3.4 Comparison of the regular H n and
3.4.2 Dimerisation in the H n chain: notion of distortion
We are now ready to compare the band structures of both the regular (Fig. 3.5) and the dimerised (Fig. 3.16) hydrogen chains. In order to make
Fig. 3.32
Comparison of the band structures for the regular (a) and dimerised (b) hydrogen chains assuming that
βi+βe 2 =β.
the comparison easier, it is convenient to generate the two systems using a repeat unit containing two hydrogen atoms and the repeat vector a (i.e.
representation 2 in Fig. 3.26). In addition we will assume that the average of the two interaction terms in the dimerised chain (βi andβe) coincides with the interaction term in the regular chain (β). The band structures of the two chains are given in Fig. 3.32.
As shown in Fig. 3.32b, the degeneracy seen at the Fermi level (at the X point) of the regular chain no longer occurs for the dimerised system. The dimerisation in the Hnchain leads to the stabilisation of some filled levels and the destabilisation of some empty levels of the regular chain in such a way that the dimerised chain is energetically preferred. This result is the equivalent for a periodic system of the stabilisation provided by a Jahn–Teller distortion in a molecule (see Section 2.4.3). We will show that the regular Hnchain is not sta- ble as a result of the coupling between the electronic motion and the vibrations of the chain. In order to understand this fact, let us consider the nature of the degenerate crystal orbitals at the Fermi level (Fig. 3.33) of the regular chain and the nature of the vibration that could break down the degeneracy.
Fig. 3.33
Crystal orbitals at the Fermi level for the regular (H2)n.
The shape of the crystal orbitals in Fig. 3.33 suggests that a vibration keeping the symmetry plane(σh)0/1and shortening one every two bonds along the chain will split the orbitals at the Fermi level by lowering the orbital for which the bonding interactions are reinforced and by raising the orbital for which the antibonding interactions are increased. This vibration may be schematically represented as the(+Q)and(−Q)displacements in Fig. 3.34.
A+Q-type displacement stabilises orbital S(X)and destabilises orbital A(X) while a−Q-type displacement leads to the opposite effect. From an electronic perspective the system is unstable with respect to this type of vibrational motion. In contrast, the nuclear repulsions tend to favour the regular geometry
Fig. 3.34
+Qand−Qdisplacements of the hydrogen atoms corresponding to the vibration splitting the degenerate orbitals at the Fermi level.
electronic energy dominates over the nuclear repulsions, something which is most likely the case. In principle, how- ever, there can be 1D metallic systems at 0 K if the nuclear repulsion term is strong enough to make any distortion too energetically costly.
From this scheme it can be predicted that atT = 0 K the system must be distorted and reside in the first vibrational state associated with one of the two potential wells. Thus, the chain is trapped in one of the two dimerised structures and the stability of this structure will increase with the magnitude ofV0. The chain will thus be structurally dimerised and electrically insulating.
However, when the temperature increases some excited states will be popu- lated. With these observations in mind we can summarise the situation in two different although equivalent ways:
• First, we can look at the double-well picture and assume that there will be a critical temperature, Tc, for which the thermal motion provides enough energy to the system to populate the states lying very near and slightly higher than the potential energy barrier,V0. In that case we will get a system which is delocalised over the double well. This means that the average structure observed for the system is that of the regular chain. The critical temperature at which the regular structure is observed thus increases with the potential energy barrier,V0.
• Another way to understand the situation consists in looking at the band structures. At T = 0 K only the lower band of the dimerised structure is filled (Fig. 3.36a). As soon as the temperature increases, the bottom levels of the upper band, lying at a higher energy than the bottom levels of the regular system upper band (Fig. 3.36b), start to be filled. In consequence, there is a critical temperature above which it is less energetically costly to adopt the regular structure and populate the bottom part of the upper band10 (Fig. 3.36c) than to populate the bottom part of the upper band of 10
Of course we talk about the bottom part of the upper band because we are adopt- ing representation 2 for the regular chain.
In the more ‘natural’ representation 1 of the regular chain, these states are those associated with absoluteka values of 14 and slightly larger.
the dimerised system. This view is summarised in Fig. 3.36.
Fig. 3.35
Schematic representation of the total energy of the hydrogen chain as a function of the deformation coordinate Q. The minima are associated with two equivalent equilibrium geometries at low temperature.
Fig. 3.36
Schematic representation of how an energy gap opens at the Fermi level as a consequence of a distortion. The partially filled states are illustrated in grey.
Therefore, there is a temperature below which the system is distorted, i.e.
dimerised in the case of the hydrogen chain. It is said that the system undergoes afirst-order Peierls distortionat the critical temperatureTc. Such a distortion results from the coupling of the electronic structure, which exhibits a partially filled 1D band, and a vibrational motion of the lattice. Such a distortion bears many points in common with the Jahn–Teller distortion in molecules (Section 2.4.3). Peierls-type distortions lead to a change in the size of the repeat unit of periodic systems as well as in their electrical behaviour. In other words, they exhibit a metal-to-insulator phase transition at the critical temperatureTc. Throughout this book we will discuss different examples of periodic systems exhibiting first-order Peierls distortions and we will show that any 1D metallic system tends to exhibit the phenomenon, which leads to a drastic change in their electrical behaviour.
Exercises
(3.1) ∗Let us assume that the number of cells in a periodic 1D system is odd.
(a) Propose an equivalence between the linear and the cyclic periodic systems for this case.
(b) Prove that if we consider a cycle containing an odd number of cells we can obtain an expression for the Bloch orbitals almost identical to that of eqn (3.2).
What is the difference? How important is such a difference knowing thatnis very large?
N.B. The solution of the exercise requires the character table of groupCnfornodd(n=n/2)given below.11 (3.2) Consider the chain An, where every atom A possesses
2pvalence orbitals. Represent schematically the Bloch orbitals BO2px at theand X points generated by the 2px orbitals. Also represent the Bloch orbitals BO2py and BO2pz at these points. Which are the more bonding and the more antibonding orbitals? Make a schematic energy diagram for these six orbitals.
11Character table of groupCnfornodd
Cn(n odd) E Cn . . . Cnm . . . Cnn−0.5
1 exp(−2iπn ) . . . exp(−2miπn ) . . . . exp(2(0.5−nn)iπ)
Correlate this result with the shape of the orbitals pre- sented in Figs 3.11 and 3.13.
(3.5) Uniform Hnchain: what is the behaviour of the two func- tions{S(±14),A(±14)}with respect to the translations associated with vectorsaand 2a? What are the symmetry operations leaving invariant each of these functions?
(3.6) Bearing in mind the treatment developed in Section 2.3 of Chapter 2, build the molecular orbitals of the cyclic H4 system. Compare these results with those obtained by replacingnby 4 in eqn (3.15).
(3.7) Consider a partially oxidised Hn chain that can be described as(H0.5+)n. In this case, every hydrogen atom contributes half an electron. What is thekavalue corre- sponding to the Fermi level of this hypothetical system?
What is the corresponding value for the(H0.3+)nsitua- tion?
(3.8) Consider a two-band system having the band structure given below. Where is the Fermi level when every unit cell contributes 1, 2, 3, or 4 electrons?
(b) Consider thatn is a multiple of 4. Determine the precise occupation of the different energy levels of the Hnsystem and thekpoint corresponding to the Fermi level.
(c) Consider now the case wherenis an even number but not a multiple of 4. Determine the precise occupation of the different energy levels of the Hnsystem and thekpoint corresponding to the Fermi level.
(d) Considering that the number of cells is very large, explain why is it possible to say that the Fermi level of Hnis associated withka= 14.
References
1. N. W. Ashcroft and N. D. Mermin,Solid State Physics, Holt, Rinehart and Winston, Philadelphia, 1976.
2. S. I. Altmann,Band Theory of Metals, Pergamon Press, Oxford, 1970.
3. S. I. Altmann,Band Theory of Solids: An Introduction from the Point of View of Symmetry, Oxford University Press, Oxford, 1991.
4. A. P. Sutton,Electronic Structure of Materials, Oxford University Press, Oxford, 1993.
First-order Peierls
distortions in periodic 1D systems
4
In Chapter 3 we saw that the Hn system is unstable with respect to a dimeri- sation removing the degeneracy at the Fermi level and thus opening an energy gap. In this chapter1 we will generalise this result showing that any 1D sys-
1The discussion in this chapter is not essential for understanding the material in the next three chapters except for the last part of Chapter 7. Thus the reader who is mostly interested in applying the concepts developed in the previous chapter to real materials can proceed to Chapters 5, 6 and 7 and come back to this chapter later on.
tem possessing a partially filled band is susceptible to a periodical structural distortion stabilising the network. [1] This distortion arises from the coupling between the conducting electrons of the undistorted metallic system and one of its lattice vibrations, and is calledelectron–phonon coupling. We will discuss in detail the link between the filling of the band and the nature of the distortion.
We will only consider distortions leading to the lifting of the degeneracy at the Fermi level as a result of the interaction between degenerate or quasi- degenerate levels. Such distortions are known asfirst-order Peierls distortions.
Other types of distortion will be discussed at the end of Chapter 7.