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)
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Electrical Conductivity p. 17.2
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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)
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Electrical Conductivity p. 17.3
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
“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.
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Electrical Conductivity p. 17.4
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.5
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.6
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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.
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Electrical Conductivity p. 17.7
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.8
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.10
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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)
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Electrical Conductivity p. 17.11
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.13
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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
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Electrical Conductivity p. 17.14
EMSE 201 — Introduction to Materials Science & Engineering © 2000 Mark R. De Guire rev.4/17/00
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%)