10.4 Co­va­lent Ma­te­ri­als

In co­va­lent ma­te­ri­als, the atoms are held to­gether by co­va­lent chem­i­cal bonds. Such bonds are strong. Note that the clas­si­fi­ca­tion is some­what vague; many crys­tals, like quartz (sil­i­con diox­ide), have partly ionic, partly co­va­lent bind­ing. An­other am­bi­gu­ity oc­curs for graphite, the sta­ble form of car­bon un­der nor­mal con­di­tion. Graphite con­sists of lay­ers of car­bon atoms arranged in a hexag­o­nal pat­tern. There are four co­va­lent bonds bind­ing each car­bon to three neigh­bor­ing atoms in the layer: three sp$\POW9,{2}$ hy­brid bonds in the plane and a fourth $\pi$-​bond nor­mal it. The $\pi$-​elec­trons are de­lo­cal­ized and will con­duct elec­tric­ity. (When rolled into car­bon nan­otubes, this be­comes a bit more com­pli­cated.) As far as the bind­ing of the solid is con­cerned, how­ever, the point is that dif­fer­ent lay­ers of graphite are only held to­gether with weak Van der Waals forces, rather than co­va­lent bonds. This makes graphite one of the soft­est solids known.

Un­der pres­sure, car­bon atoms can form di­a­mond rather than graphite, and di­a­mond is one of the hard­est sub­stances known. The di­a­mond struc­ture is a very clean ex­am­ple of purely co­va­lent bond­ing, and this sec­tion will have a look at its na­ture. Other group IV el­e­ments in the pe­ri­odic ta­ble, in par­tic­u­lar sil­i­con, ger­ma­nium, and grey tin also have the di­a­mond struc­ture. All these, of course, are very im­por­tant for en­gi­neer­ing ap­pli­ca­tions.

One ques­tion that sug­gests it­self in view of the ear­lier dis­cus­sion of met­als is why these ma­te­ri­als are not met­als. Con­sider car­bon for ex­am­ple. Com­pared to beryl­lium, it has four rather than two elec­trons in the sec­ond, L, shell. But the merged 2s and 2p bands can hold eight elec­trons, so that can­not be the ex­pla­na­tion. In fact, tin comes in two forms un­der nor­mal con­di­tions: co­va­lent grey tin is sta­ble be­low 13 $\POW9,{\circ}$C; while above that tem­per­a­ture, metal­lic white tin is the sta­ble form. It is of­ten dif­fi­cult to guess whether a par­tic­u­lar el­e­ment will form a metal­lic or co­va­lent sub­stance near the mid­dle of the pe­ri­odic ta­ble.

Fig­ure 10.14: Schematic of cross­ing bands.
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Fig­ure 10.14 gives a schematic of the en­ergy band struc­ture for a di­a­mond-type crys­tal when the spac­ing be­tween the atoms is ar­ti­fi­cially changed. When the atoms are far apart, i.e. $d$ is large, the dif­fer­ence from beryl­lium is only that car­bon has two elec­trons in 2p states ver­sus beryl­lium none. But when the car­bon atoms start com­ing closer, they have a group meet­ing and hit upon the bright idea to re­duce their en­ergy even more by con­vert­ing their one 2s and three 2p spa­tial states into four hy­brid sp$\POW9,{3}$ states. This al­lows them to share pairs of elec­trons sym­met­ri­cally in as much as four strong co­va­lent bonds. And it does in­deed work very well for low­er­ing the en­ergy of these states, filled to the gills with elec­trons. But it does not work well at all for the “anti-bond­ing” states that share the elec­trons an­ti­sym­met­ri­cally, (as dis­cussed for the hy­dro­gen mol­e­cule in chap­ter 5.2.4), and who do not have a sin­gle elec­tron to sup­port their case at the meet­ing. So a new en­ergy gap now opens up.

At the ac­tual atom spac­ing of di­a­mond, this band gap has be­come as big as 5.5 eV, mak­ing it an elec­tric in­su­la­tor (un­like graphite, which is a semi-metal). For sil­i­con how­ever, the gap is a much smaller 1.1 eV, sim­i­lar to the one for ger­ma­nium of 0.7 eV; grey tin is con­sid­er­ably smaller still; re­cent au­thor­i­ta­tive sources list it as zero. These smaller band gaps al­low no­tice­able num­bers of elec­trons to get into the empty con­duc­tion band by ther­mal ex­ci­ta­tion, so these ma­te­ri­als are semi­con­duc­tors at room tem­per­a­ture.

Fig­ure 10.15: Ball and stick schematic of the di­a­mond crys­tal.

The crys­tal struc­ture of these ma­te­ri­als is rather in­ter­est­ing. It must al­low each atom core to con­nect to 4 oth­ers to form the hy­brid co­va­lent bonds. That re­quires the rather spa­cious struc­ture sketched in fig­ure 10.15. For sim­plic­ity and clar­ity, the four hy­brid bonds that at­tach each atom core to its four neigh­bors are shown as blue or dark grey sticks rather than as a dis­tri­b­u­tion of grey tones.

Like for lithium, you can think of the spheres as rep­re­sent­ing the in­ner elec­trons. The grey gas rep­re­sents the outer elec­trons, four per atom.

To un­der­stand the fig­ure be­yond that, first note that it turns out to be im­pos­si­ble to cre­ate the di­a­mond crys­tal struc­ture from a ba­sis of a sin­gle atom. It is sim­ply not pos­si­ble to dis­trib­ute clones of a sin­gle car­bon atom around us­ing a sin­gle set of three prim­i­tive vec­tors, and pro­duce all the atoms in the di­a­mond crys­tal. A ba­sis of a pair of atoms is needed. The choice of which pair is quite ar­bi­trary, but in fig­ure 10.15 the clones of the cho­sen pair are linked by blue lines. No­tice how the en­tire crys­tal is build up from such clones. (Phys­i­cally, the choice of ba­sis is ar­ti­fi­cial, and the blue sticks in­di­cate hy­brid bonds just like the grey ones.) One pos­si­ble choice for a set of three prim­i­tive trans­la­tion vec­tors is shown in the fig­ure. The more usual choice is to take the one in the front plane to the atom lo­cated at 45 de­grees in­stead.

Now no­tice that the lower mem­bers of these pairs are lo­cated at the cor­ners and face cen­ters of the cu­bic vol­ume el­e­ments in­di­cated by the fat red lines. Yes, di­a­mond is an­other ex­am­ple of a face-cen­tered cu­bic lat­tice. What is dif­fer­ent from the NaCl case is the ba­sis; two car­bon atoms at some weird an­gle, in­stead of a na­trium and a chlo­rine ion sen­si­bly next to each other. Ac­tu­ally, if you look a bit closer, you will no­tice that in terms of the half-size cubes in­di­cated by thin red frames, the struc­ture is not that il­log­i­cal. It is again that of a three-di­men­sion­al chess board, where the cen­ters of the black cubes con­tain the up­per car­bon of a ba­sis clone, while the cen­ters of the white cubes are empty. But of course, you would not want to tell peo­ple that. They might think you spend your time play­ing games, and ter­mi­nate your sup­port.

If you look at the mas­sively cross-linked di­a­mond struc­ture, it may not come as that much of a sur­prise that di­a­mond is the hard­est sub­stance to oc­cur nat­u­rally. Un­der nor­mal con­di­tions, di­a­mond will sup­pos­edly de­gen­er­ate ex­tremely slowly into graphite, but with­out doubt, di­a­monds are for­ever.