10.5 Free-Elec­tron Gas

Chap­ter 6 dis­cussed the model of non­in­ter­act­ing elec­trons in a pe­ri­odic box. This sim­ple model, due to Som­mer­feld, is a first start­ing point for much analy­sis of solids. It was used to pro­vide ex­pla­na­tions of such ef­fects as the in­com­press­ibil­ity of solids and liq­uids, and of elec­tri­cal con­duc­tion. This sec­tion will use the model to ex­plain some of the an­a­lyt­i­cal meth­ods that are used to an­a­lyze elec­trons in crys­tals. A free-elec­tron gas is a model for elec­trons in a crys­tal when the phys­i­cal ef­fect of the crys­tal struc­ture on the elec­trons is ig­nored. The as­sump­tion is that the crys­tal struc­ture is still there, but that it does not ac­tu­ally do any­thing to the elec­trons.

The sin­gle-par­ti­cle en­ergy eigen­func­tions of a pe­ri­odic box are given by

$$\pp{\vec k}/{\skew0\vec r}/// = \frac{1}{\sqrt{{\cal V}}} ... ... = \frac{1}{\sqrt{{\cal V}}} e^{{\rm i}(k_xx+ k_y y + k_z z)}$$ (10.11)

Here the wave num­bers are re­lated to the box di­men­sions as

$$k_x = n_x \frac{2\pi}{\ell_x} \qquad k_y = n_y \frac{2\pi}{\ell_y} \qquad k_z = n_z \frac{2\pi}{\ell_z}$$ (10.12)

where the quan­tum num­bers $n_x$, $n_y$, and $n_z$ are in­te­gers. This sec­tion will use the wave num­ber vec­tor, rather than the quan­tum num­bers, to in­di­cate the in­di­vid­ual eigen­func­tions.

Note that each of these eigen­func­tions can be re­garded as a Bloch wave: the ex­po­nen­tials are the Flo­quet ones, and the pe­ri­odic parts are triv­ial con­stants. The lat­ter re­flects the fact the pe­ri­odic po­ten­tial it­self is triv­ially con­stant (zero) for a free-elec­tron gas.

Of course, there is a spin-up ver­sion $\pp{\vec k}////{\downarrow}$ and a spin-down ver­sion $\pp{\vec k}////{\uparrow}$ of each eigen­func­tion above. How­ever, spin will not be much of an is­sue in the analy­sis here.

The Flo­quet ex­po­nen­tials have not been shifted to any first Bril­louin zone. In fact, since the elec­trons ex­pe­ri­ence no forces, as far as they are con­cerned, there is no crys­tal struc­ture, hence no Bril­louin zones.

10.5.1 Lat­tice for the free elec­trons

As far as the math­e­mat­ics of free elec­trons is con­cerned, the box in which they are con­fined may as well be empty. How­ever,it is use­ful to put the re­sults in con­text of a sur­round­ing crys­tal lat­tice any­way. That will al­low some of the ba­sic con­cepts of the solid me­chan­ics of crys­tals to be de­fined within a sim­ple set­ting.

It will there­fore be as­sumed that there is a crys­tal lat­tice, but that its po­ten­tial is zero. So the lat­tice does not af­fect the mo­tion of the elec­trons. An ap­pro­pri­ate choice for this lat­tice must now be made. The plan is to keep the same Flo­quet wave num­ber vec­tors as for the free elec­trons in a rec­tan­gu­lar pe­ri­odic box. Those wave num­bers form a rec­tan­gu­lar grid in wave num­ber space as shown in fig­ure 6.17 of chap­ter 6.18. To pre­serve these wave num­bers, it is best to fig­ure out a suit­able rec­i­p­ro­cal lat­tice first.

To do so, com­pare the gen­eral ex­pres­sion for the Fourier ${\vec k}_{\vec{m}}$ val­ues that make up the rec­i­p­ro­cal lat­tice:

$${\vec k}_{\vec m} = m_1 \vec D_1 + m_2 \vec D_2 + m_3 \vec D_3$$

in which $m_1$, $m_2$, and $m_3$ are in­te­gers, with the Flo­quet ${\vec k}$ val­ues,

$${\vec k}= \nu_1 \vec D_1 + \nu_2 \vec D_2 + \nu_3 \vec D_3$$

(com­pare sec­tion 10.3.10.) Now $\nu_1$ is of the form $\nu_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $j_1$$\raisebox{.5pt}{$/$}$​$J_1$ where $j_1$ is an in­te­ger just like $m_1$ is an in­te­ger, and $J_1$ is the num­ber of lat­tice cells in the di­rec­tion of the first prim­i­tive vec­tor. For a macro­scopic crys­tal, $J_1$ will be a very large num­ber, so the con­clu­sion must be that the Flo­quet wave num­bers are spaced much more closely to­gether than the Fourier ones. And so they are in the other two di­rec­tions.

Fig­ure 10.16: As­sumed sim­ple cu­bic rec­i­p­ro­cal lat­tice, shown as black dots, in cross-sec­tion. The bound­aries of the sur­round­ing prim­i­tive cells are shown as thin red lines.

In par­tic­u­lar, if it is as­sumed that there are an equal num­ber of cells in each prim­i­tive di­rec­tion, $J_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J$, then the Fourier wave num­bers are spaced far­ther apart than the Flo­quet ones by a fac­tor $J$ in each di­rec­tion. Such a rec­i­p­ro­cal lat­tice is shown as fat black dots in fig­ure 10.16.

Note that in this sec­tion, the wave num­ber space will be shown only in the $k_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 cross-sec­tion. A full three-di­men­sion­al space, like the one of fig­ure 6.17, would get very messy when crys­tal struc­ture ef­fects are added.

A lat­tice like the one shown in fig­ure 10.16 is called a “sim­ple cu­bic lat­tice,” and it is the eas­i­est lat­tice that you can de­fine. The prim­i­tive vec­tors are or­tho­nor­mal, just a mul­ti­ple of the Carte­sian unit vec­tors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$. Each lat­tice point can be taken to be the cen­ter of a prim­i­tive cell that is a cube, and this cu­bic prim­i­tive cell just hap­pens to be the Wigner-Seitz cell too.

It is of course not that strange that the sim­ple cu­bic lat­tice would work here, be­cause the as­sumed wave num­ber vec­tors were de­rived for elec­trons in a rec­tan­gu­lar pe­ri­odic box.

How about the phys­i­cal lat­tice? That is easy too. The sim­ple cu­bic lat­tice is its own rec­i­p­ro­cal. So the phys­i­cal crys­tal too con­sists of cu­bic cells stacked to­gether. (Atomic scale ones, of course, for a phys­i­cal lat­tice.) In par­tic­u­lar, the wave num­bers as shown in fig­ure 10.16 cor­re­spond to a crys­tal that is macro­scop­i­cally a cube with equal sides $2\ell$, and that on atomic scale con­sists of $J\times{J}\times{J}$ iden­ti­cal cu­bic cells of size $d$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\ell$$\raisebox{.5pt}{$/$}$​$J$. Here $J$, the num­ber of atom-scale cells in each di­rec­tion, will be a very large num­ber, so $d$ will be very small.

In ${\vec k}$-​space, $J$ is the num­ber of Flo­quet points in each di­rec­tion within a unit cell. Fig­ure 10.16 would cor­re­spond to a phys­i­cal crys­tal that has only 40 atoms in each di­rec­tion. A real crys­tal would have many thou­sands, and the Flo­quet points would be much more densely spaced than could be shown in a fig­ure like fig­ure 10.16.

It should be pointed out that the sim­ple cu­bic lat­tice, while def­i­nitely sim­ple, is not that im­por­tant phys­i­cally un­less you hap­pen to be par­tic­u­larly in­ter­ested in polo­nium or com­pounds like ce­sium chlo­ride or beta brass. But the math­e­mat­ics is re­ally no dif­fer­ent for other crys­tal struc­tures, just messier, so the sim­ple cu­bic lat­tice makes a good ex­am­ple. Fur­ther­more, many other lat­tices fea­ture cu­bic unit cells, even if these cells are a bit larger than the prim­i­tive cell. That means that the as­sump­tion of a po­ten­tial that has cu­bic pe­ri­od­ic­ity on an atomic scale is quite widely ap­plic­a­ble.

10.5.2 Oc­cu­pied states and Bril­louin zones

The pre­vi­ous sub­sec­tion chose the rec­i­p­ro­cal lat­tice in wave num­ber space to be the sim­ple cu­bic one. The next ques­tion is how the oc­cu­pied states show up in it. As usual, it will be as­sumed that the crys­tal is in the ground state, cor­re­spond­ing to zero ab­solute tem­per­a­ture.

As shown in fig­ure 6.17, in the ground state the en­ergy lev­els oc­cu­pied by elec­trons form a sphere in wave num­ber space. The sur­face of the sphere is the Fermi sur­face. The cor­re­spond­ing sin­gle-elec­tron en­ergy is the Fermi en­ergy.

Fig­ure 10.17: Oc­cu­pied states for one, two, and three free elec­trons per phys­i­cal lat­tice cell.

Fig­ure 10.17 shows the oc­cu­pied states in $k_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 cross sec­tion if there are one, two, and three va­lence elec­trons per phys­i­cal lat­tice cell. (In other words, if there are $J^3$, 2$J^3$, and 3$J^3$ va­lence elec­trons.) For one va­lence elec­tron per lat­tice cell, the spher­i­cal re­gion of oc­cu­pied states stays within the first Bril­louin zone, i.e. the Wigner-Seitz cell around the ori­gin, though just barely. There are $J^3$ spa­tial states in a Wigner-Seitz cell, the same num­ber as the num­ber of phys­i­cal lat­tice cells, and each can hold two elec­trons, (one spin up and one spin down,) so half the states in the first Bril­louin zone are filled. For two elec­trons per lat­tice cell, there are just as many oc­cu­pied spa­tial states as there are states within the first Bril­louin zone. But since in the ground state, the oc­cu­pied free elec­tron states form a spher­i­cal re­gion, rather than a cu­bic one, the oc­cu­pied states spill over into im­me­di­ately ad­ja­cent Wigner-Seitz cells. For three va­lence elec­trons per lat­tice cell, the oc­cu­pied states spill over into still more neigh­bor­ing Wigner-Seitz cells. (It is hard to see, but the di­am­e­ter of the oc­cu­pied sphere is slightly larger than the di­ag­o­nal of the Wigner-Seitz cell cross-sec­tion.)

How­ever, these re­sults may show up pre­sented in a dif­fer­ent way in lit­er­a­ture. The rea­son is that a Bloch-wave rep­re­sen­ta­tion is not unique. In terms of Bloch waves, the free-elec­tron ex­po­nen­tial so­lu­tions as used here can be rep­re­sented in the form

$$\pp{\vec k}//// = e^{{\rm i}{\vec k}\cdot{\skew0\vec r}} \pp{{\rm p},{\vec k}}////$$

where the atom-scale pe­ri­odic part $\pp{{\rm {p}},{\vec k}}////$ of the so­lu­tion is a triv­ial con­stant. In ad­di­tion, the Flo­quet wave num­ber ${\vec k}$ can be in any Wigner-Seitz cell, how­ever far away from the ori­gin. Such a de­scrip­tion is called an “ex­tended zone scheme”.

This free-elec­tron way of think­ing about the so­lu­tions is of­ten not the best way to un­der­stand the physics. Seen within a sin­gle phys­i­cal lat­tice cell, a so­lu­tion with a Flo­quet wave num­ber in a Wigner-Seitz cell far from the ori­gin looks like an ex­tremely rapidly vary­ing ex­po­nen­tial. How­ever, all of that atom-scale physics is in the crys­tal-scale Flo­quet ex­po­nen­tial; the lat­tice-cell scale part $\pp{{\rm {p}},{\vec k}}////$ is a triv­ial con­stant. It may be bet­ter to shift the Flo­quet wave num­ber to the Wigner-Seitz cell around the ori­gin, the first Bril­louin zone. That will turn the crys­tal-scale Flo­quet ex­po­nen­tial into one that varies rel­a­tively slowly over the phys­i­cal lat­tice cell; the rapid vari­a­tion will now be ab­sorbed into the lat­tice-cell part $\pp{{\rm {p}},{\vec k}}////$. This idea is called the “re­duced zone scheme.” As long as the Flo­quet wave num­ber vec­tor is shifted to the first Bril­louin zone by whole amounts of the prim­i­tive vec­tors of the rec­i­p­ro­cal lat­tice, $\pp{{\rm {p}},{\vec k}}////$ will re­main an atom-scale-pe­ri­odic func­tion; it will just be­come non­triv­ial. This shift­ing of the Flo­quet wave num­bers to the first Bril­louin zone is il­lus­trated in fig­ures 10.18a and 10.18b. The fig­ures are for the case of three va­lence elec­trons per lat­tice cell, but with a slightly in­creased ra­dius of the sphere to avoid vi­sual am­bi­gu­ity.

Fig­ure 10.18: Re­de­f­i­n­i­tion of the oc­cu­pied wave num­ber vec­tors into Bril­louin zones.

Now each Flo­quet wave num­ber vec­tor in the first Bril­louin zone does no longer cor­re­spond to just one spa­tial en­ergy eigen­func­tion like in the ex­tended zone scheme. There will now be mul­ti­ple spa­tial eigen­func­tions, dis­tin­guished by dif­fer­ent lat­tice-scale vari­a­tions $\pp{{\rm {p}},{\vec k}}////$. Com­pare that with the ear­lier ap­prox­i­ma­tion of one-di­men­sion­al crys­tals as widely sep­a­rated atoms. That was in terms of dif­fer­ent atomic wave func­tions like the 2s and 2p ones, not a sin­gle one, that were mod­u­lated by Flo­quet ex­po­nen­tials that var­ied rel­a­tively slowly over an atomic cell. In other words, the re­duced zone scheme is the nat­ural one for widely spaced atoms: the lat­tice scale parts $\pp{{\rm {p}},{\vec k}}////$ cor­re­spond to the dif­fer­ent atomic en­ergy eigen­func­tions. And since they take care of the non­triv­ial vari­a­tions within each lat­tice cell, the Flo­quet ex­po­nen­tials be­come slowly vary­ing ones.

But you might rightly feel that the crit­i­cal Fermi sur­face is messed up pretty badly in the re­duced zone scheme fig­ure 10.18b. That does not seem to be such a hot idea, since the elec­trons near the Fermi sur­face are crit­i­cal for the prop­er­ties of met­als. How­ever, the pic­ture can now be taken apart again to pro­duce sep­a­rate Bril­louin zones. There is a con­struc­tion cred­ited to Har­ri­son that is il­lus­trated in fig­ure 10.18c. For points that are cov­ered by at least one frag­ment of the orig­i­nal sphere, (which means all points, here,) the first cov­er­ing is moved into the first Bril­louin zone. For points that are cov­ered by at least two frag­ments of the orig­i­nal sphere, the sec­ond cov­er­ing is moved into the sec­ond Bril­louin zone. And so on.

Fig­ure 10.19: Sec­ond, third, and fourth Bril­louin zones seen in the pe­ri­odic zone scheme.

Re­mem­ber that in say elec­tri­cal con­duc­tion, the elec­trons change oc­cu­pied states near the Fermi sur­faces. To sim­plify talk­ing about that, physi­cist like to ex­tend the pic­tures of the Bril­louin zones pe­ri­od­i­cally, as il­lus­trated in fig­ure 10.19. This is called the “pe­ri­odic zone scheme.” In this scheme, the bound­aries of the Wigner-Seitz cells, which are nor­mally not Fermi sur­faces, are no longer a dis­tract­ing fac­tor. It may be noted that a bit of a lat­tice po­ten­tial will round off the sharp cor­ners in fig­ure 10.19, in­creas­ing the es­thet­ics.