non-continuum energy transfer: electrons– electrons inhabit quantized (discrete) energy states...
TRANSCRIPT
AME 60634 Int. Heat Trans.
D. B. Go Slide 1
Non-Continuum Energy Transfer: Electrons
AME 60634 Int. Heat Trans.
D. B. Go Slide 2
The Crystal Lattice • The crystal lattice is the organization of atoms and/or molecules in
a solid
• The lattice constant ‘a’ is the distance between adjacent atoms in the basic structure (~ 4 Å)
• The organization of the atoms is due to bonds between the atoms – Van der Waals (~0.01 eV), hydrogen (~kBT), covalent (~1-10 eV), ionic
(~1-10 eV), metallic (~1-10 eV)
cst-www.nrl.navy.mil/lattice
NaCl Ga4Ni3
simple cubic body-centered cubic
tungsten carbide
hexagonal
a
AME 60634 Int. Heat Trans.
D. B. Go Slide 3
The Crystal Lattice • Each electron in an atom has a particular potential energy
– electrons inhabit quantized (discrete) energy states called orbitals – the potential energy V is related to the quantum state, charge, and
distance from the nucleus
• As the atoms come together to form a crystal structure, these potential energies overlap è hybridize forming different, quantized energy levels è bonds
• This bond is not rigid but more like a spring
€
V r( ) =−Znle
2
r
potential energy
AME 60634 Int. Heat Trans.
D. B. Go Slide 4
The Crystal Lattice – Electron View • The electrons of a single isolated atom occupy atomic orbitals,
which form a discrete (quantized) set of energy levels • Electrons occupy quantized electronic states characterized by four
quantum numbers – energy state (principal) è energy levels/orbitals – magnetic state (z-component of orbital angular momentum) – magnitude of orbital angular momentum – spin up or down (spin quantum number)
• Pauli exclusion principle: no 2 electrons can occupy the same exact energy level (i.e., have same set of quantum numbers)
• As atomic spacing decreases (hybridization) atoms begin to share electrons è band overlap
AME 60634 Int. Heat Trans.
D. B. Go Slide 5
Electrons - Conductors • In the atomic structure, valence electrons are in the outer most
shells – loosely bonded to the nucleus è free to move!
• In metals, there are fewer valence electrons occupying the outer shell è more places within the shell to move
• When atoms of these types come together (sharing bands as discussed before) è electrons can move from atom to atom – electrons in motion makes electricity! (must supply external force –
voltage, temperature, etc.)
• In metals the valence electrons are free to move è electrons are the energy carrier
• In insulators the valence shells are fully occupied and there’s nowhere to move è energy carriers are now the bond (spring) vibrations (phonons)
AME 60634 Int. Heat Trans.
D. B. Go Slide 6
Electrons – Free Electron Model
G. Chen
free electron
In metals, we treat these electrons as free, independent particles • free electron model, electron gas, Fermi gas • still governed by quantum mechanics and statistics
free electron gas
AME 60634 Int. Heat Trans.
D. B. Go Slide 7
Electrons – Energy and Momentum
€
−2
2m∇2Ψ
r ( ) = EΨ r ( )wave function |ψ2| can be thought of as electron probability (or likelihood of an electron being there) è Heisenberg uncertainty principle
eigenfunction of Shrödinger’s equations energy
momentum
€
p = k
The energy and momentum of a free electron is determined by Schrödinger’s equation for the electron wave function Ψ
We assume a form of the wave function
€
Ψ r ( ) =
1∀
ei k ⋅ r
€
ε k( ) =2k 2
2m
From here we determine the electron’s energy and momentum
k is again the wave vector
AME 60634 Int. Heat Trans.
D. B. Go Slide 8
Electrons – Energy and Momentum Recall phonons: we sought a relationship between energy (frequency) and momentum (wave vector) ω = f(k) (dispersion relation)
€
ω k( ) = 2 gmsin ka2( ) dispersion relation for an acoustic phonon
€
ε k( ) =2k 2
2mdispersion relation for free electron
- we assumed form of the solution:
- we set up a governing equation:
€
xna t( ) ~ e i kna−ωt( )( )
€
m d2xnadt 2
= −g 2xna − x(n−1)a − x(n+1)a[ ]
Much like phonons, from the dispersion relation we can determine the density of states, which combined with the occupation will tell us the internal energy è specific heat
AME 60634 Int. Heat Trans.
D. B. Go Slide 9
Electrons – Energy and k-space We saw with phonons that only discrete values of k (wave vectors) can occur è basically, only certain wavelengths can be supported by the atomic structure
real space k-space
Additionally, for electrons, because of the Pauli exclusion principle, each wave vector (k state) can only be occupied by 2 electrons (of opposite spin)
Recall in the analysis of electrons, the wave function was related to the wave vector
€
Ψ r ( ) =
1∀
ei k ⋅ r
It can be shown, that the wave vector may take only certain discrete states (eigenvalues)
€
kx =2πnxL;ky =
2πnyL;kz =
2πnzL
⇒ ni =1,2,3,...
AME 60634 Int. Heat Trans.
D. B. Go Slide 10
Electrons – Energy and k-space We saw with phonons that only discrete values of k (wave vectors) can occur è basically, only certain wavelengths can be supported by the atomic structure
real space k-space
We can describe the allowable momentum states in k-space which takes the form of a circle (2D) or sphere (3D)
AME 60634 Int. Heat Trans.
D. B. Go Slide 11
Electrons - Density of States • The density of states (DOS) of a system describes the number of
states (N) at each energy level that are available to be occupied – simple view: think of an auditorium where each tier represents an
energy level
http://pcagreatperformances.org/info/merrill_seating_chart/
greater available seats (N states) in this energy level
fewer available seats (N states) in this energy level
The density of states does not describe if a state is occupied only if the state exists è occupation is determined statistically
simple view: the density of states only describes the floorplan & number of seats not the number of tickets sold
AME 60634 Int. Heat Trans.
D. B. Go Slide 12
Electrons – Density of States
€
D ε( ) =1∀dNdε
=1∀dNdk
dkdε
Density of States:
The number of states is determined by examining k-space
€
dNdk
= 243πk
3( )2π
L( )3 =
k 3∀3π 2
With some manipulation, it can be shown that the 3D density of states for electrons is
€
D ε( ) =12π 2
2m2
$
% &
'
( )
32ε
€
D ω( ) =3ω 2
2π 2vg3
With some manipulation, it can be shown that the 3D density of states for phonons is
AME 60634 Int. Heat Trans.
D. B. Go Slide 13
Electrons – Fermi Levels • The number of possible electron states is simply the integral of the
density of states to the maximum possible energy level. – at T = 0 K this is the equivalent as determining the number of electrons
per unit volume – we put an electron in each state at each energy level and keep filling up
energy states until we run out
• However, the number of electrons in a solid can be determined by the atomic structure and lattice geometry è known quantity
• We call this maximum possible energy level the Fermi energy and we can similarly define the Fermi momentum, and Fermi temperature
€
ne,0K = D ε( )dε0
ε f
∫ =12π 2
2m2
%
& '
(
) *
32εdε
0
ε f
∫
€
εF =2
2m3π 2ne,0K( )
23
kF = 3π 2ne,0K( )13
TF =εFkB
AME 60634 Int. Heat Trans.
D. B. Go Slide 14
Electrons - Occupation • The occupation of energy states for T > 0 K is determined by the
Fermi-Dirac distribution (electrons are fermions)
• Electrons near the Fermi level can be thermally excited to higher energy states €
f ε( ) =1
exp ε −µkBT
$
% &
'
( ) +1
€
µ ≡ chemical potential ≈ εF
€
nelec = f ε( )D ε( )0
∞
∫ dε
electron number density
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, f(!)
electron energy, ! (eV)
ε F =
5 e
V
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electron energy, ! (eV)
1000 K
300 K
εF = 5 eV
AME 60634 Int. Heat Trans.
D. B. Go Slide 15
Electrons – Specific Heat
€
U = εf ε( )D ε( )0
∞
∫ dεtotal electron energy
specific heat
€
C =∂U∂T
≈ kB2TD εF( ) z2ez
ez +1( )2dz
−ε F kBT
∞
∫ =π 2
2neleckB
TTF~ T€
z = εkBT
If we know how many electrons (statistics), how much energy for an electron, how many at each energy level (density of states) è total energy stored by the electrons! è SPECIFIC HEAT
For total specific heat, we combine the phonon and electron contributions
€
Ctotal = Cphonon + Celectron
€
Ctotal = AT 3 + BT → T <<θD (low temperature)Ctotal = 3natomskB + BT → T >> θD (high temperature)
Basic relationships
AME 60634 Int. Heat Trans.
D. B. Go Slide 16
Electrons – Electrical & Thermal Transport • Thus far, we have determined electron energy and energy storage
by assuming a free electron model è freely moving electrons
• We can also use the free electron approach to predict electrical and thermal transport è limited applicability (what about the lattice!) – we attempt to correct for real structure by using an effective mass m*
(greater than real mass)
• We can quickly assess the electrical transport by a simple application of Newton’s law
€
m* =2
d2εdk 2
AME 60634 Int. Heat Trans.
D. B. Go Slide 17
Electrons – Electrical Transport
€
F = −q
E − m* v
τ= m* d v
dt
Newton’s 2nd Law
Coulombic force drag due to collisions
The steady-state solution gives the average electron “drift” velocity
€
v = − qτm*
E
€
µe =qτm* ≡ electron mobility
The current density is the rate of charge transport per unit area (like heat flux)
€
j = −nelecq v = nelecq
2τm*
E =σ
∇ Φ compare to Ohm’s law!
€
σ =nelecq
2τm*
Therefore, the electrical conductivity is simply
what is the relaxation time/mean free path?
AME 60634 Int. Heat Trans.
D. B. Go Slide 18
Electrons – Scattering Processes • Electrons will scatter off phonons and impurities but not the static
crystal ions
• For the time being, let’s assume we know these mean free paths. We can combine them using Matthiesen’s rule
• We now know the electrical conductivity
• What about the thermal conductivity? è kinetic theory still applies!
€
1τ
=1τ p
+1τ i
€
σ =nelecq
2τm*
AME 60634 Int. Heat Trans.
D. B. Go Slide 19
Electons – Thermal Conductivity • Recall from kinetic theory we can describe the heat flux as
• Leading to
€
qx = −vxτdNEvxdx
= −vx2τdNEdx
= −vx2τdUdx
€
qx = −13v 2τ dU
dTdTdx
= −k dTdx
Fourier’s Law
k = 13v2τC what is the mean time
between collisions?
AME 60634 Int. Heat Trans.
D. B. Go Slide 20
Electrons – Thermal Conductivity Let’s assume we know the mean free path, for the time being …
€
εF = kBTF =12m*vF
2
€
C =π 2
2neleckB
TTF
€
k =13v 2τC =
π 2kB2T3
nelecτm*
$
% &
'
( )
€
σ =nelecq
2τm*these appear
related!
Wiedemann-Franz Ratio
kσT
=π 2kB
2
3q2 = 2.45×10−8 WΩK2
- if we know (measure) one we can find the other - based on the fundamental assumption that the electrical and thermal mean free paths are equivalent - good at high and low T - based on the FREE ELECTRON MODEL
AME 60634 Int. Heat Trans.
D. B. Go Slide 21
Electrons – Free Electron Model • Limits of Free Electron Model
– poorly predicts some aspects of thermal/electrical transport – poorly predicts magnitude of specific heat – poorly predicts magnetic properties – does not explain difference between metal and insulator!
• To properly understand electrons we must account for their interactions with the lattice – we will not go into these details, but you should appreciate the
implications – this enables us to understand the different types of materials & why
computers, photovoltaics, etc. work!
Recall that free electron energy is parabolic!
€
ε k( ) =2k 2
2m€
ε
€
k
- like phonons, there is also a Brillouin zone where the dispersion relation repeats
AME 60634 Int. Heat Trans.
D. B. Go Slide 22
Electrons – Effect of Lattice
band gap!
AME 60634 Int. Heat Trans.
D. B. Go Slide 23
Electrons – Effect of Lattice
AME 60634 Int. Heat Trans.
D. B. Go Slide 24
Electrons – Band Gaps
G. Chen
AME 60634 Int. Heat Trans.
D. B. Go Slide 25
Electrons – Material Types
G. Chen
AME 60634 Int. Heat Trans.
D. B. Go Slide 26
Electrons – What We’ve Learned • Electrons are particles with quantized energy states
– store and transport thermal and electrical energy – primary energy carriers in metals – usually approximate their behavior using the Free Electron Model
• energy • wavelength (wave vector)
• Electrons have a statistical occupation, quantized (discrete) energy, and only limited numbers at each energy level (density of states)
– we can derive the specific heat!
• We can treat electrons as particles and therefore determine the thermal conductivity based on kinetic theory
– Wiedemann Franz relates thermal conductivity to electrical conductivity
• In real materials, the free electron model is limited because it does not account for interactions with the lattice – energy band is not continuous – the filling of energy bands and band gaps determine whether a material
is a conductor, insulator, or semi-conductor