Semiclassical Model of Electron Dynamics. 8.1 Description of the Semiclassical Model

Transcription

1 8. Descrpton of the Semclasscal Model Electrons n crystallne solds assume loch wave functons. The semclasscal model deals wth the dynamcs of loch electrons. Drude assumed that electrons collde wth the fxed ons. Ths pcture cannot account for very long mean free paths that had been found n metals, as well as ther temperature dependence. On the other hand, the loch theory would have predcted nfnte conductvty snce the mean velocty of a loch state, v = /ħ/, s nonvanshng. Ths can be traced to the fact that loch states are statonary solutons to the Schrödnger equaton ncorporatng the full crystal potental. So, the nteracton between the electron and the fxed perodc array of ons has been fully accounted for and the ons can no longer be sources of scatterng. However, no real sold s a perfect crystal. Furthermore, there are always mpurtes, mssng ons or other mperfectons that can scatter electrons. In fact, t s these that lmt the conductvty of metals at very low temperatures. At hgh temperatures, there are thermally excted lattce vbratons producng devatons from the perfect crystal structure, whch can scatter electrons and lmt conductvty. Whle the above reveals that Drude s pcture of electron-on scatterng s napproprate, by substtutng the scatterng events by the realstc ones defect and phonons, hs approach for formulatng the electron dynamcs s stll vald. The formulaton presented below descrbes the moton of the loch electrons n between collsons. The semclasscal model predcts how, n the absence of collsons, the poston r and wave vector of an electron evolve n the presence of externally appled electrc and magnetc felds, assumng nowledge of the electron s band structure, n, whch supposedly fully accounted for the crystal feld. In the course of tme, wth the presence of external electrc and magnetc felds Er,t and Hr,t, the poston, wave vector, and band ndex are taen to evolve accordng to the followng rules:. The band ndex n s a constant of moton. The semclasscal model gnores the possblty of nterband transton.. The tme evoluton of the poston and wave vector of an electron wth band ndex n are determned by the equatons of moton: dr n vn. 8.a dt d e[ E r, t v r, t]. 8.b dt We shall dscuss the orgn of these equatons below. 3. The wave vector of an electron s only defned to wthn an addtve recprocal lattce vector K. Therefore, all dstnctve wave vectors for a sngle band le n a sngle prmtve cell of the recprocal lattce. In thermal equlbrum, the contrbuton to the electronc densty from those electrons n the nth band wth wave vectors n the nfntesmal volume element d n the -space s gven by the usual Ferm dstrbuton: by OKC Tsu based on A&M

2 d f n d / 4 exp[ / n T ] 8.c Ponts to Note: - One should always bear n mnd that very large felds electrc or magnetc may cause nterband transtons. You are referred to A&M Ch. for the dscusson. - At or near equlbrum, bands wth all energes many T above the Ferm energy wll be unoccuped. As for those bands many T below the Ferm energy, we wll later see that they can be gored for consderaton of electronc transport propertes snce they are completely flled. Therefore, one often only needs to consder energy bands wthn a range ~ T about the Ferm energy. Ths greatly reduce the number of bands needed to be consdered. For free electrons: d dt dr dt m 8.a e[ E r, t v r, t]. 8.b The equatons of moton 8. for loch electrons wthn each band are the same as those of free electrons 8. f we adopt ħ /m for the electron energy n. However, whle ħ s the momenton of the free electrons and so eqn. 8.b s anologous to Newton s second law, t s not for the loch electrons. Instead, ħ s the crystal momentum, concerned n the momentum selecton rule for scatterng. To apprecate eqn. 8.b, notce that the total rate of change of an electron s momentum, p, s gven by the total force actng on the electron, ncludng the crystal feld. Snce the RHS of eqn. 8.b accounts for the forces due to the external E and felds wthout the crystal feld, the equaton mples that the rate of change of p ħ s exactly accountable for by the nteractons of the electron wth the crystal feld. elow s the proof: The loch wave functon s: = The expectaton value of the momentum of an electron n state s: p = < -ħ > = 8.c Consder the change n p produced by the external felds: p = 8.d It shows that p has a pece comng from the plane wave components + G n the orgnal wave functon. When the electron state changes by, the ampltudes of the by OKC Tsu based on A&M

3 plane wave components + G change as well. Physcally, ths corresponds to the electron beng reflected by the lattce and thereby causng those ampltudes to alter. If an ncdent electron wth plane wave component of momentum ħ s reflected wth momentum ħ + G, the lattce acqures the momentum -ħg. The momentum transfer to the lattce, p lat, when the state goes over to +G s: 8.e In eqn. 8.e, use has been made of the fact that the porton of the plane wave component + G beng reflected durng the state change s: Substtute eqn. 8.e n 8.d, we fnd for the total momentum change of the system electron + lattce, p tot = pp lat = ħ as mpled by eqn. 8.b. 8. Semclasscal Equaton of Moton n a DC Appled Electrc Feld The eqn. of moton, 8.a, states that the velocty of a semclasscal loch electron s the group velocty of the underlyng wave pacet, whch s perhaps readly comprehensble. ut, eqn. 8.b may not be as straghtforward to justfy. We reason t by consderng conservaton of energy for an electron traversng n the feld E =, whereby n the moton. The tme dervatve of ths equaton s: n t ert = constant 8.3 y Eqn. 8.a, Eqn. 8.4 becomes: n e r = Snce v n s generally non-zero, we must have v n [ħd/dt e] = ħd/dt = e = ee, 8.6 whch s eqn. 8.b, wthout a magnetc feld. Next, observe that eqn. 8.6 s not the only condton that gves rse to energy conservaton. For example, 8.5 would stll be by OKC Tsu based on A&M 3

4 satsfed f any term perpendcular to v n s added to the RHS of Eqn It turns out the only approprate term that may be added s ev n. Ths proves the second eqn. of moton. In most of the followng dscussons, we shall tae the electronc equlbrum dstrbuton functon to be that approprate to zero temperature. In metals, fnte temperature effects wll have neglgble nfluence on the propertes dscussed below. The sprt of the followng analyss s smlar to that of the analyss dscussed for the Drude model transport propertes. That s, we shall descrbe collsons n terms of a smple relaxaton-tme approxmaton, and focus mostly on the moton of electrons between collsons as determned by the semclasscal equatons of moton 8.a and b. Flled ands Are Inert A flled band s one n whch all the energes le below the Ferm energy. Such bands cannot contrbute to an electrc or thermal current. To see ths, notce that an nfntesmal phase space volume element d about the pont wll contrbute d/4 3 electrons per unt volume, all wth velocty v=/ħ/ to the current. Summng ths over all n the rlloun zone, we fnd that the total contrbuton to the electrc and energy thermal current denstes from a flled band are: 8.7 oth of these are zero snce the ntegral over any prmtve cell of the gradent of a perodc functon must vansh, and s perodc. Therefore, only partally flled bands need to be consdered n calculatng the electronc propertes of a sold. Ths explans why Drude s assgnment to each atom of a number of conducton electrons equal to ts valence had been successful. Clearly, a sold n whch all bands are completely flled or empty wll be an electrcal nsulator. Snce the number of levels n each band s twce the number of prmtve cells n the crystal due to the two degenerate spn states of electrons, all bands can be flled or empty only n solds wth an even number of electrons per prmtve cell. 8.3 Semclasscal Moton n an Appled DC Electrc Feld In a unform dc electrc feld, the soluton to the semclasscal equaton of moton for 8.b s: eet t Therefore, n tme t, every electron changes ts wave vector by the same amount. Accordngly, by OKC Tsu based on A&M 4

5 eet v t v If the band s completely flled, ths constant shft n the wave vector of all the electrons can have no effect on the electrc current. Ths s n contrast wth the free electron case, where v s proportonal to, and would thus grow lnearly wth tme. Fg. 8. llustrates a typcal plot of v. It s noteworthy that t decreases wth ncreasng near the two edges of the rlloun zone. In other words, the electrons of those states actually decelerate wth the externally appled feld E. Ths extraordnary behavor s a consequence of the addtonal force exerted by the crystal feld, whch, though s no longer explct n the semclasscal model, les bured n t through the dsperson relaton,. Physcally, as an electron approaches a ragg plane, the external electrc feld moves t toward levels n whch t s ncreasngly lely to be raggreflected bac n the opposte drecton. For example, t s just n the vcnty of ragg planes that the plane-wave levels wth dfferent wave vectors are most strongly mxed n the nearly free electron approxmaton. Ths effect leads to the curous observaton that electrons that are close enough to the zone boundary have been found to behave le holes. Fg. 8. and v vs. or vs. tme, va Eqn.8.8 n one dmenson or three dmensons, n a drecton parallel to a recprocal lattce vector that determnes one of the frst zone faces. Holes Here we shall provde a detaled account for how the transport propertes of electrons n some cases can be descrbed by that of postve charges called holes. y Eqn. 8.7, the contrbuton of all the electrons n a gven band to the current densty s: d j e v occuped by OKC Tsu based on A&M 5

6 where the ntegral s over all the occuped levels n the band. ut the ntegral of eqn. 8.0 over the entre band should be zero. So, d d d 0 v, v 4 v 8. 4 zone occuped unoccuped Hence we can equally well wrte Eqn. 8.0 as: j e unoccuped d v It follows that the current produced by occupyng a specfed set of levels wth electrons s precsely the same as the current that would be produced f the specfed levels were unoccuped AND all the other levels n the band were occuped but wth partcles of charge +e. Such fcttous partcles of postve charge occupyng the levels unoccuped by the electrons are called holes. When one chooses to regard a current as beng carred by postve holes rather than by negatve electrons, one should regard those states occuped by electrons to be unoccuped by holes, and vce versa. However, for any gven band, one should never adopt both pctures hole and electron for the charge carrers. Nevertheless, one may regard some bands by the electron pcture, and the other bands by the hole pcture, for one s convenence. Under an appled electrc feld, the unoccuped levels n a band evolve precsely as f they were occuped by real electrons of charge e. That s, f there are appled E and felds, the moton of both electrons and holes s governed by the same equaton: e E v H. 8.3 c Ths s because eqn. 8.3 descrbes how the occuped orbtals evolve wth tme; and the unoccuped orbtals have to evolve n the same manner because a newly occuped orbtal s necessarly accompaned by a newly empted orbtal, etc., whch requres the unoccuped states to evolve n the same manner do the occuped ones. see pp. 6-7 of A&M. Gven eqn. 8.3, how can holes be dstngushed from electrons? In classcal treatment, the RHS of eqn. 8.3 s the force actng on a charged partcle due to E and, and would have been set equal to the mass of that partcle tmes dv/dt. So, f dv/dt // d/dt, the electron orbt would resemble that of a free partcle wth negatve charge. On the other hand, f dv/dt s // -d/dt, the electron orbt would resemble that of a free partcle of postve charge. It turns out, t s more often the case that dv/dt s drected opposte to d/dt when the orbtal s unoccuped. Ths may be perceved from the followng: At equlbrum or near equlbrum whch s the condton found n most cases of nterest for electron transports, the unoccuped levels usually le near the top of the band. If the band energy has ts maxmum value at 0, say, then f s suffcently close to 0, we may expand about 0. The lnear term n 0 vanshes because 0 s a maxmum pont. If we assume that 0 s a pont wth suffcently hgh symmetry, then by OKC Tsu based on A&M 6

7 by OKC Tsu based on A&M 7, 0 0 A 8.4 where A s a postve constant. Rewrte A as:. * A m 8.5 Hence, for levels near 0, we have, * 0 m v 8.6 So,, * m v dt d a 8.7 Ths shows that the acceleraton, a, of states near the top of a band s opposte to. It follows from eqn. 8.3 that acceleraton of these states are opposte to the electrc force, mang these electrons behave le postve-charges. y regardng those near-top electrons as partcles wth charge +e, Eqn. 8.7 shows that they move as f they have an effectve mass of +m*. We can demonstrate the above pont n a more general way. Consder:, j j j dt d dt d v dt d a 8.8 Then a s opposte to f. for any vector 0 j j j 8.9 Suppose the local maxmum of s at 0. In the vcnty of 0, one can expand as: j j j It follows that Eqn. 8.9 must hold for 0 to qualfy to be a local maxmum. The quantty m* n Eqn. 8.5 determnng the dynamcs of the electrons near band maxma of hgh symmetry s nown as the hole effectve mass. More generally, one defnes an effectve mass tensor by: j j j j j v v M, 8. where the sgn s or + accordng to whether s near a band maxmum holes or mnmum electrons, respectvely. y usng eqn. 8., we can wrte:

8 dv v d a M. 8. dt dt Hence, the equaton of moton 8.3 taes the form: M a e E v H. 8.3 c If the pocet of holes or electrons s small enough, one can replace the mass tensor by ts value at the maxmum mnmum, leadng to a lnear equaton only slghtly more complcated than that for free partcles. Such equatons descrbe qute accurately the dynamcs of electrons and holes n semconductors. 8.4 Semclasscal Moton n a Unform Magnetc Feld In a unform magnetc feld, the semclasscal equatons are r v, 8.4 ev. 8.5 These equatons mmedately evdent that the component of along the feld and the electronc energy are both constants of moton. The latter s because d/dt = F v = ev v = 0. These conservaton laws dctate that electronc orbts n -space are curves gven by the ntersecton of surfaces of constant energy wth planes perpendcular to the magnetc feld Fg. 8.. The sense n whch the orbt s traversed follows by Fg. 8. In the example show, v s pontng away from = 0 and so the partcle s electron-le. Convnce yourself that the sense of rotaton of the orbt reverses when the partcle s hole-le or v s pontng towards = 0. observng that v, beng ~, ponts to hgher energes n the -space. Therefore, closed -space orbts surroundng levels of hgher energes hole orbts are traversed n the opposte sense to closed orbts surroundng levels of lower energy electron orbts. In other words, the sense of orbtal moton determnes whether the carrers are electron- or hole-le under appled magnetc feld. by OKC Tsu based on A&M 8

9 The projecton of the real space orbt n a plane perpendcular to the feld, r r ˆ ˆ r, can be found by tang the vector product of both sdes of Eqn. 8.5 wth a unt vector parallel to the feld. Ths yelds ˆ e r ˆ ˆ r e r, 8.6 whch ntegrates to: elow s a dervaton of Eqn. 8.7 and exploraton of ts physcal meanng. ˆ ˆ r v -ħ/e The above dervaton shows that the projecton of the real space orbt n a plane perpendcular to the feld s smply the -space orbt, rotated through 90 o about the feld drecton and scaled by the factor ħ/e. ut t should be noted that orbts n semclasscal generalzaton need not be closed curves Fg by OKC Tsu based on A&M 9

10 It s of nterest to express the rate at whch the orbt s traversed. Consder an orbt of energy n a partcular plane perpendcular to the feld n the -space Fg. 8.5a. The tme t taes to traverse that porton of the orbt lyng between and s 8.8 y Eqns. 8.4 and 8.5, e e 8.9 Therefore, Eqn. 8.8 can be rewrtten as: 8.30 where / s the component of / perpendcular to the feld,.e., ts projecton n the plane of the orbt. The quantty has the followng geometrcal meanng: In the plane of the orbt for electron energy, we defned the vector to be one perpendcular to the orbt at pont, and connects the pont to a neghborng orbt n the same plane.e., same of energy + Fg. 8.5b. When s very small, we have Fg. 8.3 The projecton of the r-space orbt b n a plane perpendcular to the feld s obtaned from the -space orbt a by scalng wth the factor ħ/e and rotaton through 90 o about the axs determned by. by OKC Tsu based on A&M 0

11 Fg. 8.4 Presentaton n the repeated-zone scheme of a constant-energy surface wth smple cubc symmetry, capable of gvng rse to open orbts n sutably orented magnetc felds. One such orbt s shown for a magnetc feld parallel to [0]. Fg. 8.5 The geometry of orbt dynamcs. a The tme of flght between and s gven by ħ /e d/ / b A secton of a n a plane perpendcular to contanng the orbts wth the same z. The shaded area s A, /. 8.3 Snce / s perpendcular to surfaces of constant energy, the vector / s perpendcular to the orbt and, and hence parallel to. So, Eqn. 8.3 can be wrtten as: 8.3 and Eqn becomes: 8.33 by OKC Tsu based on A&M

12 The ntegral n 8.33 gves the area of the plane between the two neghborng orbts from to Fg. 8.5b, A,. Hence f we tae the lmt of 8.33 as 0, we have Eqn s most often used n cases where the orbt s a smple closed curve, and and are chosen to gve a sngle complete crcut =. The quantty t t s then the perod T of the orbt. If A s the -space area enclosed by the orbt n ts plane, then Eqn gves 8.35 Compared to the free electron result,.e., 8.36 t s customary to defne a cyclotron effectve mass m*, z : Semclasscal Moton n Perpendcular Unform Electrc and Magnetc Felds When we dscussed the Hall effect n the Drude model, we only qualtatvely consdered the electron moton under crossed E and felds. Recall: We balanced the Lorentz force and the electrc force n the transverse drecton and thereby derved that R H = E y /j x = /ne. Here, we dscuss t n the sem-classcal model. When a unform electrc feld E s added wth a perpendcular unform magnetc feld wth E <, Eqn. 8.7 for the projecton of the real space orbt n a plane perpendcular to acqures an addtonal term 8.38 where 8.39 Note that w s the the velocty of the frame of reference n whch the electrc feld vanshes see Jacson s boo pp It follows that electrons n crossed E and felds move n a superposton of crcular orbts as f only the feld s present plus a lnear drft at the velocty w n the r-space see Fg by OKC Tsu based on A&M

13 Smaller radus when v//e. w gger radus when v//- E. Fg. 8.6 E drft of charges +e and e n crossed E and felds. Recall that d/dt = / d/dt = ħv [-e/ħv + E] = -ev E. Snce v s n general not perpendcular to E, d/dt 0 and may vary wth tme. When E and are perpendcular, one can show that the equaton of moton 8.b can be wrtten n the form see below: 8.40 where 8.4 One may apprecate the physcal orgn of these equatons by percevng that Eqn. 8.4 s the energy of a free electron n the reference frame movng wth velocty w, and Eqn s the equaton of moton an electron would have f only the magnetc feld were present, and the band structure gven by. Gven Eqns & 8.4, the -space orbts are gven by ntersectons of surfaces of constant. wth planes perpendcular to the magnetc feld. Note that n cases where E >, the orbt wll be hyperbolc and not closed elow s a dervaton of Eqn. 8.4: by OKC Tsu based on A&M 3

14 Ê The trple product would not be E-hat had E been not perpendcular to. Ths gves Hgh-Feld Hall Effect and Magnetoresstance n theor long lmt In ths secton, we shall analyse the case for crossed E and feld where s very large of the order of T or more so w = E/ << c, and dffers only slghtly from. The lmtng behavor of the current for these cases turn out to be qute dfferent dependng on whether a all occuped or all unoccuped electronc levels le on orbts that are closed curves or b some of the occuped and unoccuped levels le on orbts that do not close on themseleves, but are open n -space. a Cases where all occuped or all unoccuped orbts are closed. We shall tae the hgh magnetc feld condton to mean that these orbts can traverse many tmes between successve collsons. In the free electron case, ths reduces to the condton c >>, where s the relaxaton tme and c s the cyclotron frequency see Chapter. Suppose that the perod T s small compared wth the relaxaton tme,, for every orbt contanng occuped levels. To calculate the current, j, one uses j = ne<v>. Here, <v> s the average velocty of an electron between two collsons, whch may be taen to be from t = to t =. y Eqn. 8.38, we have 8.4 Snce we are consderng for the case where all the occuped orbts are closed, = 0 s bounded n tme. So for suffcently large, the contrbuton from the drft velocty w domnates the average velocty. Ths provdes the long- lmt to the current by OKC Tsu based on A&M 4

15 8.43 ut f t s the unoccuped levels that all le on closed orbts, the correspondng result s 8.44 Eqns and 8.44 suggest that when all relevant orbts are closed, the deflecton of the Lorentz force s so effectve n preventng electrons from acqurng energy from the electrc feld that the unform drft velocty w perpendcular to E gves the domnant contrbuton to the current. Recall that the defnton of the Hall coeffcent s the component of the electrc feld perpendcular to the current, dvded by the product of the magnetc feld and the current densty. We have R H = E transverse /j tot = E/new = /ne: 8.45 Note that the use of the symbol R s to ndcate that the expresson s vald under the condton /T. It s remarable that Eqn gves an dentcal result as that found n the free free electron case, whch preserves the noton that the hgh-feld Hall coeffcent R provdes a valuable measure of the electron or hole densty. It s also remarable that Eqn allows for the possblty of a postve Hall coeffcent. If several bands contrbute to the current densty and each of them has only closed electron or hole orbts, then Eqn or 8.44 holds separately for each band, and the total current densty n the hgh-feld lmt wll be neff e lm j E ˆ, 8.46 / T where n eff s the total densty of electrons mnus the total densty of holes. The correspondng expresson for R s R neff e Furthermore, t can be shown that the correctons to the hgh-feld current denstes Eqns & 8.44 are smaller by a factor of order / c Problem 5 of A&M Ch., the transverse magnetoresstance approaches a feld-ndependent constant n the hgh-feld lmt. b Case Some orbts are open. More specfcally, we refer to those cases where there are energy bands near the Ferm surface contanng open orbts Fg Electrons n open orbts are not forced by the magnetc feld to undergo a perodc moton along the drecton of E. Ths s n contrast wth the case of closed orbts where the electrons spend equal amount of tme travelng along and opposte to E Fg Therefore, the by OKC Tsu based on A&M 5

16 electrons may acqure energy from the E feld. In partcular, f the unbounded orbt stretches n a real space drecton nˆ, one would expect to fnd a fnte contrbuton to the current drected along nˆ and proportonal to the projecton of E along nˆ : Contrbuton from projecton of open orbt parallel to E Contrbuton from components of the open orbt perpendcular to E, and from the closed orbts, f any Fg. 8.7 llustrates the physcal orgn of how an open orbt may gve rse to a net electrc current along nˆ. The possblty of a non-vanshng term not parallel to w n the hghfeld lmt s also n accord wth the general result of Eqn For an electron traversng n an open orbt, the growth n ts wave vector due to the frst term s unbounded, enablng ts contrbuton to j to domnate that of the second term. ecause the rate at whch the orbt s traversed s proportonal to, one expects ths contrbuton to the average velocty to be ndependent of and drected along the real space drecton of the open orbt nˆ. To apprecate the lmtng behavor mpled by Eqn on the hgh-feld magnetoresstance, consder an experment n whch the drecton of current flow does not le along the drecton of the open orbt n real space nˆ whch happens when nˆ does not le parallel to the drecton where the crcut s closed, see Fg When the feld s hgh, by Eqn j can be msalgned from nˆ only f E nˆ = 0. So, f we wrte j, n v Fg by OKC Tsu based on A&M 6

17 Fg. 8.8 the E feld n the followng form: 8.49 where nˆ s a unt vector perpendcular to both nˆ and Ĥ nˆ = nˆ Ĥ, E 0 constant and E 0 as H. y defnton, the magnetoresstance,, s : 8.50 When j s not parallel to nˆ, E E 0 nˆ n the hgh-feld lmt, whch gves the correspondng lmt of : To fnd E 0 /j, subs. Eqn n Eqn. 8.48: Next, tae the dot product wth nˆ on both sdes, and use the fact that nˆ nˆ = 0, one has: whch gves by OKC Tsu based on A&M 7

18 Subs. ths result n Eqn. 8.5, we fnd: 8.55 Snce vanshes n the hgh-feld lmt, ths gves a magnetoresstance that grows wthout lmt wth ncreasng feld, and s proportonal to the square of the sne of the angle between j and the open orbt nˆ snce nˆ s perpendcular to nˆ n the same plane wth j. Therefore, the semclasscal model resolves another anomaly of free electron theory, provdng possble mechansms by whch the magnetoresstance can grow wthout lmt wth ncreasng magnetc feld. by OKC Tsu based on A&M 8