Transport of matter and charge in solids

Ilan Riess

(1)

Outline of todays talk (1): • Introduction to applications.

• Point defects in crystalline solids.

• Example: reduced ceria CeO 2-d . • Review of the contribution of point defects to the solid properties. • Equilibrium Properties: Lectures co-financed by the European Union in scope of the European Social Fund

Applications to be understood: 1. Fuel cells (FCs), 2. Solid state batteries, 3. Smart windows 4. High temperature electrolyzers, 5. Hydrocarbon generation 6. Oxygen separation membranes, 7. Memristors. While the first 6 topics are energy related the last one refers to computer memory chips. What is the connection between all of them?

Let us start with the last one !

Lectures co-financed by the European Union in scope of the European Social Fund

1 Memristive devices as memory elements in computers:

a. Originally memory elements were small magnets in the form of rings, 0 meant magnetization in one direction, 1 meant magnetization in the opposite direction.

b. Common elements today use transistors of the MOS (metal - oxide (insulator) semiconductor) type with the controlling signal being changed by charging an electrode buried in the insulator.

c. The industry is looking for faster, less energy consuming, physically smaller and long memory elements. d. One solution is to switch the state of a nano size solids between crystalline and amorphous state, provided the two states have significantly different resistance.

e. A more recent and more promising option is to change the resistance of a nano size solid by changing its composition. We concentrate on those. Lectures co-financed by the European Union in scope of the European Social Fund

Macroscopic example: Ag + • Ag dendrites are growing from the ion blocking electrode, gold, towards the silver electrode in As 2 S 3 . ( Y. Hirose, H. Hirose, J. Appl. Phys.

47

(1976) 2767 ).

• Once a full filament is formed short circuiting the electrodes, the resistance drops abruptly.

• Sizes: of the order of a mm.

Lectures co-financed by the European Union in scope of the European Social Fund

Example of the I-V relations with switching for a nano element consisting of the simple structure: metal1|oxide|metal2 Pt|TaO x |Ta J.J. Yang et al., APL, 97 (2010) 232102 • When the current is stopped the high or low resistance is maintained for a long time. • It can be probed using a low voltage (V<< 1V).

Lectures co-financed by the European Union in scope of the European Social Fund

These results raises a few questions:: 1) The oxide is an insulator (E gap ~4eV) how can it exhibit a current at room temperature?

2) Change in composition is responsible for hysteresis and memory. How can this happen?

3) What does “x” (x<<1) in TaO x mean? Is the composition not dictated by simple laws of chemistry

:

The ratio between the oxygen and metal in a compound has to be a simple number like Ta 2 O 5 ?

4) Last and not least how does a memory effect arise?

Lectures co-financed by the European Union in scope of the European Social Fund

Answers: Let us start with question (3), what does “x” (x<<1) mean: • The composition of a compound need not follow the rule mentioned before. Oxygen can leave an oxide. • For example is CeO 2 large changes in oxygen concentration up to ~10% are possible (to ~CeO 1.8

) at T> 450 o C without a phase change.

Ricken et al., J. Solid State Chem., 54 (1984) 89.

450 o C Lectures co-financed by the European Union in scope of the European Social Fund

• How is charge balanced if the ionic charges are Ce 4+ and O 2 ? • The missing oxygen ion is replaced by two electrons !

The ion O 2 leaves the oxide as a neutral atom joining another oxygen atom to form an O 2 molecule in the gas phase. The two electrons of the ion are left behind in the oxide.

• At elevated temperature the two electrons reside in the conduction band leading to increased electronic conductivity.

• At low temperature the two electrons reside on certain Ce ions turning their valence from 4+ to 3+ (Ce 4+ → Ce 3+ ) ! • The oxygen ions must move within the oxide to reach the surface.

• If they can move then they move also under an applied voltage.

Lectures co-financed by the European Union in scope of the European Social Fund

Why does the nano size play a role?

• The voltage applied is less than 10 Volts.

The (average) gradient on a 1cm solid is: E = 10 Volt/cm on a 10nm solid it is: E = 10 7 Volt/cm i.e. the driving force is very high.

• The distance the ions and electrons have to propagate is significantly shorter.

Conclusion: • Conduction of ions and electrons (and holes) in the nano size insulator (oxide) is possible at room temperature.

• This conduction is also the basis for the hysteresis and the memory.

• Ionic motion is also the feature common to the other applications discussed.

• During the course we learn to understand all this.

Lectures co-financed by the European Union in scope of the European Social Fund

2. Fuel cells for efficient energy conversion (chemical to electrical)

• Seemingly a very different topic is energy conversion, but they are close.

• E.g. the chemical energy contained in the oxidation reaction of H 2 as fuel, • 2H 2 + O 2 → 2H 2 O • If we burn hydrogen (H 2 ) only heat is generated. When converted to mechanical energy the efficiency is low due to an upper limit set by Carnot’s theorem. • In reality the efficiency of an internal combustion engine is much less than that upper limit (~25% for cars, ~ 45% for large diesel engines).

• A direct conversion of chemical energy to electrical at constant T could be much more efficient.

• It needs to be an electro-chemical reaction ! This is achieved in fuel cells.

• In a solid FC oxygen ions or protons move in a solid membrane.

Lectures co-financed by the European Union in scope of the European Social Fund

Example of applications of fuel cells (FC): • Fuel cells are considered for use on a wide range of applications: cars, stationary generators, cars, submarines, etc.

• To drive cars: replace the internal combustion engine by a FC plus electrical motors Electrical motor Fuel tank Fuel cell Lectures co-financed by the European Union in scope of the European Social Fund

• The world’s first submarine to be equipped with this unique propulsion system has already impressively proved its operating efficiency in extensive trials in a Germany Navy submarine.

• Over 15 PEM-FC submarines have been ordered by four countries.

Lectures co-financed by the European Union in scope of the European Social Fund

• Principle of operation of a fuel cell (FC):

Fuel

SE

Anode

-

+

Cathode Motor

• Example: H 2 as fuel and an oxygen ion conductor:

Fuel (H

2

) H

2

+O

2-

H

2

O + 2e

-

O

2-

O

2

+4e

-

2O

2 -

e

-

+

Motor Air Air

Lectures co-financed by the European Union in scope of the European Social Fund

3. Solid state batteries:

A solid state battery is an electrochemical device which converts the energy of a chemical reaction to electrical energy (and. depending on working conditions also to heat).

It is basically similar to a fuel cell the difference being that the chemical do not contain so called “fuel” but other reactants.

They are analogous to “regular” batteries, except that the membrane that conducts ions is a solid.

Battery Na Na + + e e Na +

+

2Na + + S Na 2 S Motor Fuel (H 2 ) Fuel cell H 2 +O 2 H 2 O + 2e e O 2 Motor

+

O 2 +4e 2O 2 Air Lectures co-financed by the European Union in scope of the European Social Fund

4. Smart windows:

• Smart windows are those the color of which can be changed by an electrical signal.

More than a matter of fashion or art it is aimed at shading the room from strong sunlight.. • The glass of the window is coated by thin layers that from a solid electrochemical system which can change color. • The principle of operation of that system is similar to that of a solid state battery, with a special choice of materials: The anode and solid electrolyte are transparent while the cathode can change color when it reacts with the ions that arrive. E.g. xLi + WO 3 → Li x WO 3 . WO 3 is transparent. For x>x c Li x WO 3 becomes dark blue, quite opaque. + Li Li +

Li x WO 3

Lectures co-financed by the European Union in scope of the European Social Fund

5 High temperature electrolyzers a. Water electrolyzer:

• One can take an solid oxide fuel cell (SOFC) and instead of extracting electrical energy from it investing fuel, one can invest electrical energy and decompose H 2 O. H 2 O H 2 Water electrolysis H 2 O + 2e O 2 H 2 +O 2 e -

+

2O 2 4e + O 2 V>V th V O 2 Fuel (H 2 ) Fuel cell H 2 +O 2 H 2 O + 2e e O 2 Motor

+

O 2 +4e 2O 2 Air

Lectures co-financed by the European Union in scope of the European Social Fund

b. CO 2 electrolyzers:

Similar to water electrolysis where H 2 O is decomposed into H 2 and O 2 , CO 2 can also be decomposed into CO and O 2 , CO 2 CO CO 2 electrolysis CO 2 + 2e O 2 CO+O 2-

+

2O 2 4e + O 2 e V>V th V O 2 Lectures co-financed by the European Union in scope of the European Social Fund

6 Hydrocarbon generation 1

• Once H 2 or CO are generated one can use an ex situ chemical reaction to produce an hydrocarbon molecule, e.g.

CO 2 +3H 2 → CH 3 OH + H 2 O • One can also use a reaction in situ (in the electrochemical cell) to generate the hydrocarbon directly from H 2 O + CO 2 , CO 2 + 2H 2 O → CH 3 OH + 3/2O 2 _________________ 1 Olah et al., J. Am. Chem. Soc.,133 (2011) 12881.

Lectures co-financed by the European Union in scope of the European Social Fund

7. Separating oxygen from air via a mixed-ionic-electronic-conductor (MIEC):

100% O Low side 2 pressure .

O 2-

Air, high pressure side

e -

• The driving force is a difference in the oxygen partial pressure. • No electrodes required.

• We use an MIEC membrane in the form of thin tubes one end closed

:

Oxygen filter, ceramic permeation membrane in the form of tubes Furnace Pump Pressurized Air

O 2 N 2

100% O 2 Oxygen depleted air Lectures co-financed by the European Union in scope of the European Social Fund

The rest of the course is aimed at understanding the different systems presented before.

Taking fuel cells which is a good example for all cases (as we shall see): • We shall learn how the oxygen ions are generated at one electrode, • react with the fuel at the opposite electrode, • exchange charge with the electrodes • are transported via the solid membrane. • We shall discuss the mechanisms, the driving forces that activate the process and the charge carrier distributions response.

Fuel (H 2 ) Fuel cell H 2 +O 2 H 2 O + 2e e O 2 Motor

+

O 2 +4e 2O 2 Air Lectures co-financed by the European Union in scope of the European Social Fund

Currents in solids

• The currents discussed are the ionic one and the electronic (electron/hole) one.

• For that we have to discuss: • the charge carrier nature • way of generation • concentration • mechanism of propagation • driving force that activate the motion of the charge carrier.

Lectures co-financed by the European Union in scope of the European Social Fund

Types of ionic point defects in crystalline solids

a. Why point defects at all?

b. Native, defects in an elemental solid A: vacancy, V A

;

interstitial, A i .

A A A A A A A A A A

V A

A A

V A

A A A A A A A i A A A A A A A A Ion/atom at the surface layer Lectures co-financed by the European Union in scope of the European Social Fund

c. Native, defects in a solid, binary, compound A n B m.

(e.g. AB): vacancy, V A , V B

;

interstitial, A i , B i

;

and misplaced, A B , B A .

A B A A B A B

V A

B A B A

V B

A B

V A

B A B A A B A i B B A A B A B A B A B A B Ion at the surface layer Lectures co-financed by the European Union in scope of the European Social Fund

d. Extrinsic defects in a binary compound: C A , C B , C i favored type of interstitial site, otherwise: C i,1 , Ci 2 …) (assuming there is only one A B A A B A B C A B A B A C B A B A B A B A A B C i A B B A B A B A B Lectures co-financed by the European Union in scope of the European Social Fund

Transport of matter and charge in solids

Ilan Riess

(2)

Outline of todays talk (2): • Electronic point defects: electrons, holes, hopping electrons in a defect band..

• Kröger-Vink notation of point defects • Example: reduced ceria CeO 2-d , YSZ.

• Example: Bi 2 O 3 .

• Example: Ag 2 S • Review of the contribution of point defects to the solid properties. • Ways to generate point defects: thermal excitation, doping, change of stoichiometry (and combination of them).

Lectures co-financed by the European Union in scope of the European Social Fund

Ricken et al., J. Solid State Chem., 54 (1984) 89.

450 o C Lectures co-financed by the European Union in scope of the European Social Fund

Equilibrium Properties: • Generation of electronic and ionic point defects in the bulk.

• Possible defect models.

• Limited solubility, solubility limit.

• Self-compensation during doping.

• Solid electrolytes and mixed ionic electronic conductors.

• The chemical potential, μ, of chemical components and of charged defects, electrochemical potential, , and electrical potential  . • Equilibrium conditions.

• Reaction between point defects and the corresponding mass action law. • Stoichiometric changes.

Lectures co-financed by the European Union in scope of the European Social Fund

Transport of matter and charge in solids

Ilan Riess

(3)

Outline of todays talk (3): • Possible defect models. Example Cu 2 O • Equilibrium vs. steady state.

• Properties of the equilibrium state. • Solubility limit.

• Self-compensation during doping.

• Solid electrolytes and mixed ionic electronic conductors.

• The chemical potential, μ, of chemical components and of charged defects, electrochemical potential, , and electrical potential • Equilibrium conditions.

  e  . • Reaction between point defects and the corresponding mass action law.

• Frenkel pairs and electron-holes production..

Lectures co-financed by the European Union in scope of the European Social Fund

References:

1. M. Ricken, J. Nölting and I. Riess, Specific Heat and Phase Diagram of Nonstoichiometric Ceria (CeO 2 ), J. Solid State Chem. 54, 89-99 (1984).

2. Y. Tsur and I. Riess, Self Compensation in Semiconductors, Phys. Rev. B 60, 8138-8146 (1999).

3. O. Porat and I. Riess, Defect Chemistry of Cu 2-y O at Elevated Temperatures. Part II. Electrical Conductivity, Thermoelectric Power and Charged Point Defects, Solid State Ionics 81, 29-41 (1995).

Lectures co-financed by the European Union in scope of the European Social Fund

Transport of matter and charge in solids Ilan Riess

(4)

• Mass action law, O 2 + H 2 . Stoichiometry change of an oxide. Transport in the bulk: • The conditions for ionic motion. • The motion of ions.

• Motion of electrons, small polarons, hopping in a defect band.

• Driving forces.

• The current density equations. • The I-V relations for an MIEC with one mobile ionic defects X + and electrons e and R e =R e (V, μ X,L , μ X,0 ), R i =R i (V, μ X,L , μ X,0 ) Lectures co-financed by the European Union in scope of the European Social Fund

1. I. Riess,

Current-Voltage Relation and Charge Distribution in Mixed Ionic Electronic Solid Conductors

, J. Phys. Chem. Solids,

47

, 129-138 (1986).

2. I. Riess,

Electrochemistry of Mixed Ionic-Electronic Conductors

, in: CRC Handbook of Solid State Electrochemistry, P.J. Gellings and H.J.M. Bouwmeester, eds. CRC Press, Inc., 1997, pp. 223-268.

3. I. Riess, D. Kalaev and J. Maier,

Currents under high driving forces

, Solid State Ionics

251

(2013) 2.

Lectures co-financed by the European Union in scope of the European Social Fund

Transport of matter and charge in solids Ilan Riess

(5)

Transport (Cont) • Example: the current density equations needed for Gd doped CeO 2-x . • I-V relations and defect distribution of the former example.

• Further, example calculations of I-V relations for another defect model.

• Defect distribution and I-V relations when taking also into consideration the space charge and electrodes contact potentials. • Electrode impedance.

Lectures co-financed by the European Union in scope of the European Social Fund

References: 1. I. Riess,

Voltage Controlled Structure of Certain p-n and p-i-n Junctions

, Phys. Rev. B35, 5740-5743 (1987).

2. D. Kalaev and I. Riess,

Rectification in solid state devices under odd conditions due to motion of ionic defects

, Solid State Ionics,

212

, 26-42 (2012).

3. I. Riess,

On the Single Chamber Solid Oxide Fuel Cells

, J. Power Sources,

175

, 325-337 (2008). Lectures co-financed by the European Union in scope of the European Social Fund

Transport of matter and charge in solids Ilan Riess

(6)

• Boundary conditions.

• The equations to be added to time varying experiments, the continuity equations and relaxation equations of internal reactions.

• Initial conditions added in time varying experiments.

• Hopping conduction in an defect band.

• Revisiting the applications mentioned in the beginning and understanding how they function.

Lectures co-financed by the European Union in scope of the European Social Fund

 X,0 ext  V th,A  V A  W,A V th =-(  X,L ext - X,0 ext )/zq V=V ext  V th,MIEC  V MIEC  W,MIEC MIEC I →  V th,C  V C  W,C  X,L ext I → V ext Lectures co-financed by the European Union in scope of the European Social Fund

1. I. Riess, M. Gödickemeier and L.J. Gauckler,

Characterization of Solid Oxide Fuel Cells Based on Solid Electrolytes or Mixed Ionic Electronic Conductors

, Solid State Ionics, 90, 91-104 (1996).

2. I. Riess and A. leshem,

Odd rectification, hysteresis and quasi swtching in solid state devices based on mixed ionic electrnic conductors

, Solid State Ionics,

225

, 161-165 (2012).

3. D. Kalaev and I. Riess,

On conditions leading to crossing of I –V curve in metal1|mixed-ionic –electronicconductor|metal2 devices

, Solid State Ionics,

241

(2013) 17 –24.

4. Y. Gil, Y. Tsur, O.M. Umurhan and I. Riess,

Properties of Solid State Devices with Significant Impurity Hopping Conduction

, J. Phys. D. Appl. Phys.,

41

, 135106 (2008).

Lectures co-financed by the European Union in scope of the European Social Fund

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