Sub­sec­tions


6.27 Op­ti­cal Ap­pli­ca­tions

This sec­tion gives a con­cise overview of op­ti­cal physics rang­ing from the x-ray spec­trum of solids to semi­con­duc­tor de­vices such as so­lar cells and light-emit­ting diodes.


6.27.1 Atomic spec­tra

Lone atoms have dis­crete elec­tron en­ergy lev­els, fig­ure 6.19. An elec­tron can tran­si­tion from one of these lev­els to an­other by emit­ting or ab­sorb­ing a pho­ton of light. The en­ergy of the pho­ton is given by the dif­fer­ence in en­ergy be­tween the lev­els. There­fore emit­ted and ab­sorbed pho­tons have very spe­cific en­er­gies, and cor­re­spond­ing very spe­cific fre­quen­cies and wave lengths.

If they are in the vis­i­ble range, they have very spe­cific col­ors. The vis­i­ble range of light cor­re­sponds to pho­tons with en­er­gies from about 1.6 eV (red) to 3.2 eV (vi­o­let). In terms of the wave length of the light, the range is from about 390 nm (vi­o­let) to 760 nm (red).

A ba­sic ex­am­ple is the red pho­ton emit­ted in an $E_3$ to $E_2$ Balmer tran­si­tion of a hy­dro­gen atom, fig­ure 4.8. Its en­ergy is 1.89 eV and its wave length is 656 nm. In gen­eral, when the light emit­ted by ex­cited lone atoms is sent through a prism, it sep­a­rates into a few dis­crete thin beams of spe­cific col­ors. The col­ors are char­ac­ter­is­tic for the type of atom that emit­ted the light.

Lone atoms can also ab­sorb pho­tons from light that passes them by. The same wave lengths that they can emit, they can also ab­sorb. Ab­sorb­ing a pho­ton puts the atoms in an ex­cited state of higher en­ergy. They may then sub­se­quently emit a pho­ton iden­ti­cal to the ab­sorbed one in a dif­fer­ent di­rec­tion. Or they may lose their ex­ci­ta­tion en­ergy in a tran­si­tion be­tween dif­fer­ent en­ergy lev­els, pro­duc­ing a pho­ton of a dif­fer­ent wave length. Or they may lose the en­ergy in col­li­sions. Ei­ther one elim­i­nates the orig­i­nal pho­ton al­to­gether. For ex­am­ple, an ex­cited hy­dro­gen atom in the $E_2$ state might ab­sorb a 656 nm pho­ton to reach the $E_3$ state. Then it may tran­si­tion di­rectly back to the $E_1$ ground state. One 656 nm pho­ton has then been elim­i­nated.

In 1817 Fraun­hofer gave a list of dark lines in the spec­trum of sun­light. His list in­cluded the red $E_2$ to $E_3$ Balmer line, as well as the blue-green $E_2$ to $E_4$ one. It was even­tu­ally dis­cov­ered that light at these fre­quen­cies is ab­sorbed by the hy­dro­gen atoms in the so­lar at­mos­phere. Other lines were due to ab­sorp­tion by other atoms like he­lium, sodium, cal­cium, ti­ta­nium, and iron. The atoms present in the so­lar at­mos­phere could be iden­ti­fied with­out hav­ing to ac­tu­ally go there in a space ship. Since the days of Fraun­hofer, spec­troscopy has be­come one of the most im­por­tant sources of in­for­ma­tion about the large-scale uni­verse.

A typ­i­cal so­lar spec­trum also in­cludes ab­sorp­tion lines due to mol­e­cules like oxy­gen and wa­ter va­por in the at­mos­phere of the earth. Mol­e­c­u­lar spec­tra tend to be more com­pli­cated than atomic ones, es­pe­cially in the in­frared re­gion. That is due to rel­a­tive mo­tion of the dif­fer­ent nu­clei. The spec­tra are also more com­pli­cated due to the larger num­ber of elec­trons in­volved.


Key Points
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Lone atoms and mol­e­cules emit and ab­sorb light at spe­cific wave lengths.

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It al­lows atoms and mol­e­cules to be rec­og­nized in the lab or far out in space.


6.27.2 Spec­tra of solids

Solids have elec­tron en­ergy lev­els arranged into con­tin­u­ous bands, fig­ure 6.19. There­fore solids do not emit dis­crete wave lengths of light like lone atoms do. When light from solids is sent through a prism, the light will spread out into bands of grad­u­ally chang­ing color. That is called broad­band ra­di­a­tion.

To be sure, tran­si­tions in­volv­ing the in­ner atomic elec­trons in solids still pro­duce ra­di­a­tion at dis­crete wave lengths. The rea­son is that the en­er­gies of the in­ner elec­trons are not sig­nif­i­cantly dif­fer­ent from the dis­crete val­ues of the cor­re­spond­ing lone atoms. But be­cause these en­er­gies are so much larger in mag­ni­tude, the pro­duced ra­di­a­tion is in the X-ray range, not in the vis­i­ble light range.


Key Points
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Solids can emit and ab­sorb elec­tro­mag­netic ra­di­a­tion in con­tin­u­ous bands.

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The X-ray range of the in­ner elec­trons is still dis­crete.


6.27.3 Band gap ef­fects

As noted above, the light from solids is not lim­ited to dis­crete wave lengths like that of lone atoms. But it is not true that solids can emit and ab­sorb all wave lengths. In par­tic­u­lar, a per­fect crys­tal of an in­su­la­tor with a large-enough band gap will be trans­par­ent to vis­i­ble light. Take di­a­mond as an ex­am­ple. Its va­lence band is com­pletely filled with elec­trons but its con­duc­tion band is empty, as sketched in fig­ure 6.36. A pho­ton of light with enough en­ergy can use its en­ergy to take take an elec­tron out of the va­lence band and put it into the con­duc­tion band. That leaves a hole be­hind in the va­lence band and elim­i­nates the pho­ton. How­ever, to do this re­quires that the pho­ton has at least the band gap en­ergy of di­a­mond, which is 5.5 eV. The pho­tons of vis­i­ble light have en­er­gies from about 1.6 eV to 3.2 eV. That is not enough. Vis­i­ble light sim­ply does not have enough en­ergy to be ab­sorbed by di­a­mond elec­trons. There­fore a per­fect di­a­mond is trans­par­ent. Vis­i­ble light passes through it un­ab­sorbed.

Fig­ure 6.36: Vicin­ity of the band gap in the elec­tron en­ergy spec­trum of an in­su­la­tor. A pho­ton of light with an en­ergy greater than the band gap can take an elec­tron from the va­lence band to the con­duc­tion band. The pho­ton is ab­sorbed in the process.
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By this rea­son­ing, all per­fect crys­tals will be trans­par­ent if their band gap ex­ceeds 3.2 eV. But ac­tu­ally, the en­ergy of the pho­ton can be some­what less than the band gap and it may still be able to ex­cite elec­trons. The model of en­ergy states for non­in­ter­act­ing elec­trons that un­der­lies spec­tra such as fig­ure 6.36 is not per­fect. The band gap in a spec­trum is re­ally the en­ergy to cre­ate a con­duc­tion band elec­tron and a va­lence band hole that do not in­ter­act. But the elec­tron is neg­a­tively charged, and the hole acts as a pos­i­tive par­ti­cle. The two at­tract each other and can there­fore form a bound state called an “ex­ci­ton.” The en­ergy of the pho­ton needed to cre­ate an ex­ci­ton is less than the band gap by the bind­ing en­ergy of the ex­ci­ton. There is some ad­di­tional slack due to vari­a­tions in this bind­ing en­ergy. In the sim­plest model, the en­ergy lev­els of lone ex­ci­tons would be dis­crete like those of the hy­dro­gen atom. How­ever, they broaden con­sid­er­ably in the less than ideal en­vi­ron­ment of the solid.

If vis­i­ble-light pho­tons do not have enough en­ergy to form elec­tron-hole pairs nor ex­ci­tons, the per­fect crys­tal will be trans­par­ent. If the blue side of the vis­i­ble spec­trum has enough en­ergy to ex­cite elec­trons, the crys­tal will be col­ored red­dish, since those com­po­nents of light will re­main un­ab­sorbed.


Key Points
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A per­fect crys­tal of a solid with a large enough band gap will be trans­par­ent.

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An ex­ci­ton is a bound state of an elec­tron and a hole.


6.27.4 Ef­fects of crys­tal im­per­fec­tions

It should be pointed out that in real life, the col­ors of most non­metals are caused by crys­tal im­per­fec­tions. For ex­am­ple, in ionic ma­te­ri­als there may be a va­cancy where a neg­a­tive ion is miss­ing. Since the va­cancy has a net pos­i­tive charge, an elec­tron can be trapped in­side it. That is called an “F-cen­ter.” Be­cause its en­ergy lev­els are rel­a­tively small, such a cen­ter can ab­sorb light in the vis­i­ble range. Be­sides va­can­cies, chem­i­cal im­pu­ri­ties are an­other com­mon cause of op­ti­cal ab­sorp­tion. A com­plete de­scrip­tion of all the dif­fer­ent types of crys­tal im­per­fec­tions and their ef­fects is be­yond the scope of this book.


Key Points
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The col­ors of most non­metals are caused by crys­tal im­per­fec­tions.

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An elec­tron bound to a va­cancy in a ionic crys­tal is a called an F-cen­ter.


6.27.5 Pho­to­con­duc­tiv­ity

For a non­metal with a suf­fi­ciently nar­row band gap, pho­tons of light may have enough en­ergy to take elec­trons to the con­duc­tion band. Then both the elec­trons in the con­duc­tion band, as well as the holes that they leave be­hind in the va­lence band, can par­tic­i­pate in elec­tri­cal con­duc­tion through the solid. In­creased elec­tri­cal con­duc­tiv­ity due to light is called “pho­to­con­duc­tiv­ity.” It is used for a va­ri­ety of light sens­ing de­vices and for Xe­rox copiers.

Note that ex­ci­tons can­not di­rectly pro­duce elec­tri­cal con­duc­tion, as the com­plete ex­ci­ton is elec­tri­cally neu­tral. How­ever, ex­ci­tons can cre­ate charge car­ri­ers by in­ter­act­ing with crys­tal im­per­fec­tions. Or pho­tons with en­er­gies less than the band gap can do so them­selves. In gen­eral, the mech­a­nisms un­der­ly­ing pho­to­con­duc­tiv­ity are highly com­plex and strongly af­fected by crys­tal im­per­fec­tions.


Key Points
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Pho­to­con­duc­tiv­ity is the in­crease in con­duc­tiv­ity of non­metals when pho­tons of light cre­ate ad­di­tional charge car­ri­ers.


6.27.6 Pho­to­voltaic cells

In the vicin­ity of a p-n junc­tion in a semi­con­duc­tor crys­tal, light can do much more than just in­crease con­duc­tiv­ity. It can cre­ate elec­tric­ity. That is the prin­ci­ple of the “pho­to­voltaic cell.” These cells are also known as so­lar cells if the source of light is sun­light.

To un­der­stand how they work, con­sider the schematic of a p-n junc­tion in fig­ure 6.33. Sup­pose that the crys­tal is ex­posed to light. If the pho­tons of light have more en­ergy than the band gap, they can knock elec­trons out of the va­lence band. For ex­am­ple, sil­i­con has a band gap of about 1.12 eV. And as noted above, the pho­tons of vis­i­ble light have en­er­gies from about 1.6 eV to 3.2 eV. So a typ­i­cal pho­ton of sun­light has plenty of en­ergy to knock a sil­i­con elec­tron out of the va­lence band.

That pro­duces a con­duc­tion band elec­tron and a va­lence band hole. The two will move around ran­domly due to ther­mal mo­tion. If they are close enough to the junc­tion, they will even­tu­ally stum­ble into its space charge re­gion, fig­ure 6.33. The elec­tric field in this re­gion will force­fully sweep elec­trons to the n side and holes to the p side. There­fore, if the p-n junc­tion is ex­posed to a con­tin­u­ous stream of light, there will be a con­tin­u­ous flow of new elec­trons to the n side and new holes to the p side. This cre­ates a us­able elec­tric volt­age dif­fer­ence be­tween the two sides: the ex­cess n-side elec­trons are will­ing to pass through an ex­ter­nal load to re­com­bine with the p-side holes.

There are lim­i­ta­tions for the ef­fi­ciency of the cre­ation of elec­tric­ity. The ex­cess en­ergy that the ab­sorbed pho­tons have above the band gap ends up as heat in­stead of as elec­tri­cal power. And pho­tons with in­suf­fi­cient en­ergy to cre­ate elec­tron-hole pairs do not con­tribute. Hav­ing p-n junc­tions with dif­fer­ent band gaps ab­sorb dif­fer­ent wave lengths of the in­com­ing light can sig­nif­i­cantly im­prove ef­fi­ciency.


Key Points
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Pho­to­voltaics is the cre­ation of elec­tric­ity by pho­tons. So­lar cells are an im­por­tant ex­am­ple.


6.27.7 Light-emit­ting diodes

In the pho­to­voltaic ef­fect, light cre­ates elec­tric­ity. But the op­po­site is also pos­si­ble. A cur­rent across a p-n junc­tion can cre­ate light. That is the prin­ci­ple of the “light-emit­ting diode” (LED) and the “semi­con­duc­tor laser.”

Con­sider again the schematic of a p-n junc­tion in fig­ure 6.33. When a for­ward volt­age is ap­plied across the junc­tion, n-side elec­trons stream into the p side. These elec­trons will even­tu­ally re­com­bine with the pre­vail­ing holes in the p side. Sim­ply put, the con­duc­tion elec­trons drop into the va­lence band holes. Sim­i­larly, p-side holes stream into the n side and even­tu­ally re­com­bine with the pre­vail­ing elec­trons at that side. Each re­com­bi­na­tion re­leases a net amount of en­ergy that is at least equal to the band gap en­ergy. In a suit­ably cho­sen semi­con­duc­tor, the en­ergy can come out as light.

As sec­tion 6.22.4 dis­cussed, sil­i­con or ger­ma­nium are not re­ally suit­able. They are what is called “in­di­rect band gap” semi­con­duc­tors. For these the en­ergy is much more likely to come out as heat rather than light. Us­ing var­i­ous tricks, sil­i­con can be made to emit some light, but the ef­fi­ciency is low. LEDs nor­mally use di­rect band gap semi­con­duc­tors. The clas­si­cal di­rect gap ma­te­r­ial is gal­lium ar­senide, which pro­duced the first patented in­frared LED. To emit vis­i­ble light, the band gap should ex­ceed about 1.6 eV. In­deed, as noted ear­lier, the pho­tons of vis­i­ble light range from about 1.6 eV (red) to 3.2 eV (vi­o­let). That re­lates the band gap of the LED to its color. (For in­di­rect gap semi­con­duc­tors a phonon is in­volved, sec­tion 6.22.4, but its en­ergy is small.) Gal­lium ar­senide, with its 1.4 eV di­rect band gap emits in­frared light with an av­er­age wave length of 940 nm. A 1.4 eV pho­ton has a wave length of 885 nm. Di­a­mond, with its 5.5 eV in­di­rect band gap emits some ul­tra­vi­o­let light with an av­er­age wave length of 235 nm. A 5.5 eV pho­ton has a wave length of 225 nm.

By the ad­di­tion of a suit­able op­ti­cal cav­ity, a “diode laser” can be con­structed that emits co­her­ent light. The cav­ity lets the pho­tons bounce a few times around through the re­gion with the con­duc­tion elec­trons and holes. Now it is one of the pe­cu­liar sym­me­tries of quan­tum me­chan­ics that pho­tons are not just good in tak­ing elec­trons out of the va­lence band, they are also good at putting them back in. Be­cause of en­ergy con­ser­va­tion, the lat­ter pro­duces more pho­tons than there were al­ready; there­fore it is called stim­u­lated emis­sion. Of course, bounc­ing the pho­tons around might just get them ab­sorbed again. But stim­u­lated emis­sion can win out over ab­sorp­tion if most elec­trons at the top of the va­lence band have been ex­cited to the bot­tom of the con­duc­tion band. That is called a pop­u­la­tion in­ver­sion. Such a sit­u­a­tion can be achieved us­ing a strong cur­rent across the junc­tion. Un­der these con­di­tions a pho­ton may pro­duce an­other pho­ton through stim­u­lated emis­sion, then the two pho­tons go on to stim­u­late the emis­sion of still more pho­tons, and so on in a run­away process. The re­sult is co­her­ent light be­cause of the com­mon ori­gin of all the pho­tons. The idea of lasers is dis­cussed in more de­tail in chap­ter 7.7.


Key Points
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A LED cre­ates light due to the re­com­bi­na­tion of elec­trons and holes near a p-n junc­tion. Nor­mally, the semi­con­duc­tor has a di­rect band gap.

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A laser diode adds an op­ti­cal cav­ity to cre­ate co­her­ent light.