6.12 Con­fine­ment and the DOS

The mo­tion of a sin­gle par­ti­cle in a con­fin­ing box was de­scribed in chap­ter 3.5.9. Non­triv­ial mo­tion in a di­rec­tion in which the box is suf­fi­ciently nar­row can be­come im­pos­si­ble. This sec­tion looks at what hap­pens to the den­sity of states for such a box. The den­sity of states gives the num­ber of sin­gle-par­ti­cle states per unit en­ergy range. It is in­ter­est­ing for many rea­sons. For ex­am­ple, for sys­tems of elec­trons the den­sity of states at the Fermi en­ergy de­ter­mines how many elec­trons in the box pick up ther­mal en­ergy if the tem­per­a­ture is raised above zero. It also de­ter­mines how many elec­trons will be in­volved in elec­tri­cal con­duc­tion if their en­ergy is raised.

By de­f­i­n­i­tion, the den­sity of states ${\cal D}$ gives the num­ber of sin­gle-par­ti­cle states ${\rm d}{N}$ in an en­ergy range from ${\vphantom' E}^{\rm p}$ to ${\vphantom' E}^{\rm p}+{\rm d}{\vphantom' E}^{\rm p}$ as

$${\rm d}N = {\cal V}{\cal D}{ \rm d}{\vphantom' E}^{\rm p}\qquad$$

where ${\cal V}$ is the vol­ume of the box con­tain­ing the par­ti­cles. To use this ex­pres­sion, the size of the en­ergy range ${\rm d}{\vphantom' E}^{\rm p}$ should be small, but still big enough that the num­ber of states ${\rm d}{N}$ in it re­mains large.

For a box that is not con­fin­ing, the den­sity of states is pro­por­tional to $\sqrt{{\vphantom' E}^{\rm p}}$. To un­der­stand why, con­sider first the to­tal num­ber of states $N$ that have en­ergy less than some given value ${\vphantom' E}^{\rm p}$. For ex­am­ple, the wave num­ber space to the left in fig­ure 6.11 shows all states with en­ergy less than the Fermi en­ergy in red. Clearly, the num­ber of such states is about pro­por­tional to the vol­ume of the oc­tant of the sphere that holds them. And that vol­ume is in turn pro­por­tional to the cube of the sphere ra­dius $k$, which is pro­por­tional to $\sqrt{{\vphantom' E}^{\rm p}}$, (6.4), so

$$N = \mbox{(some constant) } \left({\vphantom' E}^{\rm p}\right)^{3/2}$$

This gives the num­ber of states that have en­er­gies less than some value ${\vphantom' E}^{\rm p}$. To get the num­ber of states in an en­ergy range from ${\vphantom' E}^{\rm p}$ to ${\vphantom' E}^{\rm p}+{\rm d}{\vphantom' E}^{\rm p}$, take a dif­fer­en­tial:

$${\rm d}N = \mbox{(some other constant) } \sqrt{{\vphantom' E}^{\rm p}} { \rm d}{\vphantom' E}^{\rm p}$$

So the den­sity of states is pro­por­tional to $\sqrt{{\vphantom' E}^{\rm p}}$. (The con­stant of pro­por­tion­al­ity is worked out in de­riva­tion {D.26}.) This den­sity of states is shown as the width of the en­ergy spec­trum to the right in fig­ure 6.11.

Fig­ure 6.12: Se­vere con­fine­ment in the $y$-di­rec­tion, as in a quan­tum well.

Con­fine­ment changes the spac­ing be­tween the states. Con­sider first the case that the box con­tain­ing the par­ti­cles is very nar­row in the $y$-​di­rec­tion only. That pro­duces a quan­tum well, in which mo­tion in the $y$-​di­rec­tion is in­hib­ited. In wave num­ber space the states be­come spaced very far apart in the $k_y$-​di­rec­tion. That is il­lus­trated to the left in fig­ure 6.12. The red states are again the ones with an en­ergy be­low some given ex­am­ple value ${\vphantom' E}^{\rm p}$, say the Fermi en­ergy. Clearly, now the num­ber of states in­side the red sphere is pro­por­tional not to its vol­ume, but to the area of the quar­ter cir­cle hold­ing the red states. The den­sity of states changes cor­re­spond­ingly, as shown to the right in fig­ure 6.12.

Con­sider the vari­a­tion in the den­sity of states for en­er­gies start­ing from zero. As long as the en­ergy is less than that of the smaller blue sphere in fig­ure 6.12, there are no states at or be­low that en­ergy, so there is no den­sity of states ei­ther. How­ever, when the en­ergy be­comes just a bit higher than that of the smaller blue sphere, the sphere gob­bles up quite a lot of states com­pared to the small box vol­ume. That causes the den­sity of states to jump up. How­ever, af­ter that jump, the den­sity of states does not con­tinue grow like the un­con­fined case. The un­con­fined case keeps gob­bling up more and more cir­cles of states when the en­ergy grows. The con­fined case re­mains lim­ited to a sin­gle cir­cle un­til the en­ergy hits that of the larger blue sphere. At that point, the den­sity of states jumps up again. Through jumps like that, the con­fined den­sity of states even­tu­ally starts re­sem­bling the un­con­fined case when the en­ergy lev­els get high enough.

As shown to the right in the fig­ure, the den­sity of states is piece­wise con­stant for a quan­tum well. To un­der­stand why, note that the num­ber of states on a cir­cle is pro­por­tional to its square ra­dius $k_x^2+k_z^2$. That is the same as $k^2-k_y^2$, and $k^2$ is di­rectly pro­por­tional to the en­ergy ${\vphantom' E}^{\rm p}$. So the num­ber of states varies lin­early with en­ergy, mak­ing its de­riv­a­tive, the den­sity of states, con­stant. (The de­tailed math­e­mat­i­cal ex­pres­sions for the den­sity of states for this case and the ones be­low can again be found in de­riva­tion {D.26}.)

Fig­ure 6.13: Se­vere con­fine­ment in both the $y$ and $z$ di­rec­tions, as in a quan­tum wire.

The next case is that the box is very nar­row in the $z$-​di­rec­tion as well as in the $y$-​di­rec­tion. This pro­duces a quan­tum wire, where there is full free­dom of mo­tion only in the $x$-​di­rec­tion. This case is shown in fig­ure 6.13. Now the states sep­a­rate into in­di­vid­ual lines of states. The smaller blue sphere just reaches the line of states clos­est to the ori­gin. There are no en­ergy states un­til the en­ergy ex­ceeds the level of this blue sphere. Just above that level, a lot of states are en­coun­tered rel­a­tive to the very small box vol­ume, and the den­sity of states jumps way up. When the en­ergy in­creases fur­ther, how­ever, the den­sity of states comes down again: com­pared to the less con­fined cases, no new lines of states are added un­til the en­ergy hits the level of the larger blue sphere. When the lat­ter hap­pens, the den­sity of states jumps way up once again. Math­e­mat­i­cally, the den­sity of states pro­duced by each line is pro­por­tional to the rec­i­p­ro­cal square root of the ex­cess en­ergy above the one needed to reach the line.

Fig­ure 6.14: Se­vere con­fine­ment in all three di­rec­tions, as in a quan­tum dot or ar­ti­fi­cial atom.

The fi­nal pos­si­bil­ity is that the box hold­ing the par­ti­cles is very nar­row in all three di­rec­tions. This pro­duces a quan­tum dot or ar­ti­fi­cial atom. Now each en­ergy state is a sep­a­rate point, fig­ure 6.14. The den­sity of states is now zero un­less the en­ergy sphere ex­actly hits one of the in­di­vid­ual points, in which case the den­sity of states is in­fi­nite. So, the den­sity of states is a set of spikes. Math­e­mat­i­cally, the con­tri­bu­tion of each state to the den­sity of states is a delta func­tion lo­cated at that en­ergy.

(It may be pointed out that very strictly speak­ing, every den­sity of states is a set of delta func­tions. Af­ter all, the in­di­vid­ual states al­ways re­main dis­crete points, how­ever ex­tremely densely spaced they might be. Only if you av­er­age the delta func­tions over a small en­ergy range ${\rm d}{\vphantom' E}^{\rm p}$ do you get the smooth math­e­mat­i­cal func­tions of the quan­tum wire, quan­tum well, and un­con­fined box. It is no big deal, as a per­fect con­fin­ing box does not ex­ist any­way. In real life, en­ergy spikes do broaden out bit; there is al­ways some un­cer­tainty in en­ergy due to var­i­ous ef­fects.)

Key Points

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If one or more di­men­sions of a box hold­ing a sys­tem of par­ti­cles be­comes very small, con­fine­ment ef­fects show up.

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In par­tic­u­lar, the den­sity of states shows a stag­ing be­hav­ior that is typ­i­cal for each re­duced di­men­sion­al­ity.