6.20 In­tro to Elec­tri­cal Con­duc­tion

Some of the ba­sic physics of elec­tri­cal con­duc­tion in met­als can be un­der­stood us­ing a very sim­ple model. That model is a free-elec­tron gas, i.e. non­in­ter­act­ing elec­trons, in a pe­ri­odic box.

The clas­si­cal de­f­i­n­i­tion of elec­tric cur­rent is mov­ing charges. That can read­ily be con­verted to quan­tum terms for non­in­ter­act­ing elec­trons in a pe­ri­odic box. The sin­gle-par­ti­cle en­ergy states for these elec­trons have def­i­nite ve­loc­ity. That ve­loc­ity is given by the lin­ear mo­men­tum di­vided by the mass.

Con­sider the pos­si­bil­ity of an elec­tric cur­rent in a cho­sen $x$-​di­rec­tion. Fig­ure 6.18 shows a plot of the sin­gle-par­ti­cle en­ergy ${\vphantom' E}^{\rm p}$ against the sin­gle-par­ti­cle ve­loc­ity $v^{\rm {p}}_x$ in the $x$-​di­rec­tion. The states that are oc­cu­pied by elec­trons are shown in red. The par­a­bolic outer bound­ary re­flects the clas­si­cal ex­pres­sion ${\vphantom' E}^{\rm p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12{m_{\rm e}}{v^{\rm {p}}}^2$ for the ki­netic en­ergy: for the sin­gle-par­ti­cle states on the outer bound­ary, the ve­loc­ity is purely in the $x$-​di­rec­tion.

Fig­ure 6.18: Con­duc­tion in the free-elec­tron gas model.

In the sys­tem ground state, shown to the left in fig­ure 6.18, no cur­rent will flow, be­cause there are just as many elec­trons that move to­ward neg­a­tive $x$ as there are that move to­wards pos­i­tive $x$. To get net elec­tron mo­tion in the $x$-​di­rec­tion, elec­trons must be moved from states that have neg­a­tive ve­loc­ity in the $x$-​di­rec­tion to states that have pos­i­tive ve­loc­ity. That is in­di­cated to the right in fig­ure 6.18. The asym­met­ric oc­cu­pa­tion of states now pro­duces net elec­tron mo­tion in the pos­i­tive $x$-​di­rec­tion. That pro­duces a cur­rent in the neg­a­tive $x$-​di­rec­tion be­cause of the fact that the charge $\vphantom{0}\raisebox{1.5pt}{$-$}$$e$ of elec­trons is neg­a­tive.

Note that the elec­trons must pick up a bit of ad­di­tional en­ergy when they are moved from states with neg­a­tive ve­loc­ity to states with pos­i­tive ve­loc­ity. That is be­cause the Pauli ex­clu­sion prin­ci­ple for­bids the elec­trons from en­ter­ing the lower en­ergy states of pos­i­tive ve­loc­ity that are al­ready filled with elec­trons.

How­ever, the re­quired en­ergy is small. You might just briefly turn on an ex­ter­nal volt­age source to pro­duce an elec­tric field that gets the elec­trons mov­ing. Then you can turn off the volt­age source again, be­cause once set into mo­tion, the non­in­ter­act­ing elec­trons will keep mov­ing for­ever.

In phys­i­cal terms, it is not re­ally that just a few elec­trons make a big ve­loc­ity change from neg­a­tive to pos­i­tive due to the ap­plied volt­age. In quan­tum me­chan­ics elec­trons are com­pletely in­dis­tin­guish­able, and all the elec­trons are in­volved equally in the changes of state. It is bet­ter to say that all elec­trons ac­quire a small ad­di­tional drift ve­loc­ity $\Delta{v}^{\rm {p}}_x$ in the pos­i­tive $x$-​di­rec­tion. In terms of the wave num­ber space fig­ure 6.17, this shifts the en­tire sphere of oc­cu­pied states a bit to­wards the right, be­cause ve­loc­ity is pro­por­tional to wave num­ber for a free-elec­tron gas.

The net re­sult is still the en­ergy ver­sus ve­loc­ity dis­tri­b­u­tion shown to the right in fig­ure 6.18. Elec­trons at the high­est en­ergy lev­els with pos­i­tive ve­loc­i­ties go up a bit in en­ergy. Elec­trons at the high­est en­ergy lev­els with neg­a­tive ve­loc­i­ties go down a bit in en­ergy. The elec­trons at lower en­ergy lev­els move along to en­sure that there is no more than one elec­tron in each quan­tum state. The fact re­mains that the sys­tem of elec­trons picks up a bit of ad­di­tional en­ergy. (The last sub­sec­tion of de­riva­tion {D.45} dis­cusses the ef­fect of the ap­plied volt­age in more de­tail.)

Con­duc­tion elec­trons in an ac­tual metal wire be­have sim­i­lar to free elec­trons. How­ever, they must move around the metal atoms, which are nor­mally arranged in some pe­ri­odic pat­tern called the crys­tal struc­ture. The con­duc­tion elec­trons will pe­ri­od­i­cally get scat­tered by ther­mal vi­bra­tions of the crys­tal struc­ture, (in quan­tum terms, by phonons), and by crys­tal struc­ture im­per­fec­tions and im­pu­ri­ties. That kills off their or­ga­nized drift ve­loc­ity $\Delta{v}^{\rm {p}}_x$, and a small per­ma­nent elec­tric field is re­quired to re­plen­ish it. In other words, there is re­sis­tance. But it is not a large ef­fect. For one, in macro­scopic terms the con­duc­tion elec­trons in a metal carry quite a lot of charge per unit vol­ume. So they do not have to go fast. Fur­ther­more, con­duc­tion elec­trons in cop­per or sim­i­lar good metal con­duc­tors may move for thou­sands of Ångstroms be­fore get­ting scat­tered, slip­ping past thou­sands of atoms. Elec­trons in ex­tremely pure cop­per at liq­uid he­lium tem­per­a­tures may even move mil­lime­ters or more be­fore get­ting scat­tered. The av­er­age dis­tance be­tween scat­ter­ing events, or col­li­sions, is called the “free path” length $\ell$. It is very large on an atomic scale.

Of course, that does not make much sense from a clas­si­cal point of view. Com­mon sense says that a point-size clas­si­cal elec­tron in a solid should pretty much bounce off every atom it en­coun­ters. There­fore the free path of the elec­trons should be of the or­der of a sin­gle atomic spac­ing, not thou­sands of atoms or much more still. How­ever, in quan­tum me­chan­ics elec­trons are not par­ti­cles with a def­i­nite po­si­tion. Elec­trons are de­scribed by a wave func­tion. It turns out that elec­tron waves can prop­a­gate through per­fect crys­tals with­out scat­ter­ing, much like elec­tro­mag­netic waves can. The free-elec­tron gas wave func­tions adapt to the crys­tal struc­ture, al­low­ing the elec­trons to flow past the atoms with­out re­flec­tion.

It is of some in­ter­est to com­pare the quan­tum pic­ture of con­duc­tion to that of a clas­si­cal, non­quan­tum, de­scrip­tion. In the clas­si­cal pic­ture, all con­duc­tion elec­trons would have a ran­dom ther­mal mo­tion. The av­er­age ve­loc­ity $v$ of that mo­tion would be pro­por­tional to $\sqrt{{k_{\rm B}}T/m_{\rm e}}$, with $k_{\rm B}$ the Boltz­mann con­stant, $T$ the ab­solute tem­per­a­ture, and $m_{\rm e}$ the elec­tron mass. In ad­di­tion to this ran­dom ther­mal mo­tion in all di­rec­tions, the elec­trons would also have a small or­ga­nized drift ve­loc­ity $\Delta{v}^{\rm {p}}_x$ in the pos­i­tive $x$-​di­rec­tion that pro­duces the net cur­rent. This or­ga­nized mo­tion would be cre­ated by the ap­plied elec­tric field in be­tween col­li­sions. When­ever the elec­trons col­lide with atoms, they lose much of their or­ga­nized mo­tion, and the elec­tric field has to start over again from scratch.

Based on this pic­ture, a ball­park ex­pres­sion for the clas­si­cal con­duc­tiv­ity can be writ­ten down. First, by de­f­i­n­i­tion the cur­rent den­sity $j_x$ equals the num­ber of con­duc­tion elec­trons per unit vol­ume $i_{\rm {e}}$, times the elec­tric charge $\vphantom{0}\raisebox{1.5pt}{$-$}$$e$ that each car­ries, times the small or­ga­nized drift ve­loc­ity $\Delta{v}^{\rm {p}}_x$ in the $x$-​di­rec­tion that each has:

$$j_x = - i_{\rm {e}} e \Delta v^{\rm {p}}_x$$ (6.30)

The drift ve­loc­ity $\Delta{v}^{\rm {p}}_x$ pro­duced by the elec­tric field be­tween col­li­sions can be found from New­ton’s sec­ond law as the force on an elec­tron times the time in­ter­val be­tween col­li­sions dur­ing which this force acts and di­vided by the elec­tron mass. The av­er­age drift ve­loc­ity would be half that, as­sum­ing for sim­plic­ity that the drift is to­tally lost in col­li­sions, but the half can be ig­nored in the ball­park any­way. The force on an elec­tron equals $-e{\cal E}_x$ where ${\cal E}_x$ is the elec­tric field due to the ap­plied volt­age. The time be­tween col­li­sions can be com­puted as the dis­tance be­tween col­li­sions, which is the free path length $\ell$, di­vided by the ve­loc­ity of mo­tion $v$. Since the drift ve­loc­ity is small com­pared to the ran­dom ther­mal mo­tion, $v$ can be taken to be the ther­mal ve­loc­ity. The “con­duc­tiv­ity” $\sigma$ is the cur­rent den­sity per unit elec­tric field, so putting it all to­gether,

$$\sigma \sim \frac{i_{\rm e} e^2 \ell}{m_{\rm e}v}$$ (6.31)

Nei­ther the ther­mal ve­loc­ity $v$ nor the free path $\ell$ will be the same for all elec­trons, so suit­able av­er­ages have to be used in more de­tailed ex­pres­sions. The “re­sis­tiv­ity” is de­fined as the rec­i­p­ro­cal of the con­duc­tiv­ity, so as 1/$\sigma$. It is the re­sis­tance of a unit cube of ma­te­r­ial.

For met­als, things are a bit dif­fer­ent be­cause of quan­tum ef­fects. In met­als ran­dom col­li­sions are re­stricted to a small frac­tion of elec­trons at the high­est en­ergy lev­els. These en­ergy lev­els are char­ac­ter­ized by the Fermi en­ergy, the high­est oc­cu­pied en­ergy level in the spec­trum to the left in fig­ure 6.18. Elec­trons of lower en­er­gies do not have empty states nearby to be ran­domly scat­tered into. The ve­loc­ity of elec­trons near the Fermi en­ergy is much larger than the ther­mal value $\sqrt{{k_{\rm B}}T/m_{\rm e}}$, be­cause there are much too few states with ther­mal-level en­er­gies to hold all con­duc­tion elec­trons, sec­tion 6.10. The bot­tom line is that for met­als, in the ball­park for the con­duc­tiv­ity the free path length $\ell$ and ve­loc­ity $v$ of the Fermi-level elec­trons must be used. In ad­di­tion, the elec­tron mass $m_{\rm e}$ may need to be changed into an ef­fec­tive one to ac­count for the forces ex­erted by the crys­tal struc­ture on the elec­trons. That will be dis­cussed in more de­tail in sec­tion 6.22.3.

The clas­si­cal pic­ture works much bet­ter for semi­con­duc­tors, since these have much less con­duc­tion elec­trons than would be needed to fill all the quan­tum states avail­able at ther­mal en­er­gies. The mass cor­rec­tion re­mains re­quired.

Key Points

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The free-elec­tron gas can be used to un­der­stand con­duc­tion in met­als in sim­ple terms.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

In the ab­sence of a net cur­rent the elec­trons are in states with ve­loc­i­ties in all di­rec­tions. The net elec­tron mo­tion there­fore av­er­ages out to zero.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

A net cur­rent is achieved by giv­ing the elec­trons an ad­di­tional small or­ga­nized mo­tion.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

The en­ergy needed to do this is small.

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

In real met­als, the elec­trons lose their or­ga­nized mo­tion due to col­li­sions with phonons and crys­tal im­per­fec­tions. There­fore a small per­ma­nent volt­age must be ap­plied to main­tain the net mo­tion. That means that there is elec­tri­cal re­sis­tance. How­ever, it is very small for typ­i­cal met­als.