Nonlinear Resistors

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



Nonlinear Resistors Introduction (2)

NTC PTC Varistor

PTC-heater withmetalized surface

arrangement of PTC´s and radiators



U

VaristorVariable Resistor (Voltage dependent Resistor)

symbol varistor

[www.staratech.com]



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]



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



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



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)



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



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



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 ]



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 ]



Varistor

ZnO-varistors varistors for high voltage applications



ϑ

PTCPositive Temperature Coefficient Resistor

symbol PTC

[www.atpsensor.com]



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



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



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



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



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]



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



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



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]



PTCDevices

all purpose PTCoverload

protection

PTC for monitordemagnetization

temperature sensor

heating elements



PTCNonlinear I/V-Characteristic due to Self Heating Effect

[Zinke]



PTCSelf Controlled Heating Elements for Automotive Application

[Catem]PTC-characteristic

PTC heating elementwith metallized surface

Arrangement of PTC heating elements and metallic radiators



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



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



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



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



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.



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]



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

− ⎛ ⎞= −⎜ ⎟⎝ ⎠



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



NTCNegative Temperature Coefficient Resistor

ϑ

symbol NTC

[www.shibako.co.kr]



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



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



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



NTCNonlinear V/I-Characteristic due to self heating effect

self heatinglineartemperature

measurement [Zinke 1982]



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]



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



NTCGlass Coated Chip Thermistor

temperature range: -80 °C ... 300 °Clong term operation up to 225 °Cmetallization: Pt, Ir

Glass Coated Chip Thermistor Features



y

NTCSwitch-On Delay

working line

time



NTCVoltage Stabilization

Nonlinear Resistors

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