Fig. 3.5
Schematic representation of the Hn chain.
As a first application we will now consider the electronic structure of the sim- plest 1D system,the model chainHn(see Fig. 3.5). Every cell of this system may be described by using a single 1sorbital centred on the hydrogen atom.
Consequently, the crystal orbitals of this system, CO(k), may be identified with the Bloch orbitals BO1s(k) generated by the different 1s atomic orbitals (eqn (3.7)). Thencrystal orbitals of Hnare given by the following formula:
CO(k) =BO1s(k) = 1
√n
n
m=−n+1
exp(i2πmka)1sm (3.15)
withka=0,±1n,±n2, . . . ,±nn−1,12 and where the 1sm orbital is centred on the hydrogen atom of cell Mm. The different crystal orbitals only differ by their phase factors exp(i2πmka). However, for a givenm this factor is only real for the (ka=0)and X(ka= 12)points. We will start by considering the orbitals for these two points because the physical content of real orbitals is easier to grasp than that of complex ones.
3.2.1 Representation of the CO() and CO(X) functions
Fig. 3.6
Orbitals of the Hnchain at theand X points.
The phase factor at thepoint is 1 for every value ofm. The CO()orbital may thus be obtained by copying the 1s0orbital of the reference cell on every cell. At the X point the factor exp(i2πmka)is(−1)m. Thus the orbital coef- ficients in two adjacent cells have opposite signs. The CO(X)orbital exhibits alternating positive and negative 1sorbitals as schematically shown in Fig. 3.6.
The CO() orbital corresponds to the completely bonding orbital whereas the CO(X)orbital corresponds to the completely antibonding orbital. Since these are the two extreme situations, the other orbitals CO(k)
ka=0,12 are necessarily associated with intermediate energies between E() andE(X).
3.2.2 Energy of the crystal orbitals in the H¨ uckel approach
In this case, the energy E(k) associated with the crystal orbital CO(k) coin- cides with BO1s(k)) and is given by:
E(k) =
BO1s(k)|ˆh|BO1s(k)
BO1s(k)|BO 1s(k) = h1s,1s(k)
S1s,1s(k) (3.16) We will use the H¨uckel approach (see Section 2.1.5 of Chapter 2) to make an estimation of the energyE(k).
Direct calculation of the energies
Let us start estimating the numerator of eqn (3.16) by developing the Bloch orbital BO1s(k)at the left of the scalar product using eqn (3.15):
h1s,1s(k)=
n
m=−n+1
exp(i2πmk√ a)
n 1sm|ˆh|BO1s(k)
(3.17) We will denote the term between square brackets A(m). This represents the interaction between the function exp(i2√πmka)
n 1sm located in cell Mm and the Bloch orbital BO1s(k). We will first show that this term is independent ofm.
Consider,
A(m)=
n
m=−n+1
exp(i2π(−m+m)ka) n
1sm|ˆh|1sm
Using eqn (2.10) and the H¨uckel approach, this term becomes:
A(m)=
n
m=−n+1
exp(i2π(−m+m)ka)
n αδm,m
+β(δm,m+1+δm,m−1)
= 1
n(α+β (exp(i2πka)+exp(−i2πka))) A(m)= 1
n(α+2βcos(2πka)) (3.18)
Consequently, the interaction term A(m) is independent of m, i.e. on the cell Mm. This result simply originates from the fact that every cell plays an equivalent role in a periodic system. Let us emphasise that this result is independent of the method used to estimate the energies. As a result, sincem takesn different values in the summation of eqn (3.17), the interaction term h1s,1s(k) is given byntimes the A(m)value:
approach.
Interaction terms obtained through the H ¨uckel approach
We will now use the results of the previous sectionand the H¨uckel approachto reach some general results related to the calculation of the overlap, Sj j(k)and the interaction term hj j(k). The allowed energies (eqn (3.16)) are given by:
E(k) =α+2βcos(2πka) (3.20) It is easy to verify that the basis of Bloch orbitals in the H¨uckel approach is orthonormalised whatever the nature of theφj andφj
Sj j(k) =
BOj(k)|BOj(k)
=δj j (3.21)
provided that theφj andφj functions are orthonormal.
In addition, it is always possible to decompose the interaction term hj j(k) (see eqn (3.13)) through development of the Bloch orbital BOj(k)according to expression (3.15):
hj j(k) =
n
m=−n+1
exp(i2πmka)
√n (φj)m|ˆh|BOj(k)
(3.22) As a consequence of the equivalence of the different orbitals (φj)m (m= 0,±1,. . .,±(n−1), n’) (j =1,. . .,N0) imposed by the Born–von Karman boundary conditions, the term in the sum of equation (3.22) is independent of m for exactly the same reasons, making A(m) independent of m (see eqn (3.18)). The interaction term hj j(k) can be estimated by multiplying byn the interaction term
(φ
j)0
√n |ˆh|BOj(k)
involving orbital(φj)0of the reference cell:
hj j(k) =n (φj)0
√n |ˆh|BOj(k)
(3.23) The last term is very easy to evaluate in the H¨uckel approach because the(φj)0
orbital only interacts with itself and with the orbitals of adjacent atoms. This interaction term may be estimated with the help of a graphical scheme rep- resenting the interaction between the function √1
n(φj)0and the Bloch orbital BOj(k).
As an example, we will consider the interaction term h1s,1s(k)of the Hn
system through the use of eqn (3.23). In order to do this we will represent the non-zero interactions between the √1
n1s0and the Bloch orbital BO1s(k)in the H¨uckel approach (see Fig. 3.7). Three interaction terms are non-zero. They cor- respond to the ‘self-interaction’ (αterm) of the 1s0orbital and to the interaction
Fig. 3.7
Schematic representation of the orbital interaction between√1
n1s0and the Bloch orbital BO1s(k)in the H¨uckel approach.
(βterm) of the 1s0orbital with the adjacent orbitals,√1
nexp(−i2πka)1s−1and
√1
nexp(+i2πka)1s1: 1s0
√n|ˆh|BO1s(k)
= 1
n {α+β(exp(i2πka)+exp(−i2πka))} (3.24) Using eqn (3.23) we can now obtain theh1s,1s(k)interaction term:
h1s,1s(k) =n 1s0
√n|ˆh|BO1s(k)
=α+2βcos(2πka) (3.25) The approach based on the use of eqn (3.23) thus allows a fast and simple derivation of the interaction terms in the H¨uckel approach.4Later on, we will
4In the extended H¨uckel approach, besides the interactions between nearest neighbours, those implicating the second, third. . . nearest neighbours are also taken into account, up to a given point beyond which they are neglected.
use this method to evaluate the different interaction terms present in the secular determinants in order to avoid tedious calculations.
3.2.3 Band structure
Fig. 3.8
Band structure for the Hnchain. Note that the slope is nil at both theand X points.
Let us now represent the different allowed energiesE(k)(see eqn (3.19)) as a function ofka(Fig. 3.8).Let us recall that ka varies discretely and that only the energies for which ka is n (∈]−n,n])are allowed. However, since the number of cells n is very large, the allowed energy values are very close to each other. In practice we have an energy band with a width of 4β. In addition, it must be noted that all energy levels are doubly degenerate except for E() and E(X). The degeneracy of the BO(k) and BO(−k)functions is a general property,5which originates from the fact that thekand−kpoints are physi-
5In the physics literature, this property is known as ‘time reversal’.
cally equivalent. This property provides some justification for the fact that we have defined a Brillouin zone as centred on thepoint. Equation (3.8) clearly shows that the length of the Brillouin zone making possible the labelling of the different allowed functions must be equal to 2aπ, which is the norm of the reciprocal vectora∗. However, there is no compelling reason for choosing the interval ]−πa,πa] instead of ]0,2aπ] for instance. We chose the interval ]−πa,πa] because it makes transparent (and useful) the energetic equivalence of the states associated with the k and −k points, which are symmetrically placed with respect to .6 Because of the degeneracy associated with thek
6The use of a Brillouin zone centred at will be very useful for the study of 2D and 3D systems, where the exploitation of the
symmetry may be very helpful. and−kpoints, (E(k) =E(−k)), from now on we will only represent half of
Fig. 3.9
(a) Band structure of the Hnchain.
(b) Schematic representation of the allowed energy levels.
the band structure. For a 1D system we will only represent the energy levels corresponding tokavalues in the interval[0,12](Fig. 3.9). Even if we use such a representation it should not be forgotten that all energy levels E(k), except those atand X, are doubly degenerate.
3.2.4 Basis for an energy level E (± k )
Basis of real functions{σ (±k), δ(±k)}
Since the BO(k) and BO(−k)functions are associated with the same energy and are complex conjugate, it is possible to build a basis of realfunctions, {σ(±k), δ(±k)}, associated with the energy level E(±k)
σ(±k)= BO(k) +√BO(−k)
2 and δ(±k)= BO(k) −BO(−k) i√
2 (3.26)
The sign ± before k in eqn (3.26) is a reminder that we have combined two Bloch orbitals associated withkand−k in such a way that the orbitals {σ (±k), δ(±k)}are not individually adapted to the translation symmetry. For the Hnchain these functions are given by eqn (3.27):
σ (±k)= 2
n
n
m=−n+1
cos(2πmka)1sm
and δ(±k)= 2
n
n
m=−n+1
sin(2πmka)1sm (3.27) and are schematically represented in Fig. 3.10.
Fig. 3.10
Schematic representation of the basis σ (±k)andδ(±k).
These two functions have the peculiarity of being symmetric and antisym- metric, respectively, with respect to the symmetry plane(σh)0, which is per- pendicular to the chain axis and cuts the chain at the reference cell atom. Thus, we have built a basis of real functions adapted to the(σh)0symmetry plane and associated with the energy E(±k). Let us emphasise that these functions are not individually well adapted to the translational symmetry. Thus, for instance, a basis for the level E(ka= ±14) is given by eqn (3.28) and is graphically represented in Fig. 3.11.
Fig. 3.11
Schematic representation of the real functionsσ (±1/4)andδ(±1/4) associated with energyE(±1/4). A dot is used to represent a nil coefficient for a given position.
σ(±14)= 2
n
n
m=−n+1
cos(πm 2 )1sm
andδ(±14)= 2
n
n
m=−n+1
sin(πm
2 )1sm (3.28)
in the H¨uckel approach, these crystal orbitals are non-bonding and are associ- ated with an energyE(±1/4)equal toα, which is the energy of an isolated 1s orbital.
Basis of real functions{S(±k),A(±k)}
We will now explore a basis for the energy level E(±k), {S(±k),A(±k)}, made of real functions that are symmetric and antisymmetric, respectively, with respect to the symmetry plane(σh)0/1. This plane is perpendicular to the chain axis and lies equidistant from the atoms in the cells labelled 0 and 1 (see Fig. 3.12). These two functions are given by eqn (3.29) and are schematically represented in Fig. 3.12.
S(±k)= 1
√2
exp(−iπka)BO(k) +exp(iπka)BO(−k)
A(±k)= 1 i√ 2
exp(−iπka)BO(k) −exp(iπka)BO(−k)
(3.29) As an example, we show in Fig. 3.13 a basis of crystal orbitals, namely {S(±14),A(±14)}, well adapted to the symmetry plane (σh)0/1 obtained by combination of the Bloch orbitals BO1s(ka= 14)and BO1s(ka = −14).
As will be repeatedly shown throughout this book, the choice of a crystal orbital basis well adapted to the problem at hand can enormously simplify the
Fig. 3.12
Schematic representation of the basis {S(±k),A(±k)}made of functions well adapted to the symmetry plane(σh)0/1.
Fig. 3.13
Schematic representation of the real functions{S(±14),A(±14)}associated with the energyE(±14).
3.2.5 Fermi level of the Hnchain
We will now consider how thenelectrons of the Hnchain must be distributed among thencrystal orbitals of the system. The energy band of a 1D system always contains two non-degenerate levels(ka=0 andka= 12)and(n−1) doubly degenerate levels. Thus, the band is full when it is filled with 2n electrons. This means that every unit cell contributes two electrons to this band. For the Hn chain, every unit cell can contribute only one electron, so that only half the possible energy levels can be filled atT =0 K. Taking into account that the differentka allowed values are uniformly distributed in the interval ]−12,12], it is clear from the band structure in Fig. 3.8 that all crystal orbitals withkavalues between−14and14are occupied. This situation may be represented as in Fig. 3.14.
We can now draw the band structure of the Hnchain including the position of the Fermi level (Fig. 3.15). Two different ways of representing the nature of the levels at the Fermi level have been already shown in the previous section (see Figs 3.11 and 3.13).
On the basis of the band structure of Fig. 3.15 we can predict that the Hn
chain should be a metal (i.e. a conductor) since there is no gap at the Fermi level.
*As will be seen in a later chapter, the explicit consideration of the electron–
electron repulsions may lead to a dif- ferent electronic description. Throughout this book we will consider the monoelec- tronic approximation valid and thus ass- sume the progressive filling of the energy levels so that as many levels as possible are doubly filled. In Chapter 12 we will consider when this approach is expected to break down.
In the framework of the monoelectronic approximation,∗when every cell providesmelectrons and the different bands do not overlap,m/2 bands are filled in the ground state atT = 0 K. If some bands overlap, the equivalent ofm/2 bands are filled (see Exercises (3.7)–(3.9)).
Fig. 3.14
Schematic representation of the occupation of the different levels of the Hnchain.(ka)f and−(ka)f
characterise the crystal orbitals at the Fermi level.
Fig. 3.15
Band structure of the Hnchain. Note that only the states characterised by positive (and zero) values ofkaare shown in the left-hand diagram whereas all states are represented on the right.