Solids.ppt 4573KB Jun 23 2011 10:24:38 AM

Solids

Eisberg & Resnick Ch 13 & 14 RNave:

http://hyperphysics.phy-astr.gsu.edu/hbase/solcon.html#solcon

Alison Baski:

http://www.courses.vcu.edu/PHYS661/pdf/01SolidState041.ppt Carl Hepburn, “Britney Spear’s Guide to Semiconductor Physics”.

OUTLINE

• Review Ionic / Covalent Molecules

• Types of Solids (ER 13.2)

• Band Theory (ER 13.3-.4)

– basic ideas

– description based upon free electrons

– descriptions based upon nearly-free electrons

• ‘Free’ Electron Models (ER 13.5-.7)

Ionic Bonds

Ionic Bonding

Covalent Bonds

Covalent Bonding

SYM ASYM

spatial spin

ASYM SYM spatial spin

Covalent Bonding

S_tot = 1 not really parallel, but spin-symmetric

not really anti, but spin-asym

S_tot = 0

TYPES OF SOLIDS (ER 13.2)

CRYSTALINE BINDING

• molecular

• ionic

Molecular Solids

• orderly collection of molecules held together by v. d. Waals

• gases solidify only at low Temps

• easy to deform & compress

• poor conductors

Ionic Solids

• individ atoms act like closed-shell, spherical, therefore binding not so directional • arrangement so that minimize nrg for size of atoms

• tight packed arrangement  poor thermal conductors • no free electrons  poor electrical conductors

• strong forces  hard & high melting points • lattice vibrations absorb in far IR

• to excite electrons requires UV, so ~transparent visible

Covalent Solids

• 3D collection of atoms bound by shared valence

electrons

• difficult to deform because bonds are directional

• high melting points (b/c diff to deform)

• no free electrons



poor electrical conductors

• most solids adsorb photons in visible



opaque

Metallic Solids

• (weaker version of covalent bonding)

• constructed of atoms which have very weakly

bound outer electron

• large number of vacancies in orbital (not enough

nrg available to form covalent bonds)

• electrons roam around (electron gas )

• excellent conductors of heat & electricity

• absorb IR, Vis, UV



opaque

Six Closely Spaced Atoms

as fn(R)

Solid of N atoms

Two atoms Six atoms

ref: A.Baski, VCU 01SolidState041.ppt

Four Closely Spaced Atoms

Solid composed of ~N

_A

Na Atoms

as fn(R)

Sodium Bands vs Separation

Copper Bands vs Separation

Differences down a column in the Periodic Table:

IV-A Elements

Sandin

Band Spacings

in

Insulators & Conductors

electrons free to roam

electrons confined to small region

How to choose



F

and

Fermi Distribution for a selected



_F

1

1

)

(

_{( )/}





_{ kT}

F

e

How does one choose/know



F

If in unfilled band, _F is energy of highest energy electrons at T=0.

Fermions

T=0

Number of Electrons at an Energy



 



 







n

N

d

KE

Tot







0

distrib fn Number of ways

to have a particular energy

In QStat, we were doing

# states

probability of this nrg

occurring

Semiconductors

• Types

– Intrinsic – by thermal excitation or high nrg photon – Photoconductive – excitation by VIS-red or IR – Extrinsic – by doping

• n-type • p-type

• ~1 eV

Intrinsic Semiconductors

Silicon

Germanium

Doped Semiconductors

lattice

5A in 4A lattice

3A in 4A lattice

‘Free-Electron’ Models

• Free Electron Model (ER 13-5)

• Nearly-Free Electron Model (ER13-6,-7)

– Version 1 – SP221

– Version 2 – SP324

– Version 3 – SP425

• Free-Electron Model

– Spatial Wavefunctions – Energy of the Electrons – Fermi Energy

– Density of States dN/dE E&R 13.5

– Number of States as fn NRG E&R 13.5

• Nearly-Free Electron Model (Periodic Lattice Effects) – v2 E&R 13.6 • Nearly-Free Electron Model (Periodic Lattice Effects) – v3 E&R 13.6

 

         E m 0 2 2 2 





k x k y k z L

xyz  8₃ sin _x sin _y sin _z 

 

In a 3D slab of metal, e’s are free to move but must remain on the inside

Solutions are of the form:

L n_z





2 2 2



2 2

8mL nx ny nz

h   



With nrg’s:

At T = 0, all states are filled

up to the Fermi nrg



2 2 2



_max

2 2

8 x y z

fermi n n n

mL h   



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

max 2 2

2

2

n

n

n

total number of states up to an energy _fermi: 3 3 max 4 8 1 8 1

2

2

n sphereof volume 

















N

3 / 2 2

3

8















V

N

m

h

fermi





max 2 2 2

8mL n

h

fermi 



Sample Numerical Values for Copper slab

V

N

= 8.96 gm/cm3 _{1/63.6 amu 6e23 = 8.5e22 #/cm}3 _{= 8.5e28 #/m}3

_fermi = 7 eV

3 / 2 2

3

8















V

N

m

h

fermi





n_max = 4.3 e 7

Density of States

How many combinations of are there

within an energy interval  to  + d ?

3 / 2 2

3

8















V

N

m

h

fermi





2 / 3 2 8

3 _

           h mE V N  dE h m h mE V dN ₂ 2 / 1 2 8 8 2 3

3 _

           



3



1/2 1/2 3

2

8

E

m

h

V

dE

dN





    

 n N d KE

Tot _





At

T ≠ 0

the electrons will be spread out among the allowed states

How many electrons are contained in a particular energy range?

Distribution of States:

Problems with Free Electron Model

(ER13-6, -7)

 

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

1) Bragg reflection 2) .

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

Nearly-Free Electron Model

version 1 – SP221

2 / 2 2 / k

a  

2 /   k 2 / 2 2 / k

Nearly-Free Electron Model

version 2 – SP324

• Bloch Theorem

• Special Phase Conditions, k = +/- m /a • the Special Phase Condition k = +/- /a

(x) ~ u e i(kx-t)

(x) ~ u(x) e i(kx-t)

 

~~~~~~~~~~

amplitude

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

Amplitude varies with location

u(x) = u(x+a) = u(x+2a) = ….

u(x+a) = u(x)

(x+a) e -i(kx+ka-t) _(x) e -i(kx-t)

(x) ~ u(x) e i(kx-t)

(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 , a a

k  







a

k  



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

Kittel

Free-electron Nearly Free-electron

Kittel

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

2 2

m

k







Effective Mass m*

-- describing the balance between applied ext-E and lattice site reflections

2 2 2

1

*

1

k

m











m* a =  F_ext

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

* 2 2 2 2

2

2

m

k

m

k

apply perturbation fromlattice











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.

From earlier: _{Even when above barrier,}

reflection and transmission coefficients can

change in motion due to reflections is more significant than change in motion

due to applied field change in motion

due to applied field

Nearly-Free Electron Model

version 3

à la Ashcroft & Mermin, Solid State Physics

This treatment recognizes

that the reflections of electron waves off lattice sites can

right left

sum



A





B





Bloch’s Theorem defines periodicity of the wavefunctions:



x

a



e

ika _sum

 

x

sum











x

a



e

ika _sum

 

x

sum











unknown weights



x a



eika _sum

 

x

sum   





_sum



x



a





e

ika



_sum



 

x

Applying the matching conditions at x  a/2

A + B left right

A + B left right

A + B left right

A + B left right iKa iKa

e

t

e

t

r

t

ka









2

1

2

cos

2 2 m K 2 2 2   

For convenience (or tradition) set:

2 2

1t  r



i

e

t

t



_{r i r e}i





ka

t

Ka

cos

cos







Related to possible Lattice spacings Related to Energy m K 2 2 2   

al

lo

w

ed

s

ol

ut

io

n

re

gi

on

R Nave: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/supcon.html#c1

Temperature Dependence of Resistivity

Temperature Dependence of

Resistivity

A

L

• Conductors

– Resistivity  increases with increasing Temp

 Temp   but same # conduction e-’s  

• Semiconductors & Insulators

– Resistivity  decreases with increasing Temp

Superconductors.org _{Only in nanotubes}

Superconductor Classifications

• Type I

– tend to be pure elements or simple alloys –  = 0 at T < Tcrit

– Internal B = 0 (Meissner Effect) – At jinternal > jcrit, no superconductivity

– At Bext > Bcrit, no superconductivity

– Well explained by BCS theory

• Type II

– tend to be ceramic compounds

– Can carry higher current densities ~ 1010 A/m2

– Mechanically harder compounds – Higher Bcrit critical fields

Type I

Bardeen, Cooper, Schrieffer 1957, 1972 “Cooper Pairs”

Symmetry energy ~ 0.01 eV

Q: S_tot=0 or 1? L? J?

e

Sn 230 nm Al 1600 Pb 83 Nb 38

Best conductors  best ‘free-electrons’  no e – lattice interaction

 not superconducting

Popular Bad Visualizations:

Pairs are related by momentum ±p, NOT position.

More realistic 1-D billiard ball picture:

Cooper Pairs are ±k sets

Furthermore:

• Experimental Support of BCS Theory

– Isotope Effects

– Measured Band Gaps corresponding to Tcrit

predictions

– Energy Gap decreases as Temp  Tcrit

Normal Conductor

Semiconductor or

Another fact about Type I:

Type II

Q: does BCS apply ?

mixed normal/super

Yr Composition

T

_c

May 2006

InSnBa₄Tm₄Cu₆O_{18+ 150}

2004 Hg_0.8Tl_0.2Ba₂Ca₂Cu₃O_{8.33 138}

actual ~ 8 m

Type II – mixed phases

Q: does BCS apply ?

Y Ba₂ Cu₃ O₇ crystalline

La_2-x Ba_x Cu O₂ solid solution

Another fact about Type II:

Applications

OR

Other Features of Superconductors

Magnetic Levitation – Meissner Effect

Q: Why ?

Magnetic Levitation – Meissner Effect

MLX01 Test Vehicle

2003 581 km/h 361 mph 2005 80,000+ riders

2005 tested passing trains at relative 1026 km/h

Maglev in Germany (sc? idi)

32 km track

550,000 km since 1984 Design speed 550 km/h

Josephson Junction

Recall: Aharonov-Bohm Effect

-- from last semester

affects the phase of a wavefunction

Source

B

 / )

( ₂

~

e

i peA r 

/ )

( ₁

~

e

i p eA r



/

~

~ eikx eipx

SQUID

superconducting quantum interference device

left

i o

e







~



~



_o

e

iright

o

 i o

e





~

)

(

location

fn







B Bohm Aharonov loop

q

n

dl































2

q

n

B









2

2 15

10

07

.

2

)

2

(

2

m

Telsa

e











Typical B fields

MAGSAFE will be able to locate targets without flying close to the surface.

Image courtesy Department of Defence.

http://www.csiro.au/science/magsafe.html

Finding 'objects of interest' at sea with MAGSAFE

MAGSAFE is a new system for locating and identifying submarines.

Operators of MAGSAFE should be able to tell the range, depth and bearing of a target, as well as where it’s heading, how fast it’s going and if it’s diving.

Building on our extensive experience using highly sensitive magnetic sensors known as Superconducting QUantum Interference Devices (SQUIDs) for minerals exploration, MAGSAFE harnesses the power of three SQUIDs to measure slight variations in the local magnetic field.

MAGSAFE has higher sensitivity and greater immunity to external noise than conventional

Magnetic Anomaly Detector (MAD) systems. This is especially relevant to operation over shallow seawater where the background noise may 100 times greater than the noise floor of a MAD

SQUID

2 nm

1014 _{T SQUID threshold}

Heart signals 10 10 _T

• Fundamentals of superconductors:

– http://www.physnet.uni-hamburg.de/home/vms/reimer/htc/pt3.html • Basic Introduction to SQUIDs:

– http://www.abdn.ac.uk/physics/case/squids.html • Detection of Submarines

– http://www.csiro.au/science/magsafe.html

• Fancy cross-referenced site for Josephson Junctions/Josephson: – http://en.wikipedia.org/wiki/Josephson_junction

– http://en.wikipedia.org/wiki/B._D._Josephson

• SQUID sensitivity and other ramifications of Josephson’s work: – http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid2.html • Understanding a SQUID magnetometer:

– http://hyperphysics.phy-astr.gsu.edu/hbase/solids/squid.html#c1 • Some exciting applications of SQUIDs:

• Relative strengths of pertinent magnetic fields

– http://www.physics.union.edu/newmanj/2000/SQUIDs.htm • The 1973 Nobel Prize in physics

– http://nobelprize.org/physics/laureates/1973/ • Critical overview of SQUIDs

– http://homepages.nildram.co.uk/~phekda/richdawe/squid/popular/ • Research Applications

– http://boojum.hut.fi/triennial/neuromagnetic.html • Technical overview of SQUIDs:

– http://www.finoag.com/fitm/squid.html

Redraw LHS

Sn 230 nm Al 1600

Pb 83 Nb 38

Best conductors  best ‘free-electrons’  no e – lattice interaction