FEM Theory Kul 11 12

7/29/2019 FEM Theory Kul 11 12

1/40

1

Free Electron Model for Metals

Metals are very good at conducting both heat and electricity. Metals were described as behaving like a set of nuclei forming a

lattice with a sea of electrons shared between all nuclei/lattice(moving freely between them):

This is referred to as the free electron model for metals.

This model explains many of the properties of metals: Electrical Conductivity: The mobile electrons carry current.

Thermal Conductivity: The mobile electrons can also carry heat.

Malleability and Ductility: Deforming the metal still leaves eachcation surrounded by a sea of electrons, so little energy isrequired to either stretch or flatten the metal.

Opacity and Reflectance (Shininess): The electrons will have awide range of energies, so can absorb and re-emit many differentwavelengths of light.


2/40

The Free Electron Gas Model

Plot U(x) for a 1-D

crystal lattice:

Simple and

crude finite-

square-well

model:

Can we justify this model? How can one replace the entire lattice by a

constant (zero) potential?

U

U = 0


3/40

Free-Electron Model

m

k

m

p

22

222

classical description


4/40

Em

02

22

zkykxkL

xyz zyx sinsinsin8

3

In a 3D slab of metal, es are free to move

but must remain on the inside

Solutions are of the form:

L

nz

2222

2

8

zyx nnn

mL

h

With energy:

Quantum Mechanical Viewpoint


5/40

At T = 0, all states are filled

up to the Fermi energy

max

222

2

2

8zyxFo

nnnmL

h

A useful way to keep track of the states that are filled is:

max2222

nnnn zyx


6/40

Electron follow Fermi-Dirac Distribution

1

1)(

/

kTfEEeEf


7/40

total number of states up to an energy fermi:

3

3max

4

8

1

8

122

n

sphereof

volume

N

3/2

219

3/22

.10646.33

8

V

NmeVx

V

N

m

hFo

max2

2

2

8n

mL

hFo

# states/volume ~ # free es / volume


8/40

Example: Numerical Values for Copper slab

V

N

= 8.96 gm/cm3

1/63.6 amu 6e23 = 8.5e22 #/cm3

= 8.5e28 #/m3

nmax = 4.3 e 7

so we can easily pretend that theres a smooth distrib of nxnynz-states

eV

V

NmeVx

V

N

m

hFo

0.7.10646.33

8

3/2

219

3/22


9/40

Density of StatesHow many combinations of are there

within an energy interval to + d ?3/2

23

8

V

N

m

hFo

2/3

2

8

3

h

mEVN

dEhm

hmEVdN

2

2/1

288

23

3

2/12/3

2/12/13

3

2

32

8)( E

E

NEm

h

V

dE

dNEg

Fo

dgfKETot

0

Huge number of states, it can be treated in

continuous energy


10/40

At T 0 the electrons will be spread out among the allowed states

How many electrons are contained in a particular energy range?

occuringenergythis

ofyprobabilit

energyparticulara

havetowaysofnumber

1

12

8)(

/)(

2/12/13

3

kTfEeEm

h

VEn

)()()( EfEgEn


11/40

this assumes there are no other issues


12/40

Electrical transport (relaxation time) in conductor

eEv

mdt

dvm

dd j nev

ne

md

2E

Electron Conductivity

F

Fr

v

l

m

en

v

l

v

l

m

e

n

EjLawOhm

*

2

*

2

V

Nn

tyconductivi

velocityFermiv

densityelectronn

pathfreemeanelectronl

timecollisiontimerelaxation

F


13/40

Simple Kinetic Theory of Heat Conduction

tyconductivithermalCv

3

qx

x

Hot Cold

xvx

vxxvxxx xx

nEv2

1nEv

2

1q

2

( )-

( )-

xx x

x

d nEvq v

dxd nE

vdx

2- xdu dT

vdT dx

/C du dT

,

2 2 / 3xv v

2

-3

x

v dTq C

dx

2

3

Cvk

Taylor Expansion:

local thermodynamics equilibrium: u=u(T)


14/40

Assigments

Proof that at T=0 the total energy of

electron

Calculate specific heat capacity using FEM

What fraction of free electron in Cu have a

kinetic energy between 3.95 eV and 4.05 eV

at room temperature

FoNEU 5

3


15/40

Heat Capacity of the Quantum-Mechanical FEM

1

1)(

/

kTEe

Ef

where = chemical potential EF for kT


16/40

Heat Capacity of Metals: Theory vs. Expt. at low T

Very low temperature

measurements reveal:

Meta

lexpt FEG

expt/FEG =

m*/m

Li 1.63 0.749 2.18

Na 1.38 1.094 1.26

K 2.08 1.668 1.25

Cu 0.695 0.505 1.38

Ag 0.646 0.645 1.00

Au 0.729 0.642 1.14

Al 1.35 0.912 1.48

Results for simple

metals (in units

mJ/mol K) show

that the FEG values

are in reasonable

agreement with

experiment, but are

always too high:

The discrepancy is

accounted for by

defining an effectiveelectron mass m* that is

due to the neglected

electron-ion interactions


17/40

Problems with Free Electron Model

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

1) Bragg reflection

2) .

3) .


18/40

Other Problems with the Free Electron Model

graphite is conductor, diamond is insulator

variation in colors of x-A elements

temperature dependance of resistivity

resistivity can depend on orientation of crystal & current I direction frequency dependance of conductivity

variations in Hall effect parameters

resistance of wires effected by applied B-fields

.

.

.


19/40

Nearly-Free Electron Model

version 1SP221

2/2

2/k

a

2/k

2/2

2/

k

a


20/40


version 2SP324

Bloch Theorem

Special Phase Conditions, k = +/- m /a

the Special Phase Condition k = +/- /a

This treatment assumes that when

a reflection occurs, it is 100%.


21/40

(x) ~ u e i(kx-wt)

(x) ~ u(x) e i(kx-wt)

~~~~~~~~~~amplitude

In reality, lower energy waves are sensitive to the lattice:

Amplitude varies with location

u(x) = u(x+a) = u(x+2a) = .

BlochsTheorem


22/40

u(x+a) = u(x)

(x+a) e -i(kx+ka-wt) (x) e -i(kx-wt)

(x) ~ u(x) e i(kx-wt)

(x+a) e ika (x)

Something special happens with the phase when

e ika = 1

ka = +/ m m = 0 not a surprise

m = 1, 2, 3,

...,2,aa

k

What it is ?


23/40

ak

Consider a set of waves with +/ k-pairs, e.g.

k = + /a moves

k = /a moves

This defines a pair of waves moving right & left

Two trivial ways to superpose these waves are:

+ ~ e ikx + e ikx ~ e ikx e ikx

+ ~ 2 cos kx ~ 2i sin kx


24/40

+ ~ 2 cos kx ~ 2i sin kx

Kittel

|+|2 ~ 4 cos2 kx ||2 ~ 4 sin2 kx


25/40

Free-electron Nearly Free-electron

Kittel

Discontinuities occur because the lattice is impacting the movement of electrons.


26/40

Effective Mass m*

A method to force the free electron

model to work in the situations where

there are complications

ER Ch 13 p461 starting w/ eqn (13-19b)

*2

22

m

k

free electron KE functional form


27/40

Effective Mass m*-- describing the balance between applied ext-E and lattice site reflections

2

2

2

1

*

1

km

m* a = S Fext

q Eext

2)


28/40

No distinction between m & m*,

m = m*, free electron, lattice structure does

not apply additional restrictions on motion.

m = m*

greater curvature, 1/m* > 1/m > 0, m* < m

net effect of ext-E and lattice interaction

provides additional acceleration of electrons

greater |curvature| but negative,net effect of ext-E and lattice interaction

de-accelerates electrons

At inflection pt

1)

2)


29/40

*

2222

22 m

k

m

k latticefromonperturbatiapply

Another way to look at the discontinuities

Shift up implies effective mass has decreased, m* < m,

allowing electrons to increase their speed and join

faster electrons in the band.

The enhanced e-lattice interaction speeds up the electron.

Shift down implies effective mass has increased, m* > m,

prohibiting electrons from increasing their speed and making

them become similar to other electrons in the band.

The enhanced e-lattice interaction slows down the electron


30/40

From earlier: Even when above barrier,

reflection and transmission coefficients can

increase and decrease depending upon the energy.


31/40

change in motion

due to reflections

is more significantthan change in motion

due to applied field

change in motion

due to applied fieldenhanced by change in reflection coefficients

Nearly Free Electron Model


32/40


version 3 la Ashcroft & Mermin, Solid State Physics

This treatment recognizes

that the reflections of electronwaves off lattice sites can

be more complicated.


33/40

A reminder:


34/40

Waves from the left behave like:

iKxiKx

leftthe

from ere

iKx

left

thefrom et

m

K

2

22


35/40

Waves from the right behave like:

iKxiKx

rightthefrom

ere iKx

right

thefrom et

m

K

2

22


36/40

rightleftsum BA

Blochs Theorem defines periodicity of the wavefunctions:

xeax sumika

sum

xeax sumika

sum

unknown weights

Related to

Lattice spacing


37/40

xeax sumika

sum

xeax sumika

sum

Applying the matching conditions at x a/2

A + B

leftright

A + B

left right

A + B

left right

A + B

left right

iKaiKae

te

t

rtka

2

1

2cos

22

m

K

2

22

And eliminating the unknown constants A & B leaves:


38/40

For convenience (or tradition) set:

221 rt

iett ierir

kat

Ka

cos

cos


39/40

ka

t

Kacos

cos

Related to

possible

Lattice spacings

Related to

Energy

m

K

2

22

allowed solution regions


40/40

allowedsolu

tionregions

FEM Theory Kul 11 12

Documents

Transcript of FEM Theory Kul 11 12