Fyzika kondenzovaných látek I, podzimní semestr, 2013/2014
III — Lattice subsystem
III.1 Hamiltonian of the lattice and the harmonic approximation
A careful acount of the theory of lattice vibrations is given in Cely’s monograph. For
this reason, we discuss here only some crucial aspects of the theory. We refer the
reader to the monograph, where appropriate.
Adiabatic approximation
• Physical insight: Electrons move much faster than the nuclei. They can be assumed
to “feel” the current spatial conﬁguration of the nuclei (or ions). The nuclei, on the
other hand, can be assumed to feel the average conﬁguration of the electrons.
• Translated into the language of equations.
Hamiltonian function for the system of electrons and nuclei:
H = Te + Ve + Ve−n + Tn + Vn . (1)
Schrödinger equation:
HΨ = EΨ , Ψ = Ψ(r, R) . (2)
The adiabatic approximation consists in restricting the space of wavefunctions under
consideration to functions of the form
Ψ(r, R) = ψm(r, {R})χ(R) , (3)
where ψm is an eigenfunction of the electronic hamiltonian He,
He({R}) = Te + Ve + Ve−n({R}) , (4)
He({R})ψm = ǫmψm . (5)
The standard variational condition (or, alternatively, insertion of the expression 3
into the Schrödinger equation 2 and neglection of “small terms”) leads to the following
equation for the function χ
Hnχ = Eχ (6)
with
Hn = Tn + Vn + ǫm({R}) . (7)
Usually ǫm in the preceding equation is replaced with ǫGS (GS ... ground state). It can
be easily shown that this description is consistent with the physical picture of electrons
responding to the current conﬁguration of the nuclei and the nuclei responding to the
average conﬁguration of the electrons.
The problem deﬁned by the hamiltonian 1 has been splitted into two parts: the
electronic part characterized by the hamiltonian He of Eq. 4 and that of the nuclei characterized
by the hamiltonian Hn of Eq. 7, the so called hamiltonian of the crystal lattice.
The two parts are connected only by the ǫm-term in Eq. 7 which provides a substantial
contribution to the eﬀective potential energy W(R) of the nuclei deﬁned by
W(R) = Vn(R) + ǫm({R}) . (8)
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Harmonic approximation
Nuclei in a solid move around a certain equilibrium conﬁguration
{R01, R02, ... , R0N } (9)
corresponding to the global or a local minimum of W. The potential W can be expanded
into the Taylor series, see Eq. (5.2) in Cely’s monograph. The ﬁrst-order term
vanishes because the expansion is about a minimum. In the harmonic approximation
only the second order term is retained, the higher order terms neglected. Physical
picture: it is assumed that the atomic displacements are so small that they do not
bring the system out of the region, where the parabolic approximation to W is reliable.
III.2. Classical approach: EQM and the “Bloch theorem” for the crystal lattice
Classical equations of motion (see Cely’s monograph for deﬁnitions of the symbols):
MJ
d2
uα(J)
dt2
= −
N
K=1
z
β=x
Aαβ(J, K)uβ(K) , (10)
where u(J) is the displacement vector of the J-th nucleus, RJ = R0J + u(J), and
Aαβ(J, K) =
∂2
W
∂uα(J)∂uβ(K)
(11)
are the so called force constants.
For a crystal lattice, the index J can be replaced by two indices: m specifying the
position of the unit cell and µ specifying the position of the nucleus within the unit
cell. It follows from the tranlation symmetry that
Aαβ(mµ, nν) = Aαβ(Rn − Rm, µν) , (12)
where Rn is the vector connecting the origin of the coordinate system with the lattice
point labelled as n. As a consequence, it is possible to ﬁnd a complete set of solutions
of the equations 10 of the form [see Eq. (5.15) in Celý]
u(mµ, t) = uµe−iωt
eiq·Rm
. (13)
This is an analogue of the Bloch theorem. The physical meaning of the wave vector q
is the following: when going from the unit cell with the lattice vector Rm to the unit
cell with the vector Rn the phase of the atomic displacements changes by eiq·(Rn−Rm)
.
By inserting the expression on the right hand side of Eq. 13 into Eq. 10 we obtain
[cf. (5.18)]
ν,β
[Bαβ(q, µν) − ω2
Mµδαβδµν]uνβ = 0 , (14)
µ, ν = 1, 2, ..., s, where s is the number of atoms per unit cell, α, β = x, y, z, and
Bαβ(q, µν) =
h
Aαβ(Rh, µν)eiqRh
. (15)
The set of equations 14 does not represent a standard eigenvalue problem because
of the presence of the factor Mµ in the square bracket. An eigenvalue problem can
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be obtained, however, by introducing the reduced displacements wµ, µ = 1, 2, ... s,
[cf. (5.26)],
wµ = Mµuµ (16)
and the dynamical matrix Dαβ(q, µν) [cf. (5.27)],
Dαβ(q, µν) = (MµMν)−1/2
Bαβ(q, µν) . (17)
After some arrangments of Eq. 14 we obtain [cf. (5.28)]:
νβ
[Dαβ(q, µν) − ω2
δα,βδµν]wνβ = 0 or (D(q) − ω2
I3s)w = 0 , (18)
which is already a canonical eigenvalue problem. For any q (usually in the ﬁrst BZ)
we obtain a set of eigenvalues
ω1(q), ω2(q), ω3(q), ... , ω3s(q) (19)
(the corresponding functions of q are called the dispersion relations) and the corresponding
set of eigenvectors (in the 3s-dimensional space!)
w1(q), w2(q), ... , w3s(q) . (20)
The corresponding normalized eigenvectors
e1(q), e2(q), ... , e3s(q) (21)
are sometimes also called the polarization vectors. The dynamical matrix satisﬁes
the equation D(−q) = D∗
(q) and it is therefore always possible to choose a set of
polarization vectors such that ej(−q) = ej
∗
(q).
Normal coordinates, transformation of the hamiltonian
into the form corresponding to a set of harmonic oscillators
Displacement of the µ-th atom in the m-th unit cell in the state, where only one mode
(wavevector q,phonon branch j) is excited:
umµα ∼ ejµα(q)e[iqRm−ωj(q)t]
. (22)
General solution of the problem: a superposition of terms of the type 22.
Generalized or normal coordinates - see Eq. (5.33) and (5.34).
Hamiltonian in terms of the normal coordinates [cf. (5.37)]:
H =
q∈BZ
3s
j=1
1
2
P(q, j)P(−q, j) +
ω2
j (q)
2
Q(q, j)Q(−q, j) . (23)
New real coordinates Ξ(q, j) a Π(q, j) [cf. (5.41)]:
Ξ =
1
2
[Q(q, j) + Q(−q, j)] +
i
2ωj(q)
[P(q, j) − P(−q, j)] , (24)
Π =
iωj(q)
2
[−Q(q, j) + Q(−q, j)] +
1
2
[P(q, j) + P(−q, j)] . (25)
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Hamiltonian in terms of Ξ a Π [cf. (5.42)],
H =
q∈BZ
3s
j=1
1
2
Π2
(q, j) +
ω2
j (q)
2
Ξ2
(q, j) , (26)
has precisely the same form as that of a set of independent harmonic oscillators. The
quantum-mechanical treatment of the problem leading to the concept of phonons has
already been explained earlier when describing excited states of a chain of atoms.
III.3. Examples of dispersion relations and polarization vectors
• Dispersion relations of some simple materials (Pb,KBr,Ge,NaCl ...) are shown in
Kittel; Ashcroft/Mermin; Celý.
• Polarization vectors for a 2D lattice possessing two non-equivalent nuclei per unit
cell - a schematic representation.
(a)
(b)
(c)
(d)
(e)
(a) Crystal structure, the box shows the unit cell; (b) TA-mode; (c) LA-mode; (d)
TO-mode; (e) LO-mode. The polarization vectors of the long-wavelength limit are
shown in the boxes.
• Polarization vectors for some high-Tc cuprate superconductors - see the ﬁle Kova-
leva.pdf.
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