7.3 General remarks concerning Peierls distortions
8.1.2 Band structure of KCP (staggered chain)
How do the electronic properties of real KCP (Fig. 8.1a) differ from those of the eclipsed model (Fig. 8.1b)?
Band structure of the staggered chain generated by a double cell
To facilitate a comparison of the properties of the two compounds represented in Figs 8.1a and 8.1b we must generate the band structure of an eclipsed model compound with a double cell, (Pt(CN)4)2, and a repeat vectoraequal to 2a.
The band structure of the staggered compound may then easily be obtained from the band structure of the eclipsed model compound with the double cell.
In fact, this band structure can simply be obtained by folding (see Fig. 3.31) the band structure of Fig. 8.5.
The band structure of Fig. 8.7 contains ten energy bands and the bottom part of the mainly pz band. In the model eclipsed K2Pt(CN)4Br0.3ã 3H2O compound every (Pt(CN)14.7−)2unit provides 15.4 electrons. Thus, seven bands must be full while the upper band, which is mostlydz2 in character, is 70%
filled. Consequently, the Fermi level for the model eclipsed compound is characterised by ka
f =0.15. The 5dz2 and 6pz bands and their filling at T = 0 K for the model eclipsed K2Pt(CN)4Br0.3ã3H2O compound are shown in Fig. 8.8.
Band structure of KCP
We should now compare the electronic structure of the model above – the eclipsed K2Pt(CN)4Br0.3ã3H2O compound – with that of the real, staggered KCP compound. Analysis of the eclipsed system shows that the Fermi level cuts a band mostly involving the 5dz2 and 6pz orbitals of the platinum. Since the ligands do not directly interact, the only interactions occurring along the
Fig. 8.7
Band structure of the eclipsed [Pt(CN)4]nchain generated with a double unit cell. Note that the horizontal axis refers toka, i.e. the projection of thekvector on the reciprocal vectora∗, instead ofka
(a∗= a∗/2⇒ka=2ka).
chain are those associated with the platinum orbitals. Consequently, the two bands of a1 symmetry (Fig. 8.8) are independent of the orientation of the ligands. In addition, the other bands associated with the metallic orbitals that do not possess this cylindrical symmetry will be only weakly modified and their width will stay practically unaltered when moving from the eclipsed to the staggered compound. Consequently, the upper, partially filled band for the staggered compound will be in essence a dz2 band. As a result the reasoning used above applies as well and the Fermi level must be associated withka
f =0.15 (see Fig. 8.8).
Using the analysis in Section 4.4.1 we can predict that KCP must be unstable towards a 3.333-merisation of the chain represented in Fig. 8.1a sinceka
f is±3/20(2kaf = ±3/10). Such a distortion clusters an average of 6.666 platinum atoms along the chain since every repeat unit contains two platinum atoms. This deformation is at the origin of the metal-to-insulator
Fig. 8.8
dz2andpzbands of the eclipsed chain generated by a cell containing two Pt(CN)4fragments. The Fermi level shown corresponds to an eclipsed chain with formula K2Pt(CN)4Br0.3ã3H2O.
Role of doping and pressure
The analysis of the conducting properties of KCP provides an excellent exam- ple of why doping may be an efficient way to control the conductivity of certain materials. Essentially, doping a material is a way to modify the Fermi level by either oxidising or reducing the material, something which in the present case may be brought about through the use of an halogen or an alkaline metal, respectively. It is an experimental observation that the Pt–Pt distances change according to the degree of doping: for instance the Pt–Pt distance decreases as the degree of doping with the bromine anion increases. This observation may be very simply explained by noting that the oxidation of the chain empties the top part of thedz2 band, which is made up of Pt–Pt antibonding levels.
In principle, any increase of the external pressure on a crystal of KCP should lead to a decrease of the Pt–Pt distances as well as to an increase in the interactions along the chain: the bonding orbitals should be stabilised and the antibonding orbitals should be destabilised. Thus, the pressure effect should be similar to that of doping with an oxidising agent since it tends to shorten the Pt–
Pt distances. However, pressure does not modify the Fermi level. In particular, an increase of pressure or doping results in a stabilisation of the bonding BOpz()crystal orbital at the bottom of thepzband and a destabilisation of the antibonding BOdz2()crystal orbital at the top of thedz2 band (see Fig. 8.8).
When doping or pressure are low, the BOdz2()orbital is more stable, while at high pressure or doping, the BOpz()should be more stable. Now the question is: can bands 2 and 3 of Fig. 8.8 cross at high pressure or doping? The space group of KCP may be written as a direct productTn⊗D4h⊗ {E,gC8z}, where gC8z refers to the screw axis around the chain direction (gC8z =ta/2ãC8z=C8z
ãta/2). For anykpoint other thanand Xthe crystal orbitals must be adapted to the symmetry of the C4v group and thegC8z operation. Since the crystal orbitals of bands 2 and 3 of Fig. 8.8 possess the same symmetry properties with respect to these symmetry operations, the crossing of these bands is forbidden (see Exercise (8.4)). In contrast, atthe BOdz2()and BOpz()orbitals do not have the same symmetry properties with respect to theD4hgroup (in particular, with respect theσhsymmetry plane). Consequently, an accidental degeneracy between the BOd
z2()and BOpz()orbitals is allowed.
The evolution of the a1-type bands when pressure or doping increases is shown schematically in Fig. 8.9. This figure clearly demonstrates that an increase in pressure or doping leads initially to a decrease in the gap between the two bands (i.e. moving from case (a) to case (b)) and then, to an increase, when BOpz()becomes more stable than BOdz2()(i.e. moving from case (b) to case (c)). A priori, there is a pressure and doping level at which the twoa1- type bands touch at.Except for this particular point, the two bands cannot touch – the non-doped compound cannot be metallic whatever the pressure.
The compound can only become a semiconductor with a small band gap or a
Fig. 8.9
Influence of pressure or doping on the band structure of the staggered [Pt(CN)4]nchain. The dotted lines are the Bloch orbital energies whereas the continuous lines are the energies of the crystal orbitals.fandf are the Fermi levels for the non-doped and doped chains discussed in the text, respectively.
semimetal when the pressure responsible for the accidental degeneracy of the BOdz2(k)and BOpz(k)orbitals is approached.
Figure 8.9 provides an illustration of the doping effect on the chain by means of an oxidant. Doping not only partially empties the upper filled band but also decreases the Pt–Pt distances and thus makes the bands wider.3The fact that
3For very high doping levels, the width of the bands should in principle decrease as shown in Fig. 8.9c. Tight-binding calcu- lations [3] show that in the case of KCP the dispersion of bands dz2 and pz is almost optimised and that the situation is intermediate between those of cases (b) and (c) of Fig. 8.9.
the partially filled band is both partially filled and wide is at the origin of the metallic-type conductivity of KCP.
As will be shown in Chapter 12, when a band is narrow, i.e. when the intercell interactions are weak, the system may be semiconducting or even insulating even if the band is partially filled. Understanding this situation, which at first sight may appear surprising, requires consideration of the role of electronic repulsions, and thus go beyond the monoelectronic approximation used throughout this chapter. For the present case it might be expected that if doping is such that the Pt–Pt distances are long, the system may be semicon- ducting and that pressure application may render the compound metallic.