2610201811MagnetismNot only permanent ferromag.docx

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Magnetism
Not only permanent ferromagnets,
many applications
motors, transformers, imaging,
data storage (probably just as
important as semiconductors for
modern computers)
FeCo/Pt superlattices high saturation
magnetism promising for magnetic
data storage
2
• How do solids react to an external field?
• What is the cause of spontaneous magnetic ordering?
Magnetism

Magnetism is an extremely active area of research with many
still unanswered questions
Condensed matter physics uses magnetism as a testing ground
for understanding complex quantum and statistical physics
Most magnetic phenomena caused by quantum mechanical
behaviour of the electrons
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• Magnetic moments in atoms
• Weak magnetism in solids
• Magnetic ordering
Magnetic properties
Weak magnetism in solids can largely be understood by atomic
properties
magnetic ordering cannot - cannot describe it as
ordering of totally localised moments on atoms because these
have
to “talk” to each other, otherwise there is no ordering in the
first place

Ashcroft and Mermin Ch 31, 32; Oxford Basics Ch 19
4
Macroscopic description of magnetism:
Fundamental quantities
In vacuum we have:
magnetic field intensity
magnetic induction
When a material medium is placed in a magnetic field,
the medium is magnetized. This is described by the
magnetization vector M – the dipole moment per unit volume
we interpret as the “external field”
permeability of free space 4πx107 (SI units: N.A-2)
Magnetization induced by the field assume M is
proportional to H
or
- magnetic susceptibility of the medium
(Real crystals anisotropic, and susceptibility is a second-rank
tensor (ignore such effects)

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Units
• Both, and are measured in Tesla (T)
• 1 T is a strong field. The magnetic field of the earth is only
of the order of 10-5 T.
potential energy of one dipole in the external field:
Classification of materials
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All magnetic materials may be grouped into three magnetic
classes
depending on the magnetic ordering and the sign and magnitude
of the
magnetic susceptibility:
(more later)
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Diamagnetism
• Diamagnetism is caused by “currents” induced by the
external field. According to Lenz’ law, these currents always
lead to a field opposing the external field.
Potential energy U
increase in potential energy for higher field, unfavourable.
Paramagnetism
Potential energy U
the potential energy is lowered when moving the magnetized
bodies to a
higher field strength
. • Paramagnetism is caused by aligning some dipoles, which
are already present, with the magnetic field
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Ferromagnetism
A ferromagnet is a material where M can be nonzero
even in the absence of an applied magnetic field
Magnetism is said to be Spontaneous when it occurs even in the
absence of an externally applied magnetic field, as in the

case of a ferromagnet
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9
Ferromagnetism
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d-electron states/orbitals
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f-electron states/orbitals
Isolated Atoms
• The magnetic moment of a free atom has three main
sources
1. the spin of the electrons
2. the electron orbital angular momentum about the nucleus

3. the change in orbital moment induced by an applied
magnetic field
• In classical picture, electrons orbit around nucleus
• Each orbit like a loop of electric current
• A loop of current produces a magnetic field, so electrons in
an atom generates a magnetic field
• Quantum numbers n,l,m l and ms label the electrons in an
atom (alternatively called n, l, lz, σz )12
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Hund’s Rules for Isolated Atoms
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Set of Rules – Hund’s Rules that determines how electrons fill
orbitals
Recall from QM, an electron in an atomic orbital can be
labelled
by four quantum numbers:
Principle quantum number
Angular momentum
z-component of ang. mom.

z-component of spin
Sometimes the angular momentum shells
are known as and can accommodate
electrons, respectively.
Start with some fundamentals of electrons in isolated atoms
14
lz = 2
lz= 1
lz= 0
lz=-1
lz=-2
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Hund’s 0th Rule – (Aufbau Principle) shells filled starting

with lowest available energy state. An entire shell filled
before another started
Madelung Rule): energy ordering is from lowest value of n+l
to largest when two shells have the same n+l, fill one with
smallest n first
Two examples:
Nitrogen (N) 7 electrons, filled 1s shell with 2
electrons spin-up and spin-down, 2 electrons in
the 2s shell with 2 electrons spin-up and spin-down,
3 electrons in the 2p shell
Praseodymium (Pr) 59 electrons
or as
p
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The shell filling sequence is the rule which defines the overall
structure of the periodic table.
When shells are partially filled need to describe which of the
orbitals

are filled in these shells and which spin states are filled
Hund’s Rules
(1) Electrons try to align their spins, i.e. the electrons
should occupy the orbitals such that the maximum
possible value of the total spin S is realized.
Consider Pr as example - the 3 valence electrons will
have spins that point in same direction giving S=3/2
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(2) The electrons should occupy the orbitals such that
the maximum of L, consistent with S, is realized.
For Pr, this means giving
so we have and
(3) The total angular momentum J is calculated
• If the sub-shell is less than half-full J=L-S
• If the sub-shell is more than half full J=L+S

• If the sub-shell is half full, L=0 and J=S
For Pr, since shell less than half-full we use J=L-S = 6 - 3/2 =
9/2
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Preference for spins to align comes from the Coulomb
interaction
energy between the electron and nucleus
For spins anti-aligned, electrons
are closer and the nucleus is
partially screened by the
negative charge of the other
electron.
For spins aligned the electrons
repel each other and see the full
positive charge of the nucleus.
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19

• The first of Hund’s rules requires S=3/2.
• The possible l z values for the 3d shell are -2,-1,0,1,2.
Hund’s second rule requires to choose the largest possible
value of L, i.e. to choose l z =0,1,2, so L=3.
• Since the sub-band is less than half filled, J=L-S=3-3/2=3/2.
Another example: Cr 3+
Cr3+ has three electrons in the 3d sub-shell
20 Ashcroft & Mermin Ch31
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(2S+1) X J
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Coupling of electrons in atoms to an external
field
Seen how electron orbital and spin can align with each other
Now consider how electrons couple to an external magnetic
field

A is the vector potential
particle in electromagnetic field
change the momentum (operator)
First recall -
for electrons q=-e
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field
In absence of a magnetic
field the Hamiltonian for an
electron in an atom is:
V is electrostatic
potential from the
nucleus
In presence of a magnetic field:

where is the electron spin, g is the electron g-factor (about
2)
and the Bohr magneton is
Zeeman term
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Zeeman effect
(l z)
Splitting of spectral lines when atom is placed in an external
magnetic field
Predicted by Lorentz, first observed by Zeeman
Energy level splitting in the normal Zeeman effect for singlet
levels l=2 and
l=1
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field
For a uniform magnetic field, we can take and

and so
can be written as:
First two terms just Hamiltonian in absence of field,
Can rewrite 3rd term as:
where is the angular momentum of the
electron
is the Bohr magneton
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field
With 3rd term as
Can combine with 5th term of below
To obtain final expression:
paramagnetic
term
Coupling of field to total
magnetic moment of electron
diamagnetic
term

These two terms are
responsible for the paramagnetic
and diamagnetic response of atoms
to external magnetic fields
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field
Free spin (Curie or Langevin) Paramagnetism
Consider the paramagnetic term in previous equation –
generalize to multiple
electrons in the atom:
L and S, orbital and spin components
of all electrons, and
Can write as:
where
(see Oxford Solid

State Basics for
derivation or
Ashcroft and Mermin
Appendix P)
The partition function is
And the corresponding free energy is
It describes the reorientation of free spins in an atom
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field
Free spin (Curie or Langevin) Paramagnetism
Given the free energy
The magnetic moment per spin is
Assuming a density n of these atoms it can be shown that the
susceptibility is:
Curie Law
Called “Curie paramagnetism or Langevin paramagnetism”
Curie constant
Ashcroft & Mermin

Ch31
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Coupling of electrons in atoms to an external field
Curie paramagnetism dominant when
So can only observe diamagnetism when
For example, filled shell configurations like
noble gases
[Or if J=0, but L and S not equal to zero. This
occurs when shell has one electron fewer than
being half full ]
Larmor Diamagnetism
The expectation of the diamagnetic term for B in the z direction
is
The atom is rotationally symmetric:
Consider now diamagnetic term – coupling of the orbital motion
to the

magnetic field
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Coupling of electrons in atoms to an external field
Larmor Diamagnetism
The atom is rotationally symmetric:
So we have
and the magnetic moment per electron is:
Assume density of electrons, can write:
Larmor Diamagnetism
(recall M=χ H = χ B/ μ0)
= χ B/ μ0
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31
32
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Will be some amount of diamagnetism in all materials, above
good description
for core electrons – but for conduction electrons in a metal we
have
Landau-diamagnetism:
where is the susceptibility of the free
Fermi gas
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Magnetism of atoms in solids
Diamagnetism in solids
In above, Larmor diamagnetism applied to isolated atoms with
At low temperatures, noble gas atoms form weak bonds in
crystal and
description still applies, where density of electrons is put equal
to the
atomic number Z times the density of atoms, n. the radius r is
the
atomic radius
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Magnetism of atoms in solids
Curie Paramagnetism

Recall: Curie paramagnetism describes the reorientation of free
spins in an atom
Does it occur in solids?
Yes, possible,
e.g. through “crystal field splitting” where atoms are no longer
in a rotationally
symmetric environment
Also, the number of electrons on an atom can become modified
in a material,
e.g. Pr, we had 3 free electrons in valence (4f) shell (J=9/2), but
in many
compounds Pr donates two of its 6s electrons and one f electron
(J=4).
Paramagnets can have many different effective values of J –
need to know
microscopic details of bonding in system!
(in this case, L=5, S=2/2 and J=L-S = 5 -1=4)
e.g Fe iron
d 6
4μB

Module 2 - Background
SOCIAL MEDIA AND HR; BEHAVIORAL ANCHORED
RATING SCALES; SIMULATION TRAINING
Staffing
Required Material
Davenport, T. H. (2012). Case study: Social media engages
employees. FT.Com, Retrieved from the Trident Online Library.
Facebook, Blogs & the Boss: The intersection of social media &
the workplace. (2013). Retrieved
from https://www.youtube.com/watch?v=PRrJ9eINYZI
Wild About Trial (2015). Legal Smart with Alison Triessl—
Social Media & Employment. Retrieved
from https://www.youtube.com/watch?v=d26eEzr5KuI.
Wilkie, D., & Wright, A. (2014). Balance risks of screening
social media activity. HR Magazine, 59(5), 14. Retrieved from
ProQuest in the Trident Online Library.
Wright, A. Nov., 2014). How Facebook recruits. Retrieved
from http://www.shrm.org/hrdisciplines/technology/articles/pag
es/how-facebook-recruits.aspx
Optional Material
Segal, J. A. (2014). The law and social media in hiring. HR
Magazine, 59(9), 70-72. Retrieved from ProQuest in the Trident
Online Library.
Segal, J. A., & LeMay, J., S.P.H.R. (2014). Should employers
use social media to screen job applicants? HR Magazine,
59(11), 20-21. Retrieved from ProQuest in the Trident Online
Library.
Skill Boosters (2015). Top 5—Social media fails at work.
Retrieved
from https://www.youtube.com/watch?v=6TXjQt3qYwk.
Swain, K. (2017). The impact of social media in the workplace
pros and cons. Retrieved from http://work.chron.com/impact-
social-media-workplace-pros-cons-22611.html.
Walden, J. A. (2016). Integrating Social Media Into the
Workplace: A Study of Shifting Technology Use
Repertoires. Journal Of Broadcasting & Electronic

Media, 60(2), 347-363. Available in the Trident Online Library.
Wright, A. D. (2014). More states prohibit social media
snooping. HR Magazine, 59(10), 14. Retrieved from ProQuest in
the Trident Online Library.
Behaviorally Anchored Rating Scales
Required Material
Behaviorally Anchored Rating Systems—BARS. Retrieved
from http://performance-appraisals.org/appraisal-
library/Behaviorally_Anchored_Rating_Systems_-_BARS/
Govekar, P. & Christopher, J. Assessing academic advising
using behaviorally anchored rating scales (BARS). Example.
Retrieved
from http://www.westga.edu/~bquest/2007/BARS7.pdf
Optional Material
Behaviorally Anchored Rating Scale (BARS) Guide. Retrieved
from www.in.gov/spd/files/bars.doc
Simulation Training
Required Material
Abernathy, D., Allerton, H., Barron, T., & Salopek, J. (1999).
Everyday simulation. Training & Development, 53(11), 37.
Available in the Trident Online Library.
AusBusiness Traveller (2011). Inside REAL Qantas 747 Flight
Simulator HD. Retrieved
from https://www.youtube.com/watch?v=L8JUWUKXV08. (for
Discussion Forum)
(AusBusiness Traveller, 2011)
Hiringsimulation.com (2017). Why Job Simulation Works.
Optional Material
Catling, C., Hogan, R., Fox, D., Cummins, A., Kelly, M., &
Sheehan, A. (2016). Simulation workshops with first year
midwifery students. Nurse Education in Practice, 17, 109-115.
Lambert, C., and Lloyd-Jones, H. (2014). Run simulation in
your workplace. Education for Primary Care. 25(6), 357-359.
Retrieved from BBSCOHost in the Trident Online Library.
McMaster, S., Ledrick, D., Stausmire, J., & Burgard, K. (2014).

Evaluation of a simulation training program for uncomplicated
fishhook removal. Wilderness & Environmental Medicine, 25,
416-424. Available in the Trident Online Library.
Uptick in simulation training. (2013). Air Force Time, 3.
Discussion: Simulation Training/Development
Deidriaunna Priest posted Apr 22, 2020 1:57 PM
Hello Class,
Simulation training is considered to be one of the most effective
ways of learning. It provides a realistic, immersive experience
in the context of the learning job (Srivastava & Srivastava
2019). Simulation training offers visuals and scenarios that
occur in real life. Most training occurs online without hands-on
training. Having practical exercises will help all types of
learners. Companies can save money, being that there able to
assess how well their trainees are doing.
HRM professionals can objectively determine the value of
simulation training by getting real experiences, feedback, and
retention. Providing authentic experiences allows individuals to
get an idea of work functions. Reading guides and watching
webinars are not beneficial to all employees. Getting feedback
will enable employers to get a better understanding of needs and
concerns. Being that employees are training in real-life
situations, they're able to retain more information.
Simulation training has significant benefits, but it also has
disadvantages. Simulators can be costly due to updates and
maintenance. Also, training all employees should be trained
adequately on software and hardware.
Having simulation training allows trainees to participate in
activities within a safe environment. Learners are capable of
learning things from errors. Also, getting hands-on thinking
skills and effective communication will enhance real-life

situations.
References
Arias, Raphael Gonçalves. "5 Secrets to Master the Risk
Assessment Matrix." SoftExpert Excellence Blog,
31 Jan. 2020, blog.softexpert.com/en/risk-assessment-matrix-
secrets/.
Srivastava, Av, and Av Srivastava. “Simulation Training -
Definition, Learning Benefits & Top
Companies.” Learning Light, 25 Nov.
2019, www.learninglight.com/simulation-based-training-
providers/.
“The Major Benefits of Using Simulation Training in Corporate
Learning.” Designing Digitally, Inc.,
www.designingdigitally.com/blog/2018/11/major-benefits-
using-simulation-training-corporate-
learning.
23/10/2017
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Semiconductors
Semiconductors
One shouldn’t work on semiconductors, that is a
filthy mess; who knows whether any
semiconductors exist.
(Über Halbleiter soll man nicht arbeiten, das ist

eine Schweinerei; wer weiss, ob es überhaupt
Halbleiter gibt.)
Wofgang Pauli, 1931
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Why semiconductors?
• SEMICONDUCTORS: They are here, there, and everywhere
• Computers Silicon (Si) MOSFETs, ICs, CMOS
laptops, anything “intelligent”
• Cell phones, pagers Si ICs, GaAs FETs, BJTs
• CD players AlGaAs and InGaP laser diodes, Si
photodiodes
• TV remotes, mobile terminals Light emitting diodes (LEDs)
• Satellite dishes InGaAs MMICs (Monolithic Microwave
ICs)
• Fiber networks InGaAsP laser diodes, pin photodiodes

• Traffic signals, car GaN LEDs (green, blue)
taillights InGaAsP LEDs (red, amber)
• Air bags Si MEMs, Si ICs
• and, they are important, especially to Elec.Eng.& Computer
Sciences
Introduction
Semiconductors are materials whose electrical
properties lie between Conductors and Insulators.
Ex : Silicon and Germanium
Difference in conductivity
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Semiconductor Materials
• Elemental semiconductors – Si and Ge (column IV of periodic
table) –compose of single species of atoms
• Compound semiconductors – combinations of atoms of column
III and column V and some atoms from column II and VI.
(combination of two atoms results in binary compounds)
• There are also three-element (ternary) compounds (GaAsP)
and

four-elements (quaternary) compounds such as InGaAsP.
gap size
(eV)
InSb 0.18
InAs 0.36
Ge 0.67
Si 1.11
GaAs 1.43
SiC 2.3
diamond 5.5
MgF2 11
valence
band
conduction
band
Can a material with
μ in a band gap
conduct?

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Semiconductor
materials
Semiconductor Materials
• The wide variety of electronic and optical properties of these
semiconductors provides the device engineer with great
flexibility in the design of electronic and opto-electronic
functions.
• Ge was widely used in the early days of semiconductor
development for transistors and diods.
• Si is now used for the majority of rectifiers, transistors and
integrated circuits.
• Compounds are widely used in high-speed devices and
devices
requiring the emission or absorption of light.
• The electronic and optical properties of semiconductors are
strongly affected by impurities, which may be added in
precisely
controlled amounts (e.g. an impurity concentration of one part
per million can change a sample of Si from a poor conductor to
a
good conductor of electric current). This process called doping.

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Intrinsic semiconductors
• Pure, i.e. not doped, semiconductors are called intrinsic.
• For the electronic properties of a semiconductor, “pure”
means pure within 1 ppm to 1 ppb.
Intrinsic Material
A perfect semiconductor crystal with no impurities or lattice
defects is called an
intrinsic semiconductor.
At T=0 K –
No charge carriers
Valence band is filled with electrons
Conduction band is empty
At T>0
Electron-hole pairs are generated
EHPs are the only charge carriers in
intrinsic material
Since electron and holes are created in

pairs – the electron concentration in
conduction band, n (electron/cm3) is
equal to the concentration of holes in the
valence band, p (holes/cm3).
Each of these intrinsic carrier
concentrations is denoted ni.
Thus for intrinsic materials n=p=ni
Electron-hole pairs in the covalent bonding
model in the Si crystal.
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Doped semiconductors
• A very small amount of impurities can have a big
influence on the conductivity of a semiconductor.
• Controlled addition of impurities is called doping.
• There are two types of doping: n doping
(impurities increasing #electrons) and p doping

(impurities increasing #of holes).
• Typical doping levels are in the order of 1019 to
1023 impurity atoms per m3. Remember: Si has a
concentration of 5*1028 atoms per m3 and an
intrinsic carrier concentration of 1016
electrons/holes per m3 at room temperature.
Si
14
-
-
-
-
-
-
-
-
-
-
-
- -
-

However, like all
other elements it
would prefer to have
8 electrons in its
outer shell
The Silicon Atomic Structure
Silicon: our primary example and
focus
Atomic no. 14
14 electrons in three shells: 2 ) 8 ) 4
i.e., 4 electrons in the outer "bonding"
shell
Silicon forms strong covalent bonds with
4 neighbors
3s2 3p2 2s2 2p6 1s2
Si
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Si and Ge are tetravalent elements – each atom of Si (Ge) has 4

valence
electrons in crystal matrix
T=0 all electrons are bound in
covalent bonds
no carriers available for
conduction.
For T> 0 thermal fluctuations can
break electrons free creating
electron-hole pairs
Both can move throughout the lattice
and therefore conduct current.
Electrons and Holes
Excite electron from valance
band to conduction band, e.g.,
absorbing a photon or thermal
excitation.

Absence of electron in
valence
band called a “hole” – treat
holes as elementary particles.
To conserve charge, if electron
is negative, hole is positive
charged.
Electron can fall back into
hole, releasing energy ,e.g.
emitting photon, and
annihilating
electron and hole.
Holes
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Effective Mass of Electrons
As before, describe curvature at

bottom of band in terms of effective
mass.
Near bottom of conduction band,
where k=k min
And the corresponding group
velocity is
The effective mass is defined as,
Recall, for free electron
Effective Mass of Holes
valence band
convension is:
“hole”
For the top of the valence band, can
write:
And define effective mass for holes,
Energy to move hole away from top of
band is:
And corresponding hole velocity is:

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Alternative definition is to define effective mass as being the
quantity that satisfies Newton’s second law
Effective Mass
A force is applied to an electron, then work done on electron
equal to its change in energy – consider work done per unit
time
Change in energy per unit time:
Equating:
used
(chain rule)
(since )
Then:
Effective mass
as a function of
momentum

The effective mass
m
e
*/m
e
m
h
*/m
e
InSb 0.014 0.4
InAs 0.022 0.4
Ge 0.6 0.28
Si 0.43 0.54
GaAs 0.065 0.5
Na 1.2
Cu 0.99
Sb 0.85
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Electrons and Holes
Electron-hole pairs in a semiconductor.
The bottom of the conduction band
denotes as Ec and the top of the valence
band denotes as Ev.
For T>0
some electrons in the valence band receive
enough thermal energy to be excited
across the band gap to the conduction
band.
The result is a material with some electrons
in an otherwise empty conduction band and
some unoccupied states in an otherwise
filled valence band.
An empty state in the valence band is
referred to as a hole.
If the conduction band electron and the
hole are created by the excitation of a

valence band electron to the conduction
band, they are called an electron-hole
pair (EHP).
Increasing conductivity by temperature
15 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0
10 0
1 10
3
1 10
4
1 10
5
1 10
6
1 10
7
1 10
8
1 10
9
1 10
10

1 10
11
1 10
12
1 10
13
1 10
14
1 10
15
1 10
16
1 10
17
Carrier Concentration vs T emp (in Si)
T em p erature (K )
In
tr
in
si
c
C
o
n

ce
n
tr
at
io
n
(
cm
^
-3
)
ni
T
T
Therefore the conductivity of a semiconductor is influenced by
temperature
As temperature increases, the number of free electrons and
holes created increases exponentially.
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Adding Electrons or Holes with Impurities: Doping

A phosphorous atom P replaces a silicon atom. The
P atom is like an Si atom plus an extra electron.
Extra electron goes in conduction band
P is an electron donor in silicon – also called an
n-type dopant. n is symbol for electron density
n- and p-doping
donor atom acceptor atom
Analogously, an Al replacing a silicon atom. The
Al atom has one fewer electrons than Si. Gives rise to a
hole. Al is an electron acceptor in silicon – also called an
p-type dopant. p is symbol for hole density.
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Adding Electrons of Holes with Impurities: Doping
All electrons
in covalent
bond with 2

electrons
Extra
electron
Extra hole
Donor and acceptors in covalent bonding model
In the covalent bonding
model, donor and acceptor
atoms can be visualized as
shown in the Figure. An Sb
atom (column V) in the Si
lattice has the four necessary
valence electrons to complete
the covalent bonds with the
neighboring Si atoms, plus one
extra electron. This fifth
electron does not fit into the
bonding structure of the
lattice and is therefore
loosely bound to the Sb atom.
Donor and acceptor atoms
in the covalent bonding
model of a Si crystal.
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Donor and acceptors in covalent bonding model
A small amount of thermal energy
enables this extra electron to
overcome its coulombic binding to
the impurity atom and be donated
to the lattice as a whole. Thus it is
free to participate in current
conduction. This process is a
qualitative model of the excitation
of electrons out of a donor level and
into the conduction band.
Similarly, the column III impurity
Al has only three valence electrons
to contribute to the covalent
bonding, thereby leaving one bond
incomplete. With a small amount of
thermal energy, this incomplete
bond can be transferred to other
atoms as the bonding electrons
exchange positions.
Donor and acceptor atoms
in the covalent bonding
model of a Si crystal.
Consider n-type dopant, e.g. P in Si
Extra electron in conduction band acts like

a free electron with mass, m*
but also have positive charge in nucleus of P
Forms a bound state like a H atom –
Attract with potential:
Energy eigenstates of H atom
Rydberg constant
Radius of wave function
mass of electron
Analogously, for a hydrogenic
Impurity state we have:
4
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n-doping Estimate binding energy
with Bohr model:
using the modifications
phosphorus
penta-valent,

one electron too many
order of magnitude
The radius of this is quite big, 30 times the
Bohr radius
23/10/2017
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Extrinsic Material
By doping, a crystal can be altered so that it has a predominance
of either
electrons or holes. Thus there are two types of doped
semiconductors, n-type
(mostly electrons) and p-type (mostly holes). When a crystal is
doped such that
the equilibrium carrier concentrations n0 and po are different
from the intrinsic
carrier concentration ni, the material is said to be extrinsic.
Donor impurities (elements
of group V): P, Sb, As
Acceptor elements (group
III): B, Al, Ga, In

The valence and conduction bands of
silicon with additional impurity energy
levels within the energy gap.
When impurities or lattice
defects are introduced,
additional levels are created
in the energy bands
structure, usually within the
band gap.
Total number of electrons
III – Al – 13
IV – Si – 14
V - P - 15
Extrinsic Material – donation of electrons
An impurity from column V
introduces an energy level very
near the conduction band in Ge
or Si. This level is filled with
electrons at 0 K, and very little
thermal energy is required to
excite these electrons to the
conduction band. Thus, at about
50-100 K nearly all of the
electrons in the impurity level
are "donated" to the conduction
band. Such an impurity level is
called a donor level, and the

column V impurities in Ge or Si
are called donor impurities.
Donation of electrons from
a donor level to the
conduction band
n-type material
23/10/2017
16
Extrinsic Material – donation of electrons
From figure we note that the
material doped with donor
impurities can have a
considerable concentration of
electrons in the conduction band,
even when the temperature is
too low for the intrinsic EHP
concentration to be appreciable.
Thus semiconductors doped with
a significant number of donor
atoms will have n0>>(ni,p0) at
room temperature. This is n-type
material.
Donation of electrons from
a donor level to the

conduction band
n-type material
Extrinsic Material – acceptance of electrons
Acceptance of valence band
electrons by an acceptor level,
and the resulting creation of
holes.
Atoms from column III (B,
Al, Ga, and In) introduce
impurity levels in Ge or Si
near the valence band. These
levels are empty of electrons
at 0 K. At low temperatures,
enough thermal energy is
available to excite
electrons from the valence
band into the impurity level,
leaving behind holes in the
valence band.
P-type material
23/10/2017
17

Extrinsic Material – acceptance of electrons
Acceptance of valence band
electrons by an acceptor level,
and the resulting creation of
holes.
Since this type of impurity
level "accepts" electrons
from the valence band, it is
called an acceptor level, and
the column III impurities are
acceptor impurities in Ge and
Si. As figure indicates,
doping with acceptor
impurities can create a
semiconductor with a hole
concentration p0 much
greater than the conduction
band electron concentration
n0 (this is p-type material).
P-type material
Statistical Mechanics of Semiconductors
Recall from Lecture 3 – density of states for free electrons
per unit volume
Electrons in conduction band like
free electrons but with mass m*, can write:

Similarly the density of states for holes
near the top of the valence band are:
2/11/2017
1
Magnetism 2
where is the g-factor. The spin is and the Bohr
magneton is
Then the energy of an electron with spin up (same direction as
Magnetic field is:
And that with spin down is
where
Magnetic Spin Susceptibility – Pauli
Paramagnetism
Consider the response of free electrons to an externally applied

magnetic field. The electron’s motion can be curved due to the
Lorentz force, but also the spins can flip. Looking at latter
effect.
The Hamiltonian becomes:
2/11/2017
2
The spin magnetization of the system (magnetic moment per
unit volume) in direction of the field is:
With applied magnetic field the energy is lower when the spins
point down, so more of them will point down and a
magnetisation
develops in the direction of applied field – known as
Pauli paramagnetism (spin magnetization of free electron gas)
Paramagnetism
4

Paramagnetism
Find Pauli paramagnetism for T=0
For no magnetic field, electrons
are filled up to the Fermi energy with
With magnetic field, up electrons
more energetically unfavorable by
therefore will have
fewer spin up electrons
2/11/2017
3
5
Paramagnetism
That is, with the magnetic field that states with up and down
spin are
shifted in energy by and , respectively.
Hence, spin up electrons that are pushed above the Fermi energy
can lower their energies by flipping their spins to become spin

down
electrons. The total number of spins that flip (the area of the
approximately rectangular shape) is roughly
Then from:
we obtain:
and
so
6
Spontaneous magnetic order in solids
Heisenberg Hamiltonian
Model description of how spins align – assume an interaction
between
neighbouring spins – so-called “exchange interaction”
Assume an insulator, so electrons don’t hop from site to site.
Model Hamiltonian is:
is the spin on site i and B is the magnetic field experience by
the spins

is the interaction energy. Neglecting the magnetic field, and
assume each spin coupled to its neighbour with the same
strength, can
drop i,j
is the interaction energy
Factor of ½ avoids over-counting in sum
If lower energy when spins aligned; whereas if
it is lower energy
when spins are anti-aligned
2/11/2017
4
7
The Hamiltonian doesn’t indicate a
preferred spin direction.
In a real system, atoms are often in an asymmetric environment
due to the
lattice and will be directions that the spin would rather point.

Add term to the Heisenberg Hamiltonian:
called anisotropy energy as gives system a preferred direction,
here in the
or directions.
Or, for spin pointing along the orthogonal axis directions:
8
If the anisotropy term is very large in
It will force the spin to be either or
This gives the Ising Model
where only (and reintroducing the magnetic
field B)
2/11/2017
5

Experiments: MExFM
9
(a) Atomic-resolution image of an antiferromagnetic NiO(001)
surface
obtained by Non-Contact Atomic Force Microscopy (NC-AFM).
The line
section reveals an apparent height difference of 4.5 pm between
nickel
(dark) and oxygen (bright) sites.
(b) Spin-resolved image of NiO(001) with atomic resolution as
obtained
by Magnetic Exchange Force Microscopy (MExFM)
Imaging & Microscopy, Jun. 01, 2008 R. Wiesendanger,
Experiments: Spin-polarized STM
10
Differential conductance asymmetry A dI/dV. a Two magnetic
configurations in spin-STM measurements, AP and P,
corresponding to two distinct magnetic states of a bilayer Co
nanoisland, pointing up and down, respectively. b, c dI/dV
images of the Co nanoisland ‘A’ in Fig. 5 a measured at μ0 H
ext = −1 T and V b = + 0.03 V for AP (b) and P (c) states. d A

dI/dV map calculated from the dI/dV images of b and c. e Two
relative magnetization configurations of spin-STM
measurements, corresponding to two distinct magnetic states of
a bilayer Fe nanoisland, α and β. f, g dI/dV images of a Fe
nanoisland, measured at external fields of (b) 0 T and (c) a
value ≥ H sat. h A dI/dV map calculated from the dI/dV images
of f
and g.b–d
Nano Convergence 2017 4:8 Soo-hyon Pharj and Dirk Sander,
https://nanoconvergencejournal.springeropen.com/articles/10.11
86/s40580-017-0102-5#Fig5
2/11/2017
6
11
Domains and Hysteresis
In real materials there are regions “domains” with different spin
orientation.
Reduces the dipolar energy (resulting from the sum of the
individual dipole-
dipole interactions on the atoms).
Can understand like since if view as magnets; two like ends
(North/South) will

repel; lower energy by flipping one.
Boundary between domains,
call “domain wall”
Applying magnetic field
increases domain size
of that pointing in same
direction
Ising type ferromagnet
moments only up or down
12
Domains and Hysteresis
Another way of understanding why magnetic domains are
energetically
preferred is to consider the magnetic field they induce:
The magnetic field will be much lower if they are anti-aligned
as can be seen.
The magnetic field has associated energy .
Thus, minimizing the field lowers the energy of the “two

dipoles”
2/11/2017
7
13
Ferromagnetic domains: Disorder Pinning
Above: both domain walls (red) start and end at same place.
But, one on right, passes through vacancy. It therefore
has one less anti-aligned spins, so overall energy lower
(more favourable) – say, domain wall is “pinned” to the
disorder
An Ising ferromagnet
The length of the domain wall
depends on balance between J and
If large, small wall
If small, wide wall
Consider scaling of wall: if length is then each spin
twists and angle:
. Then the first term in Eq(1) can be written as:

The spin do not need, however, to only point up or down,
corresponding to
large in
The domain wall may be more like a gradual rotation of up
pointing spins
to down pointing spins like below:
14
Ferromagnetic domains: Bloch/Neel Wall
Called a Bloch Wall or Neel Wall
(1)
Small angle expansion
2/11/2017
8
15
Can see has lowest energy if so can think of the
second term being
an “energy cost”

Then for the N unit cells in the domain wall (the “energy
stiffness”), given
per unit area A (per lattice constant a):
Recall from Eq(1)
We also have 2nd term. When spins not exactly up or down, will
be energy
cost proportional to per spin, so for the N unit cells in
the domain wall:
16
Total energy cost due to anisotropy:
So with these two energy costs (penalties) we have the total
energy cost:

Minimizing this energy with respect to length L we find:
and therefore
Energy balance between cost
of domain wall formation versus
gain due to having domains
2/11/2017
9
17
Hysteresis curve
coercive
field
remanent
magnetization
saturation
magnetisation

note that here
We know from electromagnetism that ferromagnets
exhibit a hysteresis loop with applied external field.
When field is returned to zero after being applied, there
is a remanent magnetization
This is because
there is a large
activation energy for
changing the
magnetization
How to understand the activation energy barrier – consider
small crystallite with all spins aligned. The energy per volume
in an external field is:
Where M is the magnetization and is the component in the
-direction.
19

Single Domain Crystallites
=
Zeeman energy per
unit volume
Number of spins per unit volume
angle of magnetization with respect to axis
Plotting Eq(1) vs gives
parabola -
2/11/2017
10
20
Single Domain Crystallites
- minimum of energy when magnetization in plus or minus z-
direction,
corresponding to , and energy barrier in between.
For increasing
B field, there are stable and metastable states. If B field large
enough, spins
will flip – this behaviour can result in the observed hysteresis

24/10/2017
1
Semiconductors 2
Recall from Lecture 3 – density of states for free electrons
per unit volume
Electrons in conduction band like
free electrons but with mass m*, can write:
Similarly the density of states for holes
near the top of the valence band are:
24/10/2017
2
The Fermi-Dirac distribution for a semiconductor
• For a metal, the Fermi energy is the highest occupied
energy at 0 K. The chemical potential is temperature-
dependent (but not much) and so the two are essentially
the same.

• For a semiconductor, the definition of the Fermi energy
is not so clear. We better use the chemical potential.
• Some (many) people also use the term “Fermi energy” for
semiconductors but then it is temperature-dependent.
Earlier (Lecture 3) we wrote it as:
In this Section, we call it the Fermi-Dirac function to
reflect relates to
electrons – recall it gives the probability that an available
energy state E will
be occupied by an electron at absolute temperature T.
For a given chemical potential, the total number of electrons in
the conduction band as a function of temperature is:
where
For
We have
And then,
Boltzman distribution
24/10/2017

3
Want to solve integral: multiply Eq (1) by
(1)
Standard Equation for density of electrons
Similarly can get the number of holes in the valence band p as:
When substantially above the top of the valence
band we have:
and
Standard Equation for density of holes
24/10/2017
4
Number of electrons excited into conduction band must equal
number
of holes left behind in valance band so
Law of Mass Action
Intrinsic Semiconductors

Forming product of density of electrons in conduction band,
and holes in the
valence band we obtain important relation:
Depends only on band gap
Dividing the density of electrons in conduction band n(T), and
holes in the valence
band p(T) we obtain:
Intrinsic Semiconductors
Taking the log of both sides of the below
and solving for gives:
That is, an expression for the chemical potential – at T=0 gives
exactly in
the middle of the band-gap
Using the law of Mass Action above with n=p we obtain:
24/10/2017
5
Extrinsic/Doped Semiconductors
Law of Mass Action also holds for doping when we have
Concentrations n and p

From the law of mass action we have
Consider intrinsic case:
np =
Example
gap size (eV) n in m
-3
at 150 K
n in m
-3
at 300 K
InSb 0.18 2x10
22
6x10
23
Si 1.11 4x10
6
2x10
16
diamond 5.5 6x10
-68

1x10
-21
Using prefactor
24/10/2017
6
Dopants, n- and p-type
Majority and minority carriers
equal number of
electrons and holes
majority: electrons
minority: holes
24/10/2017
7
Band diagram, density of states, Fermi-Dirac distribution,
and the carrier concentrations at thermal equilibrium
Intrinsic
semiconductor

n-type
semiconductor
p-type
semiconductor
Consider a Si sample maintained at T = 300K under
equilibrium conditions, doped with Boron to a
concentration 2×1016 cm-3 : Given the intrinsic
concentration n i = p i is 1x10
10 cm-3
• What are the electron and hole concentrations (n
and p) in this sample? Is it n-type or p-type?
Example
Suppose the sample is doped additionally with Phosphorus
to a concentration 6×1016 cm-3.
• Is the material now n-type or p-type?
What are the n and p concentrations now?
21/10/2018

1
1
Dispersion of one-dimensional chain
We expect periodicity since:
In general for integer p,
The set of points in k-space which are equivalent to k=0 is
known
as the reciprocal lattice (seen this before!)
belongs to the reciprocal lattice if:
2
At shorter wavelength (larger k) we define:
speed at which a wave packet moves
speed at which maxima and minima move
21/10/2018
2
3

examples of dispersion relations
vibrations in a 1D chain
a quantum mechanical particle
k
4
light in vaccum
k
in vacuum the dispersion relation of light is linear.
Light travels with c independent of the frequency.
light in matter
21/10/2018
3
5
Recall dispersion relation for 1-D chain below:
Doesn’t hold for all k – only particular k,

k is quantized
Ashcroft & Mermin Ch 22
6
Periodic boundary conditions
Max Born and Theodore von Karman (1912)
chain with N atoms:
1
N
A finite chain with no end!
21/10/2018
4
Counting normal modes
7
1
N
chain with N atoms:

=
Must have wave ansatz satisfied
Recall:
that is, make satisfied for n n+N
must have to hold true
must then have
Length of first
Brillouin zone
a
8
Finite chain with 10 unit cells and
one atom per unit cell
• N atoms give N so-called normal modes of vibration.
• For long but finite chains, the points are very dense.
Example:
is spacing between
k values1
21/10/2018

5
9
counting normal modes.....
From boundary conditions
chain with 1 atom / unit cell and
N unit cells
N x 1 modes
(since we have N degrees of freedom)
# k-points
# eigenvalues per k-point
# k-points
N unit cells
N x 2 modes
(since we have 2xN degrees of freedom)
Will look at 2
atom chain later
10

Single harmonic oscillator: quantum model
The energy levels are
quantized
image source: wikimedia, author AllenMcC.
Quantum Modes: Phonons
In our chain, harmonic oscillator can be a collective normal
mode,
not just motion of a single particle
Correspondence: for classical harmonic system with normal
oscillation mode at frequency , corresponding quantum
system
will have eigenstates with energy:
n is an integer
http://commons.wikimedia.org/wiki/User:AllenMcC.
21/10/2018
6
11
The ground state being n=0 eigenstate, and has zero-point

energy
The lowest energy excitation is of energy greater than
the
ground state, corresponding to n=1 eigenstate.
Each excitation of this normal mode by a step up
(increasing quantum number n) is known as a phonon
A phonon is a discrete quantum of vibration
12
long chain: quantum model
• The excitations of these oscillators (normal modes) are
called phonons.
• The dispersion is often called a phonon dispersion curve.
l is an integer
21/10/2018
7
Vibrations of a 1-D Diatomic Chain
13
Assume:

or
Vibrations of a1-D Diatomic Chain
14
is quantised in units of
As for the 1-D chain with one mass
write down Newton’s equations
of motion for deviation
of the equilibrium position
Similarly, to before, propose Ansatz:
If system has N unit cells, L=Na, and using boundary conditions
as before:
As before, dividing range of k by spacing between k’s, we get N
different values of k;
one k per unit cell
21/10/2018
8
15

Substitute Ansatz into the
equation of motion:
gives:
or as an eigenvalue equation:
16
Find solutions by finding zero’s of the secular determinant
so
and the second term becomes:
21/10/2018
9
17
When phonons interact with light
(photons) it is the upper “optical”
branch, hence name
Group velocity

goes to zero at zone boundary
and for optical, at k=0
Finally, the dispersion relation is:
18
Effective spring constant
Density of chain
Expanding for small k,
can show:
Could have derived this sound velocity – recall, we had earlier:
=
21/10/2018
10
19
Acoustic mode, which has =0 is solved by,
Consider acoustic and optical phonon as , we had:

which becomes,
Says the two atoms move together for the acoustic mode in the
long
wavelength limit
20
Tells the two atoms move in opposite directions
The optical mode, at
has frequency:
As , we had
and eigenvector
21/10/2018
11
21
As for electronic states,
can unfold into the “extended zone scheme”
22

Can show that as the two atoms in cell become
identical and
dispersion becomes that of monatomic dispersion
21/10/2018
12
Vibrations/Phonons
23
Had one mass per cell, one mode per distinct value of k
(acoustic, go to zero at
k=0)
For two masses per cell, two modes per distinct value of k
(acoustic, optical)
For M atoms per cell, get M modes per distinct value of k – one
will be acoustic,
others optical.
For 1-D chain atoms only move in line, one degree of freedom
For 3-D solid atoms have three degrees of freedom
Three different acoustic modes at each k at long wavelength –
one “longitudinal

acoustic” and two “transverse acoustic”
For N atoms per cell, 3(N-1) optical modes, always 3 acoustic
modes – 3N
degrees of freedom per cell
24
Phonons in 3D crystals: Aluminium
One atom per cell, just 3 acoustic modes
21/10/2018
13
25
Phonons in 3D crystals: diamond
• We see acoustic and optical phonons. 3 branches for every
atom per unit
cell. Here two atoms, six branches, three ac, three opt (3(N-1) =
3(2-1)=3)
• important to identify Bravais lattice and basis if we want to
make
predictions as to vibrational properties
State of the art calculation + expt
26

Phonon dispersion and phonon density of states of TiC2 as
determined
by DFT calculations.
Like for the electron energy
dispersion and density of states
6 atoms per unit cell – 3(N-1)=3(6-1)=15 opt, 3 acoustic
Density of phonon
states, g(ω),
21/10/2018
14
Revision
27
28
17/10/2017
1
1

Lattice vibrations – 1D
Consider one dimensional system of atoms in a line
Recall: The potential between two neighbouring atoms has the
form above
2
In region of minimum, Taylor expansion:
At finite temperature T the atoms can oscillate between
and
Since potential is asymmetric away from minimum, this leads to
an
Average position greater than
- Thermal Expansion (though not all systems behave like this)
Handout 6
17/10/2017
2
3

Compressibility/elasticity
Hooke’s Law – quadratic potential about minimum
Applying a force to compress system
- reduces distance between atoms
Compressibility: (assuming )
In one-dimension, with L the length:
(taking = )
4
In an isotropic compressible fluid
sound waves with velocity:
For the 1-D solid take the density as where is
the mass of atom
then
bulk modulus
17/10/2017
3

5
Lattice vibrations -1D chain
Handout 6
Ch 22
Ashcroft &
Mermin
Let the position of the atom be
And the equilibrium position be
Allowing motion of atoms:
Can write total potential energy as:
6
The force on the mass
Ansatz
Solution
:

Substitute solution into (1)
(1)
17/10/2017
4
7
in general we have that ω depends on k.
ω(k) is called the dispersion relation. Periodic in
8
The first Brillouin zone
Recall: The first Brillouin zone is the region of reciprocal
space which is closer to one reciprocal lattice point than to

any other (Wigner-Seitz cell in reciprocal space).
17/10/2017
5
9
10
sound
wave
dispersion relation
For small k the sin is equal to its argument
but for k very small (lambda very long) the crystalline structure
is

unimportant and we get sound waves.
.
sound
velocity
17/10/2017
6
11
We expect periodicity since:

In general for integer p,
The set of points in k-space which are equivalent to k=0 is
known
as the reciprocal lattice (seen this before!)
belongs to the reciprocal lattice if:
12
At shorter wavelength (larger k) we define:
speed at which a wave packet moves
speed at which maxima and minima move
17/10/2017
7

13
vibrations in a 1D chain
a quantum mechanical particle
k
14
light in vaccum
k
in vacuum the dispersion relation of light is linear.
Light travels with c independent of the frequency.
light in matter

17/10/2017
8
Counting normal modes
15
1
N
A finite chain with no end!
chain with N atoms:
16
Finite chain with 10 unit cells and

one atom per unit cell
• N atoms give N so-called normal modes of vibration.
• For long but finite chains, the points are very dense.
17/10/2017
9
17
counting normal modes.....
boundary conditions
N unit cells
N x 1 modes
(since we have N degrees of freedom)
# k-points

# k-points
N unit cells
N x 2 modes
(since we have 2xN degrees of freedom)
18
Single harmonic oscillator: quantum model
The energy levels are
quantized
image source: wikimedia, author AllenMcC.

In our chain, harmonic oscillator can be a collective normal
mode,
not just motion of a single particle
Correspondence: for classical harmonic system with normal
oscillation mode at frequency , corresponding quantum
system
will have eigenstates with energy:
17/10/2017
10
19

The ground state being n=0 eigenstate, and has zero-point
energy
The lowest energy excitation is of energy greater than
the
ground state, corresponding to n=1 eigenstate.
Each excitation of this normal mode by a step up
(increasing quantum number n) is known as a phonon
A phonon is a discrete quantum of vibration
Adv.: In Handout 6, read on about effect of temperature
(Bose occupation factor) and how the heat capacity can be
obtained.

17/10/2017
1
1
Consider one dimensional system of atoms in a line
Recall: The potential between two neighbouring atoms has the
form above
2
In region of minimum, Taylor expansion:
At finite temperature T the atoms can oscillate between

and
Since potential is asymmetric away from minimum, this leads to
an
Average position greater than
- Thermal Expansion (though not all systems behave like this)
Handout 6
17/10/2017
2
3
Compressibility/elasticity
Hooke’s Law – quadratic potential about minimum

Applying a force to compress system
- reduces distance between atoms
Compressibility: (assuming )
In one-dimension, with L the length:
(taking = )
4
In an isotropic compressible fluid
sound waves with velocity:
For the 1-D solid take the density as where is
the mass of atom
then
bulk modulus

17/10/2017
3
5
Handout 6
Ch 22
Ashcroft &
Mermin
Let the position of the atom be
And the equilibrium position be
Allowing motion of atoms:

Can write total potential energy as:
6
The force on the mass
Ansatz

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