last time - purdue universityerica/545/lecture07.pdf · last time free electron model density of...
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Last Time
Free electron model
ε~gDensity of states in 3D
m
kE FF 2
22
=Fermi Surface kF =3π 2NV
⎛⎝⎜
⎞⎠⎟
1/3
( ) ( )Tfe
,1
1 εµεβ =+
= −
Fermi-Dirac Distribution Function
Debye Approximation.
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Today
Measuring the occupied density of states
Effective Mass
Electrical Conductivity
Thermal Conductivity
Wiedemann-Franz Ratio
Heat Capacity 3BTATC +=
Electrons Phonons
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Fermi-Dirac Distribution Function
http://ece-www.colorado.edu/~bart/book/distrib.htm#fermi
Becomes a step function at T=0.Low E: f ~ 1.High E: f ~ 0.
Go play with the Excel file “fermi.xls” at:
( ) ( ) 1
1,
+= −µεβεe
Tf
µ = chemical potential = “Fermi Level”
µ(T=0)=εF Fermi energy
Right at the Fermi level: f = 1/2.
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Fermi-Dirac Distribution Function
http://ece-www.colorado.edu/~bart/book/distrib.htm#fermi
Becomes a step function at T=0.Low E: f ~ 1.High E: f ~ 0.
Go play with the Excel file “fermi.xls” at:
( ) ( ) 1
1,
+= −µεβεe
Tf
µ = chemical potential = “Fermi Level”
µ(T=0)=εF Fermi energy
Right at the Fermi level: f = 1/2.
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n ε,T( ) = g ε( ) f ε,T( )
N = n ε,T( )0
∞
∫ dε = g ε( ) f ε,T( )dε = N0
∞
∫
µ T > 0( ) < εF
µ T( ) = εF 1− o T2( )⎡⎣ ⎤⎦
Number of electrons per energy range
Fermi functionDensity of states
Implicit equation for µ
N is conserved
Shaded areas are equal
⇒
0.01% @ room temp
Density of Occupied States
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n ε,T( ) = g ε( ) f ε,T( )
N = n ε,T( )0
∞
∫ dε = g ε( ) f ε,T( )dε = N0
∞
∫
µ T > 0( ) < εF
µ T( ) = εF 1− o T2( )⎡⎣ ⎤⎦
Number of electrons per energy range
Fermi functionDensity of states
Implicit equation for µ
N is conserved
Shaded areas are equal
⇒
0.01% @ room temp
Density of Occupied States
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Heat CapacityWidth of shaded region ~ kT
Room temp T ~ 300K, TF ~ 104 K→ Small width→ Few electrons thermally excited
How many electrons are excited thermally?
Shaded area ≈ triangle. Area = (base)(height)/2
Number of excited electrons: ≈ (g(εF)/2)(kT)/2 ≈ g(εF)(kT)/4
Excitation energy ≈ kT (thermal)
Total thermal energy in electrons: E ≈14g εF( )kT⎛
⎝⎜⎞⎠⎟kT( ) = 1
4g εF( ) kT( )2
C =dEdT
≈12g εF( )k2T =
34NεFk2T
C ~ THeat Capacity in a Metal
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Heat CapacityWidth of shaded region ~ kT
Room temp T ~ 300K, TF ~ 104 K→ Small width→ Few electrons thermally excited
How many electrons are excited thermally?
Shaded area ≈ triangle. Area = (base)(height)/2
Number of excited electrons: ≈ (g(εF)/2)(kT)/2 ≈ g(εF)(kT)/4
Excitation energy ≈ kT (thermal)
Total thermal energy in electrons: E ≈14g εF( )kT⎛
⎝⎜⎞⎠⎟kT( ) = 1
4g εF( ) kT( )2
C =dEdT
≈12g εF( )k2T =
34NεFk2T
C ~ THeat Capacity in a Metal
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Heat CapacityWidth of shaded region ~ kT
Room temp T ~ 300K, TF ~ 104 K→ Small width→ Few electrons thermally excited
How many electrons are excited thermally?
Shaded area ≈ triangle. Area = (base)(height)/2
Number of excited electrons: ≈ (g(εF)/2)(kT)/2 ≈ g(εF)(kT)/4
Excitation energy ≈ kT (thermal)
Total thermal energy in electrons: E ≈14g εF( )kT⎛
⎝⎜⎞⎠⎟kT( ) = 1
4g εF( ) kT( )2
C =dEdT
≈12g εF( )k2T =
34NεFk2T
C ~ THeat Capacity in a Metal
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How you would do the real calculation:
N = n ε,T( )0
∞
∫ dε
E = εn ε,T( )0
∞
∫ dε = ε f ε,T( )g ε( )0
∞
∫ dε
dEdT
= C =π 2
2g εF( )kB2T
C = AT + BT 3
Implicit equation for µ → fully determines n(ε, T)
Then
In a metallic solid,
C ~ T is one of the signatures of the metallic state
Electrons Phonons
Correct in simple metals.
Heat Capacity
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How you would do the real calculation:
N = n ε,T( )0
∞
∫ dε
E = εn ε,T( )0
∞
∫ dε = ε f ε,T( )g ε( )0
∞
∫ dε
dEdT
= C =π 2
2g εF( )kB2T
C = AT + BT 3
Implicit equation for µ → fully determines n(ε, T)
Then
In a metallic solid,
C ~ T is one of the signatures of the metallic state
Electrons Phonons
Correct in simple metals.
Heat Capacity
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Measuring n(ε, T)
X-ray Emission
(1) Bombard sample with high energy electrons to remove some core electrons
(2) Electron from condition band falls to fill “hole”, emitting a photon of the energydifference
(3) Measure the photons -- i.e. the X-ray emission spectrum
N = n ε,T( )0
∞
∫ dε = g ε( ) f ε,T( )dε = N0
∞
∫ n(ε, T) is the actual numberof electrons at ε and T
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Measuring n(ε, T)
X-ray Emission
N = n ε,T( )0
∞
∫ dε = g ε( ) f ε,T( )dε = N0
∞
∫ n(ε, T) is the actual numberof electrons at ε and T
Emission spectrum (how many X-rays come out as a function of energy) will look likethis.
Fine print: The actual spectrum is rounded by temperature, and subject to transition probabilities. Void in New Hampshire.
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EFFECTIVE MASS
Real metals: electrons still behave likefree particles, but with “renormalized” effective mass m*
E =2k2
2m*
In potassium (a metal), assuming m* =1.25m gets the correct
(measured) electronic heat capacity
Physical intuition: m* > m, due to “cloud” of phonons and other excited electrons.
Fermi Surface
At T>0, the periodic crystal and electron-electroninteractions and electron-phonon interactionsrenormalize the elementary excitation to an “electron-likequasiparticle” of mass m*
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EFFECTIVE MASS
Real metals: electrons still behave likefree particles, but with “renormalized” effective mass m*
E =2k2
2m*
In potassium (a metal), assuming m* =1.25m gets the correct
(measured) electronic heat capacity
Physical intuition: m* > m, due to “cloud” of phonons and other excited electrons.
Fermi Surface
At T>0, the periodic crystal and electron-electroninteractions and electron-phonon interactionsrenormalize the elementary excitation to an “electron-likequasiparticle” of mass m*
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Electrical Conductivity
τvm
Eedt
dvmF
*
* −−==
v = −eτm*
E
*m
ee
τµ =
Em
nevnej
*
2τ=−=
Collisions cause drag
Electric Field Accelerates charge
τ ≈ mean time between collisions
0=vSteady state solution:
⇒
=mobility
Electric current density (charge per second per area)
Units: n=N/V ~ L-3 v ~ L/S
current per area
average velocity
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Electrical Conductivity
τvm
Eedt
dvmF
*
* −−==
v = −eτm*
E
*m
ee
τµ =
Em
nevnej
*
2τ=−=
Collisions cause drag
Electric Field Accelerates charge
τ ≈ mean time between collisions
0=vSteady state solution:
⇒
=mobility
Electric current density (charge per second per area)
Units: n=N/V ~ L-3 v ~ L/S
current per area
average velocity
![Page 18: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/18.jpg)
Electrical Conductivity
Em
nevnej
*
2τ=−=
Electric current density (charge per second per area)
current per area
Ej
σ≡
*
2
m
ne τσ = Electrical Conductivity
OHM’s LAW (V = I R )
n = N/V
me = mass of electron
e = charge on electron
τ = mean time between collisions
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Electrical Conductivity
Em
nevnej
*
2τ=−=
Electric current density (charge per second per area)
current per area
Ej
σ≡
*
2
m
ne τσ = Electrical Conductivity
OHM’s LAW (V = I R )
n = N/V
me = mass of electron
e = charge on electron
τ = mean time between collisions
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What Causes the Drag?
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Bam!
Random
Collisions
On average,
I go about τ seconds between
collisions
with phonons
and impurities
electron
phonon
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Bam!
Random
Collisions
On average,
I go about τ seconds between
collisions
with phonons
and impurities
electron
phonon
![Page 23: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/23.jpg)
Scattering
It turns out that static ions do not cause collisions!
What causes the drag? (Otherwise metals would have infinite conductivity)
Electrons colliding with phonons (T > 0)
Electrons colliding with impurities
( ) ∞→= 0Tphτ
τimp is independent of T
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Mathiesen’s Rule
( ) impphtot T τττ111 +=
how often electronsscatter total how often electrons scatter
from phonons
how often electrons scatter fromimpurities
Independent scattering processes means the RATES can be added.
5 phonons per sec. + 7 impurities per sec.
= 12 scattering events per second
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Mathiesen’s Rule
( ) impphtot T τττ111 +=
how often electronsscatter total how often electrons scatter
from phonons
how often electrons scatter fromimpurities
Independent scattering processes means the RATES can be added.
5 phonons per sec. + 7 impurities per sec.
= 12 scattering events per second
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Mathiesen’s Rule ( ) impphtot T τττ111 +=
em
ne τσ2
=τ
ρ 12ne
me=
ph
eph ne
m
τρ 1
2=
imp
eimp ne
m
τρ 1
2=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+=
impph
eimpphtot ne
m
ττρρρ 11
2
Resistivity
If the rates add, then resistivities also add:
Resistivities Add
(Mathiesen’s Rule)
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Mathiesen’s Rule ( ) impphtot T τττ111 +=
em
ne τσ2
=τ
ρ 12ne
me=
ph
eph ne
m
τρ 1
2=
imp
eimp ne
m
τρ 1
2=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+=
impph
eimpphtot ne
m
ττρρρ 11
2
Resistivity
If the rates add, then resistivities also add:
Resistivities Add
(Mathiesen’s Rule)
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Thermal conductivity
⎥⎦⎤
⎢⎣⎡
⋅=
area
Ejt sec
jt = εvn
Heat current density
ε = Energy per particle
v = velocity
n = N/V
Electric current density
Heat current density
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Thermal conductivity
⎥⎦⎤
⎢⎣⎡
⋅=
area
Ejt sec
jt = εvn
Heat current density
ε = Energy per particle
v = velocity
n = N/V
Electric current density
Heat current density
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Thermal conductivityHeat current density
x
Heat Current Density jtot through the plane: jtot = jright - jleft
jrightjleft
Heat energy per particle passing through the plane
started an average of “l” away.
About half the particles are moving right, and about half to the left.
x
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Thermal conductivityHeat current density
x
Heat Current Density jtot through the plane: jtot = jright - jleft
jrightjleft
Heat energy per particle passing through the plane
started an average of “l” away.
About half the particles are moving right, and about half to the left.
x
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Thermal conductivityHeat current density
x
Limit as l goes small:
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Thermal conductivityHeat current density
x
Limit as l goes small:
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Thermal conductivityHeat current density
x
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Thermal conductivityHeat current density
x
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Thermal conductivityHeat current density
x
Tx
T ∇→∂∂ 22222 3 xzyx vvvvv =++=
Tcvj vt ∇−=
τ2
3
1Tjt ∇−= κ vcv τκ 2
3
1=
How does it depend on temperature?
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Thermal conductivityHeat current density
x
Tx
T ∇→∂∂ 22222 3 xzyx vvvvv =++=
Tcvj vt ∇−=
τ2
3
1Tjt ∇−= κ vcv τκ 2
3
1=
How does it depend on temperature?
![Page 38: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/38.jpg)
Thermal conductivityvcv τκ 2
3
1=
2
2
1FeF vmE = cv =
π 2
2nkB
TTF
⎛⎝⎜
⎞⎠⎟
κ =132EF
me
⎛⎝⎜
⎞⎠⎟τ π
2
2nkB
TTF
⎛⎝⎜
⎞⎠⎟
=π 2nkB
2Tτ3me
=κ
=π 2kB
2T3
nτme
⎛⎝⎜
⎞⎠⎟
⇒
![Page 39: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/39.jpg)
Thermal conductivityvcv τκ 2
3
1=
2
2
1FeF vmE = cv =
π 2
2nkB
TTF
⎛⎝⎜
⎞⎠⎟
κ =132EF
me
⎛⎝⎜
⎞⎠⎟τ π
2
2nkB
TTF
⎛⎝⎜
⎞⎠⎟
=π 2nkB
2Tτ3me
=κ
=π 2kB
2T3
nτme
⎛⎝⎜
⎞⎠⎟
⇒
![Page 40: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/40.jpg)
Wiedemann-Franz Ratio
⎟⎟⎠
⎞⎜⎜⎝
⎛=
e
B
m
nTk τπκ3
22
⎟⎟⎠
⎞⎜⎜⎝
⎛=
em
ne
τσ 2
28
22
1045.23 κπ
σκ Ω×=⎟
⎠⎞⎜
⎝⎛= − W
e
k
TB
Fundamental Constants !
Cu: = 2.23 × 10-8 WΩ/κ2 (Good at low Temp)
Major Assumption: τthermal = τelectronic
Good @ very hi T & very low T
(not at intermediate T)
![Page 41: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/41.jpg)
Wiedemann-Franz Ratio
⎟⎟⎠
⎞⎜⎜⎝
⎛=
e
B
m
nTk τπκ3
22
⎟⎟⎠
⎞⎜⎜⎝
⎛=
em
ne
τσ 2
28
22
1045.23 κπ
σκ Ω×=⎟
⎠⎞⎜
⎝⎛= − W
e
k
TB
Fundamental Constants !
Cu: = 2.23 × 10-8 WΩ/κ2 (Good at low Temp)
Major Assumption: τthermal = τelectronic
Good @ very hi T & very low T
(not at intermediate T)
![Page 42: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/42.jpg)
Homework Problem 3 “rs”
Radius of sphere denoting volume perconduction electron
n=N/V=density of conduction electronsVN
≡1n=43πrs
3
rs =34πn
⎛⎝⎜
⎞⎠⎟1/3
In 3D
Defines rs
![Page 43: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/43.jpg)
Homework Problem 3 “rs”
Radius of sphere denoting volume perconduction electron
n=N/V=density of conduction electronsVN
≡1n=43πrs
3
rs =34πn
⎛⎝⎜
⎞⎠⎟1/3
In 3D
Defines rs
![Page 44: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/44.jpg)
Solid State Simulations
http://www.physics.cornell.edu/sss/
Go download these and play with them!
For this week, try the simulation “Drude”
![Page 45: Last Time - Purdue Universityerica/545/Lecture07.pdf · Last Time Free electron model Density of states in 3D g~ ... t=εvn Heat current density ε = Energy per particle v = velocity](https://reader031.vdocuments.net/reader031/viewer/2022030410/5a9a0ce27f8b9adb5c8d8c7a/html5/thumbnails/45.jpg)
Today
Measuring the occupied density of states
Effective Mass
Electrical Conductivity
Thermal Conductivity
Wiedemann-Franz Ratio
Heat Capacity 3BTATC +=
Electrons Phonons