Electrical Conductivity Properties

8/6/2019 Electrical Conductivity Properties

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Electrical Conductivity p. 17.1

EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00

Electrical conductivity, σ,varies by 20 orders of magnitudeamong commonplace materials:

Typical Electrical Conductivities at Room Temperature

Material σ, (Ω m)-1 Material σ, (Ω m)-1

Pure metals Semiconductors

Ag 6.80 × 107 C 2.8 × 104

Cu 5.81 " Ge 1.7 × 100

Al 3.80 " Si 4.3 × 10-4

W 1.81 " Insulators

Alloys oxide glasses 10-10-10-14

Cu84Mn12Ni4 (manganin) 2.3 × 106 Lucite, Teflon < 10-13

Cu60Ni40 (constantan) 2.0 " Mica 10-11-10-15

Nichrome (Ni-Cr) 1.0 " SiO2 glass 1.3 × 10-18

Not only the room-temperature values of electricalconductivity, but also how it varies with temperature,distinguish the various classes of materials:

• Metals: ρ

=1σ increases linearly w/T

• Semiconductors

Insulators : ρ decreases strongly with T:

ρ ∝ exp(φ /kBT)

(Arrhenius behavior)





MACROSCOPIC DESCRIPTION: OHM’S LAW

V = IR

V: voltage drop acrossmedium [V] = [J/C]

I: current through medium [A] = [C/s]

R: resistance of medium [Ω] = [J.s/C2]

Eliminate extensive variables by substitutions:

• R = LAσ = LρA

• L: length of medium [m]

• A: cross-sectional area of medium [m2]

• σ: electrical conductivity of medium

1

Ω m =

C2

J s m• ρ: resistivity of medium; = σ-1

• Electric field, E =dVdL =

VL

J

C m

• Current density, j =IA

C

s m2

⇒ j = σE = σ dV

dLNote similarities to Fick’s first law & Fourier’s law of cooling:

J = –Ddcdx (p. 8.3)

Q.

A = – κ dTdx (p. 15.8)

Flux = (material property) × (gradient)





“MICROSCOPIC” DESCRIPTION

σ = ne|z|µ

• n: number of charge carriers per unit volume [m-3]

• e: electronic charge, 1.602 × 10-19 [C]

• z: valence on carrier [dimensionless]

• µ: mobility of carriers in medium

m2 C J s =

m2 V s

• Note similarities to thermal conductivity: κ =13 cV v δ (p. 15.9)

• Holds for all materials

• For materials with >1 type of charge carrier,σtot = ∑

iσi

where summation is over all types of charge carriers,and each σi = nie|zi|µi

Note:

• e is constant • z = 1

electrons

holesalkali ions or 2

O

=

ionsCO3= ions

∴

The wide range of σ among materials

& its T-dependence

are attributable to n and µ.





ELEMENTARY BAND THEORY

• Free atoms: discrete energy levels for electrons

• Solids: band formation

Formation ofenergy bands asisolated carbonatoms form adiamond crystal.

From B. G.Streetman, “SolidState ElectronicDevices,” 2nd ed.Prentice-Hall,Englewood Cliffs,1980.

Also see Callister,

Figs. 19.2 & 19.3

• Atoms come together to form a solid ⇒ electronicstates must shift in energy to be quantummechanically distinct (Pauli exclusion principle)

• Many atoms,

similar energies ⇒

broadening of 1-atom levels

into energy bands





ELEMENTARY BAND THEORY (cont.)

For flow of electrons (current) when an E-field is applied,

e – ’s must have access to unoccupied electronic states

Schematic electron energy level diagrams for solids

(occupancy at absolute zero)

filled states

unoccupied states

band gap (no states)

VBVB VB

CB CB CB

Metal(e.g. Cu)

Metal(e.g. Mg)

Semiconductor(e.g. Si)

Insulator(e.g. Al O )

VB: valence band

CB: conduction band

2 3

gEgE

Semiconductor : Eg < ~2.5-3 eV

Insulator : Eg > ~2.5-3 eV

Callister, Figure 19.4





CARRIER CONCENTRATION

• Metals:

• No gap ⇒ energy from applied E-field is sufficient tomove e – ’s

• Most valence e – ’s are carriers — 1-3 per atom

⇒ n ≅ 1029 m –3 and n indep. of T

• Undoped (intrinsic ) semiconductors: e – -hole pairs

(EHPs)• Thermal energy (kBT) ⇒ small fraction of valence e –

’s jump gap into CB, leaving behind a positivelycharged electron vacancy (a “hole”) in VB:

Ø →← e- + h+

[e-][h+] ∝ exp

-EgkBT (see footnote* )

• In VB and CB: e – ’s now have adjacent empty states• e – ’s and holes formed in pairs ⇒ [e – ] = [h+] ∝ exp

-Eg

2kBT in an intrinsic semiconductor

• Plot ln [e – ] or ln [h+] vs. 1/T

⇒ linear; with slope = –Eg /2kB

* The appearance of a Boltzmann factor here, and the representation ofelectron-hole pair formation as a kind of chemical reaction, might mislead oneinto thinking that the theory of electrons in solids is something that can betreated by classical mechanics. In fact, these processes require the use ofquantum mechanics to be described accurately, even though the resultsabove resemble descriptions of classical processes such as diffusion andtrue chemical reactions.





ELECTRONIC CONDUCTION in INTRINSIC SILICON

Si Si Si

Si Si Si

Si Si Si

Si Si Si

Si Si Si

Si Si Si

E-field

Hole

Free Electron

a. b.

Si Si Si

Si Si Si

Si Si Si

E-field

Hole

Free

Electron

c.

Callister, Fig. 19.10





CARRIER CONCENTRATION — Doped Semiconductors(start)

• Donors, e.g. P, As, or Sb in Si

III IV VB C NAl Si PGa Ge A sIn Sn Sb

• Extra valence e – compared to host Si atoms• This e – occupies state at Ed just below CB in Si

— little kBT needed to promote it into CB

⇒ exp

–Ed

kBT >> exp

–Eg

kBT

• Note: conduction e – created, but not a hole

(⇐ no nearby occupied states to refill the donorstate)

• T-dependence of [e – ]:

• At RT, all donor atoms “ionized” — donor“exhaustion”

• At typical dopant levels, donor e – ’soverwhelm intrinsic e – ’s

⇒ [e - ] indep. of T — “extrinsic” behavior

• At higher T’s, the number of intrinsically generatede – ’s catches up to, and exceeds, the number ofdonor atoms ⇒ “intrinsic” behavior is observed







CARRIER CONCENTRATION - Doped Semiconductors (end)

• Acceptors, e.g. B, Al, Ga, or In in Si• One less valence e – than host Si atoms

• Creates an empty state at Ea just above VB in Siinto which Si’s valence e – ’s can jump

• Note: hole created in valence band but not a free e –

(⇐ promoted e – has no nearby acceptor state toenter)

• T-dependence of [h+]:

• At a T above which all acceptor states are ionized(filled) — acceptor “saturation”

• At typical dopant levels, acceptor h+’s

overwhelm intrinsic h+

’s⇒ [h + ] indep. of T (this, too, is “extrinsic” behavior)

• At higher T’s, the number of intrinsically generatedh+’s catches up to, and exceeds, the number ofacceptor atoms ⇒ “intrinsic” behavior is observed

• Note:

• [e – ][h+] ∝ exp

–Eg

kBT true for intrinsic & extrinsic

• [e – ] ≠ [h+] for extrinsic behavior

(⇐ dopants create e – ’s or h+’s, but not both at once)





HOLE CONDUCTION in p-DOPED SILICON

Si Si Si

Si

SiSiSi

BSi

Hole

a.

Si

E-field

Si Si

Si

SiSiSi

BSi

b.

Callister, Fig. 19.13







NET EFFECTS

• Metals: — Callister, Figure 19.8

• ρtot = ρthermal + ρimpurities + ρdeformation

• ρ ~independent of T at cryogenic T’s

• ρ (= σ-1) = ρo + A(T – To) above RT

• After substitutions, σ =ne2τ

me*for metals

• Semiconductors: Callister, Figure 19.15

• ln (σ) vs. 1/T for Si

• Intrinsic

• B-doped

• Insulators

• Mobility is thermally activated: µ =|z|eDkBT

• Carrier concentration:

• Thermally activated in “intrinsic” insulators

• Not thermally activated in “extrinsic” insulators





WIEDEMANN-FRANZ RATIO, £

• Compares electrical to thermal conductivity: £ ≡ κ σT

• σ = nezµ =n e2 τme* for metals

• Recall electronic thermal conductivity, κ el =

π2nkB2Tτ3me*

• If electrons are the primary carriers of both heat andelectricity,

⇒ £ =13

πkB

e

2= 2.45 × 10-8 V2 K-2

i.e., this ratio is expected to be a constant for all metals

• For most pure metals,

2.2 < £ < 2.6 (10-8 V2 K-2)

(agreement to ±10%)

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Electrical Conductivity Properties