the need for quantum mechanics in materials …shenoy/mr301/www/drude.pdfconcepts in materials...
TRANSCRIPT
![Page 1: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/1.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 0
The Need for Quantum Mechanics inMaterials Science
![Page 2: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/2.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 1
Some Experimental Facts Regarding Metals
Dulong-Petit Law
(Kittel)
High-Temperature Molar Specific Heat = 3R
Atoms - Classical Oscillators
![Page 3: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/3.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 2
Some Experimental Facts Regarding Metals
Wiedemann-Franz Law: Ratio of thermal (κ) toelectrical conductivities (σ) depends linearly on T
κ/σ = (Const)T, (Const) ≈ 2.3 × 10−8watt-ohm/K2
(Ashcroft-Mermin)
![Page 4: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/4.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 3
Some Basics
Electron charge (e), mass(m), number density(n)
Newtons’ Lawdp
dt= F (p–momentum, F–force,
t–time)
Force on a charged particle (q), (q = −e for electron,E, B–electric and magnetic fields)
F = q(E +p
m× B)
Current density j = −enpm
Conductivities:
j = σE (Electrical), q = −κ∇T (Thermal)
![Page 5: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/5.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 4
Drude Theory
Early 19th century
Electrons treated as classical particles
Electrons collide with atoms (and other electrons)
About 1023 electrons (How to do this???)
What do you expect?
![Page 6: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/6.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 5
Drude Model
How to handle all the electrons?
Relaxation time approximation:τ – time in which an electron will definitely undergo acollision
Drude Equation
p(t + dt) =
(
1 −dt
τ
)
︸ ︷︷ ︸
prob. no coll.
(p(t) + Fdt)
=⇒dp
dt= −
p
τ+ F
![Page 7: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/7.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 6
Drude Model
Over a long time tl, electrons attain drift velocity anddo not accelerate
Thus,
1
tl
∫ tl
0
dp
dtdt = = 0
1
tl
∫ tl
0
pdt = pd
Drude Equation gives drift velocity:
pd = τF
=⇒ vd =τ
mF (drift velocity)
![Page 8: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/8.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 7
Drude Model
Drift velocity in an electric field (F = −eE)
vd = −τe
mE
Current density
j = −nevd =ne2τ
mE
Electrical conductivity
σ =ne2τ
m
![Page 9: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/9.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 8
So What?
The theory has a parameter τ
How to calculate τ?
Well, we don’t know yet!
And...so, what?
Calculate relaxation times from conductivitymeasurements
What do you think it will be?
![Page 10: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/10.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 9
Relaxation Times
Calculated relaxation times
(Ashcroft-Mermin)
![Page 11: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/11.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 10
Thermal Conductivity
Energy of an electron at temperature T , E[T ] = 3
2kBT
T(x+v ) x τ
v xτ v xτ
T(x−v ) x τ
A (one-D) body with a temperature gradient
Magnitude of velocity (speed) in x direction = vx
Heat flux from left to right =n
2vxE[T (x − vxτ)]
Heat flux from right to left =n
2vxE[T (x + vxτ)]
![Page 12: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/12.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 11
Thermal Conductivity contd.
Net heat flux towards positive x axis
q =n
2vx(E[T (x − vxτ)] − E[T (x + vxτ)]) = nvx
∂E
∂T
(
−∂T
∂xvxτ
)
= −n v2
x︸︷︷︸
kBT
m
∂E
∂T︸︷︷︸
3
2kB
∂T
∂x= −
(3nk2
BτT
2m
)∂T
∂x
Thermal conductivity:
κ =3nk2
BTτ
2m
![Page 13: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/13.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 12
And now, Wiedemann-Franz!
Ratio of thermal to electrical conductivity
κ
σ=
3
2
(kB
e
)2
T
It is linear in T !
It is independent of the metal!
What about the constant (Lorentz number)?
3
2
(kB
e
)2
= 1.11 × 10−8watt-ohm/K2
Expt. value ≈ 2.3 × 10−8watt-ohm/K2! Celebrations!
![Page 14: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/14.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 13
There is just one more thing...
Dulong-Petit say specific heat is 3R
One mole of univalent metal contains one mole of ionsand one mole of electrons
Ionic specific heat = 3R
Electronic specific heat =3
2R (ideal gas)
Total specific heat =9
2R
9
26= 3 for usual values of 2, 3, 9!
Ok, turn the music down!
![Page 15: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/15.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 14
Hall Effect
xz
y
jxE y
B
Electric field applied in the x direction – current flowsjx
Magnetic field B applied in the z direction
An electric field Ey develops in the y-direction
![Page 16: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/16.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 15
Hall Effect contd.
Hall coefficient
RH =Ey
jxB
Drude value of Hall coefficient
RDH = −
1
ne
(prove this!)
Independent of relaxation time!
![Page 17: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/17.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 16
What about experiments?
Ratio of theoretical and experimental Hall coefficients
(Ashcroft-Mermin)
Ok for some metals, but suggests that electrons arepositively charged in some metals! Oh, God!
Stop the party!
![Page 18: The Need for Quantum Mechanics in Materials …shenoy/mr301/WWW/drude.pdfConcepts in Materials Science I VBS/MRC Need for Quantum Mechanics { 0 The Need for Quantum Mechanics in Materials](https://reader031.vdocuments.us/reader031/viewer/2022030902/5b401dc97f8b9a2f138ce5c2/html5/thumbnails/18.jpg)
Concepts in Materials Science I
VBS/MRC Need for Quantum Mechanics – 17
Conclusions...and then...
Classical theory is able to explain Wiedemann-Franz(Success!)
Fails with specific heat (Ok, lets see!)
Fails miserably with Hall coefficients (Surprise!)
Well, we DO NEED Quantum Mechanics!