Order 1238142: Condensed matter
tutorthammy
Lect-1-IntroDrude.pdf
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What is Condensed Matter Physics?
Condensed matter physics is the largest topic of research in physics today. It encompasses statistical mechanics and quantum physics of solids (solid state physics), but also describes liquids (magnetic vortex liquids is my speciality) and novel states of materials.
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Condensed Matter Physics
This course
Condensed Matter Physics Normal and Advanced
19 lectures (Mon 10am; Tues 1pm; Fri 12pm)
2 Assignments (12.5% each; common and separate N and A questions)
3 Quizzes (go over in tutorials; not marked; common and separate N and A questions)
Exam (75%; common and separate N and A questions)
3 sessions of Tutoring: Friday 10am, LT5 (quizzes, course material, assignments) 5 Oct, 19 Oct, 2 Nov
Tutors also answering questions on Piazza
Lectures on Canvas
Main Books:
(i) The Oxford Solid State Basics 1st Edition by S. H. Simon (2013) – eBook in Library
(ii) Solid State Physics, N. Ashcroft and N. D. Mermin
(iii) Introduction to Solid State Physics, C. Kittel
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Condensed Matter Physics Condensed Matter Physics (also known as Solid
State Physics) explains the properties of solid materials. It is the study of the behaviour of atoms when they are placed in close proximity to one another.
The properties are expected to follow from Schrödinger’s eqn. for a collection of atomic nuclei and electrons interacting with electrostatic forces.
The fundamental laws governing the behaviour of solids are known and well tested.
“Condensed Matter” being more general than just solid state was coined by Nobel-Laureate Philip W. Anderson.
Crystalline Solids
We will deal with crystalline solids, that is solids with an atomic structure based on a regular repeated pattern.
Many important solids are crystalline.
More progress has been made in understanding the behaviour of crystalline solids than that of non- crystalline materials since the calculation are easier in crystalline materials.
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What is the point? Understanding the electrical properties of solids is
right at the heart of modern society and technology.
For example, the entire computer and electronics industry relies on the tuning of a special class of material, the semiconductor, which lies right at the metal-insulator boundary. Condensed matter physics provides a background to understand what goes on in semiconductors.
New technology for the future will inevitably involve developing and understanding new classes of materials (e.g. superconductors, spintronic, topological insulators etc)
Properties of matter depend upon how the atoms are put together:
e.g. Electrical resistivity of three states of solid matter
graphite diamond Buckminster
-fullerene
They are all just carbon!
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Lectures 1-3 Physics of solids without considering microscopic structure:
Drude Theory, Free Electron Theory
4 Types of matter and chemical bonding
5-6 Crystal structure and reciprocal lattice
7 Diffraction
8-9 Scattering and Bloch’s Theorem
10-11 Nearly free electron and Tight binding models
12-13 Quantum theory of phonons
14-15 Semiconductors
15-16 Magnetism in solids
17 N and A separate
18 N and A separate
19 N and A separate
• Basic assumptions of the classical theory • DC electrical conductivity in the Drude model • Hall effect • Thermal conduction / Wiedemann-Franz law • Shortcomings of the Drude model: heat capacity...
In this lecture, learn about ....
Classical approach (Drude theory)
A&M Ch1,pg 2-15;20-25
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Drude picture of atom
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Isolated atom Atoms in metal
Valence atoms move freely through metal Ions remain intact, act as immobile positive particles
Fig. 1.1 A&M
Drude’s classical theory
• Theory by Paul Drude in 1900, three years after the electron was discovered.
• Drude treated the (free) electrons as a classical ideal gas but the electrons collide with the stationary ions, not with each other.
so at room temp.
no interaction with each other, independent electron approximation interaction between electrons and core instantaneous and short range
rms velocity
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Drude’s classical theory
relaxation time
mean free path
(average time between scattering events)
3 assumptions: 1) electrons have a scattering/
2) scattered, electron returns to momentum=0 3) In between scattering events the electrons respond
to E and B-fields
Drude theory: electrical conductivity
In steady state the equation of motion is
integration gives
remember:
and if is the average time between collisions then the average drift speed is
for we get
Electrons in an Electric field
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Concept of drift velocity
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(a) Zero and (b) non-zero electric field In E-field the start S and finish F positions differ –
net flow of charge
Drift velocity versus time; average time between collisions is the relaxation time
Drude theory: electrical conductivity
number of electrons passing in unit time
current density
current of negatively charged electrons
and with we get
Ohm’s law
n=N/V =N/(A.dl) density of electrons
i=
j=i/A
i=Q/t =N(-e)/t
N/t =
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Drude theory: electrical conductivity
Ohm’s law
and we can define the conductivity
and the resistivity
and the mobility
Electrical conductivity of materials
Varies by 32 orders of magnitude (not including superconductivity)
From size of atom to earth-sun distance is only 22 orders of magnitude
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Drude theory: electrical conductivity
line
Experimental values increase by a factor of 10 for the low temperature; Calculated values by less than a factor of 2 The lower the temperature the worse this gets
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Drude theory: electrical conductivity
• Drude’s theory gives a reasonable picture for the phenomenon of resistance.
• Drude’s theory gives qualitatively Ohm’s law (linear relation between electric field and current density).
• It also gives reasonable quantitative values of conductivity, at least at room temperature.
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The Hall Effect
• Accumulation of charge leads to Hall field EH. • Hall field proportional to current density and B field
is called Hall coefficient
Electrons in electric and magnetic fields
The Hall coefficient
and definition
for the steady state we get
electron density form Ohm’s law
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The Hall coefficient
Ohm’s law contains e2
But for RH the sign of e is important.
Drude theory predicts wrong sign for Be, Mg, Al
What would happen for positively charged carriers?
We don’t only get the carrier density, we also get their sign!
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Questions
A uniform silver wire has a resistivity of 1.54×10–8 Ωm at room temperature. For an electric field along the wire of 1 volt cm–1, compute the average drift velocity of electron assuming that there is 5.8 × 1028 conduction electrons /m3. Also calculate the mobility.
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The Wiedemann-Franz law
• Wiedemann and Franz found in 1853 that the ratio of thermal and electrical conductivity for ALL METALS is constant at a given temperature (for room temperature and above). Later it was found by L. Lorenz that this constant is proportional to the temperature.
• Let’s try to reproduce the linear behaviour and to calculate L here.
constant
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The Wiedemann Franz law
estimated thermal conductivity (from a classical ideal gas)
or using
average speed
rms value
Comparison of the Lorenz number to experimental data
at 273 K
metal 10-8 Watt Ω K-2
Ag 2.31
Au 2.35
Cd 2.42
Cu 2.23
Mo 2.61
Pb 2.47
Pt 2.51
Sn 2.52
W 3.04
Zn 2.31
Drude prediction: 0.98 – 1.11 Watt ΩK -2
off by a factor of 2, but still very good!
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Question
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The thermal conductivity of a metal is 123.92 Wm –1 K –1. Find the electrical conductivity and Lorentz number when the metal posses relaxation time 10–14 sec at 300 K. (Density of electron = 6×1028 per m3)
Failures of the Drude model
• Despite this correct prediction, there are some serious problems with the Drude model.
• It is fortuitous – since the measured specific heat is not close to per electron
Due to two mistakes that roughly cancel – specific heat far too large, velocity far too small ! (Due to not taking Fermi statistics of the electron into account)
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Failures of the Drude model
The Drude model predicts a roughly 100 times larger value of the Peltier coefficient
The Peltier effect is that running a current through a material also transports heat
thermal current
electrical current
The ratio known as the thermopower or “Seebeck coefficient”
For most metals value is 100 times smaller
