Electrical Conduction in Solids

What is Electric Current?

What is current density?

What is Drift? … motion of charges under the influence of an Electric Field

What is drift velocity ?… average velocity of charges under an applied E-field

NOTE: many of these quantities are VECTORS !!

tq

dtdq

i

tAq

Adtdq

J

Electrical Conduction in Solids

• Which way do negative charges move with respect to the applied E-field?

• Which way do positive charges move with respect to the applied E-field?

• What type of charges are responsible for current flow in conductors (metals)?

• What type of charges are responsible for current flow in semiconductors?

• What is mobility μ ?

A

Jx

Dx

vd x

Ex

V ib ra tin g C u + io n s

ux

Ex

V

(a) (b)

Voltage Gradient and E-Field

Quick Derivation of Current Density

ΔtvΔx dxN/Vn

xNx3x2x1dx v...vvvN1v

ΔxAneΔq ΔtvAneΔq dx

(t)vne(t)J dxx

A

Jx

Dx

vd x

Ex

Important Current Density Relationships!

(t)vne(t)J dxx

μEvdx

Eμne(t)J x

Eσ(t)J x

• Mobility … expresses how “easy” electrons can move (drift) under the influence of an electric field; it is directly related to relaxation (or mean scattering) time

• Conductivity … the ability of a solid to conduct electricity (cousin of mobility!)

• ‘Mean Free Time’

pn μpeμneσor μneσ

em

e μ

J = σE, Ohm’s Law

• OHM’S LAW (V=IR!)

Temperature Dependence of Resistivity

• Electrons are scattered by the vibrating metal ions

• Therefore the time between collisions (scattering time) can be affected by the vibration frequency … which implies that the mobility and therefore the conductivity and resistivity are also affected by the vibrating atoms (ions in metals)

• What should be the relationship between frequency of vibration and time between collisions ?– The higher the freq. the shorter the time because

the metal ions are vibrating faster (i.e. higher KE) and therefore colliding more often with moving electrons

• Shorter scattering times imply lower mobility … lower conductivity and therefore higher resistivity!

• (the math/derivation is in the book! Let’s look at the expressions…)

Temperature Dependence of Mobility and Conductivity

TmCe

μe

d

nCeTm

μne1

σ1

ρ 2e

d

• IMPORTANT to remember the dependence of conductivity … mobility … etc. on T! … i.e. directly/inversely proportional

Alloys … and Matthiessen’s Rule

ILd μ1

μ1

μ1

• Previous discussion was based on the ASSUMPTION that the material was a pure metal & perfect crystal! … i.e. no impurities no point defects etc.

• When impurities are present then one needs to consider two scattering mechanisms: (1) host atoms (2) impurity atoms

• The result is:

– where subscript L is for Lattice and I is for Impurity• Therefore the effective mobility is lower (than that of a

pure metal)

• The “net” resistivity is:

which is known as Matthiessen’s Rule

ITIL

ρρμne

1μne

1ρ

tI

Strained region by impurity exerts ascattering force F = ­­d(PE) /dx

tT

Resistivity versus temperature for pure iron and 4% C steel.

Nichrome ??

Temperature Dependence of Resistivity

BATρρρ RT

• All non-intrinsic effects (impurities, crystal defects etc.) on the reisstivity can be “summed” up in a “residual” reisistivity term and the resistivity can be re-written as:

• NOTE that the thermal vibrations term is temperature dependent!! but the extrinsic component is not.

• TEMPERATURE COEFFICIENT OF RESISTIVITY α is the fractional change in resistivity per unit temperature at the reference temperature TO

• Valid over a narrow temperature range… not a bad approximation BUT be cautious !

)T-(Tα1ρρ to leads which δTδρ

ρ1

α ooooTo

O

r µ T

Tungsten

Silver

Copper

Iron

Nickel

Platinum

NiCr Heating Wire

Tin

Monel-400

Inconel-825

10

100

1000

2000

100 1000 10000Temperature (K)

Res

i sti v

it y (

n W m

)

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1 10 100 1000 10000Temperature (K)

00.5

11.5

2

2.53

3.5

0 20 40 60 80 100T­(K)

r µ T

r µ T5

r = rRr = rR

r µ T5

r µ T

r (nW m)

Res is

ti vit y

(nW

m)

Temperature Dependence of Resistivity

Solid Solutions

• What is a solid solution ?

• The second term in the above equation is NOT temperature dependent

• Therefore when forming a solid solution of two metals … i.e. one metal is the “host” (and the other the “impurity”, then the addition of the “impurity” metal will cause an increase in the resistivity and make the total resistivity less and less temperature dependent !

• NOTE: at large “impurity” amounts the material will become an ALLOY! And we need to consider “alloy effects”

etc. defects impurities ofeffect includes ρ where ρρρ RRT

1 0 0 % C u a t . % N i

C u - N i A l l o y s

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

0 2 0 4 0 6 0 8 0 1 0 0

1 0 0 % N i

Re

sis

tiv

ity

(nW

m)

Nordheim’s Rule of Solid Solutions

• A simple way to determine the effect on resistivity component – the one due to “alloying impurities”

• … and combining Matthiessen’s and Nordheim’s rules then we get

tcoefficien Nordheim the -constant a isC and solute the of fraction the is Xwhere

X)-CX(1ρ I

X)-CX(1ρρ MATRIX

Solid Solutions vs. Mixtures

bbaaeff ρχρχρ

• Solid solutions are homogenous and mixing takes place at the atomic level … Nordheim’s rule applies.

• What happens when the mixture is not homogenous? i.e. 2 phases

bbaaeff σχσχσ

Jx

a b

A

L

L

A

Jy

– First case: series mixture

– Second case: parallel mixture

Mixtures – Continuous Phase w Dispersed 2nd Phase

cdd

d

ceff ρ 10 ρfor χ1

χ21

1ρρ

cd

d

dceff ρ 0.1 ρfor

2χ1χ1

ρρ

• If we have a mixture where the “host” material is continuous (c) and the added material is dispersed (d) therein … then the following two empirical expressions apply:

L

A

Dispersed phaseContinuous phase

y

Jx

Cd

Cd

C

C

2σσσσ

χ2σσσσ

:expression general theOr

Fig 2.15

Rule ???

X 1X 2

1 0 0 % B1 0 0 % A X (% B )

L iqu id , L+L +L

T AT B

T 1

Tw o ph a se reg io n

O n e p h a s ere g io n : o n ly

T E

(a )

Tem

per

atur

e

BA

C o m p o s i t io n X (% B )0 X 1 X 2 1 0 0 % B

N o rd h e im 's R u le

M ix tu re R u le

(b )

Resi

stiv

ity

(a) The phase diagram for a binary, eutectic forming alloy. (b) Theresistivity vs composition for the binary alloy.

V ib ra tin g C u + io n sE lec tro n G as

H O T C O L DH E A T

Thermal conduction in a metal involves transferring energy from the hot region to the cold region by conduction electrons. More energetic electrons (shown with longer velocity vectors) from the hotter regions arrive at cooler regions and collide there with lattice vibrations and transfer their energy. Lengths of arrowed lines on atoms represent the magnitudes of atomic vibrations.

Thermal Conduction

Thermal Conductivity

• Good Electrical Conductors (like metals) are also good Thermal Conductors …

• Why ??

• … electrons are responsible for the electrical and thermal conductivity …

• … they pick up the thermal energy from vibrating atoms and transfer to atoms elsewhere …

• It is easy to think of Thermal Conductivity the same way we think of Electrical Conductivity …

AL

k1

θ ... AL

σ1

AL

ρR

δxδT

-AkdtdQ

Q' similarly ... δxδV

-Aσdtdq

Ior ... σEJ

Thermal Conductivity

• σ is the electrical conductivity and k the thermal conductivity

• Since electrons are responsible for both … the two are related i.e.

• … where CWFL is the Weidemann-Franz-Lorenz coefficient

WFLCσTk

Conduction of heat through a component in (a) can be modeled as athermal resistance shown in (b) where Q= T/.

H o t

L

T

(a)

Q C o ld

AQ

T

Q

(b)

Q = T /

Thermal Conductivity

• σ is related to temperature (in a certain range!) as 1/T

… the above relationship suggests that k is temperature

INDEPENDENT

• Since there are no free electrons in non-metals … how does the heat transfer take place ?

• VIBRATIONS … vibrations will increase at the hot end of a material and the vibrational energy will be transferred along the material depending on the type of bonding

WFLCσTk

0

100

200

300

400

450

0 10 20 30 40 50 60 70Electrical conductivity, s, 106 W-1 m-1

Al

Ag

Au

Cu

Brass (Cu-30Zn)

Be

Pd-40Ag

Bronze (95Cu-5Sn)

Ag-3Cu

Ag-20Cu

WMo

Hg

Mg

Steel (1080)

Ni

ks = T­CWFL

The

r mal

con

duct

ivity

, k (

W K

-1 m

-1)

Electrical Conductivity

10610310010-310-610-910-1210-1510-18 109

Semiconductors Conductors

1012

Conductivity (Wm)-1

AgGraphite NiCrTeIntrinsic Si

DegeneratelyDoped Si

Insulators

Diamond

SiO2

Superconductors

PETPVDF

AmorphousAs2Se3

Mica

Alumina

Borosilicate Pure SnO2

Inorganic Glasses

Alloys

Intrinsic GaAs

Soda silica glass

Many ceramics

MetalsPolypropylene

Charges under the influence !

• What happens when a charge finds itself under the influence of an electric field ?

E

Charges under the influence !

• What happens when a charge finds itself under the influence of a magnetic field ?

B

Right vs. Left-Handed Oriented xyz System…. And the Cross product!

yxz

xzy

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

z y x i.e.

.... order In

yzx

xyz

zxy

ˆˆˆ

ˆˆˆ

ˆˆˆ

xyz i.e.

order reverse In

The Hall EffectExample with p-type semiconductor;

i.e. holes are the majority charge carriers;

apply voltage in x direction i.e. current Ix.

apply a B-field in the z-direction. Total Force on the charge carriers

due to E and B fields is)BvEq(F

)Bvq(EF zxyy The y-component of the force is

As the holes flow in the x-direction they experience a force in the y-direction due to the B-field.

Holes will accumulate in the -y end of the bar setting up an electric field, i.e. a voltage VAB in the y direction. The net force in the y-direction becomes zero when the two components of the force i.e. due to the electric and due to the magnetic field are equal.

The Hall Effect The “setting up” of the E-field in the

y-direction is known as the Hall Effect.

The voltage VAB is known as the Hall Voltage.

This experiment is used to measure the mobility of the charge carriers as explained below:

wEV yAB

qpJνqpνJ X

XXX

zxHzx

zxyy BJRBqpJ

BvE0F

)Bvq(EF zxyy

qp1

RH AB

zx

AB

zx

y

zx

qtVBI

/w)q(Vwt)BI(

qEBJ

p

The Hall Effect

Note: the current and magnetic fields are known quantities since they are externally applied; the hall voltage can be measured.

The resistivity of the sample can also be calculated by measuring the resistance R of the bar.

wtρL

R

ρR

qR1q

ρ1

qpσ

μqpμσ H

H

pp

Hall Effect

Key Points• Direction of electric field is also the direction of conventional

current.• Hall electric field is set by the movement charge due to Lorentz

force in the presence of a magnetic field.

Right hand rule• Index finger -> Current / charge velocity• Thumb -> Lorentz force• Rest of the fingers -> Magnetic filed

Jx

vex

eEy

Jy = 0

xz

y

evexBz

B z

V

B z

A

Jx

Ey

Ex

vhx

evhxBz

+

++

+ ++

eEy

A

B

VAB>0 >0 VAB>0 <0

Application of Hall Effect

V12<0,Charge?

Charge (-)veV12 ?

Charge (-)ve, V12 <0B ?

Example #1

The resistivity of an alloy of Cu is 10-7 Ω-m; the dimensions of a rectangular bar of this material are w=4 mm, t=10 mm, and l=50 mm; the thermal conductivity of this alloy is:

Example #2• Points A, B, and C

– What phases are present– What is the composition of each phase– What is the fraction of each phase present– Max solubility of C in Fe to maintain the Ferrite

phase

Example #3

The resistivity of aluminum at 25 °C has been measured to be 2.72x10-8 Ω-m. The thermal coefficient of resistivity of aluminum at 0 °C is 4.29 x 10-3 K-1. Aluminum has a valency of 3, a density of 2.70 g cm-3, and an atomic mass of 27.

(a) what is the resistivity @ -40 °C

Example #4

The resistivity of aluminum at 25 °C has been measured to be 2.72x10-8 Ω-m. The thermal coefficient of resistivity of aluminum at 0 °C is 4.29 x 10-3 K-1. Aluminum has a valency of 3, a density of 2.70 g cm-3, and an atomic mass of 27.

(b) what is the thermal coefficient of resistivity @ -40 °C

Example #4The resistivity of aluminum at 25 °C has been measured to be 2.72x10-8 Ω-m. The thermal coefficient of resistivity of aluminum at 0 °C is 4.29 x 10-3 K-1. Aluminum has a valency of 3, a density of 2.70 g cm-3, and an atomic mass of 27.

(c) Estimate the mean free time between collisions for the conduction electrons in aluminum at 25 °C, and hence estimate their drift mobility.

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melectronsxmelectronsxx

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matomsx

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