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 CeO2-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 !
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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.
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Macroscopic example:
Ag+
• Ag dendrites are growing from the ion blocking electrode, gold, towards the silver
electrode in As2S3. (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.
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Example of the I-V relations with switching for a nano element consisting of the
simple structure: metal1|oxide|metal2
Pt|TaOx|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).
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These results raises a few questions::
1) The oxide is an insulator (Egap~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 TaOx 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 Ta2O5 ?
4) Last and not least how does a memory effect arise?
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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 CeO2 large changes in oxygen concentration up to ~10% are
possible (to ~CeO1.8) at T> 450oC without a phase change.
Ricken et al., J. Solid
State Chem., 54 (1984)
89.
450oC
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• How is charge balanced if the ionic charges are Ce4+ and O2- ?
• The missing oxygen ion is replaced by two electrons !
The ion O2- leaves the oxide as a neutral atom joining another oxygen atom
to form an O2 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+ (Ce4+ → Ce3+) !
• The oxygen ions must move within the oxide to reach the surface.
• If they can move then they move also under an applied voltage.
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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 = 107 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.
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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 H2 as fuel,
• 2H2 + O2 → 2H2O
• If we burn hydrogen (H2) 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.
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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
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• 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.
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• Principle of operation of a fuel cell (FC):
Fuel
Air
SE
Anode
Cathode
-
+
Motor
• Example: H2 as fuel and an oxygen ion conductor:
Fuel (H2)
Air
H2+O2H2O + 2e-
-
O2+4e2O2-
O2+
eMotor
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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-
+O2-
Na+
2Na+ + S
Na2S
+
e-
Fuel cell
Fuel (H2)
H2
H2O + 2e-
e-
O2+4e2O2-
O2-
+
-
Air
Motor
Motor
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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 + WO3 → LixWO3 .
WO3 is transparent. For x>xc LixWO3 becomes dark blue, quite opaque.
+
Li
Li+
LixWO3
-
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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 H2O.
Water electrolysis
H2O
H2
H2O + 2e-
O2-
2O24e- + O2
H2+O2-
-
Fuel cell
Fuel (H2)
+
H2+O2H2O + 2e-
O2
e-
O2+4e2O2-
O2-
+
-
Air
Motor
e-
V>Vth
V
Lectures co-financed by the European Union in scope of the European Social
b. CO2 electrolyzers:
Similar to water electrolysis where H2O is decomposed into H2 and O2, CO2 can
also be decomposed into CO and O2,
CO2
CO
CO2 electrolysis
CO2 + 2e-
O2-
2O24e- + O2
CO+O2-
-
O2
+
e-
V>Vth
V
Lectures co-financed by the European Union in scope of the European Social Fund
6 Hydrocarbon generation1
• Once H2 or CO are generated one can use an ex situ chemical reaction to
produce an hydrocarbon molecule, e.g.
CO2+3H2 → CH3OH + H2O
• One can also use a reaction in situ (in the electrochemical cell) to generate the
hydrocarbon directly from H2O + CO2,
CO2 + 2H2O → CH3OH + 3/2O2
_________________
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% O2.
Low
pressure
side
O2-
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:
Furnace
Pressurized Air
O2
N2
Oxygen filter, ceramic permeation
membrane in the form of tubes
Pump
100% O2
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 cell
Fuel (H2)
H2+O2H2O + 2e-
e-
Air
O2+4e2O2-
O2-
+
Motor
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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.
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Types of ionic point defects in crystalline solids
a. Why point defects at all?
b. Native, defects in an elemental solid A: vacancy, VA ; interstitial, Ai.
A
A
A
A
A
A
A
VA
A
A
A
A
A
A
A
A
A
A
Ai
A
A
A
VA
A
A
A
A
A
A
Ion/atom
at the
surface
layer
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c. Native, defects in a solid, binary, compound AnBm. (e.g. AB): vacancy, VA, VB ;
interstitial, Ai, Bi ; and misplaced, AB, BA.
A
B
A
B
A
AB
A
B
A
VB
A
B
A
B
A
B
A
B
A
B
A
B
BA
B
A
B
Ai
B
VA
B
A
B
VA
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: CA , CB , Ci (assuming there is only one
favored type of interstitial site, otherwise: Ci,1, Ci2…)
A
B
A
B
A
B
A
B
A
CB
A
B
A
B
A
B
A
A
B
B
Ci
B
CA
B
A
B
A
A
B
B
A
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 CeO2-d, YSZ.
• Example: Bi2O3.
• Example: Ag2S
• 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.
450oC
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 Cu2O
• 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 .
• What does a voltmeter measure, e?
• Equilibrium conditions.
• 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 (CeO2), 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 Cu2-yO 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, O2 + H2. 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 eand Re=Re(V,μX,L,μX,0), Ri=Ri(V,μX,L,μX,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.
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Transport of matter and charge in solids
Ilan Riess
(5)
Transport (Cont)
• Example: the current density equations needed for Gd doped CeO2-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.
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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
Vth=-(X,Lext-X,0ext)/zq
Vth,A
V=Vext
Vth,MIEC
Vth,C
VA
VMIEC
VC
W,MIEC
I→
Anode
X,0ext
MIEC
W,C
Cathode
W,A
X,Lext
I→
Vext
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
Haifa by night