PHY330 Metals, Semiconductors and Insulators
Electrons in Solids
Professor Maurice Skolnick and Dr Dmitry Krizhanovskii
S ll b
Syllabus
1. The distinction between insulators, semiconductors and metals. The
periodic table
table. Quantitative aspects.
aspects
2. Basic crystal structures. The crystalline forms of carbon.
3. Densityy of states,, Fermi-Dirac statistics. Free electron model.
4. Electrical transport. Resistivity and scattering mechanisms in metals.
Temperature dependence.
5. The nearly free electron model. The periodic lattice, Bragg diffraction,
Brillouin zones. Bloch functions.
6. Prediction of metallic,, insulatingg behaviour: pperiodic ppotential and tightg
binding descriptions.
7. Real metals, shapes of Fermi surfaces. Measurement of Fermi surfaces. De
Haas van Alphen effect and soft x-ray
x ray emission
emission.
http://www.shef.ac.uk/physics/teaching/phy330/
1
8. Semiconductors. Effective mass. Electrons and holes.
9. Optical absorption in semiconductors. Excitons. Comparison with metals.
10. Doping, donors and acceptors in semiconductors. Hydrogenic model.
11. Semiconductor statistics. Temperature dependence.
12. Temperature dependence of carrier concentration and mobility.
Compensation. Scattering mechanisms.
13 Hall
13.
H ll effect,
ff t cyclotron
l t
resonance. L
Landau
d llevels
l in
i magnetic
ti field.
fi ld
14. Plasma reflectivity in metals and semiconductors.
15 Semiconductor heterojunctions.
15.
heterojunctions Quantum wells
wells.
16. Amorphous materials.
The Nobel Prizes 2009 and 2010
2
PHY330: Some General Points
Recommended Textbooks
Solid State Physics, J R Hook and H Hall, Wiley 2nd edition
Introduction to Solid State Physics, C Kittel, Wiley 7th edition
The Solid State,, H M Rosenberg
g Oxford 1989
All the contents of the course, to a reasonable level, can be found in Hook
and Hall.
Kittel has wider coverage, and is somewhat more advanced.
Ashcroft and Mermin is a more advanced, rigorous textbook, with rigorous
proofs.
Blakemore – goodd generall textbook.
b k Similar
i il level
l l to Hookk andd Hall.
ll
3
Relation to Previous Course
The course in its present form has been given in the last five years, and will
cover approximately the same syllabus.
Assessment
The course will be assessed by an end of semester exam (85%) and two homeworks (15%) in the middle and towards the end of the semester respectively
(2 November, 14 December deadlines)
Prerequisite
ii
PHY204, Solids (L R Wilson)
L t
Lecture
N
Notes
t
The notes are organised by topic heading: these correspond to a good
approximation to lecture number
The notes provide an overview of the main points, and all important figures.
Many more details will be given during lectures.
lectures Students thus need to take
detailed notes during lectures to supplement the hand-outs.
4
Overall Aims
Electrons in solids: determine electrical and optical properties
Crystal lattice: bands, band gaps, electronic properties  metals,
semiconductors and insulators
Underpin large parts of modern technology: computer chips, light
emitting diodes,
diodes lasers,
lasers magnets,
magnets power transmission etc,
etc etc
Nanosize structures important modern development
The next slides gives some examples: there are many more
5
Electronics, computing
Integrated circuit http://www.aztex.biz/tag/integrated
‐circuits/
25nm
32nm transistors. Intel web site
Lighting, displays
Multi‐ colour LED strip light
Data storage (cd, dvd, blu‐ray)
,
Telecommunications, internet
Telecommunications laser: Bookham
6
Other major, modern-day applications from condensed matter
physics:
Magnetic materials – hard disks, data storage
Superconductors – magnets, storage ring at e.g. CERN, magnetic
levitation
Liquid crystal displays
Solar cells
M bil communications,
Mobile
i ti
satellite
t llit communications
i ti
7
Research in Semiconductor Physics
There is a highly active research group in the department in
the field of semiconductor physics
Opportunities for projects (3rd and 4th year), and PhDs
See http://ldsd.group.shef.ac.uk/ for more details, or see me
for more details
8
Topic 1: Metals,
Metals semiconductor and insulators
overview and crystal lattices
Range of electron densities
Metals:
e s: Typical
yp ca metal
e a (sodium),
(sod u ), electron
e ec o density
de s y n=2.6x10
.6 028m-3
Insulators (e.g. Diamond): electron density very small
(Eg ~ 5.6eV,
5 6eV ~5000K >>kBT at 300K)
Semiconductors: electron density controllable, and is
temperature dependent,
dependent in range ~10
1016m-3 to ~10
1025m-3
C d i i is
Conductivity
i proportional
i l to electron
l
density
d i
9
Importance of bands and band gaps
• Determine electron density
y and hence optical
p
and electronic
properties
• Understanding of origin will be important part of first 55-6
6
lectures
• Bands and band gaps arise for interaction of electrons with
periodic crystal lattice
• Th
Three schematic
h
ti diagrams
di
illustrating
ill t ti differences
diff
in
i bands,
b d gaps
and their filling in metals, semiconductors and insulators will be
given in the lecture (these are important, simple starting point
for course)
10
I
II
IV
Note also:
Transition metals
Noble metals
11
With relation to previous slide:
p
Group 1: alkali metals, partially filled bands
Group II: alkaline earths
Group IV: semiconductors, insulators, filled bands
+ transition metals, noble metals
12
Crystal Lattices
The nature of the crystal
lattice, and the number of
electrons in the outer shell
determine the conduction
properties of most elements
Periodic arrangement of
atoms
(a) Space lattice
(b) Basis, containing two
different ions
Space lattice plus basis (Fig
Kittel)
Lattice translation vector
T = u1a1 + u2a2 + u3a3
(c) Crystal structure
a1, a2, a3 lattice constants
(spacings of atoms)
Position vector r' = r +T
13
Space lattices in
two dimensions
Primitive (unit) cell
defined by
translation vectors
3D
14
Cubic lattices
Also note diamond is fcc space lattice
Primitive basis: 2 atoms for each p
point of lattice
(Kittel page 19)
15
Primitive (unit) cell: Parallelipiped defined by axes a1, a2, a3
sc, bcc
b and
d ffcc llattices
tti , lattice
l i points
i per cell
ll andd per unit
i volume
l
Simple cubic: 1 lattice point per unit cell
bcc: 2 lattice points per unit cell
fcc: 4 lattice points per unit cell
Number of lattice points per unit volume?
Number of lattice points per unit volume?
16
Periodic table and crystal structures
17
Planes and directions
18
Labelling of directions and planes (Miller indices)
(
) planes
l
[
] directions
19
The Crystalline Forms of Carbon
Diamond
http://diahttp://www theage com au
http://diahttp://www.theage.com.au
Carbon nanotube
Graphite
http://physics.berkeley.edu/research/lanzara/
Buckyball C60
http://www.azonano.com/
Graphene
2010 Nobel Prize to
G i and
Geim
d
Novoselov
http://en.wikipedia.org/wiki/Graphene
http://diahttp://www.theage.com.au
20
2010 Nobel Prize for Physics
A Geim
G i and
dKN
Novoselov
l
Graphene, single sheet of carbon atoms: high electron
motilities electrons with
motilities,
ith ne
new properties
properties, very
er strong
strong,
electronics and sensor applications potentially
21
Comparison of two crystalline forms of carbon
Key properties of diamond
Cubic (diamond) crystal lattice (see slide 20)
Very hard, high strength, insulator, chemically inert, very high
thermal conductivity, optically transparent
Key properties of graphene
Hexagonal crystal lattice (see slides 20, 21), two dimensional
plane
Very strong, metallic but conductivity can be controlled, unique
(E v k ) dispersion relations, very high thermal conductivity,
adsorbate properties
22
Topic 1 summary
1.
Distinctions between metals, semiconductors and insulators, in
particular the widely differing electron densities
2.
Impact on everyday life
3.
Importance of band gaps, and filling of bands, in controlling these
properties
4.
Periodic lattice gives rise to bands, band gaps
5
5.
The crystal structures of carbon
23
Topic 2: Free Electron Model
This is the simplest theory of conduction in metals, based on a
non interacting gas of electrons (which obey Fermi Dirac
non-interacting
statistics). It ignores the presence of the crystal lattice.
It explains
l i some b
basic
i properties,
ti
b
butt ffails
il tto accountt ffor many
others e.g. which elements are metallic, the colour of metals,
electrons and holes etc, for which we need band theory.
Based on the free electron Fermi gas
Electrons are Fermions which obey Fermi-Dirac statistics (and
the Pauli exclusion principle)
24
Fermi-Dirac distribution function
1
f (E) 
exp[E  E F ] / kT
For T→ 0,,
f(E) = 1 for E < EF
f(E) = 0 for E > EF
~kBT
f(E)
E/kB in units of 104 K
25
Free Electron Theory
Leads to condition for
allowed k-values
k values – next
page
26
Counting of States (important, needed to evaluate e.g. the
densit of states
density
states, Fermi energy
energ and other ke
key properties
Use periodic boundary conditions
Allowed values are k=0, ±2/L, ±4/L ... ±N/L (±/a)
(kx = 2n/L)
a lattice constant
L length of chain
N number of atoms (a = L/N)
Proof on next page
27
Periodic bo ndar conditions (bo side L)
Periodic boundary conditions (box, side L)  ( x  L, y , z )   ( x, y , z )
 k (r )  e
ik .r
e
i(kx xk y ykz z )
is solution
provided that kx = 0, ±2/L, ±4/L .... 2n/L, where n is a positive or negative integer
Proof: exp ik x ( x  L)  exp i
2n
( x  L)
L
2nx
 exp i
exp i 2n
L
cos 2n  i sin 2n  1  0  1
i 2nx
 exp
 exp ik x x
L
28
Dispersion Relation
 k (r )  e
Substituting into Schrödinger equation gives ik .r
2
2 2

Ek 
(k x  k y2  k z2 ) 
k2
2m
2m
Parabolic dispersion of free particle with mass m
2
p
Corresponds to E 
, with p  k
2m
p is termed the crystal momentum, and k the wavevector
29
Density of States
The Fermi energy and Fermi surface
Key
ey p
properties
ope t es of
o metals
eta s
30
Need to determine number of states in k-space up to a given
energy (the Fermi energy)
One allowed wavevector in volume element of k-space of (2/L)
3
Volume of sphere in k-space up to energy E, wavevector k is 4 k F3
3
Then calculate number of available states from E = 0 to EF, and
hence derive expression for density of states
31
Number of states,
Fermi wavevector
and Fermi energy
gy

2
EF 
3 2 n
2m
2m

23
32
Values of TF, kF, EF,
vF for sodium and
their significance
(37000K, 0.96x1010 m-1,
3.2eV, 1.07x106m/sec)
33
Topic 2 summary
1. Electrons are Fermions and obey
y Fermi-Dirac statistics and
the Pauli exclusion principle
2 States up to EF filled,
2.
filled above EF empty
3. Form of the density of states proportional to E1/2
4. Expressions and quantitative values for EF, kF, vF (these are
important!)
34
Topic 3: Conductivity
• Drude theory of conductivity based on free electron model
• Ion cores ignored, periodic lattice ignored, effective mass
• Zero frequency approximation, Ohm’s Law
• Displacement of Fermi sphere by electric field and
scattering processes
• Phonon and defect scattering, Matthiesen’s rule
35
Deduce velocity
Newton’s 2nd Law
Define mobility
dv
m
  e E  v x B 
dt

Deduce current density
density,
conductivity and Ohm’s
Law
dk
  e E  v x B 
dt
Include scattering
 dv v
m
    e E  v x B 
 dt  
 scattering time
d.c conditions , B = 0
mv

j  nev
e

m
ne 2

m
 e E
36
Fermi sea of electrons in
applied electric field, and
scattering processes
For derivation of
displacement in kkspace see next slide
37
Motion of electrons in electric field and
scattering: change in wavevector
Alternatively:
eE
vD  
m
mv  eE
k 



5 x 108 smaller
ll th
than kF
So displacement of Fermi sea by
electric field is veryy small
Scattering counters acceleration of
electrons by electric field
38
For metals two scattering
mechanisms are important
1. Lattice scattering - phonons
2. Imperfections (defects) –
impurity atoms, vacancies,
lattice defects
Scattering collisions which are
important are those which relax
momentum gained from E-field
Scattering must be across Fermi
sea
i.e. large
g k, small E
Phonon scattering
• Fermi energy ~ 3 eV
• Phonons have maximum energy
~50 meV
• Scattering
g must be to an empty
py
state
• Thus only electrons close to Fermi
surface can be scattered
• Must conserve energy and
momentum
• Collisions which relax momentum
gained
i d iin applied
li d electric
l t i fifield
ld llead
d
to resistance
• Must be across Fermi sea:
Large k small E
39
For phonons (conservation of energy and wavevector):
k  k ph  k
el
i
el
f
Eiel   ph  E elf
Situation is similar for defect scattering
• However, in this case collisions are elastic, but still with large
momentum change as for phonons
• It is again scattering with large k which is effective in leading
to resistance (as for phonon scattering)
• For phonons scattering is inelastic, but energy change is
negligible
40
Combination of two types of scattering
Phonon scattering is temperature dependent
Scattering by imperfections is temperature independent
Matthiesen’s rule (additive combination of contributions from
phonon and defect scattering)
41
Additional point (important)
Scattering of electrons is not by
ions
Instead by impurities and defects
Electrons propagate freely in periodic
structure ((see Bragg
gg scattering
g and
Bloch functions later)
Mean free path lB> 1m
 or more
lB >> interatomic spacing, so
collisions not with ions
42
Topic 3: Summary
• Theory of conductivity based on free electron model
• Ion cores ignored, periodic lattice ignored. Electrons treated with
effective mass
• Displacement of Fermi sphere by electric field and scattering
processes
• Phonon and defect scattering. Contributions are additive.
Matthiesen’s rule
• Scattering processes which relax momentum across the Fermi sea
are the important ones (in opposite direction to acceleration by field)
• Scattering
S
i iis not by
b the
h ions
i
off the
h lattice.
l i
43
Topic 4: Electrons in periodic lattice
lattice, nearly
free electron model
Many experimental observations are not explained by free electron
theory, including:
1.
2.
3.
4.
5.
6.
Existence of bands, band gaps
Existence of non-metals
Effective
ff i mass
Colours of metals
High frequency conductivity
Nature of the Hall effect
The p
periodic lattice is all important
p
in explaining
p
g these and other
phenomena
Will also discuss the 2009 Nobel Prize for Physics (and the 2010 prize to
Geim and Novoselov)
44
Periodic lattice g
gives rise to
Bragg diffraction of electron
waves
Bragg diffraction, k=nπ/a for 1D
treatment
n  n
2
 2a sin   2a
k
  90o
for waves travelling down 1D
chain
h i
Therefore k=nπ/a
Bragg condition for 1D chain
Electron wave is scattered by 2π/a
(= G) (reciprocal lattice vector)
45
O i i off band
Origin
b d gap from
f
Bragg
B
diffraction
diff
ti (following
(f ll i Kittel,
Ki l chapter
h
7, 7th edition)
di i )
Continued next 2 slides
46
See diagram next
slide
With lower and higher
energy respectively
Two solutions with different energy at same wavelength
(and hence wavevector). Leads to band gap.
47
Origin of band gap from Bragg diffraction
•Bragg diffraction
leads to band gaps,
since cos2(πx/a),
( /a)
sin2(πx/a) charge
distributions at
k
k=nπ/a
/
•Two solutions at
same wavelength
(k-vector)
•Energy gaps occur
when waves have
wavelength which is
in synchronism with
the lattice
48
As noted
A
t d earlier,
li att B
Bragg condition
diti electron
l t
wave iis scattered
tt d b
by kk
= 2π/a (= G) (reciprocal lattice vector)
Lattice
L
tti potential
t ti l (F
(Fourier
i components)
t ) mixes
i
waves att these
th
points
i t iin
dispersion in unperturbed band-structure (in (a) above), giving rise to
gaps in (b)
49
Continuing last slide
Group velocity
d 1 dE
vg 

dk  dk
is zero at zone boundary, corresponds to standing wave
50
To summarise Topic
p 4
• Bragg diffraction defines edge of Brillouin zone.
• Group velocity at Bragg condition (at zone boundary)
is zero
• Bragg diffraction, and hence band gaps, occurs for
waves (k-values)
(k l
) iin synchronism
h i
with
ith llattice
tti
periodicity
• General condition for Bragg diffraction,
k  G
• G is reciprocal lattice vector
51
Nobel Prize in Physics 2009; Strong relevance to Solid State
Physics
Charles K Kao, Optical fibres, Basis of internet data transmission
Combines semiconductor laser
sources, modulators, detectors,
knowledge of optical absorption
mechanisms in solids
52
Willard S. Boyle and George E. Smith, Charge Coupled Device Detectors
Digital imaging device in cameras, fax
machines, scanners, telescopes and
many other types of modern
i t
instrumentation.
t ti
Based
B
d on silicon
ili
integrated circuit technology and field
effect transistors
Readout of information
p
from each pixel
53
K P
Key
Points
i t off T
Topics
i 1-4
14
1. Existence of bands and band gaps vital to explain key properties of
electrons in solids
2. Band – region of allowed electron states in E(k) space
3. Band gap - region of forbidden states, no allowed states
4. Explains distinction between metals, semiconductors and insulators
5. Fermi-Dirac distribution function. States filled up to Fermi wavevector
6. Behaviour of Fermi sphere under applied electric field, small
perturbation
7. Scattering
g mechanisms. Scattering
g is not by
y ions of lattice.
8. Bragg scattering gives rise to band gaps
9. Bragg condition defines k-vectors at which Bragg scattering occurs
10. Treatment of k-vectors for which waves in synchronism with lattice
provides insight into origin of band gaps
11. General condition for Bragg diffraction
k  G
12. Outer shell electrons provide dominant contribution to conduction (see
periodic table)
54
Topic 5: Introduction to Brillouin zones,
zones half
half-filled
filled
and filled bands
• Number of states in a band
• Monovalent atoms metallic
• Insulators:
I l t
can only
l occur for
f even number
b off valence
l
electrons
l t
• Groupp II elements,, nevertheless are metallic.
• Concept of overlapping bands
55
Counting of states and filling of bands
Periodic boundary conditions
(pages 27, 28)
• Each unit cell contributes one value
of k to each Brillouin zone, and
hence to each band
• Including
I l di spin,
i 2N states
t t per b
band
d
k=0,, ±2/L,, ±4/L ... ±N/L ((±/a))
a lattice constant
L length of chain
N number of atoms (a = L/N)
Total number of states between ±/a
is N
More strictly, N is number of primitive
unit cells in chain
• If one atom per unit cell
(monovalent) then band half filled
(monovalent),
– alkali, noble metals
• Insulators can only occur for even
number of valence electrons per
primitive cell (e.g. C, Si, Ge, which
are 4 valent
valent, plus have 2 atoms per
primitive cell)
• Group II elements could be
insulators, but bands overlap, so
metals, but relatively poor metals
((also see Hall effect where there is
hole conduction)
56
Conduction in half-filled
and filled bands
57
I
II
IV
Note also:
Transition metals
Noble metals
58
Alkali metals and noble metals have one outer shell electron: partially filled band and hence metal
Group IV: semiconductors, insulators, 4 outer shell filled bands
Group II: even number of outer shell electrons, but pp g
overlapping bands. Hence metallic.
59
How bands can overlap
in
:
Ec can be less than Eb for:
Ec < Eb for
Eg 
2
2
 k
2m
And thus overlapping
g bands
i.e. energy in second band
less than that in first
60
Overlapping
pp g bands: energy
gy of state in second band lower than in first
Consequence: some of states in second band filled before
uppermost states in first
Leads to two partially filled bands Electrons and holes – anomalous Hall coefficient
61
Summary, Topic 5
• Total number of states in 1D chain, using periodic boundary conditions =
N, where N is number of atoms. Given by total number of allowed kvalues.
values
• Each unit cell contributes one value of k to each Brillouin zone, and hence
to each
h bband.
d IIncluding
l di spin
i gives
i
2N states per band
b d
• Monovalent atoms with one atom per unit cell (alkali and noble metals),
band half filled, expect metallic
• Insulators: can only
y occur for even number of valence electrons pper
primitive cell e.g. C, Si, Ge 4 valence electrons plus 2 atoms per primitive
cell
• But group II elements, the alkaline earths (metals) have even number of
electrons, expected to be insulators, but are metallic.
• Overlapping bands. Can only occur in 2 and 3D. Simple proof for 2D.
62
1. Dependence of
b d gap on atomic
band
t i
number
2. Which shells give
rise to conduction
1
With increasing atomic number,
shielding of atomic core by
outer shell electrons becomes
more effective. Hence
ionisation energy decreases.
Hence less energy required to
raise electron to conduction
band.
2
63
Summary of Bragg diffraction, Brillouin zones
1. Bragg condition defines edges of Brillouin zones
2. For one dimension, simple proof of condition k = ±/a (page 45)
3 In
3.
I generall
k  G
4. Can also be understood in terms of mixing of particular values of
k by Fourier components of periodic lattice potential (page 49) –
(Kittel pages 34-36 for rigorous treatment)
5 Dependence of band gaps on atomic number
5.
number, differing roles of
inner and outer shells.
64
Topic 6: Construction and Properties of Brillouin Zones
•
Use generalised Bragg condition to construct Brillouin Zones
•
Definition and properties of Brillouin Zones
•
Consequences for Fermi surfaces
•
Different zone schemes
•
Essential steps to understand shapes of Fermi surfaces of
real metals (and hence conduction properties)
65
Bragg Diffraction:
66
In 1D, rederivation of
Bragg’s Law
G = 2/a in 1D
67
Geometrical constructions
to obtain Brillouin Zones
Also see next slide
Hook and Hall
(p334)
Perpendicular
bisectors
of G1
68
Construction of Brillouin Zones for Square Lattice
69
Definition of
Brillouin zones
70
1. Generalised Bragg condition
2. 2k .G  G defines boundaries of Brillouin zones . k lies
on perpendicular bisector of G.
2
3. Construction of 1st, 2nd, 3rd zones
4 If Fermi surface is sufficiently large that it crosses
4.
Brillouin zone boundaries, then shape of Fermi surface
will be strongly modified.
71
Reduced
zone scheme
h
Translation vector
Hook and Hall
(p116-118)
Also see Hook and Hall
p39 for physical
discussion
By reciprocal
B
i
l llattice
tti ttranslation
l ti
(2/a), can translate points in
higher zones into first zone
72
Repeated,
R
t d reduced
d
d and
d
extended zone
schemes
Rely on reciprocal lattice
translations
One Brillouin zone → one
band in extended zone
scheme
73
Shapes of Fermi surface resulting from Brillouin Zone structure
Superimpose Fermi circle
on Brillouin Zones
using  k  G    (k )
• Additional
mechanism for
occurrence of
partially filled bands
• Complicated shapes
of Fermi surfaces
74
Summary:
y Topic
p 6
Generalised Bragg Condition: 2k .G  G
2
Brillouin zone boundaries defined by intersection of k with perpendicular
bisectors of reciprocal lattice vectors G
Reciprocal lattice vector in 1D G = 2πx/a
Generalise to 3D
First Brillouin Zone is the set of points in reciprocal space that can be reached
from origin without crossing any Bragg plane
Generalise to 2nd, nth zones
All Brillouin Zones have the same volume
 (k  G )   (k ) Basis of reduced, repeated and extended zones. Approximate
proof and consequences.
75
Topic 7: Fermi Surfaces in Metals, Their Forms and
Their Measurement
• Topic 6 has introduced effect of periodic potential and of
Brillouin zones on shapes of Fermi surface
• Topic 7 is concerned with the shapes of Fermi surfaces in real
metals, and the role of the crystal lattice potential and its
periodicity
76
Band
B
d mustt iintersect
t
t Brillouin
B ill i Zone
Z
boundary
b
d
at right angles (2D picture, also holds in 3D)
( ffor band
(as
b d att zone b
boundary
d
iin 1D)
77
Real Fermi surfaces
Fermi surface in copper
e.g.
g copper,
pp silver , g
gold
fcc lattice in real space
bcc lattice in reciprocal space
Belly, neck and dogs bone
orbits
Distortion
Di
t ti off Fermi
F
i surfaces
f
by
b
periodic potential at Brillouin
zone boundaries (as in 2D on
previous slide)
Alkali metals e.g. Na, K Fermi surface lies
inside 1st Brillouin zone, and is only very
slightly distorted
78
Intermediate summary:
1. Periodic potential produces gaps at zone
boundary
2. Fermi surface intersects zone boundary at
right angles
3. Crystal potential rounds out sharp corners in
Fermi surface
4 Total
4.
T t l volume
l e enclosed
e l ed by
b Fermi
Fe i surface
f e
depends only on electron density –
independent
p
of details of potential
p
79
area quantised
1
1.
S = (kx2 + ky2) = (n + ) 2eB/ħc
Quantum mechanically – Landau levels
(see cyclotron resonance later in course)
Condition for which
successive orbits have
same area on Fermi surface
Thus S known, and hence area of
Fermi surface perpendicular to B
80
Hence b
H
by changing
h
i magnetic
ti fifield
ld di
direction
ti and
d measuring
i
magnetic susceptibility as function of magnetic field, the
shape of the Fermi surface can be determined
This is the de Haas van Alphen effect
Extremal orbits dominate
81
2. Soft X-ray
y emission
Method to measure conduction electron
distribution in solids
1. Only outer shell electrons contribute
2 All inner shells are filled
2.
filled, and play no role e
e.g.
g
In Na 1s, 2s, 2p shells filled
3. Can measure energy distribution of conduction
electrons
l t
b
by soft
ft x-ray emission
i i
4. Use high energy electron bombardment to
create hole in one of inner shells
5. Conduction electron falls into hole. X-ray
photon emitted
6. Distribution of emitted x-rays
x rays gives measure of
conduction electron distribution
Related topics due to conduction electrons: plasmons,
plasma reflectivity see later in course. Determine Fermi
energy from plasma frequency.
82
Soft x-ray emission
spectrum
0
EF
83
Summary Topic 7
1. Periodic potential produces gaps at boundaries
2. Fermi surface must intersect zone boundary at right angles
3. C
Crystal
ys a po
potential
e a rounds
ou ds ou
out ssharp
a p co
corners
e s in Fermi
e
su
surface
ace
4. Total volume enclosed by Fermi surface depends only on number
of electrons – why. Independent of potential
5. All Brillouin zones have same volume
6. Copper,
pp , silver,, ggold,, belly,
y, neck and dog’s
g bone orbits
7. Alkali metals much simpler
8. To determine shape of Fermi surfaces: de Haas van Alphen effect
9. Soft X-Ray emission measures electron distribution of occupied
bands in solids. Complementary to conductivity, Hall effect,
plasma reflectivity
84
Origin of ‘neck’ orbits:
Energy
gy of band lowered as it approaches
pp
zone boundary
y
So states at higher k may be populated
Thus spherical Fermi surface distorted
‘Dog’s Bone’
Hole-like constant energy surface: easily visualised in
extended zone scheme
Other ways to produce holes??
85
Topic 8 Bloch Functions
• Electron wavefunction in periodic lattice potential
• Product of plane wave and function with periodicity of the
crystal lattice
• Electron wave propagates in periodic lattice without
scattering but interference leads to effective mass
scattering,
86
Bloch functions
87
• Except for particular values of k, electron wave
passes undeviated through periodic lattice
• What are these particular values of k?
• Remember at T=0, and for perfect lattice, at low
energy,
gy, electron is not scattered
• Interference which does occur leads to finite
effective mass
88
Reminder from Topic 3 that electron waves are not scattered by
the crystal lattice (except at the Bragg condition)
89
Form of Bloch
functions: simple
proof
integer
Rigorous proof:
Kittel p184/185
90
Topic 9: Tight Binding Model
1. Levels sharp in isolated atoms
2. When atoms brought together, Pauli principle does not allow
energies of electrons on different atoms to be the same.
same
3. For N atoms, bands formed to accommodate 2N electrons –
b d contains
band
t i 2N states
t t
4. Tight binding – since electrons assumed to be associated
initially with individual atoms
5. Shape of different bands different, since orbital leading to
different bands are different and have different overlap
91
Why?
S Rosenberg
See
R
b
b
book
k
92
The band structure of silicon as a real example
• Atomic levels
broaden into bands
• The band at 0 eV is
the ‘valence’ band
• The next band to
higher energy is the
‘conduction’
conduction band
• Derive from
outermost electron
states in atomic Si
S
Summary:
T
Topics
i 8,
8 9
1. Wavefunction of electron in periodic potential (Bloch function) is the
product of plane wave and a function with periodicity of lattice
2. Electrons propagate without scattering except under conditions of
Bragg diffraction at Brillouin zone boundaries
3. Wave interference is destructive except at Brillouin zone boundaries
4. Scattering is due to defects, phonons
5. Tight
g binding
g model is alternative approach
pp
to understand band
formation (intuitive approach starting from atomic orbitals)
6. Degeneracy
g
y of levels lifted due to wavefunction overlap
7. Predicts 2N states per band as does periodic potential model
94
Topic 10: Effective Mass, Electrons and Holes
• We have shown in previous topics that electrons and holes are
not scattered by the ions of the crystal lattice (except at the
B
Bragg
condition)
di i )
• However the ions and the periodic potential do lead to a
measurable change in the properties of the charge carriers: they
lead to effective masses which are not equal to the free electron
mass
• We also introduce the concept of holes in this topic
95
Derivation of
expression for
effective mass
At zone
boundary vg = ??
96
m*/m
*/ e
Variation of
effective mass
with E and k
k
97
See diagram on
previous p
p
page
g
98
Electrons
El
t
andd
holes in
electric field
99
Examples holes in
semiconductors
Partially filled bands in
metals: group II elements
See Hall effect,, cyclotron
y
resonance to determine
sign of charge carriers
Also note large range of
effective masses
100
Pictorial
representation of
motion of empty states
(holes) in electric field
Supplement
Filled band: no current
Remove one electron
Current is minus that
carried by one electron i.e.
-(-e)v = +ev
101
Summary Topic 10
1. Derivation of expression for effective mass for charge carrier
2
in energy band.
*
2 d E
m 
2. Variation of m* with k across Brillouin zone
 2 
 dk 
3 Concept
3.
C
off hholes,
l positive
i i mass, positive
i i charge
h
particle.
i l
4. Empty electron state in otherwise filled band
5. Charge transport by electrons and holes
6. Large range of effective masses
102