Non-Continuum Energy Transfer: Electrons– electrons inhabit quantized (discrete) energy states called orbitals – the potential energy V is related to the quantum state , charge

AME 60634 Int. Heat Trans.

D. B. Go Slide 1

Non-Continuum Energy Transfer: Electrons


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


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


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


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)


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


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


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


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,...


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)


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


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


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


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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occ

up

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on

, f(!)

electron energy, ! (eV)

ε F =

5 e

V

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, f(!)

electron energy, ! (eV)

1000 K

300 K

εF = 5 eV


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


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


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?


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*


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?


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


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


D. B. Go Slide 22

Electrons – Effect of Lattice

band gap!


D. B. Go Slide 23

Electrons – Effect of Lattice


D. B. Go Slide 24

Electrons – Band Gaps

G. Chen


D. B. Go Slide 25

Electrons – Material Types

G. Chen


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

Documents

Non-Continuum Energy Transfer: Electrons– electrons inhabit quantized (discrete) energy states called orbitals – the potential energy V is related to the quantum state , charge