Physics 489—Solid State notes for 10/22/2015
Nearly Free Electron Model and Examples; also Tight Binding Model.
• Plan for near future: Read ch. 7 from the text, and then ch. 9.
Chapter 7 topics -- Bloch theorem (last time), Nearly Free Electron model, formation of metals
and insulators.
Chapter 9 topics – Extended and reduced-zone scheme (last time), filling of electron states in
realistic structures such as FCC, formation of Fermi surfaces in real metals. Also hole and
electron pockets, and Tight Binding model.
Following these the plan is to cover semiconductors (ch. 8), along with selected topics related to
surfaces and nanostructures (ch. 17&18), alloys (ch. 22) and superconductors (ch. 10).
• Nearly Free Electron Model: As described last time, because of it symmetry the crystal
potential can be decomposed by Fourier components including only the reciprocal lattice vectors:


U(r ) = ∑UG eiG⋅r , while for the electron states in the most general form can be decomposed
G

ik⋅r
according to the near-continuum of discrete k states, ψ i = ∑ α k e . Plugging into the
k
Schrodinger equation we found,
2 2
 ⎧
⎫
⎞
ik⋅r ⎛  k
e
∑k ⎨⎜⎝ 2m − ε ⎟⎠ α k + ∑G UGα k−G ⎬ = 0 ,
⎩
⎭
[1]
and in general the curly bracket must itself be zero, yielding the "central equation" for each k,
⎛ 2k 2
⎞
[2]
⎜⎝ 2m − ε ⎟⎠ α k + ∑UGα k−G = 0 .
G
This led to the result of the Bloch Theorem, by which the electron states were found to have the
form of a periodic function multiplied by a plane wave:
!!
! !! ⎛
⎞ !!
[3]
ψ i = u (r )eik ⋅r = ⎜ ∑ α k ,G eiG⋅r ⎟eik ⋅r ,
⎝G
⎠
where the function u is the periodic function. With u written in the equivalent form inside the
round brackets on the right we found that these wave-functions contain Fourier components
connected by a G vector, not the entire set of possible k vectors. Thus far the results [2] and [3]
are completely general and work for any periodic solid.
(1) With U considered to be weak, in perturbation theory up through 2nd order we’ll see that the
energy is,
2
UG
2k 2
2k 2
  , ε o (k) ≡
ε=
+∑ 
.
[4]
2m
2m
G ε o (k ) − ε o (k − G)
This result is a very small deviation from parabolic free-electron states, except for the case that
Δε in the denominator is zero. This occurs at band crossings, occuring at zone boundaries.
(2) The degenerate case, at zone boundaries: When Δε = 0, eqn. [2] can be solved in parallel
for each of the states involved in the degeneracy, and as a result diagonalizing the Hamiltonian
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terms involving these states. Note that these states are always connected by one or more G

 
vectors. If only 2 states cross, with wavevectors k and k − G , we obtain a linear equation,
⎛  2 k 2 2m
⎞ ⎛ α ⎞
⎛ αk ⎞
UG
k
⎜
⎟
⋅
=
ε
⎜
⎟
⎜
⎟ , [4]
⎜⎝ α k−G ⎟⎠
⎜⎝
U −G
 2 (k − G)2 2m ⎟⎠ ⎜⎝ α k−G ⎟⎠
which can be solved readily; I attach a Mathematica notebook showing the general solution. At
the zone boundary, the energy separation between avoided-crossing states is 2 UG . This is true
independent of the size of UG , but note that in [4] we are neglecting matrix elements to all other
plane-wave states, hence the result is exact in the weak-U limit (nearly free electron case).
(3) In the attachment I also solved a more general case
in which 4 plane-wave solutions are coupled, in the first
quadrant (in k-space) of the 2D square lattice, with the
plotted results including parts of the 3rd and 4th zones.
The corresponding Brillouin zones can be seen in the
figure on the right.
[http://www.doitpoms.ac.uk/tlplib/brillouin_zones/index.php;
Creative-commons license]
• FCC-related electron band examples:
Free-electron states folded into the FCC 1st Brillouin Zone (Ashcroft & Mermin text):
8/cell
(Si or GaAs)
3/cell
(Aluminum)
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Calculated electron states for Aluminum (FCC structure, 3 electrons per cell), Silicon
(diamond structure, 8 electrons per FCC cell), and GaAs (zincblende structure, 8
electrons/FCC cell):
Aluminum (on right) from W. A. Harrison, Phys. Rev. 118, 1182 (1960).
Below are silicon band structure (on left), and GaAs (right) [LDA calculation, http://th.fhiberlin.mpg.de/th/fhi98md/doc /main/node19.html];
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• Tight Binding model: Tight Binding methods use localized basis functions, and are
particularly well-suited for states close to unperturbed atomic orbitals (insulators with weak
hybridization, or compact d- and f-orbitals in transition metal materials which remain somewhat
“atomic-like”), however they also serve as a conceptual basis for understanding materials in
general, and are computationally efficient. Tight binding methods have a historical connection to
Linear Combination of Atomic Ortbitals (LCAO) methods, and the simplified example below is
along these lines, utilizing a single s-orbital only. In empirical tight-binding parameters are
derived by fitting to experimental results; such calculations may be used for large, complex
systems for which a relatively simple computation scheme is necessary. Parameters may also be
generated from ab-initio density functional methods [the Density Functional based Tight Binding
(DFTB) method].
Example: This is silimlar to what is in Kittel. Assume a simple one-electron Bloch function,
 
1
ik ⋅ R
ψ k (r ) =
ϕ
(r
−
R)e
,
[1]
∑ n
N R
where ϕ n is a localized orbital, and n represents a particular band index plus orbital quantum
number. A more general case could include a summation of different orbitals on each site: s, p, d,
etc. Also for simplicity we assume that the overlap of neighboring orbitals is small,
ϕ n (r − Ri ) ϕ n (r − R j ) ≈ 0 . Alternatively, the orbitals can be replaced by Wannier Functions,
which are localized orbitals made to be precisely orthogonal on neighboring sites.
Thus assuming neighboring ϕ n ’s are orthogonal, if the ϕ n remain similar to atomic states, the
energy eigenvalue of [1] will not be much different than En, the atomic energy level of ϕ n :
 
E
Hψ k (r) ≈ n ∑ϕ n (r − R)e ik ⋅ R .
[2]
N R
To determine the energies in the crystal we use perturbation theory, with the first-order energy
εk = ψ k H ψ k including terms diagonal in the on-site orbital, plus terms involving nearest
neighbors, neglecting any matrix elements to more distant neighbors. Thus,
  
1
εk = En + ∑ ∑ ϕ (r − R) H ϕ (r − Ri ) + c.c. e ik ⋅ ( R − R i ) , [3]
N R i
where the sum over i includes nearest neighbors at positions Ri. In our simple case let’s consider
that the matrix elements do not depend upon orientation, as for example would occur with s
orbitals. In that case a single parameter (γ) suffices. Setting the curly bracket in [3] equal to (–γ),
and noting that the sum over R simply
gives N,

ik⋅(δ Ri )
ε k = En − γ ∑ e
,
[4]
i

where the vectors δRi point to the nearest neighbors, with respect to any given lattice site. For a
simple cubic crystal, there are six neighbors at distance ±a along each of the three axes. Plugging
in these corresponding δRi ’s, one obtains,
{
}
(
)
ε k = En − 2γ coskx a + cosky a + coskz a .
[5]
A plot of this bandstructure [5] is shown in fig. 1, along two directions in k-space, and energy
contours in the 100 plane are shown in the following figure.
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Figure 1: Tight binding energy solution, equation [5], plotted
along [100] and [111] directions (energy in units of γ).
Figure 2: constant-energy contours in [100] plane for simple tight-binding band.
The circular-looking contours near the central minimum correspond to nearly-spherical pockets
in 3D. The corner of the Brillouin zone [(π/a, π/a, π/a)] is an energy maximum, surrounded by
nearly-spherical hole pockets. The point [(π/a, π/a,0)] (corner of the [100] slice of fig. 2) is a
saddle point; 3D energy contours near that point are convoluted, connected surfaces in the
extended-zone scheme. The following figures show these contours at several energies.
Figure 3a: Electron pocket near center of zone.
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Figure 3b: Enclosed electron pocket at energy (–2.5 γ).
Figure 3c: Extended view of (–2 γ) energy surface touching zone boundary in (100) direction.
Figure 3d: Extended surface at energy – γ.
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Figure 3e: Network at energy +2 γ is electron/hole complement of Fig. 3c.
Figure 3f: Spherical hole pockets near top of band.
The density of states is given in the figure on the
right (from a web posting by Cyrille Barreteau,
at Grenoble; β corresponds to our parameter γ).
The band generated by this model is completely
symmetric about En, which set to zero in this
figure. Note the characteristic free-electron-like
behavior at the bottom and top of the band, with
g(ε ) ∝ ε − ε o
.
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