6.24 The P-N Junc­tion

The p-n junc­tion is the work horse of semi­con­duc­tor ap­pli­ca­tions. This sec­tion ex­plains its phys­i­cal na­ture, and why it can act as a cur­rent rec­ti­fier, among other things.

Fig­ure 6.33: The p-n junc­tion in ther­mal equi­lib­rium. Top: en­ergy spec­tra. Quan­tum states with elec­trons in them are in red. The mean elec­tro­sta­tic en­ergy of the elec­trons is in green. Be­low: Phys­i­cal schematic of the junc­tion. The dots are con­duc­tion elec­trons and the small cir­cles holes. The en­cir­cled plus signs are donor atoms, and the en­cir­cled mi­nus signs ac­cep­tor atoms. (Donors and ac­cep­tors are not as reg­u­larly dis­trib­uted, nor as densely, as this greatly sim­pli­fied schematic sug­gests.)

A p-n junc­tion is cre­ated by dop­ing one side of a semi­con­duc­tor crys­tal n type and the other side p type. As il­lus­trated at the bot­tom of fig­ure 6.33, the n side has a ap­pre­cia­ble amount of con­duc­tion elec­trons, shown as black dots. These elec­trons have been pro­vided by donor atoms. The donor atoms, hav­ing given up one of their neg­a­tively charged elec­trons, have be­come pos­i­tively charged and are shown as en­cir­cled plus signs.

The p side has a ap­pre­cia­ble num­ber of holes, quan­tum states that have lost their elec­trons. The holes are shown as small cir­cles in the fig­ure. Since a neg­a­tively charged elec­tron is miss­ing at a hole, the hole be­haves as a pos­i­tively charged par­ti­cle. The miss­ing elec­trons have been ab­sorbed by ac­cep­tor atoms. These atoms have there­fore ac­quired a neg­a­tive charge and are shown by en­cir­cled mi­nus signs.

The atoms are stuck in the crys­tal and can­not move. Elec­tri­cal con­duc­tion takes place by means of mo­tion of the elec­trons and holes. But un­der nor­mal con­di­tions, sig­nif­i­cant elec­tri­cal con­duc­tion can only oc­cur in one di­rec­tion. That makes the p-n junc­tion into a “diode,” a cur­rent rec­ti­fier.

To see the ba­sic rea­son is not dif­fi­cult. In the so-called for­ward di­rec­tion that al­lows a sig­nif­i­cant cur­rent, both the elec­trons in the n side and the holes in the p side flow to­wards the junc­tion be­tween the n and p sides. (Note that since elec­trons are neg­a­tively charged, they move in the di­rec­tion op­po­site to the cur­rent.) In the vicin­ity of the junc­tion, the in­com­ing n-side elec­trons can drop into the in­com­ing p-side holes. Phrased more for­mally, the elec­trons re­com­bine with the holes. That can read­ily hap­pen. A for­ward cur­rent flows freely if a suit­able for­ward-bi­ased volt­age is ap­plied.

How­ever, if a re­verse-bi­ased volt­age is ap­plied, then nor­mally very lit­tle cur­rent will flow. For a sig­nif­i­cant cur­rent in the re­verse di­rec­tion, both the elec­trons in the n side and the holes in the p side would have to flow away from the junc­tion. So new con­duc­tion elec­trons and holes would have to be cre­ated near the junc­tion to re­place them. But ran­dom ther­mal mo­tion can cre­ate only a few. There­fore there is neg­li­gi­ble cur­rent.

While this sim­ple ar­gu­ment ex­plains why a p-n junc­tion can act as a diode, it is not suf­fi­cient. It does not ex­plain the true re­sponse of the cur­rent to a volt­age. It also does not ex­plain other ap­pli­ca­tions of p-n junc­tions, such as tran­sis­tors, volt­age sta­bi­liz­ers, light-emit­ting diodes, so­lar cells, etcetera.

It turns out that in the for­ward di­rec­tion, the re­com­bi­na­tion of the in­com­ing elec­trons and holes is se­verely hin­dered by an elec­tro­sta­tic bar­rier that de­vel­ops at the con­tact sur­face be­tween the n-type and p-type ma­te­r­ial. This bar­rier is known as the “built-in po­ten­tial.” It is shown in green in fig­ure 6.33.

Con­sider first the p-n junc­tion in ther­mal equi­lib­rium, when there is no cur­rent. The junc­tion is shown in the lower part of fig­ure 6.33. The n side has an ex­cess amount of con­duc­tion elec­trons. The neg­a­tive charge of these elec­trons is bal­anced by the pos­i­tively charged donor atoms. Sim­i­larly, the p side has an ex­cess amount of holes. The pos­i­tive charge of these holes is bal­anced by the neg­a­tively charged ac­cep­tor atoms.

At the junc­tion, due to ran­dom ther­mal mo­tion the n-side elec­trons would want to dif­fuse into the p side. Sim­i­larly the p-side holes would want to dif­fuse into the n side. But that can­not go on in­def­i­nitely. These dif­fu­sion processes cause a net neg­a­tive charge to flow out of the n side and a net pos­i­tive charge out of the p side. That pro­duces the elec­tro­sta­tic bar­rier; it re­pels fur­ther n-side elec­trons from the p side and p-side holes from the n side.

The bar­rier takes the phys­i­cal form of a dou­ble layer of pos­i­tive charges next to neg­a­tive charges. This layer is called the “space charge re­gion.” It is il­lus­trated in fig­ure 6.33. Dou­ble lay­ers are com­mon at con­tact sur­faces be­tween dif­fer­ent solids. How­ever, the one at the p-n junc­tion is some­what un­usual as it con­sists of ion­ized donor and ac­cep­tor atoms. There are pre­ciously few elec­trons and holes in the space charge re­gion, and there­fore the charges of the donors and ac­cep­tors are no longer off­set by the elec­trons, re­spec­tively holes.

The rea­son for the lack of elec­trons and holes in the space charge re­gion may be un­der­stood from fig­ure 6.32: when the num­bers of elec­trons and holes be­come com­pa­ra­ble, there are not many of ei­ther. The lack of elec­trons and holes ex­plains why the space charge re­gion is also known as the “de­ple­tion layer.”

The dou­ble layer is rel­a­tively thick. It has to be, to com­pen­sate for the fact that the frac­tion of atoms that are donors or ac­cep­tors is quite small. A typ­i­cal thick­ness is 10$\POW9,{-6}$ m, but this can vary greatly with dop­ing level and any ap­plied ex­ter­nal volt­age.

An n-side elec­tron that tries to make it through the space charge re­gion is strongly pulled back by the pos­i­tive donors be­hind it and pushed back by the neg­a­tive ac­cep­tors in front of it. There­fore there is a step-up in the elec­tro­sta­tic po­ten­tial en­ergy of an elec­tron go­ing through the re­gion. This in­crease in po­ten­tial en­ergy is shown in green in fig­ure 6.33. It raises the elec­tron en­ergy lev­els in the p side rel­a­tive to the n side. In par­tic­u­lar, it makes the chem­i­cal po­ten­tials, or Fermi lev­els, of the two sides equal. It has to do so; dif­fer­ences in chem­i­cal po­ten­tial pro­duce net elec­tron dif­fu­sion, sec­tion 6.16. For the dif­fu­sion to stop, the chem­i­cal po­ten­tial must be­come every­where the same.

Fig­ure 6.34: Schematic of the op­er­a­tion of an p-n junc­tion.

There is still some flow of elec­trons and holes through the junc­tion, even in the ab­sence of net cur­rent. It is due to ran­dom ther­mal mo­tion. To sim­plify its de­scrip­tion, it will be as­sumed that there is no sig­nif­i­cant re­com­bi­na­tion of elec­trons and holes while they pass through the space charge re­gion, nor cre­ation of new elec­trons and holes. That is a stan­dard as­sump­tion, but by no means triv­ial. It re­quires great pu­rifi­ca­tion of the semi­con­duc­tor. Crys­tal de­fects can act as “re­com­bi­na­tion cen­ters,” lo­ca­tions that help the elec­trons and holes re­com­bine. For ex­am­ple, if you try to sim­ply press sep­a­rate n and p crys­tals to­gether to cre­ate a p-n junc­tion, it will not work. It will have far too many de­fects where the crys­tals meet. A proper re­com­bi­na­tion of elec­trons and holes should take place near the junc­tion, but mostly out­side the space charge re­gion.

Con­sider now first the ther­mal flow of elec­trons and holes through the junc­tion when there is no net cur­rent. It is sketched in fig­ure 6.34a. All those n-side elec­trons would love to dif­fuse into the p side, but the elec­tro­sta­tic bar­rier is hold­ing them back. Only very few elec­trons have enough en­ergy to make it through. The re­quired amount of en­ergy is the elec­tro­sta­tic en­ergy in­crease over the junc­tion. That en­ergy will be called $V_{\rm {j}}$. For n-side elec­trons to make it through the bar­rier, they need to have at least that much en­ergy above the bot­tom of the n-side con­duc­tion band. The rel­a­tive amount of elec­trons at those en­ergy lev­els is pri­mar­ily de­ter­mined by the Maxwell-Boltz­mann fac­tor (6.33). It im­plies that there are a fac­tor $e^{-V_{\rm {j}}/{k_{\rm B}}T}$ less elec­trons per quan­tum state with the ad­di­tional en­ergy $V_{\rm {j}}$ than there are at the bot­tom of the con­duc­tion band.

The cross­ings of these few very lucky elec­trons pro­duce a minis­cule cur­rent through the junc­tion. It is in­di­cated as $j_{\rm {e,maj}}$ in fig­ure 6.34a. The elec­trons are called the ma­jor­ity car­ri­ers in the n side be­cause there are vir­tu­ally no holes in that side to carry cur­rent. Note also that the fig­ure shows the neg­a­tive cur­rents for elec­trons, be­cause that gives the di­rec­tion that the elec­trons ac­tu­ally move. The cur­rents in this dis­cus­sion will be as­sumed to be per unit junc­tion area, which ex­plains why the sym­bol $j$ is used in­stead of $I$. A junc­tion twice as large pro­duces dou­ble the cur­rent, all else be­ing the same. All else be­ing the same in­cludes ig­nor­ing edge ef­fects.

The minis­cule cur­rent of the n-side ma­jor­ity elec­trons is bal­anced by an equally minis­cule but op­po­site cur­rent $j_{\rm {e,min}}$ pro­duced by p-side mi­nor­ity elec­trons that cross into the n side. Al­though the p side has very few con­duc­tion band elec­trons, the num­ber of elec­trons per state is still the same as that of n-side elec­trons with enough en­ergy to cross the bar­rier. And note that for the p-side elec­trons, there is no bar­rier. If they dif­fuse into the space charge re­gion, the elec­tro­sta­tic po­ten­tial will in­stead help them along into the n side.

For holes the story is equiv­a­lent. Be­cause they have the op­po­site charge from the elec­trons, the same bar­rier that keeps the n-side elec­trons out of the p side also keeps the p-side holes out of the n side.

The bot­tom line is that there is no net cur­rent. And there should not be; oth­er­wise you would have a bat­tery that worked for free. Bat­ter­ies must be pow­ered by a chem­i­cal re­ac­tion.

But now sup­pose that a for­ward-bias ex­ter­nal volt­age $\varphi$ is ap­plied that low­ers the bar­rier by an amount $e\varphi_{\rm {j}}$. What hap­pens then is shown in fig­ure 6.34b. The n-side ma­jor­ity elec­trons will now come pour­ing over the low­ered bar­rier, and so will the p-side ma­jor­ity holes. In­deed, the Maxwell-Boltz­mann fac­tor for the ma­jor­ity car­ri­ers that can get through the bar­rier in­creases by a fac­tor $e^{e\varphi_{\rm {j}}/{k_{\rm B}}T}$. That is a very large fac­tor if the volt­age change is big­ger than about 0.025 volt, since ${k_{\rm B}}T$ is about 0.025 eV at nor­mal tem­per­a­tures. The cur­rents of ma­jor­ity car­ri­ers ex­plode, as sketched in the fig­ure. And there­fore, so does the net cur­rent.

The cur­rents of mi­nor­ity car­ri­ers do not change ap­pre­cia­bly. What­ever mi­nor­ity car­ri­ers dif­fuse into the space charge re­gion still all pass through it. Note that the Fermi lev­els of the n and p sides do no longer match up when there is a cur­rent. If there is a cur­rent, the sys­tem is not in ther­mal equi­lib­rium.

Fig­ure 6.34c shows the case that a re­verse bias volt­age is ap­plied. The re­verse volt­age in­creases the bar­rier for the ma­jor­ity car­ri­ers. The num­ber that still have enough en­ergy to cross the junc­tion gets dec­i­mated to es­sen­tially zero. All that re­mains is a resid­ual small re­verse cur­rent of mi­nor­ity car­ri­ers through the junc­tion.

Based on this dis­cus­sion, it is straight­for­ward to write a ball­park ex­pres­sion for the net cur­rent through the junc­tion:

$$\fbox{$\displaystyle j = j_0 e^{e\varphi_{\rm{j}}/k_{\rm B}T} - j_0 $} %$$ (6.37)

The fi­nal term is the net re­verse cur­rent due to the mi­nor­ity car­ri­ers. Ac­cord­ing to the above dis­cus­sion, that cur­rent does not change with the ap­plied volt­age. The other term is the net for­ward cur­rent due to the ma­jor­ity car­ri­ers. Ac­cord­ing to the above dis­cus­sion, it dif­fers from the mi­nor­ity cur­rent pri­mar­ily by a Maxwell-Boltz­mann ex­po­nen­tial. The en­ergy in the ex­po­nen­tial is the elec­tro­sta­tic en­ergy due to the ex­ter­nal volt­age dif­fer­ence across the junc­tion.

For for­ward bias the ex­po­nen­tial ex­plodes, pro­duc­ing sig­nif­i­cant cur­rent. For re­verse bias, the ex­po­nen­tial is es­sen­tially zero and only the small re­verse mi­nor­ity cur­rent is left.

Equa­tion (6.37) is known as the “Shock­ley diode equa­tion.” It works well for ger­ma­nium but not quite that well for sil­i­con. Sil­i­con has a much larger band gap. That makes the mi­nor­ity cur­rents much smaller still, which is good. But the cor­re­spond­ingly small re­versed-bi­ased and slightly for­ward-bi­ased cur­rents are sen­si­tive to de­ple­tion layer elec­tron-hole gen­er­a­tion, re­spec­tively re­com­bi­na­tion. A fudge fac­tor called the “ide­al­ity fac­tor” is of­ten added to the ar­gu­ment of the ex­po­nen­tial to im­prove agree­ment.

Even for ger­ma­nium, the Shock­ley diode equa­tion ap­plies only over a lim­ited range. The equa­tion does not in­clude the re­sis­tance of the semi­con­duc­tor. If the cur­rent in­creases rapidly, the volt­age drop due to re­sis­tance does too, and it should be added to the volt­age drop $\varphi_{\rm {j}}$ over the junc­tion. That will even­tu­ally make the cur­rent ver­sus volt­age re­la­tion lin­ear in­stead of ex­po­nen­tial. And if the re­verse volt­age is too large, phe­nom­ena dis­cussed in sec­tion 6.26 show up.

Key Points

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The p-n junc­tion is the in­ter­face be­tween an n-type and a p-type side of a semi­con­duc­tor crys­tal.

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Un­der nor­mal con­di­tions, it will only con­duct a sig­nif­i­cant cur­rent in one di­rec­tion, called the for­ward di­rec­tion.

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In the for­ward di­rec­tion both the n-side elec­trons and the p-side holes move to­wards the junc­tion.

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The Shock­ley diode equa­tion de­scribes the cur­rent ver­sus volt­age re­la­tion of p-n junc­tions, but only in a lim­ited range.

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At the junc­tion a space-charge re­gion ex­ists. It pro­vides a bar­rier for the ma­jor­ity car­ri­ers. How­ever, it ac­cel­er­ates the mi­nor­ity car­ri­ers pass­ing through the junc­tion.