la evidencia a partir del observatorio bancaja-Ivie

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Figure 10.3: Left: If each atom has valence 1, then the band is half-filled. The states that are shaded are filled with both up and down spin electrons. The Fermi surface is the boundary between the filled and unfilled states. Right: When a small electric field is applied, at only a small cost of energy, the Fermi sea can shift slightly thus allowing current to run.

On the other hand, if each atom in our model were di-valent (donates two electrons to the band) then the band would be entirely full of electrons. In fact, it does not matter if we think about this as being a full band where every k-state |ki is filled with two electrons (one up and one down), or a filled band where every site |ni is filled — these two statements describe the same multi-electron wavefunction. In fact, there is a single unique wavefunction that describes this completely filled band.

In the case of the filled band, were one to apply a small electric field to this system, the system cannot respond at all. There is simply no freedom to repopulate the occupation of k-states because every state is already filled. We conclude an important principle,

Principle: A filled band carries no current.

Thus our example of a di-valent tightbinding model is an insulator. (This type of insulator is known as a band insulator). Indeed, many systems of di-valent atoms are insulators (although in a moment we will discuss how di-valent atoms can also form metals).

10.4

Multiple Bands

In the above model, we considered only the case where there is a single atom in the unit cell and a single orbital per atom. However, more generally we might consider a case where we have multiple orbitals per unit cell.

One possibility is to consider one atom per unit cell, but several orbitals per atom5. Anal- ogous to what we found with the above tight binding model, when the atoms are very far apart, one has only the atomic orbitals on each atom. However, as the atoms are moved closer together, the orbitals merge together and the energies spread to form bands6_{. Analogous to Fig. 10.2 we}

5_{Each atom actually has an infinite number of orbitals to be considered. But only a small number of them are}

filled, and within our level of approximation, we can only consider very few of them.

6_{This picture of atomic orbitals in the weak hopping limit merging together to form bands does not depend on}

the fact that the crystal of atoms is ordered. Glasses and amorphous solids can have this sort of band structure as well!

94 CHAPTER 10. TIGHT BINDING CHAIN (INTERLUDE AND PREVIEW) have shown how this occurs for the two band case in Fig. 10.4.

Inter-atomic distance

2

Metal-Insulator Transition

Figure 10.4: Caricature of Bands for a Two-Band Model as a Function of Interatomic Spacing . In the atomic limit, the orbitals have energies 1

atomic and 2atomic. If the system has valence one

(per unit cell), then in the atomic limit, the lower orbital is filled and the upper orbital is empty. When the atoms are pushed together, the lower band will remain filled, and the upper will remain empty, until the bands start to overlap, whereupon we may have two bands both partially filled, which becomes a metal.

A very similar situation occurs when we have two atoms per unit cell but only one orbital per atom. We will do a problem like this for homework7_{. However, the general result will be quite}

analogous to what we found for vibrations of a diatomic chain in chapter 9.

In Fig. 10.5 we show the spectrum of a tight-binding model with two different atoms per unit cell – each having a single orbital. We have shown results here in both the reduced and extended zone schemes.

As for the case of vibrations, we see that there are now two possible energy eigenstates at each value of k. In the language of electrons, we say that there are two bands (we do not use the words “acoustic” and “optical” for electrons, but the idea is similar). Note that there is a gap between the two bands where there are simply no energy eigenstates.

Let us think for a second about what should result in this situation. If each atom (of either type) were divalent, then the two electrons donated would completely fill the single orbital on each site. In this case, both bands would be completely filled with both spin-up and spin-down electrons.

On the other hand, if each atom (of either type) is monovalent, then this means exactly half of the states of the system should be filled. However, here, when one fills half of the states of the system, then all of the states of the lower band are completely filled (with both spins) but all of

7_{The homework problem is sufficiently simplified that the bands do not overlap as they do in figure 10.4. One}

can obtain overlapping bands by including second-neighbor hopping as well as neighbor hopping. (If you are brave you might try it!).

10.5. SUMMARY OF TIGHT BINDING CHAIN 95