Page 1
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 1, 11.02.2006Nonlinear Resistors.ppt,
Nonlinear Resistors Introduction (1)
spinells e.g. (Ni,Mn)3O4
σ ≈ 10-2 S/mhopping condution in metal oxidesself heating effects
NTC
n-doped BaTiO3σ ≈ 102...103 S/m
grain boundary phenomena in semiconducting ferroelectric ceramics self heating effects
PTC
n-doped ZnOgrain boundary phenomena in semiconducting ceramicsy
varistor
materialphysical effecttype
[Schaumburg 1990]
Thermistor: thermally sensitive resistorNTC: negative temperature coefficientPTC: positive temperature coefficientVaristor: variable resistor
glossary
Page 2
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 2, 11.02.2006Nonlinear Resistors.ppt,
Nonlinear Resistors Introduction (2)
NTC PTC Varistor
PTC-heater withmetalized surface
arrangement of PTC´s and radiators
Page 3
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 3, 11.02.2006Nonlinear Resistors.ppt,
U
VaristorVariable Resistor (Voltage dependent Resistor)
symbol varistor
[www.staratech.com]
Page 4
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 4, 11.02.2006Nonlinear Resistors.ppt,
VaristorNonlinear I/V-Characteristic
varistor: variable resistorvoltage dependent resistorwith symmetrical I/V-chracteristic
I K U α= ± ⋅
nonlinearity coefficient α
geometryparameter K
materials: SiC, ZnOtypical α-values:• SiC: 5...7• ZnO: 30...70
U
I
UK
IKΔUK
ΔIKK
[Heywang 1984]
Page 5
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 5, 11.02.2006Nonlinear Resistors.ppt,
VaristorEquivalent Circuit
[Ein 1982]
equivalentcircuity
equivalent circuit
2 RK
CP
CP
RP
RP
CD
CD
RD
RD
path I path II
CD ≅ 240 nF/cm²ρK ≅ 1 ΩcmρP ≅ 80 kΩcmCP ≅ 90 nF/cm²
varistor leakage current
grain- (RK) and grain boundary (varistor) (RD) resistance RD and CD in parallel
RD and CD are voltage dependent
leakage currentRP, CP in parallel
grain
secondaryphase
grainboundary
microstructure
path I path II
Page 6
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 6, 11.02.2006Nonlinear Resistors.ppt,
y
VaristorDefects and Band-Structure in ZnO-Varistoren
[Heywang 1984 ]
point defects in ZnO(2-dimensional view)
VO•, VO
••: O-vacancies (ED1, ED2) D: donorVZn´, VZn´´: Zn-vacancies (EA1, EA2) A: acceptore´, (n): electron (-concentration)h•, (p): hole (-concentration)
n + [VZn´] + 2[VZn´´] = p + [VO•] + 2[VO
••]
EC
EV
Eg =3,2 eV
EA2 =2,8 eV
EA1 =0,7 eV
ED1 =0,05 eV
ED2 =2,0 eV
band structure with energy levelsof schottky-defects (EA, ED)
VZn´
VO••e´
electroneutrality
Page 7
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 7, 11.02.2006Nonlinear Resistors.ppt,
VaristorGrain Boundary Properties
ZnO-grain controlledcooling process
n
x
processing (sintering temperature ≈ 1300 °C)
n
x
operation (operating temperature < 100 °C)
oxygen vacancy excess
charge carrier: n = [VO•] + 2·[VO
••]
→ high temperature equilibrium:
„frozen“high temperature
equilibriumn - type
low temperature equilibrium only at the
grain boundaries
→ low temperature equilibrium:
[ VZn´ ] + 2[ VZn´´ ] ≈ [ VO• ] + 2[ VO
•• ]
⇒ n <<
ZnO-grain
(+ additionaldoping of grainboundaries)
Page 8
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 8, 11.02.2006Nonlinear Resistors.ppt,
VaristorBand Structure of the Grain Boundary in a ZnO-Varistor (1)
n
pcharge carrierconcentrations
band structure (no voltage applied)
100 nm
„microvaristor“pair of grains
grain boundarysingle barrier layer
charge carrier concentrationat grain boundaries
[Heywang 1984 ]
EC
EV
EF
acceptors
Page 9
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 9, 11.02.2006Nonlinear Resistors.ppt,
Varistor Band Structure of the Grain Boundary in a ZnO-Varistor (2)
100 nm
> 3 eV
no voltage applied voltage applied
[Fasching 1994]
EC EC
EV EV
EF EF
acceptors tunneling effect
thermalactivation
Page 10
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 10, 11.02.2006Nonlinear Resistors.ppt,
VaristorSummary
• varistor effect in doped ZnO-ceramic is related to grain boundary properties, ZnO-grains exhibit a high conductivity
• varistor effect only at grains exhibiting a direct contact
• varistor effect starts at a voltage of ~ 3,0 ± 0,5 V / grain boundary
• grain boundary forms a potential barrier of 0,6 eV.
• time constants of several nanoseconds
• technical varistor systems: ZnO + Bi2O3 + MnO2 + Co3O4 + Sb2O3 + Cr2O3 + additional additives (up to 10 components to optimize secondary properties)
[Ein 1982 ]
Page 11
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 11, 11.02.2006Nonlinear Resistors.ppt,
VaristorApplication: Overvoltage Protection
logarithmic graph of U(I)• at operating voltage• at overvoltage US = 9,8 KV
UB = 200 V
Ri = 10 Ω
varistor
I
US = 9,8 KV
the voltage peak US(t) increasesthe voltage U for a short time
U(t) = US(t) + UB - Ri · I(t)
consumer
U
without varistor
with SiC-varistor
with ZnO-varistor
operating voltage
current / A →
volta
geU
/ V→
3 %
25 %
100 %
104
103
102
10-1 101 103
[Heywang 1984 ]
Page 12
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 12, 11.02.2006Nonlinear Resistors.ppt,
Varistor
ZnO-varistors varistors for high voltage applications
Page 13
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 13, 11.02.2006Nonlinear Resistors.ppt,
ϑ
PTCPositive Temperature Coefficient Resistor
symbol PTC
[www.atpsensor.com]
Page 14
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 14, 11.02.2006Nonlinear Resistors.ppt,
PTCTemperature Dependency of a Resistor
Rmin, Rmaxminimum and maximum resistance, change of sign of the temperature coefficient
Rmax/Rmineffective resistance increasemax. 107
TBnominal temperatureof the device at whichRB = 2 RminTB–values: -30...+250 °C
TKR: temperature coefficient of resistance , TKR –values of 5...70% K-1= ⋅1
RdRTK
R dT
TB0 100 200 300
RB
Rmin
Rmax
temperature / °C →
resi
stan
ceR
/ Ω
→
100
102
104
106
[Heywang 1984]
PTC-behaviorresistance increasesseveral orders of magnitude
Page 15
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 15, 11.02.2006Nonlinear Resistors.ppt,
PTCDefect Chemistry of Barium-Titanate (BaTiO3)
[ ] // •• •Ba O Ba2 V 2 V Lae ⎡ ⎤ ⎡ ⎤ ⎡ ⎤′ + ≈ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦
/ //Ba 2 Ba
1Ba 2 O ( ) V BaO2
e g+ + ↓ + ↑
system BaTiO3
-
oxygen ion O2-
barium ion Ba2+
--
--
La-doping on the Ba-site
BaLa•
//BaV
OV••
barium vacancies
oxygen vacancies
x /O O 2
1O V 2 O ( )2
e g•• + + ↑
electroneutrality
acceptors donors
Page 16
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 16, 11.02.2006Nonlinear Resistors.ppt,
PTCElectrical Properties of Grain Boundaries in Barium-Titanate
x
dono
r con
cent
ratio
n
acce
ptor
con
cent
ratio
n
grain
insulating grain boundary area
[LaBa•]
2[VBa´´]
n
slow coolingdown
barium vacancy concentration and profile is determined by the temperature profile during cooling down
fast cooling down thinner insulating grain boundary area
charge carrierconcentration
BaTiO3
fast coolingdown
Page 17
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 17, 11.02.2006Nonlinear Resistors.ppt,
PTCCharge Carrier Transport at Grain Boundaries
20
0
[ ]8
Δ = gbB
r
e AE
nε ε
x
E
ΔEB
conductivity (tunneling current)
--
EV
EC
~ exp Δ⎛ ⎞−⎜ ⎟⋅⎝ ⎠BE
k Tσ
n
x
++ +
+
+ ++ +
+
-
---
----
grainn-typedoping
grain boundary (KG) p-type doping
reduced chargecarrier (electrons)concentration
~ exp Δ⎛ ⎞⎜ ⎟⋅⎝ ⎠
Bgb
ERk T
hight of the schottky-barrier Δ WB at the grain boundary
[Agb]: surface charge at the grain boundary
AKG
grain boundary resistance
grainn-typedoping
Page 18
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 18, 11.02.2006Nonlinear Resistors.ppt,
PTCPhase Transition in the PTC Temperature Range
~ exp⎛ ⎞⎜ ⎟⋅⎝ ⎠
Bgb
ERk T
0 100 200 300temperature / °C →
resi
stan
ceR
/ Ω
→
100
102
104
106
PTCNTC NTC
temperature dependency of the grain boundary resistance does not explain the observed temperature dependence.
phase transition in the PTC temperature range
[Siemens]
Page 19
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 19, 11.02.2006Nonlinear Resistors.ppt,
250
PTCCompensation of the GB Potential Barrier due to spontaneous Polarization
T / °C50 150
EBEB
TC
negative grain boundary charge compensated by polarisation charge
→ EB <<
negative grain boundary charge compensated by space charge
→ EB >>
grain boundary
-
---
----
T > TC
Ps
grain boundary
Ps
-
-
-- -
---
-----
+++
++
++
+
---
T < TC
Page 20
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 20, 11.02.2006Nonlinear Resistors.ppt,
PTCEquivalent Circuit
RgrainRgb Rgrain
Cgrain Cgb Cgrain
Ps
grain boundary
Ps
-
-
-- -
---
-----
+++
++
++
+
---
summary
PTC´s consist of polycrystallineferroelectric compounds, mostlymetal-oxides like BaTiO3.
The PTC-effect is based on theinsulating grain boundary (schottky-barrier) and their interaction with the temperature dependent ferroelectric porperties.
The negative grain boundary charge is compensated either by a space charge (paraelectric region → high resistivity) or by a polarisation charge(ferroelectric region → low resistivity).
equivalent circuit
Page 21
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 21, 11.02.2006Nonlinear Resistors.ppt,
PTCApplication
sensors controller heater
heat from the environment self heating effect heat supply
• electrical thermometer
• overtemperature protection
• current stabilization• switch time delay• level monitoring • current measurement
• heating elements self controlled
[Schau 94 / 203]
Page 22
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 22, 11.02.2006Nonlinear Resistors.ppt,
PTCDevices
all purpose PTCoverload
protection
PTC for monitordemagnetization
temperature sensor
heating elements
Page 23
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 23, 11.02.2006Nonlinear Resistors.ppt,
PTCNonlinear I/V-Characteristic due to Self Heating Effect
[Zinke]
Page 24
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 24, 11.02.2006Nonlinear Resistors.ppt,
PTCSelf Controlled Heating Elements for Automotive Application
[Catem]PTC-characteristic
PTC heating elementwith metallized surface
Arrangement of PTC heating elements and metallic radiators
Page 25
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 25, 11.02.2006Nonlinear Resistors.ppt,
PTCPTC for Flow Measurements
elP U I= ⋅ ( )th uP A T Tα= ⋅ ⋅ −
( )u
PTC
A T TI
Uα ⋅ ⋅ −
=
PTC
II
αα
Δ Δ= ~ vα
[Tränkler]
v = flow speedA = PTC surface areaPel = electrical powerU = voltageI = currentPth = thermal powerα = heat transfer coefficientT = temperature of PTCTu = ambient temperature
Page 26
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 26, 11.02.2006Nonlinear Resistors.ppt,
Hopping ConductivityEnergy Band Structure in Solids
single atom solid state
vacuum potential
a0 a0
a
a >> a0
3p3s2p2s
E E
3p
2p3s
2s
discreteenergy levels
smallinteraction
energy bandwith free electrons(metal)
orbitals:discrete
energy levels
a0 lattice parameter
Page 27
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 27, 11.02.2006Nonlinear Resistors.ppt,
Hopping ConductivityTransport Mechanisms
hopping conductivityEven if the thermal energy of the charge carriers is not sufficient for the transport in the conduction band, some materials show an electric conductivity. The transport mechanism is called hopping conductivity because the charge carriers jump from one trap to another.The conductivity depends on the concentration and energetic level of the traps.
transport in the conduction band
hopping condcutivity
[Heywang 1984]
place
Page 28
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 28, 11.02.2006Nonlinear Resistors.ppt,
elements Ti V Cr Mn Fe Co Ni Cu Znvalence states +2 +2 +2 +2 +2 +2 +2 +2
+3 +3 +3 +3 +3 +3 +3+4 +4 +4 +4 +4
+5
Hopping ConductivityValence Exchange
crystal lattice: A2+ B3+2 O2-
4 (spinel)
typical compounds: Ni2+ Mn3+2 O2-
4
Materials: metal oxide e.g. Mn, Fe, Co, Ni, Cu, Zn (transition metals)
Bond type: predominantly or partially heteropolar (ionic) bond(metal: cation, oxygen: anion)
Semiconducting properties can result from valence exchange of the cations in the oxide.Charge transport through hopping-mechanism (valence exchange)
valence exchangehopping conduction
Mn2+ Mn3+ Mn2+ Mn3+- -
thermal activated
Page 29
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 29, 11.02.2006Nonlinear Resistors.ppt,
Hopping Conductivity Transport of Electrons by Valence Exchange
hopping of electrons in Fe2O3
gradientd(eU)/dx
Fe2+Fe3+
Fe3+
Fe3+
place x
pote
ntia
l ene
rgy
W
AE
gradientd(eU)/dx
Fe3+Fe2+
Fe3+
Fe3+
pote
ntia
l ene
rgy
W
place x
AE
For the hopping transport, an activation energy EA is required. For a valency exchange the thermal energy of the electron has to exceed the activation energy.
Page 30
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 30, 11.02.2006Nonlinear Resistors.ppt,
Hopping ConductivityPolarons
In case of a light orbital overlapping the electrons interact with the phonons.
The electrons polarize their surrounding, which results in a lattice distortion.
The electron and the surrounding distortion field are called polaron.
For an electron transport this state must be broken by thermal energy supplied to the electron.
[Maier 2000]
Page 31
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 31, 11.02.2006Nonlinear Resistors.ppt,
Hopping ConductivityMobility of Charge Carriers
[Heywang 1984]
EA : activation energy1-c : amount of non occupied sitesa : jump distanceν0 : Debye-frequency
q Dk T
μ ⋅=
⋅
( ) 201 exp AED c a
kTν ⎛ ⎞= − ⋅ −⎜ ⎟
⎝ ⎠
diffusion coefficient of polarons
Einstein-relation:
mobility
( ) 201
e x p Aq c a Ek T k T
νμ
− ⎛ ⎞= −⎜ ⎟
⎝ ⎠
in case of a constant concentration of point defects:
( ) ( )01 exp AET ze n TT kT
σ μ ⎛ ⎞= ⋅ −⎜ ⎟⎝ ⎠
∼
q : chargeT : absolut temperature
( ) 201
exp Aq c a EkT kT
νμ
− ⎛ ⎞= −⎜ ⎟⎝ ⎠
Page 32
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 32, 11.02.2006Nonlinear Resistors.ppt,
Hopping ConductivityConduction Mechanisms
~ ( )AE
kTD f T eμ
−
bandμD ~ T-3/2interaction with
phonons anddefects
e.g. metals,semiconductors102 .. 104 ∼ 0
hopping
μD ~ T-1/2large polaron e.g. BaTiO3, SrTiO3, InSb100 .. 102 ∼ 0,1
small polaron* e.g. NaCl, Fe2O3, ZnO2
∼ 0,510-4 .. 10-2
EA [eV]mechanism
* also in some insulators and ionic conductors
mobility μ [cm2/Vs]conduction
Page 33
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 33, 11.02.2006Nonlinear Resistors.ppt,
NTCNegative Temperature Coefficient Resistor
ϑ
symbol NTC
[www.shibako.co.kr]
Page 34
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 34, 11.02.2006Nonlinear Resistors.ppt,
NTCTemperature Dependency of the Conductivity of different Fe-Oxides
( )( ) 2
0= ≈ ⋅
⋅ ⋅ ⋅
BThR T A e
e n T dμ
general equation for thetemperature dependency of an NTC
A,B material dependent constantsh lengthd2 cross section areaR resistanceT temperature in K
σ(S
cm-1
)
temperature
pure magnetit
Page 35
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 35, 11.02.2006Nonlinear Resistors.ppt,
NTCNormalized R(T)-Characteristic of NTCs
( ) 1 1expNN
R T R BT T
⎡ ⎤⎛ ⎞= ⋅ ⋅ −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
temperature dependency related to RN
B material constantR resistanceRN resistance at T = TNT temperature in KTN reference temperatureα temperature coefficient
21 dR BR dT T
α = ⋅ = −
temperature coefficient:
temperature
rela
tive
resi
stan
ce
Page 36
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 36, 11.02.2006Nonlinear Resistors.ppt,
NTCV/I-Characteristic including self heating effect
thermal equilibrium: Pel = PK
PK
U
( ) ( )0K K U th UP A T T G T Tα= ⋅ ⋅ − = ⋅ −
heat losses due to convection
temperature of the NTC temperature of the environmentheat conductivityheat transference numbersurface area of the NTC
T :TU :Gth : αK :A0 :
valid for T < 350 °C(heat losses due to emission not considered)
22 ( ) ( )
( ) th UUU I I R T G T T
R T⋅ = = ⋅ = ⋅ −
I
ϑ
Page 37
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 37, 11.02.2006Nonlinear Resistors.ppt,
NTCNonlinear V/I-Characteristic due to self heating effect
self heatinglineartemperature
measurement [Zinke 1982]
Page 38
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 38, 11.02.2006Nonlinear Resistors.ppt,
NTCTime Dependency of the Temperature during Self Heating
0 Uth c m
Pϑ ϑ
τ−
= ⋅ ⋅
P: el. power (W)τ : time constant (s)ϑU: environmental temperature (K)ϑ0: final temperature (K)m: mass of the NTC (g)c: heat capacitance (Ws/gk)
0.7 Ws/gK for many NTC-materials
[Zinke 1982]
Page 39
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 39, 11.02.2006Nonlinear Resistors.ppt,
NTCChip Thermistor
fast response time (< 1 s in liquids)• 1%, 2%, 5%, 10% accuracy• 1 mW/K losses (in air at 25 °C)• R25°C=100, 1000 ... 1 MΩ• temperature coefficient α25°C= - 3,68 ; B0/50°C=3263• metallization: Ag, Au
[BetaTherm]
Chip Thermistor Features
Page 40
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 40, 11.02.2006Nonlinear Resistors.ppt,
NTCGlass Coated Chip Thermistor
temperature range: -80 °C ... 300 °Clong term operation up to 225 °Cmetallization: Pt, Ir
Glass Coated Chip Thermistor Features
Page 41
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 41, 11.02.2006Nonlinear Resistors.ppt,
y
NTCSwitch-On Delay
working line
time
Page 42
Institut fürWerkstoffe der Elektrotechnik IWEMaterials and Devices in
Electrical Engineeringslide: 42, 11.02.2006Nonlinear Resistors.ppt,
NTCVoltage Stabilization