y1

The two electrons of the O2- ion are shown left behind

More realistic picture, where the two electrons are delocalised on neighbouring cations

“Normal” chemistry:

Defect chemistry:

)g(O)s(MO)s(MO 22y

y22

Oxygen deficiency

)g(OvO 221x

OxO

The two electrons of the O2- ion are shown left behind

The two electrons are loosely bonded since the nuclear charge of the former O2- ion is gone. They get a high energy close to the state of the reduced cations…the conduction band. The vacancy is a donor.

Ionisation of the oxygen vacancy donor

/O

/O

xO e2vevv

Electrons excited to conduction band delocalised over entire crystal, mainly in orbitals of reduced cation

Oxygen deficiency – overall reaction

)g(Oe2vO 221/

OxO

Defect reactions involving foreign elements

Substituents

Dopants

Foreign elements; some terminology

Foreign elements are often classified as ◦ impurities – non-intentionally present◦ dopants – intentionally added in small amounts ◦ substituents – intentionally substituted for a host component

(we tend to call it all doping and dopants)

They may dissolve interstitially or substitutionally

Substitutionally dissolved foreign elements may be

◦ homovalent – with the same valency as the host it replaces

◦ heterovalent – with a different valency than the host it replaces.

Also called aliovalent

Heterovalent metals Higher valent metals will sometimes be denoted Mh (h for higher valent). Lower valent metals will sometimes be denoted Ml (l for lower valent).

Doping of semiconductors

In covalently bonded semiconductors, the valence electrons will strive to satisfy the octet rule for each atom.

As example, we add P or B to Si.

Si has 4 valence electrons and forms 4 covalent bonds.

Phosphorous P has 5 valence electrons. When dissolved in the Si structure it thus easily donates its extra electron to the conduction band in order to become isoeletronic with Si.

Boron B has 3 valence electrons. When dissolved in the Si structure it thus easily accepts the lacking electron from the valence band in order to become isoeletronic with Si.

Doping of semiconductors

Doping of Si with P (donor) or B (acceptor)

/Si

xSi ePPP hBBB /

SixSi

Doping of ionic compounds: M1-xO doped substitutionally with Mh2O3

Mh3+ substituting M2+ will constitute donor-doping, giving effectively positive dopants.

M1-xO contains M vacancies and electron holes.

The doping may thus be compensated by producing M vacancies:

or – less relevant - by consuming electron holes. This is a reduction reaction and releases oxygen:

xO

//MM32 O3vMh2OMh

)g(OO2Mh2h2OMh 221x

OM32

Ni1-xO doped substitutionally with Li2O

h2O2Li2)g(O)s(OLi xO

/Ni22

12

xO

/Ni

//Ni2 OLi2v)s(OLi

Li+ and Ni2+ are similar in size, so Li+ may substitute Ni2+. This will constitute acceptor-doping with effectively negative dopants. (This is utilised in Li-doped NiO for p-type conducting electrodes for fuel cells, batteries etc.)

Ni1-xO contains nickel vacancies and electron holes.

The doping may thus be compensated by consuming Ni vacancies

or – better - by producing electron holes. This is an oxidation reaction and requires uptake of oxygen

ZrO2-y doped substitutionally with Y2O3

Y3+ will form effectively negative defects when substituting Zr4+ and thus acts as an acceptor. It must be compensated by a positive defect.

ZrO2-y contains oxygen vacancies and electrons

The doping is thus most relevantly written in terms of forming oxygen vacancies:

xOO

/Zr32 O3vY2OY

ZrO2-y doped substitutionally with Y2O3

xOO

/Zr32 O3vY2OY

Note: Electrons donated from oxygen vacancy are accepted by Y dopants; no electronic defects in the bands.

Hydration – dissolution of protons from H2O

OxOO2 OH2Ov)g(OH

Water as source of protons. Equivalent to other oxides as source of foreign elements.

Example: Hydration of acceptor-doped MO2, whereby oxygen vacancies are annihilated, and protons dissolved as hydroxide ions.

The acceptor dopants are already in, and are not visible in the hydration reaction in this case

Ternary and higher compounds

With ternary and higher compounds the site ratio conservation becomes a little more troublesome to handle, that’s all.

For instance, consider the perovskite CaTiO3. To form Schottky defects in this we need to form vacancies on both cation sites, in the proper ratio:

And to form e.g. metal deficiency we need to do something similar:

(But oxygen deficiency or excess would be just as simple as for binary oxides, since the two cations sites are not affected in this case …)

O////Ti

//Ca 3vvv0

h6O3vv)g(O xO

////Ti

//Ca22

3

Doping of ternary compounds

The same rule applies: Write the doping as you imagine the synthesis is done: If you are doping by substituting one component, you have to remove some of the component it is replacing, and thus having some left of the other component to react with the dopant.

For instance, to make undoped LaScO3, you would probably react La2O3 and Sc2O3 and you could write this as:

Now, to dope it with Ca2+ substituting La3+ you would replace some La2O3 with CaO and let that CaO react with the available Sc2O3:

The latter is thus a proper doping reaction for doping CaO into LaScO3, replacing La2O3.

xO

xSc

xLa322

1322

1 O3ScLaOScOLa

O21x

O25x

Sc/La322

1 vOScCaOScCaO

Back to where we started: Conductivity

Product of charge, charge mobility, and concentration

s can be a constituent or a defect

ssss ceuzσ

Which ions can we use? Let us look at what are the fuels for fuel cells

H2

CH4

Diesel

CH3OH

C2H5OH

CH3OCH3

NH3

They will all be reformed or cracked into H2 externally or at the anode catalyst.

H2 is effectively the fuel at all fuel cell anodes – to be oxidised to H2O by O2 in air, via an ionic conduction transport

Fuel cells – types according to electrolyte

Aqueouspolymer

Aqueous Molten Solid

SOFC O2-

PCFC H+

MCFC CO32-

AFC OH-

PAFC H3O+

PEMFC H3O+

PC-SOFC (PCFC)BaCe0.9Y0.1O3-d

SAFCCsHSO4

HT-PEMFCPhosphonated PBI

No solid OH- conductors

No good H+ conductors among hydroxides

No solid OH- conductors

No good H+ conductors among hydroxides

Conductivity of electrolyte

Must be fully ionic tion > 0.99

Requires large band gaps, typically > 3 eV

Preferably > 0.01 S/cm

Requires ionic disorder (high concentration, high mobility)◦ Liquid state (aqueous solution, molten salt)

◦ Solid state crystals with intrinsic disorder

◦ Solid state doped crystals

Conductivity requirements for the electrodes

The job is to transport electrons

But transport of ions is also welcome

What kind of materials can we use?

Conductivity requirements for the interconnects

The job is to transport electrons only

What kind of materials can we use?

Stability

What are the stability issues for our three materials classes?

Electrolyte

Electrodes◦ Anode

◦ Cathode

Interconnect

Microstructure

What are the microstructural requirements for our three materials classes?

Electrolyte

Electrodes◦ Anode

◦ Cathode

Interconnect

How can we now refine the selection criteria for

Electrolyte?

Electrode?

Interconnects?

Any suggestions for actual materials?

At 80°C for the PEMFCAt 800°C for the SOFC?