Order 1252715: Condensed Matter physics
tutorthammy
Lect-13-Phonons_CMP.pdf
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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:
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Dispersion of one-dimensional chain
At shorter wavelength (larger k) we define:
speed at which a wave packet moves
speed at which maxima and minima move
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examples of dispersion relations
vibrations in a 1D chain
a quantum mechanical particle
k
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examples of dispersion relations
light in vaccum
k
in vacuum the dispersion relation of light is linear.
Light travels with c independent of the frequency.
light in matter
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Dispersion of one-dimensional chain
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Recall dispersion relation for 1-D chain below:
Doesn’t hold for all k – only particular k,
k is quantized
Ashcroft & Mermin Ch 22
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Periodic boundary conditions Max Born and Theodore von Karman (1912)
chain with N atoms:
1
N
A finite chain with no end!
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Counting normal modes
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Periodic boundary conditions
1
N
Max Born and Theodore von Karman (1912)
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
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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
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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
# eigenvalues per k-point
chain with 2 atom / unit cell and
N unit cells N x 2 modes
(since we have 2xN degrees of freedom) Will look at 2
atom chain later
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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
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Quantum Modes: Phonons
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
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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
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Vibrations of a 1-D Diatomic Chain
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Assume:
or
Vibrations of a1-D Diatomic Chain
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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
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Vibrations of a 1-D Diatomic Chain
Substitute Ansatz into the
equation of motion:
gives:
or as an eigenvalue equation:
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Vibrations of a1-D Diatomic Chain
Find solutions by finding zero’s of the secular determinant
so
and the second term becomes:
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Vibrations of a 1-D Diatomic Chain
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:
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Vibrations of a 1-D Diatomic Chain
Effective spring constant
Density of chain
Expanding for small k,
can show:
Could have derived this sound velocity – recall, we had earlier:
=
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Vibrations of a 1-D Diatomic Chain
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
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Vibrations of a 1-D Diatomic Chain
Tells the two atoms move in opposite directions
The optical mode, at
has frequency:
As , we had
and eigenvector
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Vibrations of a1-D Diatomic Chain As for electronic states,
can unfold into the “extended zone scheme”
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Vibrations of a 1-D Diatomic Chain Can show that as the two atoms in cell become identical and
dispersion becomes that of monatomic dispersion
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Vibrations/Phonons
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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
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Phonons in 3D crystals: Aluminium
One atom per cell, just 3 acoustic modes
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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
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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),
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Revision
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