Introduction to Condensed Matter · PDF fileDiatomic linear chain ... normal modes. We may...
Transcript of Introduction to Condensed Matter · PDF fileDiatomic linear chain ... normal modes. We may...
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M.P. Vaughan
Introduction to Condensed
Matter Physics
Phonons II
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Overview
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Overview of phonons II
• Monatomic linear chain
• The harmonic potential
• 3D monatomic crystal
• The dynamical matrix
• 3D crystal with a basis
• Measuring dispersion relations
• Macroscopic properties
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The Monatomic Linear
Chain
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Diatomic linear chain
In the previous lecture, we analysed the monatomic linear
chain in terms of classical modes.
![Page 6: Introduction to Condensed Matter · PDF fileDiatomic linear chain ... normal modes. We may obtain these from the 3 couled ... PY3105 Introduction to Condensed Matter PhysicsAuthors:](https://reader031.fdocuments.in/reader031/viewer/2022030416/5aa1cf2d7f8b9a07758c31af/html5/thumbnails/6.jpg)
Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Diatomic linear chain
This yielded two coupled differential equations with
eigenvalues of the angular frequency
,
cos1211
2/1
22
2
2
1
2
2
2
12
2
2
1
2
qa
where
.and2
2
2
1
2
1m
k
m
k SS
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Diatomic linear chain
The wavevectors q were given by
where N is the number of links (containing 2 atoms) in the
chain.
,2
N
n
aq
Plotting the eigenvalues against wavevector, we obtained
graphs illustrating two phonon branches:
1. acoustic phonons
2. optical phonons
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Dispersion relations
Optical modes
Acoustic modes
Linear approximation for
acoustic branch
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Monatomic linear chain
The solution for the monatomic linear chain is less
involved. For completeness, we give it here.
In this case, we have just one differential equation
.11 nnSnnSn uukuukum
Looking for solutions of the form
,tqnai
n Aeu
we have
.112 iqa
S
iqa
S ekekm
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Monatomic linear chain
Writing this in terms of
we have
.cos12 2
1
2 qa
,2
1m
kS
Note that this is only an acoustic mode solution – there are
no optical modes.
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The Harmonic Potential
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
Limiting the expansion of the potential energy of the ring to
terms quadratic in un, the potential energy of the monatomic
ring may be written
.2
2
1
2
10 n
nnnnS uuuu
kUU
Differentiating this with respect to un gives
.11
nnSnnS
n
uukuuku
U
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
Multiplying by -1 then just gives the force on the mass at
position n, so
which reproduces the differential equation solved earlier.
,11
nnSnnS
n
n uukuuku
Uum
The same strategy may be applied to the diatomic linear
chain. In this case, the harmonic potential is
.2
2
,2,1
2
1,2,10
n
nnnnS uuuu
kUU
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
.,2,11,2,1
,1
nnSnnS
n
uukuuku
U
Differentiating this with respect to u1,n gives
Meanwhile, differentiating this with respect to u2,n gives
.,1,21,1,2
,2
nnSnnS
n
uukuuku
U
(Tip: consider the n+1 terms in the summation in U).
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
n
nu
Uum
,1
,11
Hence, putting
and
,,2
,22
n
nu
Uum
reproduces the coupled equations
.
,
,21,1,1,2,22
,1,21,2,1,11
nnSnnSn
nnSnnSn
uukuukum
uukuukum
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
Let us consider an alternative form for the harmonic potential
of a monatomic chain. Putting U0 = 0, we can write the
contribution to U due to a single link at position n as
.2
111 nnSnnn uukuuU
Putting U0 = 0 and extending the summation beyond nearest
neighbour interactions, we have
nn
nnnnnn uukuuU'
'''4
1
(the additional factor of ½ ensures that we do not double
count distinct links).
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
The total potential is usually written in the form
.2
1
'
''nn
nnnn uDuU
.4
1
4
1
'
''
'
' nn
nnnn
nn
nnnn ukuukuU
,'' nnnn kk To justify this we note that, since we can write the
potential as
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
.'''
''',' nn
n
nnnnnn kkD
We now define
(Note that the second term is not included in the
summation). Inserting this into the expression for U
.4
1
,4
1
'
'''
''
''',
'
'''
nn
nnnnn
n
nnnnn
nn
nnnnnn
ukuuku
uukuuU
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The harmonic potential
,4
1
4
1
,4
1
4
1
'
''
'
'
'
''
''
''
nn
nnnn
nn
nnnn
nn
nnnn
n
n
n
nnn
ukuuku
ukuukuU
Separating this into two multiple summations and employing
the Kronecker delta,
which reproduces the original form of the potential.
The utility of the new form for the potential becomes
apparent when we move to 3D crystals.
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The 3D monatomic crystal
Generalising to 3D, the displacements may now be
expressed as vectors u associated with a lattice point Rn.
,2
1
'
''nn
nnn
T
nU uDu
The potential may now be written in matrix form as
where Dnn’ is a matrix and the T superscript denotes the
operation of transposition.
We shall not attempt to solve this system exactly here, just
highlight features of the general solution.
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The 3D monatomic crystal
We now require solutions of the form
where n is the polarisation vector giving the direction of the
ionic displacements.
The equations of motion for a monatomic crystal are
,
ti
nne
Rq
nu
,,
,
n
nu
Uum
where labels the component of un.
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The dynamical matrix
From the expression for U, we have
Hence
.'
''
,
n
nnn
nu
UuD
is known as the dynamical matrix.
,2nDn qm
where
'
''
n
i
nnne
Rq
q DD
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The 3D monatomic crystal
For a monatomic crystal of N atoms, we would then have 3N
normal modes. We may obtain these from the 3 couled
differential equations.
As in the case of the monatomic linear chain, the 3D
monatomic crystal yields only acoustic modes.
These modes may be differentiated on the basis of
polarisation. We have
1. One longitudinal mode (LA)
2. Two transverse modes (TA), orthogonal to the
direction of propagation and to each other
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
The 3D crystal with a basis
Restricting our consideration of the number of modes to a
particular value of q, the number of modes for the 3D
monatomic crystal is 3.
For a 3D crystal with a basis of p atoms, the number of
modes per q is 3p. These are categorised as follows
• 1 longitudinal acoustic (LA) mode
• 2 transverse acoustic (TA) modes
• p -1 longitudinal optical (LO) modes
• 2(p -1) transverse optical (TO) modes
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Measuring dispersion
relations
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Measuring dispersion relations (1)
• Neutron Inelastic Scattering
• Due to penetrative power of neutrons, can
be used to measure bulk phonons
• Electron Energy Loss Spectroscopy
(EELS)
• Uses beam of electrons with narrow range
of energies. Used for surface phonons.
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Measuring dispersion relations (2)
• Inelastic Atomic Scattering
• Due to low penetrability of (He) atoms,
limited to surface phonons
• Raman Scattering
• Inelastic scattering of light due to
absorption (sometimes emission) of energy
by optical modes.
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Brillouin zone of diamond / zinc blende
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Diamond
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Silicon
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
GaAs
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Macroscopic properties
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Coláiste na hOllscoile Corcaigh, Éire
University College Cork, Ireland ROINN NA FISICE Department of Physics
PY3105 Introduction to
Condensed Matter Physics
Macroscopic properties
• The properties of phonons are
related to the macroscopic properties
of a material. A few of these are
• Elastic properties
• Thermal conductivity
• Electrical resistance