Sub­sec­tions


6.21 In­tro to Band Struc­ture

Quan­tum me­chan­ics is es­sen­tial to de­scribe the prop­er­ties of solid ma­te­ri­als, just as it is for lone atoms and mol­e­cules. One well-known ex­am­ple is su­per­con­duc­tiv­ity, in which cur­rent flows with­out any re­sis­tance. The com­plete ab­sence of any re­sis­tance can­not be ex­plained by clas­si­cal physics, just like su­per­flu­id­ity can­not for flu­ids.

But even nor­mal elec­tri­cal con­duc­tion sim­ply can­not be ex­plained with­out quan­tum the­ory. Con­sider the fact that at or­di­nary tem­per­a­tures, typ­i­cal met­als have elec­tri­cal re­sis­tiv­i­ties of a few times 10$\POW9,{-8}$ ohm-m (and up to a hun­dred thou­sand times less still at very low tem­per­a­tures), while Wikipedia lists a re­sis­tance for teflon of up to 10$\POW9,{24}$ ohm-m. (Teflon’s one-minute re­sis­tiv­ity can be up to 10$\POW9,{19}$ ohm-m.) That is a dif­fer­ence in re­sis­tance be­tween the best con­duc­tors and the best in­su­la­tors by over thirty or­ders of mag­ni­tude!

There is sim­ply no way that clas­si­cal physics could even be­gin to ex­plain it. As far as clas­si­cal physics is con­cerned, all of these ma­te­ri­als are quite sim­i­lar com­bi­na­tions of pos­i­tive nu­clei and neg­a­tive elec­trons.

Con­sider an or­di­nary sewing nee­dle. You would have as lit­tle trou­ble sup­port­ing its tiny 60 mg weight as a metal has con­duct­ing elec­tric­ity. But mul­ti­ply it by 10$\POW9,{30}$. Well, don’t worry about sup­port­ing its weight. Worry about the en­tire earth com­ing up over your ears and en­gulf­ing you, be­cause the nee­dle now has ten times the mass of the earth. That is how widely dif­fer­ent the elec­tri­cal con­duc­tiv­i­ties of solids are.

Only quan­tum me­chan­ics can ex­plain why it is pos­si­ble, by mak­ing the elec­tron en­ergy lev­els dis­crete, and more im­por­tantly, by group­ing them to­gether in bands.


Key Points
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Even ex­clud­ing su­per­con­duc­tiv­ity, the elec­tri­cal con­duc­tiv­i­ties of solids vary enor­mously.


6.21.1 Met­als and in­su­la­tors

To un­der­stand elec­tri­cal con­duc­tion in solids re­quires con­sid­er­a­tion of their elec­tron en­ergy lev­els.

Typ­i­cal en­ergy spec­tra are sketched in fig­ure 6.19. The spec­trum of a free-elec­tron gas, non­in­ter­act­ing elec­trons in a box, is shown to the left. The en­ergy ${\vphantom' E}^{\rm p}$ of the sin­gle-par­ti­cle states is shown along the ver­ti­cal axis. The en­ergy lev­els al­lowed by quan­tum me­chan­ics start from zero and reach to in­fin­ity. The en­ergy lev­els are spaced many or­ders of mag­ni­tude more tightly to­gether than the hatch­ing in the fig­ure can in­di­cate. For al­most all prac­ti­cal pur­poses, the en­ergy lev­els form a con­tin­uum. In the ground state, the elec­trons fill the low­est of these en­ergy lev­els, one elec­tron per state. In the fig­ure, the oc­cu­pied states are shown in red. For a macro­scopic sys­tem, the num­ber of elec­trons is prac­ti­cally speak­ing in­fi­nite, and so is the num­ber of oc­cu­pied states.

Fig­ure 6.19: Sketch of elec­tron en­ergy spec­tra in solids at ab­solute zero tem­per­a­ture. (No at­tempt has been made to pic­ture a den­sity of states). Far left: the free-elec­tron gas has a con­tin­u­ous band of ex­tremely densely spaced en­ergy lev­els. Far right: lone atoms have only a few dis­crete elec­tron en­ergy lev­els. Mid­dle: ac­tual met­als and in­su­la­tors have en­ergy lev­els grouped into densely spaced bands sep­a­rated by gaps. In­su­la­tors com­pletely fill up the high­est oc­cu­pied band.
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How­ever, the free-elec­tron gas as­sumes that there are no forces on the elec­trons. In­side a solid, this would only be true if the elec­tric charges of the nu­clei and fel­low elec­trons would be ho­mo­ge­neously dis­trib­uted through­out the en­tire solid. In that case the forces come equally from all di­rec­tions and can­cel each other out per­fectly. In a true solid, forces from dif­fer­ent di­rec­tions do tend to can­cel each other out, but this is far from per­fect. For ex­am­ple, an elec­tron very close to one par­tic­u­lar nu­cleus ex­pe­ri­ences a strong at­trac­tion from that nu­cleus, much too strong for the rest of the solid to can­cel.

The di­a­met­ri­cal op­po­site of the free-elec­tron gas pic­ture is the case that the atoms of the solid are spaced so far apart that they are es­sen­tially lone atoms. In that case, of course, the solid would not phys­i­cally be a solid at all, but a thin gas. Lone atoms do not have a con­tin­uum of elec­tron en­ergy lev­els, but dis­crete ones, as sketched to the far right in fig­ure 6.19. One ba­sic ex­am­ple is the hy­dro­gen spec­trum shown in fig­ure 4.8. Every lone atom in the sys­tem has the ex­act same dis­crete en­ergy lev­els. Widely spaced atoms do not con­duct elec­tric­ity, as­sum­ing that not enough en­ergy is pro­vided to ion­ize them. While for the free-elec­tron gas con­duc­tion can be achieved by mov­ing a few elec­trons to slightly higher en­ergy lev­els, for lone atoms there are no slightly higher en­ergy lev­els.

When the lone atoms are brought closer to­gether to form a true solid, how­ever, the dis­crete atomic en­ergy lev­els broaden out into bands. In par­tic­u­lar, the outer elec­trons start to in­ter­act strongly with sur­round­ing atoms. The dif­fer­ent forms that these in­ter­ac­tions can take pro­duce vary­ing en­er­gies, caus­ing ini­tially equal elec­tron en­er­gies to broaden into bands. The re­sult is sketched in the mid­dle of fig­ure 6.19. The higher oc­cu­pied en­ergy lev­els spread out sig­nif­i­cantly. (The in­ner atomic elec­trons, hav­ing the most neg­a­tive net en­er­gies, do not in­ter­act sig­nif­i­cantly with dif­fer­ent atoms, and their en­ergy lev­els do not broaden much. This is not just be­cause these elec­trons are far­ther from the sur­round­ing atoms, but also be­cause the in­ner elec­trons have much greater ki­netic and po­ten­tial en­ergy lev­els to start with.)

For met­als, con­duc­tion now be­comes pos­si­ble. Elec­trons at the high­est oc­cu­pied en­ergy level, the Fermi en­ergy, can be moved to slightly higher en­ergy lev­els to pro­vide net mo­tion in a par­tic­u­lar di­rec­tion. That is just like they can for a free-elec­tron gas as dis­cussed in the pre­vi­ous sec­tion. The net mo­tion pro­duces a cur­rent.

In­su­la­tors are dif­fer­ent. As sketched in fig­ure 6.19, they com­pletely fill up the high­est oc­cu­pied en­ergy band. That filled band is called the “va­lence band.” The next higher and empty band is called the “con­duc­tion band.”

Now it is no longer pos­si­ble to prod elec­trons to slightly higher en­ergy lev­els to cre­ate net mo­tion. There are no slightly higher en­ergy lev­els avail­able; all lev­els in the va­lence band are al­ready filled with elec­trons.

To cre­ate a state with net mo­tion, some elec­trons would have to be moved to the con­duc­tion band. But that would re­quire large amounts of en­ergy. The min­i­mum en­ergy re­quired is the dif­fer­ence be­tween the top of the va­lence band and the bot­tom of the con­duc­tion band. This en­ergy is ap­pro­pri­ately called the “band gap” en­ergy ${\vphantom' E}^{\rm p}_{\rm {gap}}$. It is typ­i­cally of the or­der of elec­tron volts, com­pa­ra­ble to atomic po­ten­tials for outer elec­trons. That is in turn com­pa­ra­ble to ion­iza­tion en­er­gies, a great amount of en­ergy on an atomic scale.

Re­sis­tance is de­ter­mined for volt­ages low enough that Ohm’s law ap­plies. Such volt­ages do not pro­vide any­where near the en­ergy re­quired to move elec­trons to the con­duc­tion band. So the elec­trons in an in­su­la­tor are stuck. They can­not achieve net mo­tion at all. And with­out net mo­tion, there is no cur­rent. That makes the re­sis­tance in­fi­nite. In this way the band gaps are re­spon­si­ble for the enor­mous dif­fer­ence in re­sis­tance be­tween met­als and in­su­la­tors.

Note that a nor­mal ap­plied volt­age will not have a sig­nif­i­cant ef­fect on the band struc­ture. Atomic po­ten­tial en­er­gies are in terms of eV or more. For the ap­plied volt­age to com­pete with that would re­quire a volt­age drop com­pa­ra­ble to volts per atom. On a mi­cro­scopic scale, the ap­plied po­ten­tial does not change the states.


Key Points
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Quan­tum me­chan­ics al­lows only dis­crete en­ergy lev­els for the elec­trons in a solid, and these lev­els group to­gether in bands with gaps in be­tween them.

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If the elec­trons fill the spec­trum right up to a gap be­tween bands, the elec­trons are stuck. It will re­quire a large amount of en­ergy to ac­ti­vate them to con­duct elec­tric­ity or heat. Such a solid is an in­su­la­tor at ab­solute zero tem­per­a­ture.

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The filled band is called the va­lence band, and the empty band above it the con­duc­tion band.


6.21.2 Typ­i­cal met­als and in­su­la­tors

If a ma­te­r­ial com­pletely fills up its va­lence band with elec­trons, it is an in­su­la­tor. But what ma­te­ri­als would do that? This sub­sec­tion gives a few rules of thumb.

One im­por­tant rule is that the el­e­ments to­wards the left in the pe­ri­odic ta­ble fig­ure 5.8 are met­als. A rel­a­tively small group of el­e­ments to­wards the right are non­metals.

Con­sider first the al­kali met­als found in group I to the far left in the ta­ble. The lone atoms have only one va­lence elec­tron per atom. It is in an atomic s state that can hold two elec­trons, chap­ter 5.9.4. Every spa­tial state, in­clud­ing the s state, can hold two elec­trons that dif­fer in spin.

Now if the lone atoms are brought closer to­gether to form a solid, the spa­tial states change. Their en­ergy lev­els broaden out into a band. How­ever, the to­tal num­ber of states does not change. One spa­tial state per atom stays one spa­tial state per atom. Since each spa­tial state can hold two elec­trons, and there is only one, the band formed from the s states is only half filled. There­fore, like the name says, the al­kali met­als are met­als.

In he­lium the spa­tial 1s states are com­pletely filled with the two elec­trons per atom. That makes solid he­lium an in­su­la­tor. It should be noted that he­lium is only a solid at very low tem­per­a­tures and very high pres­sures. The atoms are barely held to­gether by very weak Van der Waals forces.

The al­ka­line met­als found in group II of the pe­ri­odic ta­ble also have two va­lence elec­trons per atom. So you would ex­pect them to be in­su­la­tors too. How­ever, like the name says, the al­ka­line met­als are met­als. What hap­pens is that the filled band orig­i­nat­ing from the atomic s states merges with an empty band orig­i­nat­ing from the atomic p states. That pro­duces a par­tially filled com­bined band.

This does not ap­ply to he­lium be­cause there are no 1p states. The low­est empty en­ergy states for he­lium are the 2s ones. Still, com­pu­ta­tions pre­dict that he­lium will turn metal­lic at ex­tremely high pres­sures. Com­press­ing a solid has the pri­mary ef­fect of in­creas­ing the ki­netic en­ergy of the elec­trons. Roughly speak­ing, the ki­netic en­ergy is in­versely pro­por­tional to the square of the elec­tron spac­ing, com­pare the Fermi en­ergy (6.16). And in­creas­ing the ki­netic en­ergy of the elec­trons brings them closer to a free-elec­tron gas.

A case re­sem­bling that of he­lium is ionic ma­te­ri­als in which the ions have a no­ble-gas elec­tron struc­ture. A ba­sic ex­am­ple is salt, sodium chlo­ride. These ma­te­ri­als are in­su­la­tors, as it takes sig­nif­i­cant en­ergy to take apart the no­ble-gas elec­tron con­fig­u­ra­tions. See how­ever the dis­cus­sion of ionic con­duc­tiv­ity later in this sec­tion.

An­other case that re­quires ex­pla­na­tion is hy­dro­gen. Like the al­kali met­als, hy­dro­gen has only one va­lence elec­tron per atom. That is not enough to fill up the en­ergy band re­sult­ing from the atomic 1s states. So you would ex­pect solid hy­dro­gen to be a metal. But ac­tu­ally, hy­dro­gen is an in­su­la­tor. What hap­pens is that the en­ergy band pro­duced by the 1s states splits into two. And the lower half is com­pletely filled with elec­trons.

The rea­son for the split­ting is that in the solid, the hy­dro­gen atoms com­bine pair­wise into mol­e­cules. In an hy­dro­gen mol­e­cule, there are not two sep­a­rate spa­tial 1s states of equal en­ergy, chap­ter 5.2. In­stead, there is a low­ered-en­ergy two-elec­tron spa­tial state in which the two elec­trons are sym­met­ri­cally shared. There is also a raised-en­ergy two-elec­tron spa­tial state in which the two elec­trons are an­ti­sym­met­ri­cally shared. So there are now two en­ergy lev­els with a gap in be­tween them. The two elec­trons oc­cupy the lower-en­ergy sym­met­ric state with op­po­site spins. In the solid, the hy­dro­gen mol­e­cules are barely held to­gether by weak Van der Waals forces. The in­ter­ac­tions be­tween the mol­e­cules are small, so the two mol­e­c­u­lar en­ergy lev­els broaden only slightly into two thin bands. The gap be­tween the filled sym­met­ric states and the empty an­ti­sym­met­ric ones re­mains.

Note that shar­ing elec­trons in pairs in­volves a non­triv­ial in­ter­ac­tion be­tween the two elec­trons in each pair. The truth must be stretched a bit to fit it within the band the­ory idea of non­in­ter­act­ing elec­trons. Truly non­in­ter­act­ing elec­trons would have the spa­tial states of the hy­dro­gen mol­e­c­u­lar ion avail­able to them, chap­ter 4.6. Here the lower en­ergy state is one in which a sin­gle elec­tron is sym­met­ri­cally shared be­tween the atoms. And the higher en­ergy state is one in which a sin­gle elec­tron is an­ti­sym­met­ri­cally shared. In the model of non­in­ter­act­ing elec­trons, both elec­trons oc­cupy the lower-en­ergy sin­gle-elec­tron spa­tial state, again with op­po­site spins. One prob­lem with this pic­ture is that the sin­gle-elec­tron states do not take into ac­count where the other elec­tron is. There is then a sig­nif­i­cant chance that both elec­trons can be found around the same atom. In the cor­rect two-elec­tron state, the elec­trons largely avoid that. Be­ing around the same atom would in­crease their en­ergy, since the elec­trons re­pel each other.

Note also that us­ing the ac­tual hy­dro­gen mol­e­c­u­lar ion states may not be the best ap­proach. It might be bet­ter to ac­count for the pres­ence of the other elec­tron ap­prox­i­mately us­ing some nu­clear shield­ing ap­proach like the one used for atoms in chap­ter 5.9. An im­proved, but still ap­prox­i­mate way of ac­count­ing for the sec­ond elec­tron would be to use a so-called Hartree-Fock method. More gen­er­ally, the most straight­for­ward band the­ory ap­proach tends to work bet­ter for met­als than for in­su­la­tors. Al­ter­na­tive nu­mer­i­cal meth­ods ex­ist that work bet­ter for in­su­la­tors. At the time of writ­ing there is no sim­ple magic bul­let that works well for every ma­te­r­ial.

Group IV el­e­ments like di­a­mond, sil­i­con, and ger­ma­nium pull a sim­i­lar trick as hy­dro­gen. They are in­su­la­tors at ab­solute zero tem­per­a­ture. How­ever, their 4 va­lence elec­trons per atom are not enough to fill the merged band aris­ing from the s and p states. That band can hold 8 elec­trons per atom. Like hy­dro­gen, a gap forms within the band. First the s and p states are con­verted into hy­brids, chap­ter 5.11.4. Then states are cre­ated in which elec­trons are shared sym­met­ri­cally be­tween atoms and states in which they are shared an­ti­sym­met­ri­cally. There is an en­ergy gap be­tween these states. The lower en­ergy states are filled with elec­trons and the higher en­ergy states are empty, pro­duc­ing again an in­su­la­tor. But un­like in hy­dro­gen, each atom is now bonded to four oth­ers. That turns the en­tire solid into es­sen­tially one big mol­e­cule. These ma­te­ri­als are much stronger and more sta­ble than solid hy­dro­gen. Like he­lium, hy­dro­gen is only a solid at very low tem­per­a­tures.

It may be noted that un­der ex­tremely high pres­sures, hy­dro­gen might be­come metal­lic. Not only that, as the small­est atom of them all, and in the ab­sence of 1p atomic states, metal­lic hy­dro­gen is likely to have some very un­usual prop­er­ties. It makes metal­lic hy­dro­gen the holy grail of high pres­sure physics.

It is in­struc­tive to ex­am­ine how the band the­ory of non­in­ter­act­ing elec­trons ac­counts for the fact that hy­dro­gen is an in­su­la­tor. Un­like the dis­cus­sion above, band the­ory does not ac­tu­ally look at the num­ber of va­lence elec­trons per atom. For one, a solid may con­sist of atoms of more than one kind. In gen­eral, crys­talline solids con­sist of el­e­men­tary build­ing blocks called prim­i­tive cells that can in­volve sev­eral atoms. Band the­ory pre­dicts the solid to be a metal if the num­ber of elec­trons per prim­i­tive cell is odd. If the num­ber of elec­trons per prim­i­tive cell is even, the ma­te­r­ial may be an in­su­la­tor. In solid hy­dro­gen each prim­i­tive cell holds a com­plete mol­e­cule, so there are two atoms per prim­i­tive cell. Each atom con­tributes an elec­tron, so the num­ber of elec­trons per prim­i­tive cell is even. Ac­cord­ing to band the­ory, that al­lows hy­dro­gen to be an in­su­la­tor. In a sim­i­lar way group V el­e­ments can fill up their va­lence bands with an odd num­ber of va­lence elec­trons per atom. And like hy­dro­gen, di­a­mond, sil­i­con, and ger­ma­nium have two atoms per prim­i­tive cell, re­flect­ing the gap that forms in the merged s and p bands.

Of course, that can­not be the com­plete story. It does not ex­plain why atoms to­wards the right in the pe­ri­odic ta­ble would group to­gether into prim­i­tive cells that al­low them to be in­su­la­tors. Why don't the atoms to the left in the pe­ri­odic ta­ble do the same? Why don't the al­kali met­als group to­gether in two-atom mol­e­cules like hy­dro­gen does? Qual­i­ta­tively speak­ing, met­als are char­ac­ter­ized by va­lence elec­trons that are rel­a­tively loosely bound. Sup­pose you com­pare the size of the 2s state of a lithium atom with the spac­ing of the atoms in solid lithium. If you do, you find that on av­er­age the 2s va­lence elec­tron is no closer to the atom to which it sup­pos­edly be­longs than to the neigh­bor­ing atoms. There­fore, the elec­trons are what is called de­lo­cal­ized. They are not bound to one spe­cific lo­ca­tion in the atomic crys­tal struc­ture. So they are not re­ally in­ter­ested in help­ing bond their par­tic­u­lar atom to its im­me­di­ate neigh­bors. On the other hand, to the right in the pe­ri­odic ta­ble, in­clud­ing hy­dro­gen and he­lium, the va­lence elec­trons are much more tightly held. To de­lo­cal­ize them would re­quire that the atoms would be squeezed much more tightly to­gether. That does not hap­pen un­der nor­mal pres­sures be­cause it pro­duces very high ki­netic en­ergy of the elec­trons.

Where hy­dro­gen re­fuses to be a metal with one va­lence elec­tron per atom, boron re­fuses to do so with three. How­ever, boron is very am­biva­lent about it. It does not re­ally feel com­fort­able with ei­ther metal­lic or co­va­lent be­hav­ior. A bit of im­pu­rity can read­ily turn it metal­lic. That great sen­si­tiv­ity to im­pu­rity makes the el­e­ment very hard to study. At the time of writ­ing, it is be­lieved that boron has a co­va­lent ground state un­der nor­mal pres­sures. The con­vo­luted crys­tal struc­ture is be­lieved to have a unit cell with ei­ther 12 or 106 atoms, de­pend­ing on pre­cise con­di­tions.

In group IV, tin is metal­lic above 13 $\POW9,{\circ}$C, as white tin, but co­va­lent be­low this tem­per­a­ture, as grey tin. It is of­ten dif­fi­cult to pre­dict whether an el­e­ment is a metal or co­va­lent near the mid­dle of the pe­ri­odic ta­ble. Lead, of course, is a metal.

It should fur­ther be noted that band the­ory can be in er­ror be­cause it ig­nores the in­ter­ac­tions be­tween the elec­trons. “Mott in­su­la­tors” and “charge trans­fer in­su­la­tors” are, as the name says, in­su­la­tors even though con­ven­tional band the­ory would pre­dict that they are met­als.


Key Points
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In the pe­ri­odic ta­ble, the group I, II, and III el­e­ments are nor­mally met­als.

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Hy­dro­gen and he­lium are non­metals. Don’t ask about boron.

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The group IV el­e­ments di­a­mond, sil­i­con, and ger­ma­nium are in­su­la­tors at ab­solute zero tem­per­a­ture.


6.21.3 Semi­con­duc­tors

Tem­per­a­ture can have sig­nif­i­cant ef­fects on elec­tri­cal con­duc­tion. As the pre­vi­ous sec­tion noted, higher tem­per­a­ture de­creases the con­duc­tion in met­als, as there are more crys­tal vi­bra­tions that the mov­ing elec­trons can get scat­tered by. But a higher tem­per­a­ture also changes which en­ergy states the elec­trons oc­cupy. And that can pro­duce semi­con­duc­tors.

Fig­ure 6.19 showed which en­ergy states the elec­trons oc­cupy at ab­solute zero tem­per­a­ture. There are no elec­trons with en­er­gies above the Fermi level in­di­cated by the red tick mark. Fig­ure 6.20 shows how that changes for a nonzero tem­per­a­ture. Now ran­dom ther­mal mo­tion al­lows elec­trons to reach en­ergy lev­els up to roughly ${k_{\rm B}}T$ above the Fermi level. Here $k_{\rm B}$ is the Boltz­mann con­stant and $T$ the ab­solute tem­per­a­ture. This change in elec­tron en­er­gies is de­scribed math­e­mat­i­cally by the Fermi-Dirac dis­tri­b­u­tion dis­cussed ear­lier.

Fig­ure 6.20: Sketch of elec­tron en­ergy spec­tra in solids at a nonzero tem­per­a­ture.
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It does not make much dif­fer­ence for a free-elec­tron gas or a metal. How­ever, for an in­su­la­tor it may make a dra­matic dif­fer­ence. If the band gap is not too large com­pared to ${k_{\rm B}}T$, ran­dom ther­mal mo­tion will put a few very lucky elec­trons in the pre­vi­ously empty con­duc­tion band. These elec­trons can then be prod­ded to slightly higher en­er­gies to al­low some elec­tric cur­rent to flow. Also, the cre­ated “holes” in the va­lence band, the states that have lost their elec­trons, al­low some elec­tric cur­rent. Va­lence band elec­trons can be moved into holes that have a pre­ferred di­rec­tion of mo­tion from states that do not. These elec­trons will then leave be­hind holes that have the op­po­site di­rec­tion of mo­tion.

It is of­ten more con­ve­nient to think of the mov­ing holes in­stead of the elec­trons as the elec­tric cur­rent car­ri­ers in the va­lence band. Since a hole means that a neg­a­tively charged elec­tron is miss­ing, a hole acts much like a pos­i­tively charged par­ti­cle would.

Be­cause both the elec­trons in the con­duc­tion band and the holes in the va­lence band al­low some elec­tri­cal con­duc­tion, the orig­i­nal in­su­la­tor has turned into what is called a “semi­con­duc­tor.”

The pre­vi­ous sec­tion men­tioned that a clas­si­cal pic­ture of mov­ing elec­trons sim­ply does not work for met­als. Their mo­tion is much too much re­strained by a lack of avail­able empty en­ergy states. How­ever, the con­duc­tion band of semi-con­duc­tors is largely empty. There­fore a clas­si­cal pic­ture works much bet­ter for the mo­tion of the elec­trons in the con­duc­tion band of a semi­con­duc­tor.


Key Points
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For semi­con­duc­tors, con­duc­tion can oc­cur be­cause some elec­trons from the va­lence band are ther­mally ex­cited to the con­duc­tion band.

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Both the elec­trons that get into the con­duc­tion band and the holes they leave be­hind in the va­lence band can con­duct elec­tric­ity.


6.21.4 Semi­met­als

One ad­di­tional type of elec­tron en­ergy spec­trum for solids should be men­tioned. For a “semi­metal,” two dis­tinct en­ergy bands over­lap slightly at the Fermi level. In terms of the sim­plis­tic spec­tra of fig­ure 6.19, that would mean that semi­met­als are met­als. In­deed they do al­low con­duc­tion at ab­solute zero tem­per­a­ture. How­ever, their fur­ther be­hav­ior is no­tice­ably dif­fer­ent from true met­als be­cause the over­lap of the two bands is only small. One dif­fer­ence is that the elec­tri­cal con­duc­tion of semi­met­als in­creases with tem­per­a­ture, un­like that of met­als. Like for semi­con­duc­tors, for semi­met­als a higher tem­per­a­ture means that there are more elec­trons in the up­per band and more holes in the lower band. That ef­fect is sketched to the far right in fig­ure 6.20.

The clas­si­cal semi­met­als are ar­senic, an­ti­mony, and bis­muth. Ar­senic and an­ti­mony are not just semi­met­als, but also met­al­loids, a group of el­e­ments whose chem­i­cal prop­er­ties are con­sid­ered to be in­ter­me­di­ate be­tween met­als and non­metals. But semi­metal and met­al­loid are not the same thing. Semi­met­als do not have to con­sist of a sin­gle el­e­ment. Con­versely, met­al­loids in­clude the semi­con­duc­tors sil­i­con and ger­ma­nium.

A semi­metal that is re­ceiv­ing con­sid­er­able at­ten­tion at the time of writ­ing is graphite. Graphite con­sists of sheets of car­bon atoms. A sin­gle sheet of car­bon, called graphene, is right on the bound­ary be­tween semi­metal and semi­con­duc­tor. A car­bon nan­otube can be thought of as a strip cut from a graphene sheet that then has its long edges at­tached to­gether to pro­duce a cylin­der. Car­bon nan­otubes have elec­tri­cal prop­er­ties that are fun­da­men­tally dif­fer­ent de­pend­ing on the di­rec­tion in which the strip is cut from the sheet. They can ei­ther be metal­lic or non­metal­lic.


Key Points
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Semi­met­als have prop­er­ties in­ter­me­di­ate be­tween met­als and semi­con­duc­tors.


6.21.5 Elec­tronic heat con­duc­tion

The va­lence elec­trons in met­als are not just very good con­duc­tors of elec­tric­ity, but also of heat. In in­su­la­tors elec­trons do not as­sist in heat con­duc­tion; it takes too much en­ergy to ex­cite them. How­ever, atomic vi­bra­tions in solids can con­duct heat too. For ex­am­ple, di­a­mond, an ex­cel­lent elec­tri­cal in­su­la­tor, is also an ex­cel­lent con­duc­tor of heat. There­fore the dif­fer­ences in heat con­duc­tion be­tween solids are not by far as large as those in elec­tri­cal con­duc­tion. Be­cause atoms can con­duct sig­nif­i­cant heat, no solid ma­te­r­ial will be a truly su­perb ther­mal in­su­la­tor. Prac­ti­cal ther­mal in­su­la­tors are highly porous ma­te­ri­als whose vol­ume con­sists largely of voids.


Key Points
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Elec­trons con­duct heat very well, but atoms can do it too.

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Prac­ti­cal ther­mal in­su­la­tors use voids to re­duce atomic heat con­duc­tion.


6.21.6 Ionic con­duc­tiv­ity

It should be men­tioned that elec­trons do not have an ab­solute mo­nop­oly on elec­tri­cal con­duc­tion in solids. A dif­fer­ent type of elec­tri­cal con­duc­tion is pos­si­ble in ionic solids. These solids con­sist of a mix­ture of pos­i­tively and neg­a­tively charged ions. Pos­i­tive ions, or cations, are atoms that have lost one or more elec­trons. Neg­a­tive ions, or an­ions, are atoms that have ab­sorbed one or more ad­di­tional elec­trons. A sim­ple ex­am­ple of a ionic solid is salt, which con­sists of $\rm {N}a^{+}$ sodium cations and $\rm {C}l^{-}$ chlo­rine an­ions. For ionic solids a small amount of elec­tri­cal con­duc­tion may be pos­si­ble due to mo­tion of the ions. This re­quires de­fects in the atomic crys­tal struc­ture in or­der to give the atoms some room to move.

Typ­i­cal de­fects in­clude va­can­cies, in which an atom is miss­ing from the crys­tal struc­ture, and in­ter­sti­tials, in which an ad­di­tional atom has been forced into one of the small gaps be­tween the atoms in the crys­tal. Now if a ion gets re­moved from its nor­mal po­si­tion in the crys­tal to cre­ate a va­cancy, it must go some­where. One pos­si­bil­ity is that it gets squeezed in be­tween the other atoms in the crys­tal. In that case both a va­cancy and an in­ter­sti­tial have been pro­duced at the same time. Such a com­bi­na­tion of a va­cancy and an in­ter­sti­tial is called a “Frenkel de­fect.” An­other pos­si­bil­ity oc­curs in, for ex­am­ple, salt; along with the orig­i­nal va­cancy, a va­cancy for a ion of the op­po­site kind is cre­ated. Such a com­bi­na­tion of two op­po­site va­can­cies is called a “Schot­tky de­fect.” In this case there is no need to squeeze an atom in the gaps in the crys­tal struc­ture; there are now equal num­bers of ions of each kind to fill the sur­round­ing nor­mal crys­tal sites. Cre­at­ing de­fects in Frenkel or Schot­tky pairs en­sures that the com­plete crys­tal re­mains elec­tri­cally neu­tral as it should.

Im­pu­ri­ties are an­other im­por­tant de­fect. For ex­am­ple, in salt a $\rm {C}a^{2+}$ cal­cium ion might be sub­sti­tuted for a $\rm {N}a^{+}$ sodium ion. The cal­cium ion has the charge of two sodium ions, so a sodium va­cancy en­sures elec­tric neu­tral­ity of the crys­tal. In yt­tria-sta­bi­lized zir­co­nia, (YSZ), oxy­gen va­can­cies are cre­ated in zir­co­nia, $\rm {Zr}O{}_2$, by re­plac­ing some $\rm {Z}r^{4+}$ zir­co­nium ions with $\rm {Y}^{3+}$ yt­trium ones. Cal­cium ions can also be used. The oxy­gen va­can­cies al­low mo­bil­ity for the oxy­gen ions. That is im­por­tant for ap­pli­ca­tions such as oxy­gen sen­sors and solid ox­ide fuel cells.

For salt, the main con­duc­tion mech­a­nism is by na­trium va­can­cies. But the ionic con­duc­tiv­ity of salt is al­most im­mea­sur­ably small at room tem­per­a­ture. That is due to the high en­ergy needed to cre­ate Schot­tky de­fects and for na­trium ions to mi­grate into the na­trium va­can­cies. In­deed, what­ever lit­tle con­duc­tion there is at room tem­per­a­ture is due to im­pu­ri­ties. Heat­ing will help, as it in­creases the ther­mal en­ergy avail­able for both de­fect cre­ation and ion mo­bil­ity. As seen from the Maxwell-Boltz­mann dis­tri­b­u­tion dis­cussed ear­lier, ther­mal ef­fects in­crease ex­po­nen­tially with tem­per­a­ture. Still, even at the melt­ing point of salt its con­duc­tiv­ity is eight or­ders of mag­ni­tude less than that of met­als.

There are how­ever ionic ma­te­ri­als that have much higher con­duc­tiv­i­ties. They can­not com­pete with met­als, but some ionic solids can com­pete with liq­uid elec­trolytes. These solids may be re­ferred to as “solid elec­trolytes, “fast ion con­duc­tors,” or “su­pe­ri­onic con­duc­tors.” They are im­por­tant for such ap­pli­ca­tions as bat­ter­ies, fuel cells, and gas sen­sors. Yt­tria-sta­bi­lized zir­co­nia is an ex­am­ple, al­though un­for­tu­nately only at tem­per­a­tures around 1,000 $\POW9,{\circ}$C. In the best ionic con­duc­tors, the crys­tal struc­ture for one kind of ion be­comes so ir­reg­u­lar that these ions are ef­fec­tively in a molten state. For ex­am­ple, this hap­pens for the sil­ver ions in the clas­si­cal ex­am­ple of hot sil­ver io­dide. Throw in 25% of ru­bid­ium chlo­ride and $\rm {R}bAg{}_4Cl{}_5$ stays su­pe­ri­onic to room tem­per­a­ture.

Crys­tal sur­faces are also crys­tal de­fects, in a sense. They can en­hance ionic con­duc­tiv­ity. For ex­am­ple, nanoion­ics can greatly im­prove the ionic con­duc­tiv­ity of poor ionic con­duc­tors by com­bin­ing them in nanoscale lay­ers.


Key Points
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In ionic solids, some elec­tri­cal con­duc­tion may oc­cur through the mo­tion of the ions in­stead of in­di­vid­ual elec­trons.

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It is im­por­tant for ap­pli­ca­tions such as bat­ter­ies, fuel cells, and gas sen­sors.