10.3 Metals

Metals are unique in the sense that there is no true molecular equivalent to the way the atoms are bound together in metals. In a metal, the valence electrons are shared on crystal scales, rather than between pairs of atoms. This and subsequent sections will discuss what this really means in terms of quantum mechanics.

10.3.1 Lithium

The simplest metal is lithium. Before examining solid lithium, first consider once more the free lithium atom. Figure 10.4 gives a more realistic picture of the atom than the simplistic analysis of chapter 5.9 did. The atom is really made up of two tightly bound electrons in “$\big\vert\rm {1s}\big\rangle$” states very close to the nucleus, plus a loosely bound third “valence” electron in an expansive “$\big\vert\rm {2s}\big\rangle$” state. The core, consisting of the nucleus and the two closely bound 1s electrons, resembles an helium atom that has picked up an additional proton in its nucleus. It will be referred to as the “atom core.” As far as the 2s electron is concerned, this entire atom core is not that much different from an hydrogen nucleus: it is compact and has a net charge equivalent to one proton.

Figure 10.4: The lithium atom, scaled more correctly than before.

One obvious question is then why under normal circumstances lithium is a solid metal and hydrogen is a thin gas. The quanti­tative difference is that a single-charge core has a favorite distance at which it would like to hold its electron, the Bohr radius. In the hydrogen atom, the electron is about at the Bohr radius, and hydrogen holds onto it tightly. It is willing to share electrons with one other hydrogen atom, but after that, it is satisfied. It is not looking for any other hydrogen molecules to share electrons with; that would weaken the bond it already has. On the other hand, the 2s electron in the lithium atom is only loosely attached and readily given up or shared among multiple atoms.

Figure 10.5: Body-centered-cubic (BCC) structure of lithium.

Now consider solid lithium. The perfect lithium crystal would look as sketched in figure 10.5. The atom cores arrange themselves in a regular, repeating, pattern called the “ crystal structure.” As indicated in the figure by the thick red lines, you can think of the total crystal volume as consisting of many identical little cubes called “(unit) cells.”. There are atom cores at all eight corners of these cubes and there is an additional core in the center of the cubic cell. In solid mechanics, this arrangement of positions is referred to as the “body-centered cubic” (BCC) lattice. The crystal “basis” for lithium is a single lithium atom, (or atom core, really); if you put a single lithium atom at every point of the BCC lattice, you get the complete lithium crystal.

Around the atom cores, the 2s electrons form a fairly homoge­neous electron density distribution. In fact, the atom cores get close enough together that a typical 2s electron is no closer to the atom core to which it supposedly “belongs” than to the surrounding atom cores. Under such conditions, the model of the 2s electrons being associated with any particular atom core is no longer really meaningful. It is better to think of them as belonging to the solid as a whole, moving freely through it like an electron “gas.”

Under normal conditions, bulk lithium is “poly-crystalline,” meaning that it consists of many microscopi­cally small crystals, or “grains,“ each with the above BCC structure. The “grain boundaries“ where different crystals meet are crucial to understand the mechanical properties of the material, but not so much to understand its electrical or heat properties, and their effects will be ignored. Only perfect crystals will be discussed.

Key Points

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Lithium can meaningfully be thought of as an atom core, with a net charge of one proton, and a 2s valence electron around it.

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In the solid, the cores arrange themselves into a “body-centered cubic” (BCC) lattice.

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The 2s electrons form an “electron gas” around the cores.

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Normally the solid, like other solids, does not have the same crystal lattice throughout, but consists of microscopic grains, each crystalline, (i.e. with its lattice oriented its own way).

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The grain structure is critical for mechanical properties like strength and plasticity. But that is another book.

10.3.2 One-di­mensional crystals

Even the quantum mechanics of a perfect crystal like the lithium one described above is not very simple. So it is a good idea to start with an even simpler crystal. The easiest example would be a “crystal” consisting of only two atoms, but two lithium atoms do not make a lithium crystal, they make a lithium molecule.

Figure 10.6: Fully periodic wave function of a two-atom lithium “crystal.”

Fortunately, there is a dirty trick to get a “crystal” with only two atoms: assume that nature keeps repeating itself as indicated in figure 10.6. Mathemati­cally, this is called “using periodic boundary conditions.” It assumes that after moving towards the left over a distance called the period, you are back at the same point as you started, as if you are walking around in a circle and the period is the circumference.

Of course, this is an outrageous assumption. If nature repeats itself at all, and that is doubtful at the time of this writing, it would be on a cosmo­logical scale, not on the scale of two atoms. But the fact remains that if you make the assumption that nature repeats, the two-atom model gives a much better description of the mathematics of a true crystal than a two-atom molecule would. And if you add more and more atoms, the point where nature repeats itself moves further and further away from the typical atom, making it less and less of an issue for the local quantum mechanics.

Key Points

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Periodic boundary conditions are very artificial.

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Still, for crystal lattices, periodic boundary conditions often work very well.

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And nobody is going to put any real grain boundaries into any basic model of solids anyway.

10.3.3 Wave functions of one-di­mensional crystals

To describe the energy eigen­states of the electrons in one-di­mensional crystals in simple terms, a further assumption must be made: that the detailed inter­actions between the electrons can be ignored, except for the exclusion principle. Trying to correctly describe the complex inter­actions between the large numbers of electrons found in a macroscopic solid is simply impossible. And it is not really such a bad assumption as it may appear. In a metal, electron wave functions overlap greatly, and when they do, electrons see other electrons in all directions, and effects tend to cancel out. The equivalent in classical gravity is where you go down far below the surface of the earth. You would expect that gravity would become much more important now that you are surrounded by big amounts of mass at all sides. But they tend to cancel each other out, and gravity is actually reduced. Little gravity is left at the center of the earth. It is not recommended as a vacation spot anyway due to excessive pressure and temperature.

In any case, it will be assumed that for any single electron, the net effect of the atom cores and smeared-out surrounding 2s electrons produces a periodic potential that near every core resembles that of an isolated core. In particular, if the atoms are spaced far apart, the potential near each core is exactly the one of a free lithium atom core. For an electron in this two atom “crystal,” the intuitive eigen­functions would then be where it is around either the first or the second core in the 2s state, (or rather, taking the periodicity into account, around every first or every second core in each period.) Alternatively, since these two states are equivalent, quantum mechanics allows the electron to hedge its bets and to be about each of the two cores at the same time with some proba­bility.

Figure 10.7: Flip-flop wave function of a two-atom lithium “crystal.”

But as soon as the atoms are close enough to start noticeably affecting each other, only two true energy eigen­functions remain, and they are ones in which the electron is around both cores with equal proba­bility. There is one eigen­function that is exactly the same around both of the atom cores. This eigen­function is sketched in figure 10.6; it is periodic from core to core, rather than merely from pair of cores to pair of cores. The second eigen­function is the same from core to core except for a change of sign, call it a flip-flop eigen­function. It is shown in figure 10.7. Since the grey-scale electron proba­bility distribution only shows the magnitude of the wave function, it looks periodic from atom to atom, but the actual wave function is only the same after moving along two atoms.

To avoid the grey fading away, the shown wave functions have not been normalized; the darkness level is as if the 2s electrons of both the atoms are in that state.

As long as the atoms are far apart, the wave functions around each atom closely resemble the isolated-atom $\big\vert\rm {2s}\big\rangle$ state. But when the atoms get closer together, differen­ces start to show up. Note for example that the flip-flop wave function is exactly zero half way in between two cores, while the fully periodic one is not. To indicate the deviations from the true free-atom $\big\vert\rm {2s}\big\rangle$ wave function, parenthetical super­scripts will be used.

Figure 10.8: Wave functions of a four-atom lithium “crystal.” The actual picture is that of the fully periodic mode.

A one-di­mensional crystal made up from four atoms is shown in figure 10.8. Now there are four energy eigen­states. The energy eigenstate that is the same from atom to atom is still there, as is the flip-flop one. But there is now also an energy eigenstate that changes by a factor ${\rm i}$ from atom to atom, and one that changes by a factor $\vphantom0\raisebox{1.5pt}{$-$}$${\rm i}$. They change more slowly from atom to atom than the flip-flop one: it takes two atom distances for them to change sign. Therefore it takes a distance of four atoms, rather than two, for them to return to the same values.

Key Points

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The electron energy eigen­functions in a metal like lithium extend over the entire crystal.

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If the cores are relatively far apart, near each core the energy eigen­function of an electron still resembles the 2s state of the free lithium atom.

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However, the magnitude near each core is of course much less, since the electron is spread out over the entire crystal.

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Also, from core to core, the wave function changes by a factor of magnitude one.

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The extreme cases are the fully periodic wave function that changes by a factor one (stays the same) from core to core, versus the flip-flop mode that changes sign completely from one core to the next.

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The other eigen­functions change by an amount in between these two extremes from core to core.

10.3.4 Analysis of the wave functions

There is a pattern to the wave functions of one-di­mensional crystals as discussed in the previous subsection. First of all, while the spatial energy eigen­functions of the crystal are different from those of the individual atoms, their number is the same. Four free lithium atoms would each have one $\big\vert\rm {2s}\big\rangle$ spatial state to put their one 2s electron in. Put them in a crystal, and there are still four spatial states to put the four 2s electrons in. But the four spatial states in the crystal are no longer single atom states; each now extends over the entire crystal. The atoms share all the electrons. If there were eight atoms, the eight atoms would share the eight 2s electrons in eight possible crystal-wide states. And so on.

To be very precise, a similar thing is true of the inner 1s electrons. But since the $\big\vert\rm {1s}\big\rangle$ states remain well apart, the effects of sharing the electrons are trivial, and describing the 1s electrons as belonging pair-wise to a single lithium nucleus is fine. In fact, you may recall that the anti­symmetr­ization requirement of electrons requires every electron in the universe to be slightly present in every occupied state around every atom. Obviously, you would not want to consider that in the absence of a non­trivial need.

The reason that the energy eigen­functions take the form shown in figure 10.8 is relatively simple. It follows from the fact that the Hamiltonian commutes with the “transl­ation operator” that shifts the entire wave function over one atom spacing $\vec{d}$. After all, because the potential energy is exactly the same after such a transl­ation, it does not make a difference whether you evaluate the energy before or after you shift the wave function over.

Now commuting operators have a common set of eigen­functions, so the energy eigen­functions can be taken to be also eigen­functions of the transl­ation operator. The eigenvalue must have magnitude one, since periodic wave functions cannot change in overall magnitude when translated. So the eigenvalue describing the effect of an atom-spacing transl­ation on an energy eigen­function can be written as $e^{{\rm i}2\pi\nu}$ with $\nu$ a real number. (The factor $2\pi$ does nothing except rescale the value of $\nu$. Apparently, crystallo­graphers do not even put it in. This book does so that you do not feel short-changed because other books have factors $2\pi$ and yours does not.)

This can be verified for the example energy eigen­functions shown in figure 10.8. For the fully periodic eigen­function $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, making the transl­ation eigenvalue $e^{{\rm i}2\pi\nu}$ equal to one. So this eigen­function is multi­plied by one under a transl­ation by one atom spacing $d$: it is the same after such a transl­ation. For the flip-flop mode, $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12$; this mode changes by $e^{{\rm i}\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$1 under a transl­ation over an atom spacing $d$. That means that it changes sign when translated over an atom spacing $d$. For the two inter­mediate eigen­functions $\nu$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pm\frac14$, so, using the Euler formula (2.5), they change by factors $e^{\pm{\rm i}\pi/2}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pm{\rm i}$ for each transl­ation over a distance $d$.

In general, for an $J$-atom periodic crystal, there will be $J$ values of $\nu$ in the range $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$. In particular for an even number of atoms $J$:

$$\nu = \frac{j}{J} \quad\mbox{for}\quad j = -\frac{J}{2}+1,... ...2,\; -\frac{J}{2}+3,\; \ldots,\; \frac{J}{2}-1,\; \frac{J}{2}$$

Note that for these values of $\nu$, if you move over $J$ atom spacings, $e^{{\rm i}2\pi\nu{J}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 as it should; according to the imposed periodic boundary conditions, the wave functions must be the same after $J$ atoms. Also note that it suffices for $j$ to be restricted to the range $\vphantom0\raisebox{1.5pt}{$-$}$$J$$\raisebox{.5pt}{$/$}$2 $\raisebox{.3pt}{$<$}$ $j$ $\raisebox{-.3pt}{$\leqslant$}$ $J$$\raisebox{.5pt}{$/$}$2, hence $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$: if $j$ is outside that range, you can always add or subtract a whole multiple of $J$ to bring it back in that range. And changing $j$ by a whole multiple of $J$ does absolutely nothing to the eigenvalue $e^{{\rm i}2\pi\nu}$ since $e^{{\rm i}2\pi{J}/J}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{{\rm i}2\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.

10.3.5 Floquet (Bloch) theory

Mathemati­cally it is awkward to describe the energy eigen­functions piecewise, as figure 10.8 does. To arrive at a better way, it is helpful first to replace the axial Cartesian coordinate $z$ by a new “crystal coordinate” $u$ defined by

$$\fbox{$\displaystyle z {\hat k}= u \vec d $}$$ (10.2)

where $\vec{d}$ is the vector shown in figure 10.8 that has the length of one atom spacing $d$. Material scientists call this vector the “primitive transl­ation vector” of the crystal lattice. Primitive vector for short.

The advantage of the crystal coordinate $u$ is that if it changes by one unit, it changes the $z$-position by exactly one atom spacing. As noted in the previous subsection, such a transl­ation should multiply an energy eigen­function by a factor $e^{{\rm i}2\pi\nu}$. A continuous function that does that is the exponential $e^{{\rm i}2\pi\nu{u}}$. And that means that if you factor out that exponential from the energy eigen­function, what is left does not change under the transl­ation; it will be periodic on atom scale. In other words, the energy eigen­functions can be written in the form

$$\pp//// = e^{{\rm i}2\pi\nu u} \pp{\rm p}////$$

where $\pp{\rm {p}}////$ is a function that is periodic on the atom scale $d$; it is the same in each successive inter­val $d$.

This result is part of what is called “Floquet theory:”

If the Hamiltonian is periodic of period $d$, the energy eigen­functions are not in general periodic of period $d$, but they do take the form of exponentials times functions that are periodic of period $d$.

In physics, this result is known as “Bloch’s theorem,” and the Floquet-type wave function solutions are called “Bloch functions” or “Bloch waves,” because Floquet was just a mathema­tician, and the physicists’ hero is Bloch, the physicist who succeeded in doing it too, half a century later. {N.20}.

The periodic part $\pp{\rm {p}}////$ of the energy eigen­functions is not the same as the $\big\vert\rm {2s}\big\rangle ^{(.)}$ states of figure 10.8, because $e^{{\rm i}2\pi\nu{u}}$ varies continuously with the crystal position $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $ud$, unlike the factors shown in figure 10.8. However, since the magnitude of $e^{{\rm i}2\pi\nu{u}}$ is one, the magnitudes of $\pp{\rm {p}}////$ and the $\big\vert\rm {2s}\big\rangle ^{(.)}$ states are the same, and therefore, so are their grey scale electron proba­bility pictures.

It is often more convenient to have the energy eigen­functions in terms of the Cartesian coordinate $z$ instead of the crystal coordinate $u$, writing them in the form

$$\fbox{$\displaystyle \pp{k}//// = e^{{\rm i}k z}\pp{{\rm... ...$\pp{{\rm p},k}////$\ periodic on the atom scale $d$} $} %$$ (10.3)

The constant $k$ in the exponential is called the wave number, and subscripts $k$ have been added to $\pp////$ and $\pp{\rm {p}}////$ just to indicate that they will be different for different values of this wave number. Since the exponential must still equal $e^{{\rm i}2\pi\nu{u}}$, clearly the wave number $k$ is propor­tional to $\nu$. Indeed, substituting $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $ud$ into $e^{{\rm i}{k}z}$, $k$ can be traced back to be

$$\fbox{$\displaystyle k = \nu D \qquad D = \frac{2\pi}{d}... ...\raisebox{-.7pt}{$\leqslant$}}{\textstyle\frac{1}{2}} $} %$$ (10.4)

10.3.6 Fourier analysis

As the previous subsection explained, the energy eigen­functions in a crystal take the form of a Floquet exponential times a periodic function $\pp{{\rm {p}},k}////$. This periodic part is not normally an exponential. However, it is generally possible to write it as an infinite sum of exponentials:

$$\fbox{$\displaystyle \pp{{\rm p},k}//// = \sum_{m=-\inft... ...k_m z} \qquad k_m = m D \mbox{ for $m$\ an integer} $} %$$ (10.5)

where the $c_{km}$ are constants whose values will depend on $x$ and $y$, as well as on $k$ and the integer $m$.

Writing the periodic function $\pp{{\rm {p}},k}////$ as such a sum of exponentials is called “Fourier analysis,” after another French mathema­tician. That it is possible follows from the fact that these exponentials are the atom-scale-periodic eigen­functions of the $z$-momentum operator $p_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar\partial$$\raisebox{.5pt}{$/$}$${\rm i}\partial{z}$, as is easily verified by straight substitution. Since the eigen­functions of an Hermitian operator like $p_z$ are complete, any atom-scale-periodic function, including $\pp{{\rm {p}},k}////$, can be written as a sum of them. See also {D.8}.

10.3.7 The reciprocal lattice

As the previous two subsections discussed, the energy eigen­functions in a one-di­mensional crystal take the form of a Floquet exponential $e^{{\rm i}{k}z}$ times a periodic function $\pp{{\rm {p}},k}////$. That periodic function can be written as a sum of Fourier exponentials $e^{{\rm i}{k_m}z}$. It is a good idea to depict all those $k$-values graphi­cally, to keep them apart. That is done in figure 10.9.

Figure 10.9: Reciprocal lattice of a one-di­mensional crystal.

The Fourier $k$ values, $k_m$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mD$ with $m$ an integer, form a lattice of points spaced a distance $D$ apart. This lattice is called the “reciprocal lattice.” The spacing of the reciprocal lattice, $D$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi$$\raisebox{.5pt}{$/$}$$d$, is propor­tional to the reciprocal of the atom spacing $d$ in the physical lattice. Since on a macroscopic scale the atom spacing $d$ is very small, the spacing of the reciprocal lattice is very large.

The Floquet $k$ value, $k$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\nu{D}$ with $-\frac12$ $\raisebox{.3pt}{$<$}$ $\nu$ $\raisebox{-.3pt}{$\leqslant$}$ $\frac12$, is somewhere in the grey range in figure 10.9. This range is called the first “Brillouin zone.’ It is an inter­val, a unit cell if you want, of length $D$ around the origin. The first Brillouin zone is particularly important in the theory of solids. The fact that the Floquet $k$ value may be assumed to be in it is but one reason.

To be precise, the Floquet $k$ value could in principle be in an inter­val of length $D$ around any wave number $k_m$, not just the origin, but if it is, you can shift it to the first Brillouin zone by splitting off a factor $e^{{\rm i}{k_m}z}$ from the Floquet exponential $e^{{\rm i}{k}z}$. The $e^{{\rm i}{k_m}z}$ can be absorbed in a redefinition of the Fourier series for the periodic part $\pp{{\rm {p}},k}////$ of the wave function, and what is left of the Floquet $k$ value is in the first zone. Often it is good to do so, but not always. For example, in the analysis of the free-electron gas done later, it is critical not to shift the $k$ value to the first zone because you want to keep the (there trivial) Fourier series intact.

The first Brillouin zone are the points that are closest to the origin on the $k$-axis, and similarly the second zone are the points that are second closest to the origin. The points in the inter­val of length $D$$\raisebox{.5pt}{$/$}$2 in between $k_{-1}$ and the first Brillouin zone make up half of the second Brillouin zone: they are closest to $k_{-1}$, but second closest to the origin. Similarly, the other half of the second Brillouin zone is given by the points in between $k_1$ and the first Brillouin zone. In one dimension, the boundaries of the Brillouin zone fragments are called the “Bragg points.” They are either reciprocal lattice points or points half way in between those.

10.3.8 The energy levels

Valence band. Conduction band. Band gap. Crystal. Lattice. Basis. Unit cell. Primitive vector. Bloch wave. Fourier analysis. Reciprocal lattice. Brillouin zones. These are the jargon of solid mechanics; now they have all been defined. (Though certainly not fully discussed.) But jargon is not physics. The physi­cally inter­esting question is what are the energy levels of the energy eigen­functions.

Figure 10.10: Schematic of energy bands.

For the two-atom crystal of figures 10.6 and 10.7, the answer is much like that for the hydrogen molecular ion of chapter 4.6 and hydrogen molecule of chapter 5.2. In particular, when the atom cores are far apart, the $\big\vert\rm {2s}\big\rangle ^{(.)}$ states are the same as the free lithium atom wave function $\big\vert\rm {2s}\big\rangle$. In either the fully periodic or the flip-flop mode, the electron is with 50% proba­bility in that state around each of the two cores. That means that at large spacing $d$ between the cores, the energy is the 2s free lithium atom energy, whether it is the fully periodic or flip-flop mode. That is shown in the left graph of figure 10.10.

When the distance $d$ between the atoms decreases so that the 2s wave functions start to noticeably overlap, things change. As the same left graph in figure 10.10 shows, the energy of the flip-flop state increases, but that of the fully periodic state initially decreases. The reasons for the latter are similar to those that gave the symmetric hydrogen molecular ion and hydrogen molecule states lower energy. In particular, the electrons pick up more effective space to move in, decreasing their uncertainty-principle demanded kinetic energy. Also, when the electron clouds start to merge, the repulsion between electrons is reduced, allowing the electrons to lose potential energy by getting closer to the nuclei of the neighboring atoms. (Note however that the simple model used here would not faithfully reproduce that since the repulsion between the electrons is not correctly modeled.)

Next consider the case of a four-atom crystal, as shown in the second graph of figure 10.10. The fully periodic and flip flop states are unchanged, and so are their energies. But there are now two additional states. Unlike the fully periodic state, these new states vary from atom, but less rapidly than the flip flop mode. As you would then guess, their energy is somewhere in between that of the fully periodic and flip-flop states. Since the two new states have equal energy, it is shown as a double line in 10.10. The third graph in that figure shows the energy levels of an 8 atom crystal, and the final graph that of a 24 atom crystal. When the number of atoms increases, the energy levels become denser and denser. By the time you reach a one hundredth of an inch, one-million atom one-di­mensional crystal, you can safely assume that the energy levels within the band have a continuous, rather than discrete distribution.

Now recall that the Pauli exclusion principle allows up to two electrons in a single spatial energy state. Since there are an equal number of spatial states and electrons, that means that the electrons can pair up in the lowest half of the states. The upper states will then be unoccupied. Further, the actual separation distance between the atoms will be the one for which the total energy of the crystal is smallest. The energy spectrum at this actual separation distance is found inside the vanishingly narrow vertical frame in the rightmost graph of figure 10.10. It shows that lithium forms a metal with a partially-filled band.

The partially filled band means that lithium conducts electricity well. As was already discussed earlier in chapter 6.20, an applied voltage does not affect the band structure at a given location. For an applied voltage to do that, it would have to drop an amount comparable to volts per atom. The current that would flow in a metal under such a voltage would vaporize the metal instantly. Current occurs because electrons get excited to states of slightly higher energy that produce motion in a preferential direction.

10.3.9 Merging and splitting bands

The explan­ation of electrical conduction in metals given in the previous subsection is incom­plete. It incorrectly seems to show that beryllium, (and similarly other metals of valence two,) is an insulator. Two valence electrons per atom will completely fill up all 2s states. With all states filled, there would be no possi­bility to excite electrons to states of slightly higher energy with a preferential direction of motion. There would be no such states. All states would be red in figure 10.10, so nothing could change.

What is missing is consid­eration of the 2p atom states. When the atoms are far enough apart not to affect each other, the 2p energy levels are a bit higher than the 2s ones and not involved. However, as figure 10.11 shows, when the atom spacing decreases to the actual one in a crystal, the widening bands merge together. With this influx of 300% more states, valence-two metals have plenty of free states to excite electrons to. Beryllium is actually a better conductor than lithium.

Figure 10.11: Schematic of merging bands.

Hydrogen is a more complicated story. Solid hydrogen consists of molecules and the attractions between different molecules are weak. The proper model of hydrogen is not a series of equally spaced atoms, but a series of pairs of atoms joined into molecules, and with wide gaps between the molecules. When the two atoms in a single molecule are brought together, the energy varies with distance between the atoms much like the left graph in figure 10.10. The wave function that is the same for the two atoms in the current simple model corre­sponds to the normal covalent bond in which the electrons are symmetri­cally shared; the flip-flop function that changes sign describes the “anti-bonding” state in which the two electrons are anti-symmetri­cally shared. In the ground state, both electrons go into the state corre­sponding to the covalent bond, and the anti-bonding state stays empty. For multiple molecules, each of the two states turns into a band, but since the inter­actions between the molecules are weak, these two bands do not fan out much. So the energy spectrum of solid hydrogen remains much like the left graph in figure 10.10, with the bottom curve becoming a filled band and the top curve an empty one. An equivalent way to think of this is that the 1s energy level of hydrogen does not fan out into a single band like the 2s level of lithium, but into two half bands, since there are two spacings involved; the spacing between the atoms in a molecule and the spacing between molecules. In any case, because of the band gap energy required to reach the empty upper half 1s band, hydrogen is an insulator.

10.3.10 Three-di­mensional metals

The ideas of the previous subsections gener­alize towards three-di­mensional crystals in a relatively straight­forward way.

Figure 10.12: A primitive cell and primitive transl­ation vectors of lithium.

As the lithium crystal of figure 10.12 illustrates, in a three-di­mensional crystal there are three “primitive transl­ation vectors.” The three-di­mensional Cartesian position ${\skew0\vec r}$ can be written as

$$\fbox{$\displaystyle {\skew0\vec r}= u_1 \vec d_1 + u_2 \vec d_2 + u_3 \vec d_3 $} %$$ (10.6)

where if any of the “crystal coordinates” $u_1$, $u_2$, or $u_3$ changes by exactly one unit, it produces a physi­cally completely equivalent position.

Note that the vectors $\vec{d}_1$ and $\vec{d}_2$ are two bottom sides of the “cubic unit cell” defined earlier in figure 10.5. However, $\vec{d}_3$ is not the vertical side of the cube. The reason is that primitive transl­ation vectors must be chosen to allow you to reach any point of the crystal from any equivalent point in whole steps. Now $\vec{d}_1$ and $\vec{d}_2$ allow you to step from any point in a horizontal plane to any equivalent point in the same plane. But if $\vec{d}_3$ was verti­cally upwards like the side of the cubic unit cell, stepping with $\vec{d}_3$ would miss every second horizontal plane. With $\vec{d}_1$ and $\vec{d}_2$ defined as in figure 10.12, $\vec{d}_3$ must point to an equivalent point in an immediately adjacent horizontal plane, not a horizontal plane farther away.

Despite this requirement, there are still many ways of choosing the primitive transl­ation vectors other than the one shown in figure 10.12. The usual way is to choose all three to extend towards adjacent cube centers. However, then it gets more difficult to see that no lattice point is missed when stepping around with them.

The parallel­epiped shown in figure 10.12, with sides given by the primitive transl­ation vectors, is called the “primitive cell.” It is the smallest building block that can be stacked together to form the total crystal. The cubic unit cell from figure 10.5 is not a primitive cell since it has twice the volume. The cubic unit cell is instead called the “conven­tional cell.”

Since the primitive vectors are not unique, the primitive cell they define is not either. These primitive cells are purely mathematical quantities; an arbitrary choice for the smallest single volume element from which the total crystal volume can be build up. The question suggests itself whether it would not be possible to define a primitive cell that has some physical meaning; whose definition is unique, rather than arbitrary. The answer is yes, and the unambiguously defined primitive cell is called the “Wigner-Seitz cell.” The Wigner-Seitz cell around a lattice point is the vicinity of locations that are closer to that lattice point than to any other lattice point.

Figure 10.13: Wigner-Seitz cell of the BCC lattice.

Figure 10.13 shows the Wigner-Seitz cell of the BCC lattice. To the left, it is shown as a wire frame, and to the right as an opaque volume element. To put it within context, the atom around which this Wigner-Seitz cell is centered was also put in the center of a conven­tional cubic unit cell. Note how the Wigner-Seitz primitive cell is much more spherical than the parallel­epiped-shaped primitive cell shown in figure 10.12. The outside surface of the Wigner-Seitz cell consists of hexagonal planes on which the points are just on the verge of getting closer to a corner atom of the conven­tional unit cell than to the center atom, and of squares on which the points are just on the verge of getting closer to the center atom of an adjacent conven­tional unit cell. The squares are located within the faces of the conven­tional unit cell.

The reason that the entire crystal volume can be build up from Wigner-Seitz cells is simple: every point must be closest to some lattice point, so it must be in some Wigner-Seitz cell. When a point is equally close to two nearest lattice points, it is on the boundary where adjacent Wigner-Seitz cells meet.

Turning to the energy eigen­functions, they can now be taken to be eigen­functions of three transl­ation operators; they will change by some factor $e^{{\rm i}2\pi\nu_1}$ when translated over $\vec{d}_1$, by $e^{{\rm i}2\pi\nu_2}$ when translated over $\vec{d}_2$, and by $e^{{\rm i}2\pi\nu_3}$ when translated over $\vec{d}_3$. All that just means that they must take the Floquet (Bloch) function form

$$\pp//// = e^{{\rm i}2\pi(\nu_1u_1+\nu_2u_2+\nu_3u_3)} \pp{\rm p}////,$$

where $\pp{\rm {p}}////$ is periodic on atom scales, exactly the same after one unit change in any of the crystal coordinates $u_1$, $u_2$ or $u_3$.

It is again often convenient to write the Floquet exponential in terms of normal Cartesian coordinates. To do so, note that the relation giving the physical position ${\skew0\vec r}$ in terms of the crystal coordinates $u_1$, $u_2$, and $u_3$,

$${\skew0\vec r}= u_1 \vec d_1 + u_2 \vec d_2 + u_3 \vec d_3$$

can be inverted to give the crystal coordinates in terms of the physical position, as follows:

$$\fbox{$\displaystyle u_1 = \frac1{2\pi} \vec D_1 \cdot {... ...ad u_3 = \frac1{2\pi} \vec D_3 \cdot {\skew0\vec r} $} %$$ (10.7)

(Again, factors $2\pi$ have been thrown in merely to fully satisfy even the most demanding quantum mechanics reader.) To find the vectors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$, simply solve the expression for ${\skew0\vec r}$ in terms of $u_1$, $u_2$, and $u_3$ using linear algebra procedures. In particular, they turn out to be the rows of the inverse of matrix $(\vec{d}_1,\vec{d}_2,\vec{d}_3)$.

If you do not know linear algebra, it can be done geometri­cally: if you dot the expression for ${\skew0\vec r}$ above with $\vec{D}_1$$\raisebox{.5pt}{$/$}$$2\pi$, you must get $u_1$; for that to be true, the first three conditions below are required:

$$\fbox{$\displaystyle \begin{array}{lll} \vec d_1 \cdot... ... 0, & \vec d_3 \cdot \vec D_3 = 2\pi. \end{array} $} %$$ (10.8)

The second set of three equations is obtained by dotting with $\vec{D}_2$$\raisebox{.5pt}{$/$}$$2\pi$ to get $u_2$ and the third by dotting with $\vec{D}_3$$\raisebox{.5pt}{$/$}$$2\pi$ to get $u_3$. From the last two equations in the first row, it follows that vector $\vec{D}_1$ must be orthogonal to both $\vec{d}_2$ and $\vec{d}_3$. That means that you can get $\vec{D}_1$ by first finding the vectorial cross product of vectors $\vec{d}_2$ and $\vec{d}_3$ and then adjusting the length so that $\vec{d}_1\cdot\vec{D}_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi$. In similar ways, $\vec{D}_2$ and $\vec{D}_3$ may be found.

If the expressions for the crystal coordinates are substituted into the exponential part of the Bloch functions, the result is

$$\fbox{$\displaystyle \pp{\vec k}//// = e^{{\rm i}{\vec k... ... k}= \nu_1 \vec D_1 + \nu_2 \vec D_2 + \nu_3 \vec D_3 $} %$$ (10.9)

So, in three dimensions, a wave number $k$ becomes a “wave number vector” ${\vec k}$.

Just like for the one-di­mensional case, the periodic function $\pp{{\rm {p}},{\vec k}}////$ too can be written in terms of exponentials. Converted from crystal to physical coordinates, it gives:

$$\fbox{$\displaystyle \begin{array}[b]{c} \displaystyle... ...2$, and $m_3$\ integers}\strut^{\strut} \end{array} $} %$$ (10.10)

If these wave number vectors ${\vec k}_{\vec{m}}$ are plotted three-di­mensionally, it again forms a lattice called the “reciprocal lattice,” and its primitive vectors are $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$. Remarkably, the reciprocal lattice to lithium’s BCC physical lattice turns out to be the FCC lattice of NaCl fame!

And now note the beautiful symmetry in the relations (10.8) between the primitive vectors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$ of the reciprocal lattice and the primitive vectors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$ of the physical lattice. Because these relations involve both sets of primitive vectors in exactly the same way, if a physical lattice with primitive vectors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$ has a reciprocal lattice with primitive vectors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$, then a physical lattice with primitive vectors $\vec{D}_1$, $\vec{D}_2$, and $\vec{D}_3$ has a reciprocal lattice with primitive vectors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$. Which means that since NaCl’s FCC lattice is the reciprocal to lithium’s BCC lattice, lithium’s BCC lattice is the reciprocal to NaCl’s FCC lattice. You now see where the word “reciprocal” in reciprocal lattice comes from. Lithium and NaCl borrow each other’s lattice to serve as their lattice of wave number vectors.

Finally, how about the definition of the “Brillouin zones” in three dimensions? In particular, how about the first Brillouin zone to which you often prefer to move the Floquet wave number vector ${\vec k}$? Well, it is the magnitude of the wave number vector that is important, so the first Brillouin zone is defined to be the Wigner-Seitz cell around the origin in the reciprocal lattice. Note that this means that in the first Brillouin zone, $\nu_1$, $\nu_2$, and $\nu_3$ are not simply numbers in the range from $-\frac12$ to $\frac12$ as in one dimension; that would give a parallel­epiped-shaped primitive cell instead.

Solid state physicists may tell you that the other Brillouin zones are also reciprocal lattice Wigner-Seitz cells, [27, p. 38], but if you look closer at what they are actually doing, the higher zones consist of fragments of reciprocal lattice Wigner-Seitz cells that can be assembled together to produce a Wigner-Seitz cell shape. Like for the one-di­mensional crystal, the second zone are again the points that are second closest to the origin, etcetera.

The boundaries of the Brillouin zone fragments are now planes called “Bragg planes.” Each is a perpendicular bisector of a lattice point and the origin. That is so because the locations where points stop being first/, second/, third/, ...closest to the origin and become first/, second/, third/, ...closest to some other reciprocal lattice point must be on the bisector between that lattice point and the origin. Sections 10.5.1 and 10.6 will give Bragg planes and Brillouin zones for a simple cubic lattice.

The quali­tative story for the valence electron energy levels is the same in three dimensions as in one. Sections 10.5 and 10.6 will look a bit closer at them quanti­tatively.