1.6
Band-filling and materials properties
Due to the fermionic character of electrons each of the band states |n, k, si can be occupied
with one electron taking also the spin quantum number into account with spin s =" and #
(Pauli exclusion principle). The count of electrons has profound implications on the properties
of materials. Here we would like to look at the most simple classification of materials based on
independent electrons.
1.6.1
Electron count and band filling
We consider here a most simple band structure in the one-dimensional tight-binding model with
nearest-neighbor coupling. The lattice has N sites (N even) and we assume periodic boundary
conditions. The Hamiltonian is given by
H=
t
N X
X
j=1 s=",#
{ĉ†j+1,s ĉj,s + ĉ†j,s ĉj+1,s }
(1.90)
where we impose the equivalence j + N = j (periodic boundary conditions). This Hamiltonian
can be diagonalized by the Fourier transform
1 X
ĉj,s = p
âk,s eiRj k
N k
leading to
H=
X
✏k â†k,s âk,s
with ✏k =
(1.91)
2t cos ka .
(1.92)
k,s
Now we request
eiRj k = ei(Rj +L)k
)
Lk = N ak = 2⇡n
)
k=
2⇡
2⇡ n
n=
L
a N
(1.93)
with the pseudo-momentum k within the first Brillouin zone ( ⇡a < k < ⇡a ) and n being an
integer. On the real-space lattice an electron can take 2N di↵erent states. Thus, for k we find
that n should take the values, n + N/2 = 1, 2, . . . , N 1, N . Note that k = ⇡/a and k = +⇡/a
di↵er by a reciprocal lattice vector G = 2⇡/a and are therefore identical. This provides the
same number of states (2N ), since per k we have two spins (see Fig.1.7).
We can fill these states with electrons following the Pauli exclusion principle. In Fig.1.7 we show
the two typical situations: (1) N electrons corresponding to half of the possible electrons which
can be accommodated leading to a half-filled band and (2) 2N electrons exhausting all possible
states representing a completely filled band. In the case of half-filling we define the Fermi energy
as the energy ✏F of the highest occupied state, here ✏F = 0. This corresponds to the chemical
potential, the energy necessary to add an electron to the system at T = 0K.
An important di↵erence between (1) and (2) is that the former allows for many di↵erent states
which may be separated from each other by a very small energy. For example, considering
the ground state (as in Fig.1.7) and the excited states obtained by moving one electron from
k = ⇡/2a (n = N/4) to k 0 = 2⇡(1 + N/4)/N a (n = 1 + N/4), we find the energy di↵erence
E = ✏k 0
✏k ⇡ 2t
2⇡
1
/
N
N
(1.94)
which shrinks to zero for N ! 1. On the other hand, for case (2) there is only one electron
configuration possible and no excitations within the one-band picture.
24
half filled band
-3
-2
-1
0
1
2
3
4
n
completely filled band
Figure 1.7: One-dimensional tight-binding model with N = 8 and periodic boundary conditions.
The dispersion has eight di↵erent k-levels whereby it has to be noticed that +⇡/a and ⇡/a
are equivalent. The condition of half-filling and complete filling are shown, where for half-filling
a ground state configuration is shown (note there are 4 degenerate states). For the completely
filled band all k levels are occupied by two electrons of opposite spin. This means in real space
that also all sites are occupied by two electrons. This is a non-degenerate state.
1.6.2
Metals, semiconductors and insulators
The two situations depicted in Fig.1.7 are typical as each atom (site) in a lattice contributes an
integer number of electrons to the system. So we distinguish the case that there is an odd or
an even number of electrons per unit cell. Note that the unit cell may contain more than one
atom, unlike the situation shown in our tight-binding example.
• The bands can be either completely filled or empty when the number of electrons per
atom (unit cell) is even. Thus taking the complete set of energy bands into account, the
chemical potential cannot be identified with a Fermi energy but lies within the energy
gap separating highest filled and the lowest empty band (see Fig.1.8). There is a finite
energy needed to add, to remove or to excite an electron. If the band gap Eg is much
smaller than the bandwidth, we call the material a semiconductor. for Eg of the order of
the bandwidth, it is an (band) insulator. In both cases, for temperatures T ⌧ Eg /kB the
application of a small electric voltage will not produce an electric transport. The highest
filled band is called valence band, whereas the lowest empty band is termed conduction
band. Examples for insulators are C as diamond and for semiconductors Si and Ge. They
have diamond lattice structure with two atoms per unit cell. As these atoms belong to
the group IV in the periodic table, each provides an even number of electrons suitable for
completely filling bands. Note that we will encounter another form of an insulator, the
Mott insulator, whose insulating behavior is not governed by a band structure e↵ect, but
by a correlation e↵ect through strong Coulomb interaction.
• If the number of electrons per unit cell is odd, the uppermost non-empty band is half
filled (see Figure 1.8). Then the system is a metal, in which electrons can move and
25
insulator
semiconductor
E
metal
semimetal
metal
E
E
EF
EF
EF
filled
filled
filled
k
k
k
Figure 1.8: Material classes according to band filling: left panel: insulator or semiconductor
(partially filled bands with the Fermi level in band gap); center panel: metal (Fermi level inside
band); right panel: metal or semimetal (Fermi level inside two overlapping bands).
excitations with arbitrarily small energies are possible. The electrons remain mobile down
to arbitrarily low temperatures. The standard example of a metal are the Alkali metals
in the first column of the periodic table (Li, Na, K, Rb, Cs), as all of them have the
configuration [noble gas] (ns)1 , i.e., one mobile electron per ion.
• In general, band structures are more complex. Di↵erent bands need not to be separated
by energy gaps, but can overlap instead. In particular, this happens, if di↵erent orbitals
are involved in the structure of the bands. In these systems, bands can have any fractional
filling (not just filled or half-filled). The earth alkaline metals are an example for this
(second column of the periodic table, Be, Mg, Ca, Sr, Ba), which are metallic despite
having two (n, s)-electrons per unit cell. Systems, where two bands overlap at the Fermi
energy but the overlap is small, are termed semi-metals. The extreme case, where valence
and conduction band touch in isolated points so that there are no electrons at the Fermi
energy and still the band gap is zero, is realized in graphene.
The electronic structure is also responsible for the cohesive forces necessary for the formation
of a regular crystal. We may also classify materials according to relevant forces. We distinguish
four major types of crystals:
Molecular crystals are formed from atoms or molecules with closed-shell atomic structures
such as the noble gases He, Ne etc. which become solid under pressure. Here the van der
Waals forces generate the binding interactions.
Ionic crystals combine di↵erent atoms, A and B, where A has a small ionization energy while
B has a large electron affinity. Thus, electrons are transferred from A to B giving a
positively charged A+ and a negatively charge B . In a regular (alternating) lattice the
energy gained through Coulomb interaction can overcome the energy expense for the charge
transfer stabilizing the crystal. A famous example is NaCl where one electron leaves Na
([Ne] 3s) and is added to Cl ([Ne] 3s2 3p5 ) as to bring both atoms to closed-shell electronic
configuration.
Covalent-bonded crystals form through chemical binding, like in the case of the H2 , where
neighboring atoms share electrons through the large overlap of the electron orbital wavefunction. Insulators like diamond C or semiconductors like Si or GaAs are important
26
examples of this type as we will discuss later. Note that electrons of covalent bonds are
localized between the atoms.
Metallic bonding is based on delocalized electrons (in contrast to the covalent bonds) stripped
from their atoms. The stability of simple metals like the alkaly metals Li, Na, K etc will
be discussed later. Note that many metals, such as the noble metals Au or Pt, can also
involve aspects of covalent or molecular bonding through overlapping but more localized
electronic orbitals.
1.7
Dynamics of band electrons - semiclassical approach
In quantum mechanics, the Ehrenfest theorem shows that the expectation values of the position
and momentum operators obey equations similar to the equation of motion in Newtonian mechanics.8 An analogous formulation holds for electrons in a periodic potential, where we assume
that the electron may be described as a wave packet of the form
X
0
gk (k0 )eik ·r i✏k0 t ,
(1.98)
k (r, t) =
k0
where gk (k0 ) is centered around k in reciprocal space and has a width of k. k should be
much smaller than the size of the Brillouin zone for this Ansatz to make sense, i.e., k ⌧ 2⇡/a.
Therefore, the wave packet is spread over many unit cells of the lattice since Heisenberg’s
uncertainty principle ( k)( x) > 1 implies x
a/2⇡. In this way, the pseudo-momentum k
of the wave packet remains well defined. Furthermore, the applied electric and magnetic fields
have to be small enough not to induce transitions between di↵erent bands. The latter condition
is not very restrictive in practice.
1.7.1
Semi-classical equations of motion
We introduce the rules of the semi-classical motion of electrons with applied electric and magnetic
fields without proof:
• The band index of an electron is conserved, i.e., there are no transitions between the bands.
8
Ehrenfest theorem for free electrons with the Hamiltonian,
H=
b2
p
+ V (b
r)
2m
(1.95)
states for the expectation values for a particle represented by a wave packet,
and
hb
pi
d
i
b]i =
hb
r i = h[H, r
dt
~
m
d
i
b]i = hrV (b
hb
pi = h[H, p
r )i
dt
~
which has a form similar to Newtons equations with the restriction hrV (b
r )i 6= rV (hb
r i).
27
(1.96)
(1.97)