Download PHY380 Solid State Physics

PHY380 Solid State Physics
Professor Maurice Skolnick, Dr Dmitry Krizhanovskii and Professor David Lidzey
Syllabus
1. The distinction between insulators, semiconductors and metals. The periodic table.
Quantitative aspects.
2. Basic crystal structures. The crystalline forms of carbon.
3. Density 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.
6. Prediction of metallic, insulating behaviour: periodic potential and tight-binding
descriptions.
7. Real metals, shapes of Fermi surfaces.
8. Soft x-ray emission.
http://www.sheffield.ac.uk/physics/teaching/phy380
1
8. 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 effect, cyclotron resonance. Landau levels in magnetic field.
14. Plasma reflectivity in metals and semiconductors.
15. Magnetism (6 lectures)
The Nobel Prizes 2009 and 2010
2
PHY380: 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 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.
3
Relation to Previous Courses
This course amalgamates much of the previous PHY330 and the magnetism
section of PHY331.
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
(1 November, 13 December deadlines)
Prerequisite
PHY250, 251, Solids (L R Wilson)
Lecture Notes
The notes provide an overview of the main points, and all important figures.
Many more details will be given during 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, lasers, magnets, power transmission 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: Oclaro
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
Mobile communications, satellite communications
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, semiconductor and insulators
overview and crystal lattices
Range of electron densities
Metals: Typical metal (sodium), electron density n=2.6x1028m-3
Insulators (e.g. diamond): electron density very small
(Eg ~ 5.6eV, ~5000K >>kBT at 300K)
Semiconductors: electron density controllable, and is
temperature dependent, in range ~1016m-3 to ~1025m-3
Conductivity is proportional to electron density
9
Importance of bands and band gaps
• Determine electron density and hence optical and electronic
properties
• Understanding of origin will be important part of first 7 lectures
• Bands and band gaps arise for interaction of electrons with
periodic crystal lattice
• Three schematic diagrams illustrating differences in bands, 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:
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
Space Lattice
plus basis
(a) Space lattice
This figure and
slide 14 not
covered in lecture
– here for extra
(useful) information
(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
Lattice points defined by translation vectors
Also note diamond is fcc space lattice
Primitive basis: 2 atoms for each point of lattice
(Kittel page 19)
15
Primitive (unit) cell: Parallelipiped defined by axes a1, a2, a3
sc, bcc and fcc lattices, lattice points per cell and per unit volume
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?
16
Periodic table and crystal structures
17
Planes and directions
(
) planes, [
] directions (covered in 2nd year)
18
The Crystalline Forms of Carbon
Diamond
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
Geim and
Novoselov
http://en.wikipedia.org/wiki/Graphene
http://diahttp://www.theage.com.au
19
2010 Nobel Prize for Physics
A Geim and K Novoselov
Graphene, single sheet of carbon atoms: high electron
motilities, electrons with new properties, very strong,
electronics and sensor applications potentially
20
Comparison of two crystalline forms of carbon
Key properties of diamond
Cubic (diamond) crystal lattice (see slide 19)
Very hard, high strength, insulator, chemically inert, very high
thermal conductivity, optically transparent
Key properties of graphene
Hexagonal crystal lattice (see slides 19, 20), two dimensional
plane
Very strong, metallic but conductivity can be controlled, unique
linear dispersion relations (E v k ), very high thermal
conductivity, adsorbate properties
21
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.
The crystal structures of carbon
22
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
statistics). It ignores the presence of the crystal lattice.
It explains some basic properties, but fails to account for 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)
23
Fermi-Dirac distribution function
1
f (E) =
exp[(E − EF )] / 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
24
Free Electron Theory
Leads to condition for
allowed k-values – next
two pages
25
Periodic boundary conditions (box, side L) – to count states
ψ ( x + L, y , z ) = ψ ( x, y , z )
ψ k (r ) = e
ik .r
=e
i(kx x+k y y+kz z )
is travelling wave solution
provided that kx = 0, ±2π/L, ±4π/L .... 2πn/L, where n is a positive or
negative integer
Proof:
exp ik x ( x + L) = exp i
2πn
( x + L)
L
2πnx
= exp i
exp i 2πn
L
= exp
cos 2πn + i sin 2πn = 1 + 0 = 1
i 2πnx
= exp ik x x
L
26
Counting of States (important, needed to evaluate e.g. the
density of states, Fermi energy and other key properties)
Allowed values of k are thus kx = 0, ±2π/L, ±4π/L .... 2πn/L
In one dimension, one allowed value of k for range of k of 2π/L
27
Dispersion Relation
ik .r
Substituting ψ k (r ) = e
into Schrödinger equation gives
2
2 2

Ek =
(k x + k y2 + k z2 ) =
k2
2m
2m
Parabolic dispersion of free particle with mass m
Corresponds to
p2
E=
, with p = k
2m
p is termed the crystal momentum, and k the wavevector
28
Density of States
The Fermi energy and Fermi surface
Key properties of metals
29
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 (2π/L)
3
Volume of sphere in k-space up to energy E, wavevector k is 4 πk F3
3
Then calculate number of available states from E = 0 to EF, and
hence derive expression for density of states
+ the Pauli exclusion principle
30
Number of states,
Fermi wavevector
and Fermi energy
(
2
EF =
3π 2 n
2m
)
23
31
Values of TF, kF, EF,
vF for sodium and
their significance
(37000K, 0.96x1010 m-1,
3.2eV, 1.07x106m/sec)
32
Topic 2 summary
1. Electrons are Fermions and obey Fermi-Dirac statistics and
the Pauli exclusion principle
2. States up to EF 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!)
33
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
34
Deduce velocity
Newton’s 2nd Law
Define mobility
dv
= − e( E + v x B )
m
dt

Deduce current density,
conductivity and Ohm’s
Law
dk
= − e( E + v x B )
dt
j = nev
Include scattering
 dv v
m
+  = − e( E + v x B )
 dt τ 
τ scattering time
d.c conditions , B = 0
mv
τ
eτ
µ=
m
ne 2τ
σ=
m
= −e E
35
Fermi sea of electrons in
applied electric field, and
scattering processes
For derivation of
displacement in kspace see next slide
36
Motion of electrons in electric field and
scattering: change in wavevector
Alternatively:
eτE
vD = −
m
mv − eEτ
δk =
=


5 x 108 smaller than kF
So displacement of Fermi sea by
electric field is very small
Scattering counters acceleration of
electrons by electric field
37
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 k, small E
Phonon scattering
• Fermi energy ~ 3 eV
• Phonons have maximum energy
~50 meV
• Scattering must be to an empty
state
• Thus only electrons close to Fermi
surface can be scattered
• Must conserve energy and
momentum
• Collisions which relax momentum
gained in applied electric field lead
to resistance
• Must be across Fermi sea:
Large k small E
38
For phonons (conservation of energy and wavevector):
el
i
k + k ph = k
E + ω ph = E
el
i
el
f
el
f
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
39
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)
40
Additional point (important)
Scattering of electrons is not by
ions
Instead by impurities and defects
Electrons propagate freely in periodic
structure (see Bragg scattering later)
Mean free path lB> 1µm or more
lB >> interatomic spacing, so
collisions not with ions
41
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 is not by the ions of the lattice.
42
Topic 4: Electrons in periodic lattice, nearly
free electron model
Many experimental observations are not explained by free electron
theory, including:
1.
2.
3.
4.
5.
6.
7.
Existence of bands, band gaps
Existence of non-metals
Effective mass
Colours of metals
High frequency conductivity
Existence of holes
Nature of the Hall effect
The periodic lattice is all important in explaining these and other
phenomena
43
Periodic lattice gives rise to
Bragg diffraction of electron
waves
44
1D chain
nλ = n
2π
= 2a sin θ = 2a
k
θ = 90o
for waves travelling down 1D
chain
Therefore k=±nπ/a
Bragg condition for 1D chain
Electron wave is scattered by 2π/a
(= G) (reciprocal lattice vector)
45
Origin of band gap from Bragg diffraction (following Kittel, chapter 7, 7th edition)
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),
sin2(πx/a) charge
distributions at
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 earlier, at Bragg condition electron wave is scattered by ∆k
= 2π/a (= G) (reciprocal lattice vector)
Lattice potential (Fourier components) mixes waves at these points in
dispersion in unperturbed band-structure (in (a) above), giving rise to
gaps in (b)
49
Continuing last slide
Group velocity
dω 1 dE
vg =
=
dk  dk
is zero at zone boundary, corresponds to standing wave
50
To summarise Topic 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) in synchronism with lattice
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
instrumentation. Based on silicon
integrated circuit technology and field
effect transistors
Readout of information
from each pixel
53
Key Points of Topics 1-4
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 mechanisms. Scattering is not by 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
Atomic configuration of sodium:
1s2, 2s2, 2p6, 3s1
n = 1, 2 shells tightly bound and give rise to lower energy
valence bands. Do not contribute to conduction.
3s electron is weakly bound and leads to conduction. Half
filled band
55
Topic 5: Introduction to Brillouin zones, half-filled
and filled bands
• Number of states in a band
• Monovalent atoms metallic
• Insulators: can only occur for even number of valence electrons
• Group II elements, nevertheless are metallic.
• Concept of overlapping bands
56
Counting of states and filling of bands
Periodic boundary conditions
(following from pages 26, 27)
k = 0, ±2π/L, ±4π/L .... 2πn/L
L is length of chain of atoms, n is an
integer
If N is number of atoms, the lattice
constant a is equal to L/N
Total number of states between ±π/a
is N
More strictly, N is number of primitive
unit cells in chain
• Each unit cell contributes one value
of k to each Brillouin zone, and
hence to each band
• Including spin, 2N states per band
• If one atom per unit cell
(monovalent), then band half filled
– 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, 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)
57
Conduction in half-filled
and filled bands
58
I
II
IV
Note also:
Transition metals
Noble metals
59
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
overlapping bands. Hence metallic.
60
How bands can overlap
in
:
Ec can be less than Eb for:
Ec < Eb for
Eg <
2
2
 k
2m
And thus overlapping bands
i.e. energy in second band
less than that in first
61
Overlapping bands: energy 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
62
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.
• Each unit cell contributes one value of k to each Brillouin zone, and hence
to each band. Including spin gives 2N states per band
• Monovalent atoms with one atom per unit cell (alkali and noble metals),
band half filled, expect metallic
• Insulators: can only occur for even number of valence electrons per
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.
63
Summary of Bragg diffraction, Brillouin zones
1. Bragg condition defines edges of Brillouin zones
2. For one dimension, simple proof of condition k = ±π/a (page 45)
3. In general
∆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, 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 = 2π/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
Brillouin zone boundaries, then shape of Fermi surface
will be strongly modified.
71
Reduced
zone scheme
Translation vector
Hook and Hall
(p116-118)
Also see Hook and Hall
p39 for physical
discussion
By reciprocal lattice translation
(2π/a), can translate points in
higher zones into first zone
72
Repeated, reduced and
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 ψ (k + G ) = ψ (k )
• Additional
mechanism for
occurrence of
partially filled bands
• Complicated shapes
of Fermi surfaces
74
Summary: Topic 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
ψ (k + G ) = ψ (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
Will discuss silicon band structure
in next lecture - download
76
Band must intersect Brillouin Zone boundary
at right angles (2D picture, also holds in 3D)
(as for band at zone boundary in 1D)
77
Real Fermi surfaces
Fermi surface in copper
e.g. copper, silver , gold
Repeated zone scheme
fcc lattice in real space
bcc lattice in reciprocal space
Belly, neck and dogs bone
orbits
Distortion of Fermi surfaces by
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
Origin of ‘neck’ orbits:
Energy of band lowered as it approaches zone boundary
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??
79
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 volume enclosed by Fermi surface
depends only on electron density –
independent of details of potential
80
Soft X-ray emission
Method to measure conduction electron
distribution in solids
1. Only outer shell electrons contribute
2. All inner shells are filled, and play no role e.g.
In Na 1s, 2s, 2p shells filled (p55)
3. Can measure energy distribution of conduction
electrons by soft x-ray emission
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 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.
81
Soft x-ray emission
spectrum
0
EF
82
Summary Topic 7
1. Periodic potential produces gaps at boundaries
2. Fermi surface must intersect zone boundary at right angles
3. Crystal potential rounds out sharp corners in Fermi surface
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, silver, gold, belly, neck and dog’s bone orbits
7. Alkali metals much simpler
8. Soft X-Ray emission measures electron distribution of occupied
bands in solids. Complementary to conductivity, Hall effect,
plasma reflectivity
83
Topic 8: 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.
3. For N atoms, bands formed to accommodate 2N electrons –
band contains 2N states
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
84
Why?
See Rosenberg book
85
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’ band
• Derive from
outermost electron
states in atomic Si
Summary: Topic 8
1. Tight binding model is alternative approach to understand band
formation (intuitive approach starting from atomic orbitals)
2. Degeneracy of levels lifted due to wavefunction overlap
3. Predicts 2N states per band as does periodic potential model
87
Topic 9: 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
Bragg condition)
• 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
88
Derivation of
expression for
effective mass
At zone
boundary vg = ??
89
m*/me
Variation of
effective mass
with E and k
k
90
See diagram on
previous page
91
Electrons and
holes in
electric field
92
Examples holes in
semiconductors
Partially filled bands in
metals: group II elements
See Hall effect, cyclotron
resonance to determine
sign of charge carriers
Also note large range of
effective masses
93
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
94
Summary Topic 9
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 of holes, positive mass, positive charge particle.
4. Empty electron state in otherwise filled band
5. Charge transport by electrons and holes
6. Large range of effective masses
95