Sub­sec­tions


6.22 Elec­trons in Crys­tals

A mean­ing­ful dis­cus­sion of semi­con­duc­tors re­quires some back­ground on how elec­trons move through solids. The free-elec­tron gas model sim­ply as­sumes that the elec­trons move through an empty pe­ri­odic box. But of course, to de­scribe a real solid the box should re­ally be filled with the count­less atoms around which the con­duc­tion elec­trons move.

This sub­sec­tion will ex­plain how the mo­tion of elec­trons gets mod­i­fied by the atoms. To keep things sim­ple, it will still be as­sumed that there is no di­rect in­ter­ac­tion be­tween the elec­trons. It will also be as­sumed that the solid is crys­talline, which means that the atoms are arranged in a pe­ri­odic pat­tern. The atomic pe­riod should be as­sumed to be many or­ders of mag­ni­tude shorter than the size of the pe­ri­odic box. There must be many atoms in each di­rec­tion in the box.

Fig­ure 6.21: Po­ten­tial en­ergy seen by an elec­tron along a line of nu­clei. The po­ten­tial en­ergy is in green, the nu­clei are in red.
\begin{figure}\centering
\epsffile{bwpotclmb.eps}
\end{figure}

The ef­fect of the crys­tal is to in­tro­duce a pe­ri­odic po­ten­tial en­ergy for the elec­trons. For ex­am­ple, fig­ure 6.21 gives a sketch of the po­ten­tial en­ergy seen by an elec­tron along a line of nu­clei. When­ever the elec­tron is right on top of a nu­cleus, its po­ten­tial en­ergy plunges. Close enough to a nu­cleus, a very strong at­trac­tive Coulomb po­ten­tial is seen. Of course, on a line that does not pass ex­actly through nu­clei, the po­ten­tial will not plunge that low.

Fig­ure 6.22: Po­ten­tial en­ergy seen by an elec­tron in the one-di­men­sional sim­pli­fied model of Kro­nig & Pen­ney.
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Kro­nig & Pen­ney de­vel­oped a very sim­ple one-di­men­sion­al model that ex­plains much of the mo­tion of elec­trons through crys­tals. It as­sumes that the po­ten­tial en­ergy seen by the elec­trons is pe­ri­odic on some atomic-scale pe­riod $d_x$. It also as­sumes that this po­ten­tial con­sists of square dips, like in fig­ure 6.22. You might think of the re­gions of low­ered po­ten­tial en­ergy as the im­me­di­ate vicin­ity of the nu­clei. This is the model that will be ex­am­ined. The atomic pe­riod $d_x$ is as­sumed to be much smaller than the pe­ri­odic box size $\ell_x$, i.e. the size of the com­plete crys­tal. In par­tic­u­lar, the box should con­tain a large and whole num­ber of atomic pe­ri­ods.

Three-di­men­sion­al Kro­nig & Pen­ney quan­tum states can be formed as prod­ucts of one-di­men­sion­al ones, com­pare chap­ter 3.5.8. How­ever, such states are lim­ited to po­ten­tials that are sums of one-di­men­sion­al ones. In any case, this sec­tion will re­strict it­self mostly to the one-di­men­sion­al case.


6.22.1 Bloch waves

This sub­sec­tion ex­am­ines the sin­gle-par­ti­cle quan­tum states, or en­ergy eigen­func­tions, of elec­trons in one-di­men­sion­al solids.

For free elec­trons, the en­ergy eigen­func­tions were given in sec­tion 6.18. In one di­men­sion they are:

\begin{displaymath}
\pp{n_x}/x/// = C e^{{\rm i}k_x x}
\end{displaymath}

where in­te­ger $n_x$ merely num­bers the eigen­func­tions and $C$ is a nor­mal­iza­tion con­stant that is not re­ally im­por­tant. What is im­por­tant is that these eigen­func­tions do not just have def­i­nite en­ergy ${\vphantom' E}^{\rm p}_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\hbar^2k_x^2$$\raisebox{.5pt}{$/$}$$2m_{\rm e}$, they also have def­i­nite lin­ear mo­men­tum $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_x$. Here $m_e$ is the elec­tron mass and $\hbar$ the re­duced Planck con­stant. In clas­si­cal terms, the elec­tron ve­loc­ity is given by the lin­ear mo­men­tum as $v^{\rm {p}}_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p_x$$\raisebox{.5pt}{$/$}$$m_{\rm e}$.

To find the equiv­a­lent one-di­men­sion­al en­ergy eigen­func­tions $\pp{n_x}/x///$ in the pres­ence of a crys­tal po­ten­tial $V_x(x)$ is messy. It re­quires so­lu­tion of the one-di­men­sion­al Hamil­ton­ian eigen­value prob­lem

\begin{displaymath}
-\frac{\hbar^2}{2m_{\rm e}} \frac{\partial^2\psi^{\rm p}}{\...
...2}
+ V_x \psi^{\rm p} = {\vphantom' E}^{\rm p}_x \psi^{\rm p}
\end{displaymath}

where ${\vphantom' E}^{\rm p}_x$ is the en­ergy of the state. The so­lu­tion is best done on a com­puter, even for a po­ten­tial as sim­ple as the Kro­nig & Pen­ney one, {N.9}.

How­ever, it can be shown that the eigen­func­tions can al­ways be writ­ten in the form:

\begin{displaymath}
\fbox{$\displaystyle
\pp{n_x}/x/// = \pp{{\rm p},n_x}/x/// e^{{\rm i}k_x x}
$} %
\end{displaymath} (6.32)

in which $\pp{{\rm {p}},n_x}/x///$ is an pe­ri­odic func­tion on the atomic pe­riod. Note that as long as $\pp{{\rm {p}},n_x}/x///$ is a sim­ple con­stant, this is ex­actly the same as the eigen­func­tions of the free-elec­tron gas in one di­men­sion; mere ex­po­nen­tials. But if the pe­ri­odic po­ten­tial $V_x(x)$ is not a con­stant, then nei­ther is $\pp{{\rm {p}},n_x}/x///$. In that case, all that can be said a pri­ori is that it is pe­ri­odic on the atomic pe­riod.

En­ergy eigen­func­tions of the form (6.32) are called “Bloch waves.” It may be pointed out that this form of the en­ergy eigen­func­tions was dis­cov­ered by Flo­quet, not Bloch. How­ever, Flo­quet was a math­e­mati­cian. In nam­ing the so­lu­tions af­ter Bloch in­stead of Flo­quet, physi­cists cel­e­brate the physi­cist who could do it too, just half a cen­tury later.

The rea­son why the en­ergy eigen­func­tions take this form, and what it means for the elec­tron mo­tion are dis­cussed fur­ther in chap­ter 7.10.5. There are only two key points of in­ter­est for now. First, the pos­si­ble val­ues of the wave num­ber $k_x$ are ex­actly the same as for the free-elec­tron gas, given in (6.28). Oth­er­wise the eigen­func­tion would not be pe­ri­odic on the pe­riod of the box. Sec­ond, the elec­tron ve­loc­ity can be found by dif­fer­en­ti­at­ing the sin­gle par­ti­cle en­ergy ${\vphantom' E}^{\rm p}_x$ with re­spect to the “crys­tal mo­men­tum” $p_{{\rm {cm}},x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_x$. That is the same as for the free-elec­tron gas. If you dif­fer­en­ti­ate the one-di­men­sion­al free-elec­tron gas ki­netic en­ergy ${\vphantom' E}^{\rm p}_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $({\hbar}k_x)^2$$\raisebox{.5pt}{$/$}$$2m_{\rm e}$ with re­spect to $p_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_x$, you get the ve­loc­ity.


Key Points
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In the pres­ence of a pe­ri­odic crys­tal po­ten­tial, the en­ergy eigen­func­tions pick up an ad­di­tional fac­tor that has the atomic pe­riod.

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The wave num­ber val­ues do not change.

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The ve­loc­ity is found by dif­fer­en­ti­at­ing the en­ergy with re­spect to the crys­tal mo­men­tum.


6.22.2 Ex­am­ple spec­tra

As the pre­vi­ous sec­tion dis­cussed, the dif­fer­ence be­tween met­als and in­su­la­tors is due to dif­fer­ences in their en­ergy spec­tra. The one-di­men­sion­al Kro­nig & Pen­ney model can pro­vide some in­sight into it.

Fig­ure 6.23: Ex­am­ple Kro­nig & Pen­ney spec­tra.
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Find­ing the en­ergy eigen­val­ues is not dif­fi­cult on a com­puter, {N.9}. A cou­ple of ex­am­ple spec­tra are shown in fig­ure 6.23. The ver­ti­cal co­or­di­nate is the sin­gle-elec­tron en­ergy, as usual. The hor­i­zon­tal co­or­di­nate is the elec­tron ve­loc­ity. (So the free elec­tron ex­am­ple is the one-di­men­sion­al ver­sion of the spec­trum in fig­ure 6.18, but the axes are much more com­pressed here.) Quan­tum states oc­cu­pied by elec­trons are again in red.

The ex­am­ple to the left in fig­ure 6.23 tries to roughly model a metal like lithium. The depth of the po­ten­tial drops in fig­ure 6.22 was cho­sen so that for lone atoms, (i.e. for widely spaced po­ten­tial drops), there is one bound spa­tial state and a sec­ond mar­gin­ally bound state. You might think of the bound state as hold­ing lithium’s two in­ner 1s elec­trons, and the mar­gin­ally bound state as hold­ing its loosely bound sin­gle 2s va­lence elec­tron.

Note that the 1s state is just a red dot in the lower part of the left spec­trum in fig­ure 6.23. The en­ergy of the in­ner elec­trons is not vis­i­bly af­fected by the neigh­bor­ing atoms. Also, the ve­loc­ity does not budge from zero; elec­trons in the in­ner states would hardly move even if there were un­filled states. These two ob­ser­va­tions are re­lated, be­cause as men­tioned ear­lier, the ve­loc­ity is the de­riv­a­tive of the en­ergy with re­spect to the crys­tal mo­men­tum. If the en­ergy does not vary, the ve­loc­ity is zero.

The sec­ond en­ergy level has broad­ened into a half-filled con­duc­tion band. Like for the free-elec­tron gas in fig­ure 6.18, it re­quires lit­tle en­ergy to move some Fermi-level elec­trons in this band from neg­a­tive to pos­i­tive ve­loc­i­ties to achieve net elec­tri­cal con­duc­tion.

The spec­trum in the mid­dle of fig­ure 6.23 tries to roughly model an in­su­la­tor like di­a­mond. (The one-di­men­sion­al model is too sim­ple to model an al­ka­line metal with two va­lence elec­trons like beryl­lium. The spec­tra of these met­als in­volve dif­fer­ent en­ergy bands that merge to­gether, and merg­ing bands do not oc­cur in the one-di­men­sion­al model.) The volt­age drops have been in­creased a bit to make the sec­ond en­ergy level for lone atoms more solidly bound. And it has been as­sumed that there are now four elec­trons per atom, so that the sec­ond band is com­pletely filled.

Now the only way to achieve net elec­tri­cal con­duc­tion is to move some elec­trons from the filled va­lence band to the empty con­duc­tion band above it. That re­quires much more en­ergy than a nor­mal ap­plied volt­age could pro­vide. So the crys­tal is an in­su­la­tor.

The rea­sons why the spec­tra look as shown in fig­ure 6.23 are not ob­vi­ous. Note {N.9} ex­plains by ex­am­ple what hap­pens to the free-elec­tron gas en­ergy eigen­func­tions when there is a crys­tal po­ten­tial. A much shorter ex­pla­na­tion that hits the nail squarely on the head is “That is just the way the Schrö­din­ger equa­tion is.”


Key Points
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A pe­ri­odic crys­tal po­ten­tial pro­duces en­ergy bands.


6.22.3 Ef­fec­tive mass

The spec­trum to the right in fig­ure 6.23 shows the one-di­men­sion­al free-elec­tron gas. The re­la­tion­ship be­tween ve­loc­ity and en­ergy is given by the clas­si­cal ex­pres­sion for the ki­netic en­ergy in the $x$-​di­rec­tion:

\begin{displaymath}
{\vphantom' E}^{\rm p}_x = {\textstyle\frac{1}{2}} m_{\rm e}{v^{\rm p}_x}^2
\end{displaymath}

This leads to the par­a­bolic spec­trum shown.

It is in­ter­est­ing to com­pare this spec­trum to that of the metal to the left in fig­ure 6.23. The oc­cu­pied part of the con­duc­tion band of the metal is ap­prox­i­mately par­a­bolic just like the free-elec­tron gas spec­trum. To a fair ap­prox­i­ma­tion, in the oc­cu­pied part of the con­duc­tion band

\begin{displaymath}
{\vphantom' E}^{\rm p}_x - {\vphantom' E}^{\rm p}_{{\rm c},x} = {\textstyle\frac{1}{2}} m_{{\rm eff},x} {v^{\rm p}_x}^2
\end{displaymath}

where ${\vphantom' E}^{\rm p}_{{\rm {c}},x}$ is the en­ergy at the bot­tom of the con­duc­tion band and $m_{{\rm {eff}},x}$ is a con­stant called the “ef­fec­tive mass.”

This il­lus­trates that con­duc­tion band elec­trons in met­als be­have much like free elec­trons. And the sim­i­lar­ity to free elec­trons be­comes even stronger if you de­fine the zero level of en­ergy to be at the bot­tom of the con­duc­tion band and re­place the true elec­tron mass by an ef­fec­tive mass. For the metal shown in fig­ure 6.23, the ef­fec­tive mass is 61% of the true elec­tron mass. That makes the parabola some­what flat­ter than for the free-elec­tron gas. For elec­trons that reach the con­duc­tion band of the in­su­la­tor in fig­ure 6.23, the ef­fec­tive mass is only 18% of the true mass.

In pre­vi­ous sec­tions, the va­lence elec­trons in met­als were re­peat­edly ap­prox­i­mated as free elec­trons to de­rive such prop­er­ties as de­gen­er­acy pres­sure and thermionic emis­sion. The jus­ti­fi­ca­tion was given that the forces on the va­lence elec­trons tend to come from all di­rec­tions and av­er­age out. But as the ex­am­ple above now shows, that ap­prox­i­ma­tion can be im­proved upon by re­plac­ing the true elec­tron mass by an ef­fec­tive mass. For the va­lence elec­trons in cop­per, the ap­pro­pri­ate ef­fec­tive mass is about one and a half times the true elec­tron mass, [41, p. 257]. So the use of the true elec­tron mass in the ex­am­ples was not dra­mat­i­cally wrong.

And the agree­ment be­tween con­duc­tion band elec­trons and free elec­trons is even deeper than the sim­i­lar­ity of the spec­tra in­di­cates. You can also use the den­sity of states for the free-elec­tron gas, as given in sec­tion 6.3, if you sub­sti­tute in the ef­fec­tive mass.

To see why, as­sume that the re­la­tion­ship be­tween the en­ergy ${\vphantom' E}^{\rm p}_x$ and the ve­loc­ity $v^{\rm {p}}_x$ is the same as that for a free-elec­tron gas whose elec­trons have the ap­pro­pri­ate ef­fec­tive mass. Then so is the re­la­tion­ship be­tween the en­ergy ${\vphantom' E}^{\rm p}_x$ and the wave num­ber $k_x$ the same as for that elec­tron gas. That is be­cause the ve­loc­ity is merely the de­riv­a­tive of ${\vphantom' E}^{\rm p}_x$ with re­spect to ${\hbar}k_x$. You need the same ${\vphantom' E}^{\rm p}_x$ ver­sus $k_x$ re­la­tion to get the same ve­loc­ity. (This as­sumes that you mea­sure both the en­ergy and the wave num­ber from the lo­ca­tion of min­i­mum con­duc­tion band en­ergy.) And if the ${\vphantom' E}^{\rm p}_x$ ver­sus $k_x$ re­la­tion is the same as for the free-elec­tron gas, then so is the den­sity of states. That is be­cause the quan­tum states have the same wave num­ber spac­ing re­gard­less of the crys­tal po­ten­tial.

It should how­ever be pointed out that in three di­men­sions, things get messier. Of­ten the ef­fec­tive masses are dif­fer­ent in dif­fer­ent crys­tal di­rec­tions. In that case you need to de­fine some suit­able av­er­age to use the free-elec­tron gas den­sity of states. In ad­di­tion, for typ­i­cal semi­con­duc­tors the en­ergy struc­ture of the holes at the top of the va­lence band is highly com­plex.


Key Points
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The elec­trons in a con­duc­tion band and the holes in a va­lence band are of­ten mod­eled as free par­ti­cles.

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The er­rors can be re­duced by giv­ing them an ef­fec­tive mass that is dif­fer­ent from the true elec­tron mass.

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The den­sity of states of the free-elec­tron gas can also be used.


6.22.4 Crys­tal mo­men­tum

The crys­tal mo­men­tum of elec­trons in a solid is not the same as the lin­ear mo­men­tum of free elec­trons. How­ever, it is sim­i­larly im­por­tant. It is re­lated to op­ti­cal prop­er­ties such as the dif­fer­ence be­tween di­rect and in­di­rect gap semi­con­duc­tors. Be­cause of this im­por­tance, spec­tra are usu­ally plot­ted against the crys­tal mo­men­tum, rather than against the elec­tron ve­loc­ity. The Kro­nig & Pen­ney model pro­vides a sim­ple ex­am­ple to ex­plain some of the ideas.

Fig­ure 6.24: Spec­trum against wave num­ber in the ex­tended zone scheme.
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Fig­ure 6.24 shows the sin­gle-elec­tron en­ergy plot­ted against the crys­tal mo­men­tum. Note that this is equiv­a­lent to a plot against the wave num­ber $k_x$; the crys­tal mo­men­tum is just a sim­ple mul­ti­ple of the wave num­ber, $p_{{\rm {cm}},x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_x$. The fig­ure has nondi­men­sion­al­ized the wave num­ber by mul­ti­ply­ing it by the atomic pe­riod $d_x$. Both the ex­am­ple in­su­la­tor and the free-elec­tron gas are shown in the fig­ure.

There is how­ever an am­bi­gu­ity in the fig­ure:

The crys­tal wave num­ber, and so the crys­tal mo­men­tum, is not unique.
Con­sider once more the gen­eral form of a Bloch wave,

\begin{displaymath}
\pp{n_x}/x/// = \pp{{\rm p},n_x}/x/// e^{{\rm i}k_x x}
\end{displaymath}

If you change the value of $k_x$ by a whole mul­ti­ple of $2\pi$$\raisebox{.5pt}{$/$}$$d_x$, it re­mains a Bloch wave in terms of the new $k_x$. The change in the ex­po­nen­tial can be ab­sorbed in the pe­ri­odic part $\pp{{\rm {p}},n_x}////$. The pe­ri­odic part changes, but it re­mains pe­ri­odic on the atomic scale $d_x$.

There­fore there is a prob­lem with how to de­fine a unique value of $k_x$. There are dif­fer­ent so­lu­tions to this prob­lem. Fig­ure 6.24 fol­lows the so-called “ex­tended zone scheme.” It takes the wave num­ber to be zero at the min­i­mum en­ergy and then keeps in­creas­ing the mag­ni­tude with en­ergy. This is a good scheme for the free-elec­tron gas. It also works nicely if the po­ten­tial is so weak that the en­ergy states are al­most the free-elec­tron gas ones.

Fig­ure 6.25: Spec­trum against wave num­ber in the re­duced zone scheme.
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A sec­ond ap­proach is much more com­mon, though. It uses the in­de­ter­mi­nacy in $k_x$ to shift it into the range $\vphantom{0}\raisebox{1.5pt}{$-$}$$\pi$ $\raisebox{-.3pt}{$\leqslant$}$ $k_xd_x$ $\raisebox{-.3pt}{$\leqslant$}$ $\pi$. That range is called the “first Bril­louin zone.” Re­strict­ing the wave num­bers to the first Bril­louin zone pro­duces fig­ure 6.25. This is called the “re­duced zone scheme.” Es­thet­i­cally, it is clearly an im­prove­ment in case of a non­triv­ial crys­tal po­ten­tial.

But it is much more than that. For one, the dif­fer­ent en­ergy curves in the re­duced zone scheme can be thought of as mod­i­fied atomic en­ergy lev­els of lone atoms. The cor­re­spond­ing Bloch waves can be thought of as mod­i­fied atomic states, mod­u­lated by a rel­a­tively slowly vary­ing ex­po­nen­tial $e^{{{\rm i}}k_xx}$.

Sec­ond, the re­duced zone scheme is im­por­tant for op­ti­cal ap­pli­ca­tions of semi­con­duc­tors. In par­tic­u­lar,

A lone pho­ton can only pro­duce an elec­tron tran­si­tion along the same ver­ti­cal line in the re­duced zone spec­trum.
The rea­son is that crys­tal mo­men­tum must be con­served. That is much like lin­ear mo­men­tum must be pre­served for elec­trons in free space. Since a pho­ton has neg­li­gi­ble crys­tal mo­men­tum, the crys­tal mo­men­tum of the elec­tron can­not change. That means it must stay on the same ver­ti­cal line in the re­duced zone scheme.

To see why that is im­por­tant, sup­pose that you want to use a semi­con­duc­tor to cre­ate light. To achieve that, you need to some­how ex­cite elec­trons from the va­lence band to the con­duc­tion band. How to do that will be dis­cussed in sec­tion 6.27.7. The ques­tion here is what hap­pens next. The ex­cited elec­trons will even­tu­ally drop back into the va­lence band. If all is well, they will emit the en­ergy they lose in do­ing so as a pho­ton. Then the semi­con­duc­tor will emit light.

Fig­ure 6.26: Some one-di­men­sional en­ergy bands for a few ba­sic semi­con­duc­tors.
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It turns out that the ex­cited elec­trons are mostly in the low­est en­ergy states in the con­duc­tion band. For var­i­ous rea­sons. that tends to be true de­spite the ab­sence of ther­mal equi­lib­rium. They are cre­ated there or evolve to it. Also, the holes that the ex­cited elec­trons leave be­hind in the va­lence band are mostly at the high­est en­ergy lev­els in that band.

Now con­sider the en­ergy bands of some ac­tual semi­con­duc­tors shown to the left in fig­ure 6.26. In par­tic­u­lar, con­sider the spec­trum of gal­lium ar­senide. The ex­cited elec­trons are at the low­est point of the con­duc­tion band. That is at zero crys­tal mo­men­tum. The holes are at the high­est point in the va­lence band, which is also at zero crys­tal mo­men­tum. There­fore, the ex­cited elec­trons can drop ver­ti­cally down into the holes. The crys­tal mo­men­tum does not change, it stays zero. There is no prob­lem. In fact, the first patent for a light emit­ting diode was for a gal­lium ar­senide one, in 1961. The en­ergy of the emit­ted pho­tons is given by the band gap of gal­lium ar­senide, some­what less than 1.5 eV. That is slightly be­low the vis­i­ble range, in the near in­frared. It is suit­able for re­mote con­trols and other non­vis­i­ble ap­pli­ca­tions.

But now con­sider ger­ma­nium in fig­ure 6.26. The high­est point of the va­lence band is still at zero crys­tal mo­men­tum. But the low­est point of the con­duc­tion band is now at max­i­mum crys­tal mo­men­tum in the re­duced zone scheme. When the ex­cited elec­trons drop back into the holes, their crys­tal mo­men­tum changes. Since crys­tal mo­men­tum is con­served, some­thing else must ac­count for the dif­fer­ence. And the pho­ton does not have any crys­tal mo­men­tum to speak of. It is a phonon of crys­tal vi­bra­tion that must carry off the dif­fer­ence in crys­tal mo­men­tum. Or sup­ply the dif­fer­ence, if there are enough pre-ex­ist­ing ther­mal phonons. The re­quired in­volve­ment of a phonon in ad­di­tion to the pho­ton makes the en­tire process much more cum­ber­some. There­fore the en­ergy of the elec­tron is much more likely to be re­leased through some al­ter­nate mech­a­nism that pro­duces heat in­stead of light.

The sit­u­a­tion for sil­i­con is like that for ger­ma­nium. How­ever, the low­est en­ergy in the con­duc­tion band oc­curs for a dif­fer­ent di­rec­tion of the crys­tal mo­men­tum. The spec­trum for that di­rec­tion of the crys­tal mo­men­tum is shown to the right in fig­ure 6.26. It still re­quires a change in crys­tal mo­men­tum.

At the time of writ­ing, there is a lot of in­ter­est in im­prov­ing the light emis­sion of sil­i­con. The rea­son is its preva­lence in semi­con­duc­tor ap­pli­ca­tions. If sil­i­con it­self can be made to emit light ef­fi­ciently, there is no need for the com­pli­ca­tions of in­volv­ing dif­fer­ent ma­te­ri­als to do it. One trick is to min­i­mize processes that al­low elec­trons to drop back into the va­lence band with­out emit­ting pho­tons. An­other is to use sur­face mod­i­fi­ca­tion tech­niques that pro­mote ab­sorp­tion of pho­tons in so­lar cell ap­pli­ca­tions. The un­der­ly­ing idea is that at least in ther­mal equi­lib­rium, the best ab­sorbers of elec­tro­mag­netic ra­di­a­tion are also the best emit­ters, sec­tion 6.8.

Gal­lium ar­senide is called a “di­rect-gap semi­con­duc­tor” be­cause the elec­trons can fall straight down into the holes. Sil­i­con and ger­ma­nium are called “in­di­rect-gap semi­con­duc­tors” be­cause the elec­trons must change crys­tal mo­men­tum. Note that these terms are ac­cu­rate and un­der­stand­able, a rar­ity in physics.

Con­ser­va­tion of crys­tal mo­men­tum does not just af­fect the emis­sion of light. It also af­fects its ab­sorp­tion. In­di­rect-gap semi­con­duc­tors do not ab­sorb pho­tons very well if the pho­tons have lit­tle more en­ergy than the band gap. They ab­sorb pho­tons with enough en­ergy to in­duce ver­ti­cal elec­tron tran­si­tions a lot bet­ter.

It may be noted that con­ser­va­tion of crys­tal mo­men­tum is of­ten called “con­ser­va­tion of wave vec­tor.” It is the same thing of course, since the crys­tal mo­men­tum is sim­ply $\hbar$ times the wave vec­tor. How­ever, those pesky new stu­dents of­ten have a fairly good un­der­stand­ing of mo­men­tum con­ser­va­tion, and the term mo­men­tum would leave them in­suf­fi­ciently im­pressed with the bril­liance of the physi­cist us­ing it.

(If you won­der why crys­tal mo­men­tum is pre­served, and how it even can be if the crys­tal mo­men­tum is not unique, the an­swer is in the dis­cus­sion of con­ser­va­tion laws in chap­ter 7.3 and its note. It is not re­ally mo­men­tum that is con­served, but the prod­uct of the sin­gle-par­ti­cle eigen­val­ues $e^{{{\rm i}}k_xd_x}$ of the op­er­a­tor that trans­lates the sys­tem in­volved over a dis­tance $d_x$. These eigen­val­ues do not change if the wave num­bers change by a whole mul­ti­ple of $2\pi$$\raisebox{.5pt}{$/$}$$d_x$, so there is no vi­o­la­tion of the con­ser­va­tion law if they do. For a sys­tem of par­ti­cles in free space, the po­ten­tial is triv­ial; then you can take $d_x$ equal to zero to elim­i­nate the am­bi­gu­ity in $k_x$ and so in the mo­men­tum. But for a non­triv­ial crys­tal po­ten­tial, $d_x$ is fixed. Also, since a pho­ton moves so fast, its wave num­ber is al­most zero on the atomic scale, giv­ing it neg­li­gi­ble crys­tal mo­men­tum. At least it does for the pho­tons in the eV range that are rel­e­vant here.)

Fig­ure 6.27: Spec­trum against wave num­ber in the pe­ri­odic zone scheme.
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Re­turn­ing to the pos­si­ble ways to plot spec­tra, the so-called “pe­ri­odic zone scheme” takes the re­duced zone scheme and ex­tends it pe­ri­od­i­cally, as in fig­ure 6.27. That makes for very es­thetic pic­tures, es­pe­cially in three di­men­sions.

Of course, in three di­men­sions there is no rea­son for the spec­tra in the $y$ and $z$ di­rec­tions to be the same as the one in the $x$-​di­rec­tion. Each can in prin­ci­ple be com­pletely dif­fer­ent from the other two. Re­gard­less of the dif­fer­ences, valid three-di­men­sion­al Kro­nig & Pen­ney en­ergy eigen­func­tions are ob­tained as the prod­uct of the $x$, $y$ and $z$ eigen­func­tions, and their en­ergy is the sum of the eigen­val­ues.

Sim­i­larly, typ­i­cal spec­tra for real solids have to show the spec­trum ver­sus wave num­ber for more than one crys­tal di­rec­tion to be com­pre­hen­sive. One ex­am­ple was for sil­i­con in fig­ure 6.26. A more com­plete de­scrip­tion of the one-di­men­sion­al spec­tra of real semi­con­duc­tors is given in the next sub­sec­tion.


Key Points
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The wave num­ber and crys­tal mo­men­tum val­ues are not unique.

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The ex­tended, re­duced, and pe­ri­odic zone schemes make dif­fer­ent choices for which val­ues to use.

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The re­duced zone scheme lim­its the wave num­bers to the first Bril­louin zone.

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For a pho­ton to change the crys­tal mo­men­tum of an elec­tron in the re­duced zone scheme re­quires the in­volve­ment of a phonon.

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That makes in­di­rect gap semi­con­duc­tors like sil­i­con and ger­ma­nium un­de­sir­able for some op­ti­cal ap­pli­ca­tions.


6.22.5 Three-di­men­sional crys­tals

A com­plete de­scrip­tion of the the­ory of three-di­men­sion­al crys­tals is be­yond the scope of the cur­rent dis­cus­sion. Chap­ter 10 pro­vides a first in­tro­duc­tion. How­ever, be­cause of the im­por­tance of semi­con­duc­tors such as sil­i­con, ger­ma­nium, and gal­lium ar­senide, it may be a good idea to ex­plain a few ideas al­ready.

Fig­ure 6.28: Schematic of the zinc blende (ZnS) crys­tal rel­e­vant to im­por­tant semi­con­duc­tors in­clud­ing sil­i­con.
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Con­sider first a gal­lium ar­senide crys­tal. Gal­lium ar­senide has the same crys­tal struc­ture as zinc sul­fide, in the form known as zinc blende or spha­lerite. The crys­tal is sketched in fig­ure 6.28. The larger spheres rep­re­sent the nu­clei and in­ner elec­trons of the gal­lium atoms. The smaller spheres rep­re­sent the nu­clei and in­ner elec­trons of the ar­senic atoms. Be­cause ar­senic has a more pos­i­tively charged nu­cleus, it holds its elec­trons more tightly. The fig­ure ex­ag­ger­ates the ef­fect to keep the atoms vi­su­ally apart.

The grey gas be­tween these atom cores rep­re­sents the va­lence elec­trons. Each gal­lium atom con­tributes 3 va­lence elec­trons and each ar­senic atom con­tributes 5. That makes an av­er­age of 4 va­lence elec­trons per atom.

As the fig­ure shows, each gal­lium atom core is sur­rounded by 4 ar­senic ones and vice-versa. The grey sticks in­di­cate the di­rec­tions of the co­va­lent bonds be­tween these atom cores. You can think of these bonds as some­what po­lar sp$\POW9,{3}$ hy­brids. They are po­lar since the ar­senic atom is more elec­troneg­a­tive than the gal­lium one.

It is cus­tom­ary to think of crys­tals as be­ing build up out of sim­ple build­ing blocks called “unit cells.” The con­ven­tional unit cells for the zinc blende crys­tal are the lit­tle cubes out­lined by the thicker red lines in fig­ure 6.28. Note in par­tic­u­lar that you can find gal­lium atoms at each cor­ner of these lit­tle cubes, as well as in the cen­ter of each face of them. That makes zinc blende an ex­am­ple of what is called a “face-cen­tered cu­bic” lat­tice. For ob­vi­ous rea­sons, every­body ab­bre­vi­ates that to FCC.

You can think of the unit cells as sub­di­vided fur­ther into 8 half-size cubes, as in­di­cated by the thin­ner red lines. There is an ar­senic atom in the cen­ter of every other of these smaller cubes.

The sim­ple one-di­men­sion­al Kro­nig & Pen­ney model as­sumed that the crys­tal was pe­ri­odic with a pe­riod $d_x$. For real three-di­men­sion­al crys­tals, there is not just one pe­riod, but three. More pre­cisely, there are three so-called “prim­i­tive trans­la­tion vec­tors” $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$. A set of prim­i­tive trans­la­tion vec­tors for the FCC crys­tal is shown in fig­ure 6.28. If you move around by whole mul­ti­ples of these vec­tors, you ar­rive at points that look iden­ti­cal to your start­ing point.

For ex­am­ple, if you start at the cen­ter of a gal­lium atom, you will again be at the cen­ter of a gal­lium atom. And you can step to what­ever gal­lium atom you like in this way. At least as long as the whole mul­ti­ples are al­lowed to be both pos­i­tive and neg­a­tive. In par­tic­u­lar, sup­pose you start at the gal­lium atom with the Ga la­bel in fig­ure 6.28. Then $\vec{d}_1$ al­lows you to step to any other gal­lium atom on the same line go­ing to­wards the right and left. Vec­tor $\vec{d}_2$ al­lows you to step to the next or pre­vi­ous line in the same hor­i­zon­tal plane. And vec­tor $\vec{d}_3$ al­lows you to step to the next higher or lower hor­i­zon­tal plane.

The choice of prim­i­tive trans­la­tion vec­tors is not unique. In par­tic­u­lar, many sources pre­fer to draw the vec­tor $\vec{d}_1$ to­wards the gal­lium atom in the front face cen­ter rather than to the one at the right. That is more sym­met­ric, but mov­ing around with them gets harder to vi­su­al­ize. Then you would have to step over $\vec{d}_1$, $\vec{d}_2$, and $-\vec{d}_3$ just to reach the atom to the right.

You can use the prim­i­tive trans­la­tion vec­tors also to men­tally cre­ate the zinc blende crys­tal. Con­sider the pair of atoms with the Ga and As la­bels in fig­ure 6.28. Sup­pose that you put a copy of this pair at every point that you can reach by step­ping around with the prim­i­tive trans­la­tion vec­tors. Then you get the com­plete zinc blende crys­tal. The pair of atoms is there­fore called a “ba­sis” of the zinc blende crys­tal.

This also il­lus­trates an­other point. The choice of unit cell for a given crys­tal struc­ture is not unique. In par­tic­u­lar, the par­al­lelepiped with the prim­i­tive trans­la­tion vec­tors as sides can be used as an al­ter­na­tive unit cell. Such a unit cell has the small­est pos­si­ble vol­ume, and is called a prim­i­tive cell.

The crys­tal struc­ture of sil­i­con and ger­ma­nium, as well as di­a­mond, is iden­ti­cal to the zinc blende struc­ture, but all atoms are of the same type. This crys­tal struc­ture is ap­pro­pri­ately called the di­a­mond struc­ture. The ba­sis is still a two-atom pair, even if the two atoms are now the same. In­ter­est­ingly enough, it is not pos­si­ble to cre­ate the di­a­mond crys­tal by dis­trib­ut­ing copies of a sin­gle atom. Not as long as you step around with only three prim­i­tive trans­la­tion vec­tors.

For the one-di­men­sion­al Kro­nig & Pen­ney model, there was only a sin­gle wave num­ber $k_x$ that char­ac­ter­ized the quan­tum states. For a three-di­men­sion­al crys­tal, there is a three-di­men­sion­al wave num­ber vec­tor ${\vec k}$ with com­po­nents $k_x$, $k_y$, and $k_z$. That is just like for the free-elec­tron gas in three di­men­sions as dis­cussed in ear­lier sec­tions.

In the Kro­nig & Pen­ney model, the wave num­bers could be re­duced to a fi­nite in­ter­val

\begin{displaymath}
- \frac{\pi}{d_x} \mathrel{\raisebox{-.7pt}{$\leqslant$}}k_x < \frac{\pi}{d_x}
\end{displaymath}

This in­ter­val was called the first Bril­louin zone. Wave num­bers out­side this zone are equiv­a­lent to ones in­side. The gen­eral rule was that wave num­bers a whole mul­ti­ple of $2\pi$$\raisebox{.5pt}{$/$}$$d_x$ apart are equiv­a­lent.

Fig­ure 6.29: First Bril­louin zone of the FCC crys­tal.
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In three di­men­sions, the first Bril­louin zone is no longer a one-di­men­sion­al in­ter­val but a three-di­men­sion­al vol­ume. And the sep­a­ra­tions over which wave num­ber vec­tors are equiv­a­lent are no longer so sim­ple. In­stead of sim­ply tak­ing an in­verse of the pe­riod $d_x$, as in $2\pi$$\raisebox{.5pt}{$/$}$$d_x$, you have to take an in­verse of the ma­trix formed by the three prim­i­tive trans­la­tion vec­tors $\vec{d}_1$, $\vec{d}_2$, and $\vec{d}_3$. Next you have to iden­tify the wave num­ber vec­tors clos­est to the ori­gin that are enough to de­scribe all quan­tum states. If you do all that for the FCC crys­tal, you will end up with the first Bril­louin zone shown in fig­ure 6.29. It is shaped like a cube with its 8 cor­ners cut off.

The shape of the first Bril­louin zone is im­por­tant for un­der­stand­ing graphs of three-di­men­sion­al spec­tra. Every sin­gle point in the first Bril­louin zone cor­re­sponds to mul­ti­ple Bloch waves, each with its own en­ergy. To plot all those en­er­gies is not pos­si­ble; it would re­quire a four-di­men­sion­al plot. In­stead, what is done is plot the en­er­gies along rep­re­sen­ta­tive lines. Such plots will here be in­di­cated as one-di­men­sion­al en­ergy bands. Note how­ever that they are one-di­men­sion­al bands of true three-di­men­sion­al crys­tals. They are not just Kro­nig & Pen­ney model bands.

Typ­i­cal points be­tween which one-di­men­sion­al bands are drawn are in­di­cated in fig­ure 6.29. You and I would prob­a­bly name such points some­thing like F (face), E (edge), and C (cor­ner), with a clar­i­fy­ing sub­script as needed. How­ever, physi­cists come up with names like K, L, W, and X, and de­clare them stan­dard. The cen­ter of the Bril­louin zone is the ori­gin, where the wave num­ber vec­tor is zero. Nor­mal peo­ple would there­fore in­di­cate it as O or 0. How­ever, physi­cists are not nor­mal peo­ple. They in­di­cate the ori­gin by $\Gamma$ be­cause the shape of this Greek let­ter re­minds them of a gal­lows. Physi­cists just love gal­lows hu­mor.

Fig­ure 6.30: Sketch of a more com­plete spec­trum of ger­ma­nium. (Based on re­sults of the VASP 5.2 com­mer­cial com­puter code.)
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Com­puted one-di­men­sion­al en­ergy bands be­tween the var­i­ous points in the Bril­louin zone can be found in the plot to the left in fig­ure 6.30. The plot is for ger­ma­nium. The zero level of en­ergy was cho­sen as the top of the va­lence band. The var­i­ous fea­tures of the plot agree well with other ex­per­i­men­tal and com­pu­ta­tional data.

The ear­lier spec­trum for ger­ma­nium in fig­ure 6.26 showed only the part within the lit­tle frame in fig­ure 6.30. That part is for the line be­tween zero wave num­ber and the point L in fig­ure 6.29. Un­like fig­ure 6.30, the ear­lier spec­trum fig­ure 6.26 showed both neg­a­tive and pos­i­tive wave num­bers, as its left and right halves. On the other hand, the ear­lier spec­trum showed only the high­est one-di­men­sion­al va­lence band and the low­est one-di­men­sion­al con­duc­tion band. It was suf­fi­cient to show the top of the va­lence band and the bot­tom of the con­duc­tion band, but lit­tle else. As fig­ure 6.30 shows, there are ac­tu­ally four dif­fer­ent types of Bloch waves in the va­lence band. The en­ergy range of each of the four is within the range of the com­bined va­lence band.

The com­plete va­lence band, as well as the lower part of the con­duc­tion band, is sketched in the spec­trum to the right in fig­ure 6.30. It shows the en­ergy plot­ted against the den­sity of states ${\cal D}$. Note that the com­puted den­sity of states for the con­duc­tion elec­trons is a mess when seen over its com­plete range. It is nowhere near par­a­bolic as it would be for elec­trons in empty space, fig­ure 6.1. Sim­i­larly the den­sity of states ap­plic­a­ble to the va­lence band holes is nowhere near an in­verted parabola over its com­plete range. How­ever, typ­i­cally only about 1/40th of an eV be­low the top of the va­lence band and above the bot­tom of the con­duc­tion band is rel­e­vant for ap­pli­ca­tions. That is very small on the scale of the fig­ure.

An in­ter­est­ing fea­ture of fig­ure 6.30 is that two dif­fer­ent en­ergy bands merge at the top of the va­lence band. These two bands have the same en­ergy at the top of the va­lence band, but very dif­fer­ent cur­va­ture. And ac­cord­ing to the ear­lier sub­sec­tion 6.22.3, that means that they have dif­fer­ent ef­fec­tive mass. Physi­cists there­fore speak of light holes and heavy holes to keep the two types of quan­tum states apart. Typ­i­cally even the heavy holes have ef­fec­tive masses less than the true elec­tron mass, [28, pp. 214-216]. Di­a­mond is an ex­cep­tion.

The spec­trum of sil­i­con is not that dif­fer­ent from ger­ma­nium. How­ever, the bot­tom of the con­duc­tion band is now on the line from the ori­gin $\Gamma$ to the point X in fig­ure 6.29.


Key Points
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Sil­i­con and ger­ma­nium have the same crys­tal struc­ture as di­a­mond. Gal­lium ar­senide has a gen­er­al­ized ver­sion, called the zinc blende struc­ture.

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The spec­tra of true three-di­men­sion­al crys­tals are con­sid­er­ably more com­plex than those of the one-di­men­sion­al Kro­nig & Pen­ney model.

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In three di­men­sions, the pe­riod turns into three prim­i­tive trans­la­tion vec­tors.

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The first Bril­louin zone be­comes three-di­men­sion­al.

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There are light holes and heavy holes at the top of the va­lence band of typ­i­cal semi­con­duc­tors.