6.13 Fermi-Dirac Dis­tri­b­u­tion

The pre­vi­ous sec­tions dis­cussed the ground state of a sys­tem of fermi­ons like elec­trons. The ground state cor­re­sponds to ab­solute zero tem­per­a­ture. This sec­tion has a look at what hap­pens to the sys­tem when the tem­per­a­ture be­comes greater than zero.

For nonzero tem­per­a­ture, the av­er­age num­ber of fermi­ons $\iota^{\rm {f}}$ per sin­gle-par­ti­cle state can be found from the so-called

$$\fbox{$\displaystyle \mbox{Fermi-Dirac distribution:}\quad... ...c{1}{e^{({\vphantom' E}^{\rm p}- \mu)/{k_{\rm B}}T} + 1} $} %$$ (6.19)

This dis­tri­b­u­tion is de­rived in chap­ter 11. Like the Bose-Ein­stein dis­tri­b­u­tion for bosons, it de­pends on the en­ergy ${\vphantom' E}^{\rm p}$ of the sin­gle-par­ti­cle state, the ab­solute tem­per­a­ture $T$, the Boltz­mann con­stant $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.38 10$\POW9,{-23}$ J/K, and a chem­i­cal po­ten­tial $\mu$. In fact, the math­e­mat­i­cal dif­fer­ence be­tween the two dis­tri­b­u­tions is merely that the Fermi-Dirac dis­tri­b­u­tion has a plus sign in the de­nom­i­na­tor where the Bose-Ein­stein one has a mi­nus sign. Still, that small change makes for very dif­fer­ent sta­tis­tics.

The biggest dif­fer­ence is that $\iota^{\rm {f}}$ is al­ways less than one: the Fermi-Dirac dis­tri­b­u­tion can never have more than one fermion in a given sin­gle-par­ti­cle state. That fol­lows from the fact that the ex­po­nen­tial in the de­nom­i­na­tor of the dis­tri­b­u­tion is al­ways greater than zero, mak­ing the de­nom­i­na­tor greater than one.

It re­flects the ex­clu­sion prin­ci­ple: there can­not be more than one fermion in a given state, so the av­er­age per state can­not ex­ceed one ei­ther. The Bose-Ein­stein dis­tri­b­u­tion can have many bosons in a sin­gle state, es­pe­cially in the pres­ence of Bose-Ein­stein con­den­sa­tion.

Note in­ci­den­tally that both the Fermi-Dirac and Bose-Ein­stein dis­tri­b­u­tions count the dif­fer­ent spin ver­sions of a given spa­tial state as sep­a­rate states. In par­tic­u­lar for elec­trons, the spin-up and spin-down ver­sions of a spa­tial state count as two sep­a­rate states. Each can hold one elec­tron.

Con­sider now the sys­tem ground state that is pre­dicted by the Fermi-Dirac dis­tri­b­u­tion. In the limit that the tem­per­a­ture be­comes zero, sin­gle-par­ti­cle states end up with ei­ther ex­actly one elec­tron or ex­actly zero elec­trons. The states that end up with one elec­tron are the ones with en­er­gies ${\vphantom' E}^{\rm p}$ be­low the chem­i­cal po­ten­tial $\mu$. Sim­i­larly the states that end up empty are the ones with ${\vphantom' E}^{\rm p}$ above $\mu$.

To see why, note that for ${\vphantom' E}^{\rm p}-\mu$ $\raisebox{.3pt}{$<$}$ 0, in the limit $T\to0$ the ar­gu­ment of the ex­po­nen­tial in the Fermi-Dirac dis­tri­b­u­tion be­comes mi­nus in­fin­ity. That makes the ex­po­nen­tial zero, and $\iota^{\rm {f}}$ is then equal to one. Con­versely, for ${\vphantom' E}^{\rm p}-\mu$ $\raisebox{.3pt}{$>$}$ 0, in the limit $T\to0$ the ar­gu­ment of the ex­po­nen­tial in the Fermi-Dirac dis­tri­b­u­tion be­comes pos­i­tive in­fin­ity. That makes the ex­po­nen­tial in­fi­nite, and $\iota^{\rm {f}}$ is then zero.

The cor­rect ground state, as pic­tured ear­lier in fig­ure 6.11, has one elec­tron per state be­low the Fermi en­ergy ${\vphantom' E}^{\rm p}_{\rm {F}}$ and zero elec­trons per state above the Fermi en­ergy. The Fermi-Dirac ground state can only agree with this if the chem­i­cal po­ten­tial at ab­solute zero tem­per­a­ture is the same as the Fermi en­ergy:

$$\fbox{$\displaystyle \mu = {\vphantom' E}^{\rm p}_{\rm{F}} \quad \mbox{at}\quad T = 0 $}$$ (6.20)

Fig­ure 6.15: A sys­tem of fermi­ons at a nonzero tem­per­a­ture.

Next con­sider what hap­pens if the ab­solute tem­per­a­ture is not zero but a bit larger than that. The story given above for zero tem­per­a­ture does not change sig­nif­i­cantly un­less the value of ${\vphantom' E}^{\rm p}-\mu$ is com­pa­ra­ble to ${k_{\rm B}}T$. Only in a en­ergy range of or­der ${k_{\rm B}}T$ around the Fermi en­ergy does the av­er­age num­ber of par­ti­cles in a state change from its value at ab­solute zero tem­per­a­ture. Com­pare the spec­trum at ab­solute zero tem­per­a­ture as sketched to the right in fig­ure 6.11 to the one at a nonzero tem­per­a­ture shown in fig­ure 6.15. The sharp tran­si­tion from one par­ti­cle per state, red, be­low the Fermi en­ergy to zero par­ti­cles per state, grey, above it smooths out a bit. As the wave num­ber space to the left in fig­ure 6.15 il­lus­trates, at nonzero tem­per­a­ture a typ­i­cal sys­tem en­ergy eigen­func­tion has a few elec­trons slightly be­yond the Fermi sur­face. Sim­i­larly it has a few holes (states that have lost their elec­tron) im­me­di­ately be­low the Fermi sur­face.

Put in phys­i­cal terms, some elec­trons just be­low the Fermi en­ergy pick up some ther­mal en­ergy, which gives them an en­ergy just above the Fermi en­ergy. The af­fected en­ergy range, and also the typ­i­cal en­ergy that the elec­trons in this range pick up, is com­pa­ra­ble to ${k_{\rm B}}T$.

You may at first hardly no­tice the ef­fect in the wave num­ber space shown in fig­ure 6.15. And that fig­ure greatly ex­ag­ger­ates the ef­fect to en­sure that it is vis­i­ble at all. Re­call the ball­park Fermi en­ergy given ear­lier for cop­per. It was equal to a ${k_{\rm B}}T$ value for an equiv­a­lent tem­per­a­ture of 33 000 K. Since the melt­ing point of cop­per is only 1 356 K, ${k_{\rm B}}T$ is still neg­li­gi­bly small com­pared to the Fermi en­ergy when cop­per melts. To good ap­prox­i­ma­tion, the elec­trons al­ways re­main like they were in their ground state at 0 K.

One of the mys­ter­ies of physics be­fore quan­tum me­chan­ics was why the va­lence elec­trons in met­als do not con­tribute to the heat ca­pac­ity. At room tem­per­a­ture, the atoms in typ­i­cal met­als were known to have picked up an amount of ther­mal en­ergy com­pa­ra­ble to ${k_{\rm B}}T$ per atom. Clas­si­cal physics pre­dicted that the va­lence elec­trons, which could ob­vi­ously move in­de­pen­dently of the atoms, should pick up a sim­i­lar amount of en­ergy per elec­tron. That should in­crease the heat ca­pac­ity of met­als. How­ever, no such in­crease was ob­served.

The Fermi-Dirac dis­tri­b­u­tion ex­plains why: only the elec­trons within a dis­tance com­pa­ra­ble to ${k_{\rm B}}T$ of the Fermi en­ergy pick up the ad­di­tional ${k_{\rm B}}T$ of ther­mal en­ergy. This is only a very small frac­tion of the to­tal num­ber of elec­trons, so the con­tri­bu­tion to the heat ca­pac­ity is usu­ally neg­li­gi­ble. While clas­si­cally the elec­trons may seem to move freely, in quan­tum me­chan­ics they are con­strained by the ex­clu­sion prin­ci­ple. Elec­trons can­not move to higher en­ergy states if there are al­ready elec­trons in these states.

To dis­cour­age the ab­sence of con­fu­sion, some or all of the fol­low­ing terms may or may not in­di­cate the chem­i­cal po­ten­tial $\mu$, de­pend­ing on the physi­cist: Fermi level, Fermi brim, Fermi en­ergy, and elec­tro­chem­i­cal po­ten­tial. It is more or less com­mon to re­serve Fermi en­ergy to ab­solute zero tem­per­a­ture, but to not do the same for Fermi level or “Fermi brim.” In any case, do not count on it. This book will oc­ca­sion­ally use the term Fermi level for the chem­i­cal po­ten­tial where it is com­mon to do so. In par­tic­u­lar, a Fermi-level elec­tron has an en­ergy equal to the chem­i­cal po­ten­tial.

The term elec­tro­chem­i­cal po­ten­tial needs some ad­di­tional com­ment. The sur­faces of solids are char­ac­ter­ized by un­avoid­able lay­ers of elec­tric charge. These charge lay­ers pro­duce an elec­tro­sta­tic po­ten­tial in­side the solid that shifts all en­ergy lev­els, in­clud­ing the chem­i­cal po­ten­tial, by that amount. Since the charge lay­ers vary, so does the elec­tro­sta­tic po­ten­tial and with it the value of the chem­i­cal po­ten­tial. It would there­fore seem log­i­cal to de­fine some in­trin­sic chem­i­cal po­ten­tial, and add to it the elec­tro­sta­tic po­ten­tial to get the to­tal, or elec­tro­chem­i­cal po­ten­tial.

For ex­am­ple, you might con­sider defin­ing the in­trin­sic chem­i­cal po­ten­tial $\mu_{\rm {i}}$ of a solid as the value of the chem­i­cal po­ten­tial $\mu$ when the solid is elec­tri­cally neu­tral and iso­lated. Now, when you bring dis­sim­i­lar solids at a given tem­per­a­ture into elec­tri­cal con­tact, dou­ble lay­ers of charge build up at the con­tact sur­faces be­tween them. These lay­ers change the elec­tro­sta­tic po­ten­tials in­side the solids and with it their to­tal elec­tro­chem­i­cal po­ten­tial $\mu$.

In par­tic­u­lar, the strengths of the dou­ble lay­ers ad­just so that in ther­mal equi­lib­rium, the elec­tro­chem­i­cal po­ten­tials $\mu$ of all the solids (in­trin­sic plus ad­di­tional elec­tro­sta­tic con­tri­bu­tion due to the changed sur­face charge lay­ers) are equal. They have to; solids in elec­tri­cal con­tact be­come a sin­gle sys­tem of elec­trons. A sin­gle sys­tem should have a sin­gle chem­i­cal po­ten­tial.

Un­for­tu­nately, the as­sumed in­trin­sic chem­i­cal po­ten­tial in the above de­scrip­tion is a some­what du­bi­ous con­cept. Even if a solid is un­charged and iso­lated, its chem­i­cal po­ten­tial is not a ma­te­r­ial prop­erty. It still de­pends un­avoid­ably on the sur­face prop­er­ties: their con­t­a­m­i­na­tion, rough­ness, and an­gu­lar ori­en­ta­tion rel­a­tive to the atomic crys­tal struc­ture. If you men­tally take a solid at­tached to other solids out to iso­late it, then what are you to make of the con­di­tion of the sur­faces that were pre­vi­ously in con­tact with other solids?

Be­cause of such con­cerns, nowa­days many physi­cists dis­dain the con­cept of an in­trin­sic chem­i­cal po­ten­tial and sim­ply re­fer to $\mu$ as the chem­i­cal po­ten­tial. Note that this means that the ac­tual value of the chem­i­cal po­ten­tial de­pends on the de­tailed con­di­tions that the solid is in. But then, so do the elec­tron en­ergy lev­els. The lo­ca­tion of the chem­i­cal po­ten­tial rel­a­tive to the spec­trum is well de­fined re­gard­less of the elec­tro­sta­tic po­ten­tial.

And the chem­i­cal po­ten­tials of solids in con­tact and in ther­mal equi­lib­rium still line up.

The Fermi-Dirac dis­tri­b­u­tion is also known as the “Fermi fac­tor.” Note that in proper quan­tum terms, it gives the prob­a­bil­ity that a state is oc­cu­pied by an elec­tron.

Key Points

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The Fermi-Dirac dis­tri­b­u­tion gives the num­ber of elec­trons, or other fermi­ons, per sin­gle-par­ti­cle state for a macro­scopic sys­tem at a nonzero tem­per­a­ture.

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Typ­i­cally, the ef­fects of nonzero tem­per­a­ture re­main re­stricted to a, rel­a­tively speak­ing, small num­ber of elec­trons near the Fermi en­ergy.

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These elec­trons are within a dis­tance com­pa­ra­ble to ${k_{\rm B}}T$ of the Fermi en­ergy. They pick up a ther­mal en­ergy that is also com­pa­ra­ble to ${k_{\rm B}}T$.

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Be­cause of the small num­ber of elec­trons in­volved, the ef­fect on the heat ca­pac­ity can usu­ally be ig­nored.

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When solids are in elec­tri­cal con­tact and in ther­mal equi­lib­rium, their (elec­tro)chem­i­cal po­ten­tials / Fermi lev­els / Fermi brims / what­ever line up.