电子材料电性能1
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3.225 © E.A. Fitzgerald-1999
8
4
Equation of Motion - Impact of Collisions
Assume: • probability of collision in time dt = dt/τ • time varying field F(t) v(t+dt) = (1- dt/τ) {v(t) +dv} = (1- dt/τ) {v(t) + (F(t)dt)/m} ≈ v(t) + (F(t)dt)/m - v(t) dt/τ (for small dt) ⇒ dv(t)/dt + v(t)/τ = F(t)/m Note: Term proportional to velocity corresponds to frictional damping term
3.225 © H.L. Tuller-2001
1
What Features Distinguish Different Conductors?
• Magnitude: varies by over 25 orders of magnitude! • metal; semiconductor; insulator • Carrier type: • electrons vs ions; • negative vs positive • Mechanism: • wave-like • activated hopping • Field Dependence: • Linear vs non-linear
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Does this Microscopic Picture of Metals Give us Ohm’s Law? E F=-eE F=ma m(dv/dt)=-eE v =-(eE/m)t v,J,σ,I t E t
Constant E gives ever-increasing v
No, Ohm’s law can not be only from electric force on electron!
• # charges crossing plane per unit time and area = j
j = n v d dtA (− e ) dtA = − nevd = (ne 2τ m )E
j = σ E ⇒ σ = (ne 2τ m ) = j E
(
)
• Ohm’s Law:
Dimensional analysis: (A/cm2)/(V/cm)=A/(V-cm)= (ohm-cm)-1 = Siemens/cm-(S/cm)
Schematic model of a crystal of sodium metal.
From: C. Kittel, Introduction to Solid State Physics, 3rd Ed., Wiley (1967) p. 198.
3.225
© E.A. Fitzgerald-1999
How much current does a 100 W bulb draw? I = 100W/115V = 0.87A
3.225
© H.L. Tuller-2001
13
Predicting Conductivity using Drude
ntheory from the periodic table (# valence e- and the crystal structure) ntheory=AVZρm/A, where AV is 6.023x1023 atoms/mole ρm is the density Z is the number of electrons per atom A is the atomic weight For metals, ntheory~1022 cm-3 If we assume that this is correct, we can extract τ
3.225
© E.A. Fitzgerald-1999
14
7
Extracting Typical τ for Metals
•
τ~10-14 sec for metals in Drude model
3.225
© E.A. Fitzgerald-1999
15
Thermal Velocity
• So far we have discussed drift velocity vD and scattering time τ related to the applied electric field x • Thermal velocity vth is much greater than vD
10
5
Mean-free Time Between Collisions, Electron Mobility
In steady state,
dp(t ) =0 dt
p(t ) = p∞ (1 − e )
p∞ = − eEτ
−t τ
p
-eEτ
τ
t
If the environment has a lot of collisions, mvavg=-eEτ vavg=-eEτ/m Define v
Electronic Materials
Silicon Age: Pervasive technology • Communications • Computation • Automation • Defense • ……….. Factors: • Reproducibility/Reliability • Miniaturization • Functionality • Cost • …………..
x
1 2 3 mvth = kT 2 2
L=vDτ
x
vth =
3kT m
Thermal velocity is much greater than drift velocity
3.225
© E.A. Fitzgerald-1999
16
8
Resistivity/Conductivity-- Pessimist vs Optimist
V t W L R = ρ L/Wt = ρ L/A ⇒ ρ(οhm-cm) σ = 1/ρ ⇒ σ (οhm-cm)-1 ⇒ σ (Siemens/cm) (Test your dimensions: R=V/I; σ=E/j=neµ)
Ohms/square ⇒ Note, if L=W, then R= ρ /t independent of magnitude of L and W. Useful for working with films of thickness, t.
3.225
© H.L. Tuller-2001
12
6
Energy Dissipation - Joule Heating
Frictional damping term leads to energy losses: • Power absorbed by particle from force F: P = W/t = (F•d)/t = F•v • Electron gas: P/vol= n(-eE)•(-eτE/m) = ne2τE2/m = σ E2 = jE = (I/A)(V/l) = IV/vol • Total power absorbed: P = IV = V2/R = I2R
dp(t ) p( t ) ≈− − eE τ dt
Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms
3.225
© E.A. Fitzgerald-1999
3.225
= − µE
µ=
eτ m
© E.A. Fitzgerald-1999
11
What is the Current Density ?
E vd n (#/vol) j = I/A dx A
• # electrons crossing plane in time dt = n(dxA) = n(vddtA)
3.225
© H.L. Tuller-2001
9
Hydrodynamic Representation of e- Motion
p=momentum=mv
dp( t ) p( t ) =− + F1 ( t ) + F2 ( t ) +... τ dt
Response (ma) Drag Driving Force Restoring Force...
3.225
© H.L. Tuller-2001
4
2
Origin of Conduction Range of Resistivity
Why?
3.225
© E.A. Fitzgerald-1999
5
Response of Material to Applied Potential
I V=f(I)
Rectification, Non-linear, Non-Ohmic Linear, Ohmic
3.225 © H.L. Tuller-2001
3
What is the Application?
• • • •
Interconnect Resistor Insulator Non-ohmic device
– diode, transistor
8
4
Equation of Motion - Impact of Collisions
Assume: • probability of collision in time dt = dt/τ • time varying field F(t) v(t+dt) = (1- dt/τ) {v(t) +dv} = (1- dt/τ) {v(t) + (F(t)dt)/m} ≈ v(t) + (F(t)dt)/m - v(t) dt/τ (for small dt) ⇒ dv(t)/dt + v(t)/τ = F(t)/m Note: Term proportional to velocity corresponds to frictional damping term
3.225 © H.L. Tuller-2001
1
What Features Distinguish Different Conductors?
• Magnitude: varies by over 25 orders of magnitude! • metal; semiconductor; insulator • Carrier type: • electrons vs ions; • negative vs positive • Mechanism: • wave-like • activated hopping • Field Dependence: • Linear vs non-linear
7
Does this Microscopic Picture of Metals Give us Ohm’s Law? E F=-eE F=ma m(dv/dt)=-eE v =-(eE/m)t v,J,σ,I t E t
Constant E gives ever-increasing v
No, Ohm’s law can not be only from electric force on electron!
• # charges crossing plane per unit time and area = j
j = n v d dtA (− e ) dtA = − nevd = (ne 2τ m )E
j = σ E ⇒ σ = (ne 2τ m ) = j E
(
)
• Ohm’s Law:
Dimensional analysis: (A/cm2)/(V/cm)=A/(V-cm)= (ohm-cm)-1 = Siemens/cm-(S/cm)
Schematic model of a crystal of sodium metal.
From: C. Kittel, Introduction to Solid State Physics, 3rd Ed., Wiley (1967) p. 198.
3.225
© E.A. Fitzgerald-1999
How much current does a 100 W bulb draw? I = 100W/115V = 0.87A
3.225
© H.L. Tuller-2001
13
Predicting Conductivity using Drude
ntheory from the periodic table (# valence e- and the crystal structure) ntheory=AVZρm/A, where AV is 6.023x1023 atoms/mole ρm is the density Z is the number of electrons per atom A is the atomic weight For metals, ntheory~1022 cm-3 If we assume that this is correct, we can extract τ
3.225
© E.A. Fitzgerald-1999
14
7
Extracting Typical τ for Metals
•
τ~10-14 sec for metals in Drude model
3.225
© E.A. Fitzgerald-1999
15
Thermal Velocity
• So far we have discussed drift velocity vD and scattering time τ related to the applied electric field x • Thermal velocity vth is much greater than vD
10
5
Mean-free Time Between Collisions, Electron Mobility
In steady state,
dp(t ) =0 dt
p(t ) = p∞ (1 − e )
p∞ = − eEτ
−t τ
p
-eEτ
τ
t
If the environment has a lot of collisions, mvavg=-eEτ vavg=-eEτ/m Define v
Electronic Materials
Silicon Age: Pervasive technology • Communications • Computation • Automation • Defense • ……….. Factors: • Reproducibility/Reliability • Miniaturization • Functionality • Cost • …………..
x
1 2 3 mvth = kT 2 2
L=vDτ
x
vth =
3kT m
Thermal velocity is much greater than drift velocity
3.225
© E.A. Fitzgerald-1999
16
8
Resistivity/Conductivity-- Pessimist vs Optimist
V t W L R = ρ L/Wt = ρ L/A ⇒ ρ(οhm-cm) σ = 1/ρ ⇒ σ (οhm-cm)-1 ⇒ σ (Siemens/cm) (Test your dimensions: R=V/I; σ=E/j=neµ)
Ohms/square ⇒ Note, if L=W, then R= ρ /t independent of magnitude of L and W. Useful for working with films of thickness, t.
3.225
© H.L. Tuller-2001
12
6
Energy Dissipation - Joule Heating
Frictional damping term leads to energy losses: • Power absorbed by particle from force F: P = W/t = (F•d)/t = F•v • Electron gas: P/vol= n(-eE)•(-eτE/m) = ne2τE2/m = σ E2 = jE = (I/A)(V/l) = IV/vol • Total power absorbed: P = IV = V2/R = I2R
dp(t ) p( t ) ≈− − eE τ dt
Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms
3.225
© E.A. Fitzgerald-1999
3.225
= − µE
µ=
eτ m
© E.A. Fitzgerald-1999
11
What is the Current Density ?
E vd n (#/vol) j = I/A dx A
• # electrons crossing plane in time dt = n(dxA) = n(vddtA)
3.225
© H.L. Tuller-2001
9
Hydrodynamic Representation of e- Motion
p=momentum=mv
dp( t ) p( t ) =− + F1 ( t ) + F2 ( t ) +... τ dt
Response (ma) Drag Driving Force Restoring Force...
3.225
© H.L. Tuller-2001
4
2
Origin of Conduction Range of Resistivity
Why?
3.225
© E.A. Fitzgerald-1999
5
Response of Material to Applied Potential
I V=f(I)
Rectification, Non-linear, Non-Ohmic Linear, Ohmic
3.225 © H.L. Tuller-2001
3
What is the Application?
• • • •
Interconnect Resistor Insulator Non-ohmic device
– diode, transistor
