Lecture note on Solid State Physics de Haas-van Alphen effect

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1 Ltur not on Solid Stt Phsis d Hs-vn Alphn fft Mstsugu Suzuki nd Itsuko S. Suzuki Stt Univrsit of Nw York t Binghmton Binghmton Nw York 39-6 (April 6 6) ABSTRACT Hr th phsis on th d Hs-vn Alphn (dhva) fft is prsntd. Thr hv bn mn ltur nots (in Wb sits) on th dhva fft. Mn of thm hv bn writtn b thorists who hv no prin on th msurmnt of th dhva fft. On of th uthors (M.S.) hs studid th frqun miing fft (dhva) nd th stti skin fft (Shubnikov-d Hs fft) of bismuth (Bi) s prt of his Ph.D. Thsis (Phsis) (Univrsit of Toko 977) undr th instrution of Prof. Si-ihi Tnum (Ph.D. dvisor). Around 974 Prof. Dvid Shonbrg (th lt) visitd th Univrsit of Toko nd gv n llnt tlk on th dhva fft of oppr t th Phsis Colloquium (Prof Rogo Kubo ws lso prsnt). Whn h plind th dhva priod rltd to th dog s bon h pronound it in Jpns inu no hon. His tlk ws vr imprssiv nd grtl ntrtind th udin of th Phsis Dprtmnt. Bfor his tlk Prof. Shonbrg lso visitd th Institut of Solid Stt Phsis t th Univrsit of Toko. At tht tim M.S. msurd th dhva fft of oppr to min th possibilit of th zon osilltion fft. Prof. Shonbrg gv invlubl suggstions to M.S. on th primnt (unfortuntl this primnt hs fild) nd grtl nourgd M.S. to ontinu to do th dhva primnts. This ltur not is writtn bsd on th prin of M.S. during his Ph.D. work on th dhva fft. Not tht th pionring works on th dhva of Bi wr don b Prof. Shonbrg [Pro. Ro. So. A (936) Pro. Ro. So. A 56 7 (936) Pro. Ro. So. A 7 34 (936)]. Numril lultions (lthough th r vr simpl lultions) r md b Mthmti 5.. For onvnin on progrm is lso givn in th Appndi. Nottions: : Plnk onstnt : vloit of th light -: hrg of ltron m : mss of fr ltron m : lotron mss m: mss of ltron (in thor) ω : lotron frqun ( /( m)) µ B Bohr mgnton ( µ /( )) Φ : quntum fluoid (Φ 7 /.678 Guss m ) B B: mgnti fild l: mgnti lngth ( l / B) T: Tsl ( T 4 O) O unit of th mgnti fild ( Guss) ε F : th Frmi nrg S : trml ross-stionl r of th Frmi surf in pln norml to th mgnti fild.

2 Contnts. Introdution. Frmi surf of Bi. Enrg disprsion rltion. Brillouin zon nd Frmi surf of Bi 3. Thniqus for th msurmnt of dhva 3. Fild modultion mthod 3. Torqu mthod 4. Rsuls of dhva in Bi 4. Rsult from modultion mthod 4. Rsult from torqu d Hs 4.3 Rsult from dhva fft (Bhrgrv) 5. Chng of Frmi nrg s funtion of mgnti fild 6. Thortil bkground 6.. Th dnsit of stts: dgnr of th Lndu lvl 6.. Smilssil quntiztion of orbits in mgnti fild 6.3 Quntum mhnis 6.3. Lndu gug smmtri gug nd gug trnsformtion 6.3. Oprtors in quntum mhnis Shrödingr qution (Lndu gug) Anothr mthod 6.4 Th Zmn splitting of th Lndu lvl du to th spin mgnti momnt 6.5 Numril lultions using Mthmti Enrg disprsion rltion of th Lndu lvl 6.5. Solution of Shrödingr qution (Lndu gug) Wv funtions 7. Gnrl form of th osilltor mgntiztion (Lifshitz-Kosvih) 8. Simpl modl to undrstnd th dhva fft 9. Drivtion of th osilltor bhvior in D modl. Totl nrg vs B. Mgntiztion M vs B. Conlusion REFERENCES Appndi Mthmti progrm. Introdution Th d Hs-vn Alphn (dhva) fft is n osilltor vrition of th dimgnti susptibilit s funtion of mgnti fild strngth (B). Th mthod provids dtils of th trml rs of Frmi surf. Th first primntl obsrvtion of this bhvior ws md b d Hs nd vn Alphn (93). Th hv msurd mgntiztion M of smimtl bismuth (Bi) s funtion of th mgnti fild (B) in high filds t 4. K nd found tht th mgnti susptibilit M/B is priodi funtion of th riprol of th mgnti fild (/B). This phnomnon is obsrvd onl t low

3 tmprturs nd high mgnti filds. Similr osilltor bhvior hs bn lso obsrvd in mgntorsistn (so lld th Shubnikov-d Hs fft). Th dhva phnomnon ws plind b Lndu s dirt onsqun of th quntiztion of losd ltroni orbits in mgnti fild nd thus s dirt obsrvtionl mnifsttion of purl quntum mhnis. Th phnomnon bm of vn grtr intrst nd importn whn Onsgr pointd out tht th hng in /B through singl priod of osilltion ws dtrmind b th rmrkbl simpl rltion P ( ) () F B S whr P is th priod (Guss - ) of th dhva osilltion in /B F is th dhva frqun (Guss) nd S is n trml ross-stionl r of th Frmi surf in pln norml to th mgnti fild. If th z is is tkn long th mgnti fild thn th r of Frmi surf ross stion t hight k z is S(k z ) nd th trml rs S r th vlus of S(k z ) t th k z whr ds ( k z ) / dk z. Thus mimum nd minimum ross stions r mong th trml ons. Sin ltring th mgnti fild dirtion brings diffrnt trml rs into pl ll trml rs of th Frmi surf n b mppd out. Whn thr r two trml ross-stionl r of th Frmi surf in pln norml to th mgnti fild nd ths two priods r nrl qul bt phnomnon of th two priods will b obsrvd. Eh priod must b disntngld through th nlsis of th Fourir trnsform. Fig. Frmi surf of th hol pokt for Bi. Th mgnti fild (dnotd b rrows) is in th YZ pln. Fig. Frmi surf of th ltron () pokt for Bi. Th mjor is of th llipsoid is tiltd b 6.5º from th bistri is. Eprimntll th vlu of S (m - ) n b dtrmind from mor onvnint form 3

4 S P Φ P 7 (Guss - m - )/P(Guss - ) [m - ] () whr P is in unit of Guss - 7 nd Φ ( /.678 Guss m ) is th quntum fluoid. Th dhva fft n b obsrvd in vr pur mtls onl t low tmprturs nd in strong mgnti filds tht stisf ε F >> ω >> k B T. (3) Th first inqulit mns tht th ltron sstm is quntum-mhnill dgnrt vn though s rquird b th sond inqulit th mgnti fild is suffiintl strong. On th othr hnd th obsrvtion of dhva osilltion is dtrmind b B ω 4. (4) B ε F Tht is for th obsrvtion of osilltions th flututions Β in n mgnti fild should b smll nd th ltron dnsit should not b too high bus th priod dpnds on th rtio ω / ε. F. Frmi surf of Bi 3-. Enrg disprsion rltion Bismuth is tpil smimtl. Th modl of th bnd strutur of Bi onsists of st of thr quivlnt ltron llipsoids t th L point nd singl hol llipsoid t th T point (s th Brillouin zon in S.). In on of th ltron llipsoids (-pokt) th nrg E is rltd to th momntum p in th bsn of mgnti fild b E E( ) p m * p (5) EG m (L modl 5 or llipsoidl non-prboli modl) whr E G is th nrg gp to th nt lowr bnd nd m* is th fftiv mss tnsor in units of th fr ltron mss m. Th fftiv mss tnsor m * is of th form m m * m m (6) m 4 m 4 3 whr nd 3 rfr to th binr (X) th bistri (Y) nd th trigonl (Z) s rsptivl. Th othr two ltron llipsoids (b nd pokts) r obtind b rottions of ±º bout th trigonl is rsptivl. Th fftiv mss tnsors m b * for th b pokt nd m * for th pokt r givn b ( m m ) m 3m 3 3m4 ± ± 4 4 3( m m ) 3m m m4 m b* ±. (7) 4 4 3m4 m4 ± m3 4

5 For th hols th nrg momntum rltionship in th bsn of mgnti fild is tkn to b E E p M * p (8) m whr E is th nrg of th top of th hol bnd rltiv to th bottom of th ltron bnd nd th fftiv mss tnsor M* for th hol pokt is M M * M. (9) M 3 Th Frmi surf onsists of on hol llipsoid of rvolution nd thr ltron llipsoids. On ltron llipsoid hs its mjor is tiltd b smll positiv ngl ( 6.5º) from th bistri dirtion. Tbl I Bi bnd prmtrs usd b Tkno nd Kwmur 8. Brillouin zon nd Frmi surf of Bi Th Brillouin zon nd th Frmi surf of Bi r shown hr. 5

6 Fig.3 Brillouin zon of bismuth 3- Fig.4 Frmi surf of bismuth: binr is (X) bistri (Y) nd trigonl (Z). b r th ltron pokt (Frmi surfs) nd h is th hol pokt. 6

7 3. Thniqus for th msurmnt of dhva Thr r two mjor thniqus to msur th dhva osilltions: () fild modultion mthod using lok-in mplifir. () torqu mthod. Bus of th Frmi surf in Bi is so smll th dhva fft n b obsrvd in quit smll filds s low s O t.3 K nd t firl high tmprturs up to or 3 K t filds of fw ko). It is in ft th mtl in whih th dhva fft ws first disovrd nd hv probbl bn mor studid vr sin thn n othr mtl. 3. Fild-modultion mthod Th sstm onsists of dtting oil ompnstion oil nd fild modultion oil. Th stti mgnti fild B (supronduting mgnt or ion or mgnt) is modultd b smll AC fild h osωt (ω is ngulr frqun) gnrtd b th fild modultion oil. Th dirtion of th AC fild is prlll to tht of stti mgnti fild B. Th voltg indud in th pik-up oil is givn b M M v ω { h sin( ωt) h sin(ω t)...} () h h whr h << B. Th signl obtind from th pik-up oil is phs snsitivl dttd t th first hrmoni or sond hrmoni mods with lok-in mplifir. Th DC signl is M M proportionl to ω h for th first-hrmoni mod nd ωh for th sondhrmoni mod. Ths signls r priodi in /B. Th Fourir nlsis lds to th h h dhva frqun F (or th dhva priod P /F). Fig.5 Th blok digrm of th pprtus for th msurmnt of th dhva fft b mns of th fild modultion mthod. 7

8 3. Torqu mthod Whn n trnl mgnti fild is pplid to th smpl thr is torqu on th smpl givn M BV whr M is th omponnt of M prpndiulr to B nd V is th volum. Using this mthod th bsolut vlu of th mgntiztion n b tl dtrmind. Not tht th torqu is qul to zro whn th dirtion of th mgnti fild is prlll to th smmtri dirtion of th smpl. Fig.6 Th blok digrm of th pprtus for msuring th dhva fft b Torqu d Hs mthod Rsults of dhva fft in Bi 4. Rsult from th modultion mthod (Suzuki.9 ) W show tpil mpls of th dhva fft in Bi nd th Fourir sptr for th dh vh priods. 8

9 Fig.7 Th dhva fft of Bi in th YZ pln. T.5 K. This signl orrsponds to th first hrmonis ( M / h ). Fig.8 Th dhva fft of Bi in th YZ pln. T.5 K. Th signl orrsponds to th sond hrmonis ( M/ h ). 4. Rsult of torqu d Hs (Suzuki 9 ) W show tpil mpls of th torqu d Hs in Bi. 9

10 Fig.9 Angulr dpndn of th torqu d Hs in th YZ pln. Th torqu is zro t th smmtr s (Y nd Z). B 5 ko. T 4. K. Fig. Th torqu d Hs in th YZ pln. T.5 K.

11 Fig. Th Fourir sptrum of th dhva osilltion. Th mgnti fild is orintd in th YZ pln. Th Z is orrsponds to º. Th brnhs A B nd C orrspond to th - b- nd -ltron pokts rsptivl. Th brnh E orrsponds to th frqun miing du to th quntum osilltion of th Frmi nrg (s S.5). Fig. Th Fourir sptrum of th dhva osilltion. Th mgnti fild is orintd to mk -36º from th Z is in th YZ pln. Th brnhs A B nd C orrspond to th - b- nd -ltron pokts rsptivl.

12 Fig.3 Th ngulr dpndn of th dhva frqunis in th YZ pln. Th brnhs A B nd C orrspond to th - b- nd -ltron pokts rsptivl. Th dhva frqun F F is pproimtl qul to F 3A nd F D nd F E oinid with F A F BC. Not tht th b- nd - pokts sprt into two brnhs in th rng of th fild ngls from -48º to -7º nd this might b rsult of th ft tht th dirtion of mgnti fild dos not tl li in th YZ pln. Not tht th frqun of α-osilltion is dnotd s F α whr α mns A BC D E A or 3A. 4.3 Rsult of dhva fft in Bi (Bhrgrv 7 ) Tbl II Th summr of rsults of dhva fft in Bi. 7 : binr : bistri 3: trigonl

13 Fig.4 Th ngulr dpndn of ltron dhva priod P in th XY pln for Bi. Th solid lin is fit ssuming n llipsoidl Frmi surf nd using th msurd vlus of priods in th rstl is nd tilt ngl of 6.5º. 7 Fig.5 Th ngulr dpndn of ltron dhva priods in th YZ pln. Th tilt ngl msurd is 6.5±.5º. Th shdd r shows th rgion whr ltron priods wr nvr rportd. Th solid lin is fit using n llipsoidl Frmi surf Chng of Frmi nrg s funtion of mgnti fild Th dimnsion of th Frmi surf of Bi is vr smll omprd with tht of ordinr mtls. Thrfor th quntum numbr of th Lndu lvl t th Frmi nrg hs smll vlu vn t low mgnti fild. Th Frmi nrg vris with mgnti fild in qusi osilltor w sin th Lndu lvl intrvls of th hol nd ltrons r gnrll diffrnt to h othr. Th Frmi nrg is dtrmind from th hrg b nutrlit ondition tht Nh ( B) N ( B) N ( B) N ( B). Th fild dpndn of th Frmi nrg in Bi is shown blow whn B is prlll to th binr bistri nd trigonl s rsptivl. W not tht th dhva frqun miing hs bn obsrvd in Bi b Suzuki t l.. Th Frmi nrg hngs t mgnti filds whr th Lndu lvl rosss th Frmi 3

14 nrg so tht th Frmi nrg shows psudo priodi vrition with th fild. This vrition is rmrkbl vn t low mgnti fild in Bi. Th obsrvd frqun miing is du to this fft. () B // th binr is (X) Fig.6 Th mgnti fild dpndn of th Frmi nrg (B //X T K). Th dottd nd solid lins orrspond to th Lndu lvls of th ltron nd hol rsptivl. Th urv of E F vs B hibits kinks t th filds whr th Lndu lvls ross th Frmi nrg. BCn±: th Lndu lvl of th ltron b- nd pokts with th quntum numbr n nd th spin up () (down (-))-stt. hn±: th Lndu lvl of th hol pokts with th quntum numbr n nd th spin up () (down (-))-stt. E ( n σ ) ω ( n ν sσ ) whr ν s is spin-splitting ftor dfind in S.6.4 nd σ ±. Th prssion of E(n σ)will b disussd ltr. Th ground Lndu lvl is dsribd b ithr Brff 6 modl (dnotd B) or L 5 modl (dnotd b L). (b) B // th bistri (Y) Fig.7 Th mgnti fild dpndn of th Frmi nrg (T K). Mgnti fild is long th Y is (bistri). 9 4

15 () B //th trigonl is (Z) Fig.8 Th mgnti fild dpndn of th Frmi nrg (T K). Mgnti fild is long th Z is (trigonl) Thortil Bkground Th dnsit of stts: dgnr of th Lndu lvl Th ltrons in ubi sstm with sid L r hrtrizd b thir quntum numbr k with omponnts whr k (k k k z ) /L (n n n z ) nd n n nd n z r intgrs. Th nrg of th sstm is givn b E ( k) k m whr m is th mss of ltrons (w ssum m instd of m in th thor for onvnin). Th k sp ontours of onstnt nrg r sphrs nd for givn k n ltron hs vloit givn b v k E( k). () k Wht hppns in mgnti fild to th distribution of orbitls in k sp? Whn mgnti fild B is pplid long th z is th ltron motion in this dirtion is unfftd b this fild but in th ( ) pln th Lorntz for indus irulr motion of th ltrons. Th Lorntz for uss rprsnttiv point in k sp to rott in th (k k ) pln with frqun ω B/m (w us this nottion in this Stion) whr - is th hrg of ltron. This frqun whih is known s th lotron frqun is indpndnt of k so th whol sstm of th rprsnttiv points rott bout n is (prlll to B) through th origin of k sp. This rgulr priodi motion introdus nw quntiztion of th nrg lvls (Lndu lvls) in th (k k ) pln orrsponding to thos of hrmoni osilltor with frqun ω nd nrg ε n ω ( n ) k () m whr k is th mgnitud of th in-pln wv vtor nd th quntum numbr n tks intgr vlus 3... Eh Lndu ring is ssoitd with n r of k sp. Th r S n is th r of th orbit n with th rdius k k S n B kn ( n ). (3) n 5

16 Thus in mgnti fild th r of th orbit in k sp is quntizd. Th r btwn two djnt Lndu rings is B Sn Sn Sn (l: th mgnti lngth) (4) l Fig.9 Quntiztion shm for fr ltrons. Eltron stts r dnotd b points in th k sp in th bsn nd prsn of trnl mgnti fild B. Th stts on h irl r dgnrt. () Whn B thr is on stt pr r (/L). (b) Whn B th ltron nrg is quntizd into Lndu lvls. Eh irl rprsnts Lndu lvl with nrg E ω ( n / ). Th dgnr of quntum numbr n (th numbr of stts) is Sn B L BL D B ρ L or ρ (5) L 6. Smilssil quntiztion of orbits in mgnti fild Th Onsgr-Lifshitz id ws bsd on simpl smi-lssil trtmnt of how ltrons mov in mgnti fild using th Bohr-Sommrfld ondition to quntiz th motion. Th dhva frqun F (i.. th riprol of th priod in /B) is dirtl proportionl to th trml ross-stion r S of th Frmi surf. Th Lgrngin of th ltron in th prsn of ltri nd mgnti fild is givn b L mv q( φ v A) (6) whr m nd q r th mss nd hrg of th prtil. Cnonil momntum: L q p mv A. (7) v Mhnil momntum: q mv p A. (8) n 6

17 Th Hmiltonin: q q H p v L ( mv A) v L mv qφ ( p A) qφ. (9) m Th Hmiltonin formlism uss th vtor potntil A nd th slr potntil φ nd not E nd B dirtl. Th rsult is tht th dsription of th prtil dpnds on th gug hosn. W ssum tht th orbits in mgnti fild r quntizd b th Bohr-Sommrfld rltion q m v k p A p A. () p dr ( n γ ). () γ / for fr ltron. whr q - (>) is th hrg of ltron n is n intgr nd γ is th phs orrtion: p dr k dr A dr ( n γ ). () Th qution of motion of n ltron in mgnti fild is givn b dk v B. (3) dt This mns tht th hng in th vtor k is norml to th dirtion of B nd is lso norml to v (norml to th nrg surf). Thus k must b onfind to th orbit dfind b th intrstion of th Frmi surf with norml to B. Sin v ( / ) k εk d r / dt k r r ) B (4) ( whr r [( )] is th position vtor of th ntr of th orbit (guiding ntr): k k B (5) B In th ompl pln w hv th rltion i / ( ) i( ) ( k ik ). (6) B This mns tht th mgnitud of th position vtor r r ( - ) of th ltron is rltd to tht of th wv vtor k (k k ) b sling ftor η l / B. Th phs of th position vtor is diffrnt from tht of th wv vtor b / for th ltron Frmi surf. l is so-lld mgnti lngth. 7

18 Fig. Th orbitl motion of ltron in th prsn of B (B is dirtd out of pg) in th k-sp is similr to tht in th r-sp but sld b th ftor η nd through /. Not w ssum r in this figur. k dr r B dr B ( r dr) B An Φ. (7) whr ( r d r) (r nlosd within th orbit) n (gomtril rsult) nd Φ is th mgnti flu ontind within th orbit in rl sp Φ B An. On th othr hnd A dr ( A) d B d Φ (8) b th Stoks thorm. Thn w hv p dr Φ Φ Φ ( n γ ). (9) It follows tht th orbit of n ltron is quntizd in suh w tht th flu through it is Φn ( n γ ) Φ( n γ ) (Onsgr rltion) (3) whr Φ is quntum fluoid nd is givn b h 7 Φ.678 Guss m. (3) In th dhva w nd th r of th orbit in th k-sp. W dfin S n (r) s n r nlosd b th orbit in th rl sp (r) nd S n (k) s n r nlosd b th orbit in th k-sp. Thn w hv rltion S n( r) Sn( k) l Sn( k). (3) B Th quntizd mgnti flu is givn b 8

19 Φn BSn( r ) Bl Sn( k) ( n γ ) Φ( n γ ) (33) or B S n ( k ) ( n γ ) ( n γ ) B. (34) B Not tht this qution n lso b drivd from th orrspondn prinipl. Th frqun for motion long losd orbit is ω B m (35) whr ω is dfind s S m (36) ε In th smilssil limit on should obtin quidistnt lvls with sprtion ε qul to ω. Hn B ε ω (37) ( S / ε ) or S B ε S. (38) ε In th Frmi surf primnts w m b intrstd in th inrmnt Β for whih two sussiv orbits n nd n hv th sm r in th k-sp on th Frmi surf Sn ( k) Sn ( k) S( k) ( n γ ) Bn ( n γ ) Bn S( k) ( n γ ) S( k) ( n γ ) (39) Bn Bn or S( k )( ). (4) B B n n 6.3 Quntum mhnis 6.3. Lndu gug smmtri gug nd gug trnsformtion q H ( p A) qφ ( p A) φ. (4) m m In th prsn of th mgnti fild B (onstnt) w n hoos th vtor potntil s z A ( B r) B ( B B) (smmtri gug). (4) z Hr w dfin gug trnsformtion btwn th vtor potntils A nd A A' A χ 9

20 whr χ B. Sin χ B( ) (43) th nw vtor potntil A ' is obtind s A ' ( B) (Lndu gug). (44) Th orrsponding gug trnsformtion for th wv funtions iqχ ib ψ '( r ) p( ) ψ ( r) p( ) ψ ( r) (45) with q - (>) Oprtors in quntum mhnis W bgin b th rltion p A. [ ] [ p A p A ] [ p A ] [ p A ] (46) A A Bz i i i or [ ] Bz (47) i A A whr Bz. Similrl w hv [ z] B nd [ z ] B (48) i i Sin A ommut with r (A is funtion of r ) [ ] [ p ] i [ ] [ p ] i [ z ] [ z p ] i. z z [ ] [ p A] [ ] [ p A ] (49) Whn B (B) or B z B B [ ] [ z ] [ z ] (5) i Not tht B [ ] i (5) i whr l is lld s mgnti lngth nd it is lotron rdius for th ground stt Lndu lvl: / B Hr w dfin th oprtors X nd Y for th guiding-ntr oordints.

21 l B X l Y (5) Th ommuttion rltion is givn b 4 ] [ ] [ ] [ ] [ ] [ il l l l l l Y X ] [ ] [ ] [ ] [ l l X ] [ ] [ ] [ ] [ l l Y. (53) Whn th unrtintis X nd Y r dfind b > < ) ( X X nd > < ) ( Y Y rsptivl w hv th unrtint rltion 4 4) (/ ] [ 4) (/ ) ( ) ( l Y X Y X or ) (/ ) )( ( l Y X. Th Hmiltonin Ĥ is givn b ) ( ) ( m m H A p (54) W dfin th rtion nd nnihiltion oprtors ) ( i ) ( i (55)(56) or ) ( ) ( i (57) ) ( ] [ ] [ ] [ i i i i i ) ( ) ( ] ) ( ) [( Thus w hv ) ( ) ( m H ω (58) whr m B B m m ) / ( ω. Whn N th Hmiltonin is dsribd b ) ( N H ω. (59) W thus find th nrg lvls for th fr ltrons in homognous mgnti fild- lso known s Lndu lvls Shrödingr qution (Lndu gug) W onsidr th Hmiltonin givn b

22 H [ p ( ) p B pz ] m (6) p p B (6) Th guiding-ntr oordints r X l l l l ( p B) p Y p (6) Th Hmiltonin Ĥ ommuts with p nd p z. [ H ] nd [ H ] p Th Hmiltonin Ĥ lso ommuts with X : [ H X ]. H n k k E n k k z n z p z nd p n k k z k n k k z nd p z n k kz kz n k kz p n k kz k n k kz z p n k k z k n k k z or n k kz k n k kz z n k kz kz z n k kz i i z Shrödingr qution [( ) ( B) ( ) ] ψ ( z) εψ ( z) (63) m i i i ik ik z z ψ ( z) φ( ) (64) ξ mω with β B nd β ω B m k ξ β k k. B B W ssum th priodi boundr ondition long th is. ψ ( L z) ψ ( z) (65) or or ik L k ( / L ) n (n : intgrs) (66) Thn w hv. φ"( ξ ) [( ξ ξ) ( me kz )] φ( ξ ) B W put or kz E ω ( n ) (Lndu lvl) (67) m

23 B me kz m ω ( n ) kz ( n ) φ "( ξ ) [( ξ ξ) (n )] φ( ξ ). Finll w gt diffrntil qution for φ (ξ ). φ"( ξ ) [n ( ξ ξ) ] φ( ξ ). Th solution of this diffrntil qution is ( ξ ξ ) n φ ( ) (!) / n ξ n H n( ξ ξ) (68) with ξ k k B B ξ ξ k β Th oordint is th ntr of orbits. Suppos tht th siz of th sstm long th is is L. Th oordint should stisf th ondition < <L. Sin th nrg of th sstm is indpndnt of this stt is dgnrt. ξ < ξ k < L (69) β or k n < L L or LL n <. Thus th dgnr is givn b th numbr of llowd k vlus for th sstm. L L A A BA Φ g (7) Φ Φ B whr 7 Φ.678 Guss m. Th nrg disprsion is plottd s funtion of k z for h Lndu lvl with th ind n. kz E( n kz ) ω ( n ). (7) m Anothr mthod H ( p A) [ p A ( p A A p)] m m p A A p p A p A p A A p A p A p z z z z 3

24 Thn w hv H [ p m [ p A ] [ p A ] [ p A ] A p z z A A p. i A ( A A p )] i ( p A A A p ) m i Sin A B B H ( p A A p ) p p m m m m whr B B ω mω B m m m B m H ω p ω p p ω p. m m m Th first nd sond trms of this Hmiltonin r tht of th simpl hrmonis long th is. This Hmiltonin Ĥ ommuts with p nd p z. Thus th wv funtion n b dsribd b th form i( k k z z ) ψ ( z) φ ( ). n 6.4 Th Zmn splitting of th Lndu lvl Hr w onsidr th fft of th spin mgnti momnt on th Lndu lvl. Fig. Spin ngulr momntum S nd spin mgnti momnt µ s for fr ltron. S /. ( S / ). µ / m (Bohr mgntron). s µ B B Th spin mgnti momnt µ s is givn b µ s g µ B( S / ) ( gµ B / ) whr µ /( m B ) (Bohr mgnton). Th ftor g is lld th Lndé-g ftor nd is qul to g.3 for fr ltrons. In th prsn of mgnti fild B long th z is th Zmn nrg is givn b 4

25 g µ B m gσ s B Bσ ω ( ) ων sσ (7) m whr ν s gm / m nd σ ±. Thus w hv th splitting of th Lndu lvl in th prsn of mgnti fild s E ( n σ ) ω ( n ν sσ ). (73) whr ν s is muh smllr thn for Bi. 6.5 Numril lultions using Mthmti ((Mthmti 5.-)) Enrg disprsion of th Lndu lvl W onsidr th nrg disprsion of th Lndu lvl with th quntum numbr n s funtion of k z. Hr w ssum tht ω nd m for numril lultions. n. G n_ : n m rul{ m kz } { m } GG[n]/.rul kz n PlotEvlutTblGn kz PlotStl TblHu.i i Prolog AbsolutThiknss Bkground GrLvl.5 AsLbl z " "k z " "Enk PlotRng 8 8 z Enk k z Grphis Fig. Enrg disprsion of th Lndu lvls with n nd k z for 3D ltron gs in th prsn of mgnti fild long th z is. 5

26 6.5.. ((Mthmti 5.-)) Solution of Shrödingr qution (*Lndu lvl*) : D # & # B : D# & [ z [z]] B z B! B z B! z: z B D # & z [ [ [z]]]//simplif Nst[ [z]]//simplif z"! z"! z## z## f$ % Nst&' ) )* m (& z Nst&' z) ) * (& Nst&' z) ) z) --. z(& / E(& Simplif z 3 m B. 3 /45 z6 B/45 3. /45 z7/45 z8889 E/ z (*W ssum th form of wv funtion [z](ep[: : ; k kz z] [] *) rul{ (Ep[: : ; k # kz #3] [#]&)} #E &FG ff/.rul//simplif HIJKkLkzzM m z7 NNO B P BkQ P N Em O N k P kz RQRRST UP QSVVT URW X qy ZZ[ \ B Bk] \ m[ \ ^]^^_` \ ]_bb`^ d f Z E Zk kz B Bkg f d d kf hghhi j kf gillj k m Em kz qn Solvoqpqqorr ss ss Simplif Flttn tuvvw Bk } z { Em} z k}kz~ ~~ uw q3 ƒ ƒ.q { zb 6

27 B Bk Em k vhngeq z_ f_: f n_nst Eq.D Df zd# z&z z!!"########## hng % &'''''''' ( of vribl m$ % B kz n ) B m is dimnsionlss sq*vhngq3 5: --./FullSimplif 9;Bk58: CD <8<<<456< 9;Em:9k :kz B J K L PkQSTTTTTTTT sqsq/.rul//simplif klmbhnok L B MNOMNOPBQLRMNOEmPMNOk Rkz UVWLUVWUVWXYQZRB LX[[YQZUVW\ sq3]solv^sq_``^bbsimplifflttn dffghij l mpemok pkhqrrrrrrr s tuvntuvtuvghi mk okz B Š B Œ Bn w sq4zz{ }~zz{ }.sq3 FullSimplif ƒ Š ŽEm Žk Œ ƒ k kz B BŒ œœœœœœœœ š šk B B k B rul3žÿk B EFG HIIIIIIIII B šk B œœœœœœœœ B š l Š 7

28 k B BEmkz sq5sq4/.rul3//simplif B E Th nrg E B n kz m m rul4e B "! # n! kz $E% 'n() m ' * m B& kz ) -./ m m sq6sq5/.rul4//simplif 7789:;< 4-./ n DSolv[sq65[6]6] C9;HrmitH9n:A:;B C9;HprgomtriFCAn D:A:EFGG ((Mthmti 5.-3)) Plot of th Lndu wv funtion s funtion of ξ whr ξ. (*Simpl Hrmonis wv funtion*) (*plot of Hn[6]*) onjugtrulijcomplkr_ im_lmcomplkrnimlo; UnprottKSuprStrL;SuprStrP:p_Q:IpP. onjugtrul; ProttKSuprStrL {SuprStr} _ HrmitHSnTU RSn_T_U:VWnXYWX4Zn[\WX Ep]^T pt[n_]:plot[evlut[`[n6]]{6-66}plotlbl{n}plotpointsplotrngalldisplfun tionidntitfrmtru] ptevlut[tbl[pt[n]{n8}]];show[grphisarr[prti tion[pt]]] 8

29 GrphisArr Plot[Evlut[Tbl[ `[n6]{n6}]]{6-66}prologabsolutthiknss[]plotstltbl[hu[. i]{i8}]aslbl{"6"" [n6]"}frmtrubkgroundgrlvl[.5]] 9

30 n Grphis Fig.3 Plot of th wv funtion φ n (ξ ) with ξ s funtion of ξ. n nd Gnrl form of th osilltor mgntiztion (Lifshitz-Kosvih) Th prssion of th osilltor mgntiztion is drivd b Lifshitz nd Kosvih s / 3 / T ( / ) S Tm S m M S p( )sin( ± ) os( ) (74) 3 / B / pz B B 4 m whr th sum ovr tnds ll trml ross-stionl r of th Frmi surf th phs /4 if S / pz> (minimum) nd -/4 if S / pz < (mimum) m is mss of fr ltron nd m ( / ) S / ε. Th trm os( m / m ) riss from th Zmn splitting of spins. Th mgntiztion osilltions r priodi in /B. Th priod is ( ) (75) H S Th influn of ltron sttring is not tkn into ount in th drivtion givn bov. Its fft is sil stimtd. A propr ount of th influn of ollisions givs ris to n dditionl ftor. If th mn tim btwn ollisions is τ th orrsponding unrtint in ltron nrgis / τ is quivlnt to tmprtur so-lld Dingl tmprtur m kbtd m p( ) p( ) (76) τh H whr T d is th Dingl tmprtur nd is dfind b T d. kbτ 8. Simpl modl to undrstnd th dhva fft 36 Considr th figur showing Lndu lvls ssoitd with sussiv vlus of n s. Th uppr grn lin rprsnts th Frmi lvl ε F. Th lvls blow ε F r filld thos bov r mpt. Sin ε F is muh lrgr thn th lvl-sprtion ω th 3

31 numbr n s of oupid lvls is vr lrg. Lt us ssum tht th mgnti fild is inrsd slightl. Th lvl sprtion will inrs nd on of th lowr lvls will vntull ross th Frmi lvl. Th rsulting distribution of lvls is similr to th originl on pt tht th numbr of filld lvls blow ε F is now n s- instd of n s. Sin n is lrg this diffrn is ssntill ngligibl so tht on pts th nw stt to b quivlnt to th originl on. This implis priodi dpndn of th mgntiztion. Fig.4 Shmti nrg digrm of D fr ltron gs in th bsn nd prsn of B. At B th stts blow ε F r oupid. Th nrg lvls r split into th Lndu lvls with () n nd s for spifid fild nd (b) n nd s- for nothr spifid fild. Th totl nrg of th ltrons is th sm s in th bsn of mgnti fild. ((Mthmti 5.-4)) Shmti nrg digrm s funtion of /B This figur shows th shmti digrm of th lotion of h Lndu lvls s funtion of ε / ω. Whn ε / ω s (intgr) thr r s Lndu lvls blow th F Frmi lvl ε F. F Plot Evlut Tbl i n UnitStp n UnitStp n n 3 i n 3 Prolog AbsolutThiknss 3 PlotStl Tbl Hu.j j PlotRng 3 3

32 Grphis Fig.5 Shmti digrm for th sprtion of th Lndu lvl s funtion of /B. Th is is s N/(ρB). Th is is qul to th nrg normlizd b th Frmi nrg ε F. Th numbr of th Lndu lvls blow ε F is qul to s t s. 9. Drivtion of th osilltor bhvior in D modl. Th nrg lvl of h Lndu lvl is givn b ( n / ) whr n.. Eh on of th Lndu lvl is dgnrt nd ontins ρb stts. W now onsidr svrl ss. ω (A) Th n s- stts r oupid. n s stt is mpt. Fig.6 ε F ω s. 3

33 N ρbs. Th totl nrg is onstnt s U U ρb( n ) ω ωρb[ s( s ) s] ωρb s ε F N. (77) n (B) Th s whr th n s stt is not filld. W now onsidr th s whn ω drss. This orrsponds to th drs of B. (i) ε < ω / whr ε is th nrg diffrn dfind b th figur blow. Fig.7 (ii) ω / < ε < ω Fig.8 ε ω ε F s 33

34 with < ε < ω. Th n (s-) lvls r oupid nd th n s lvl is not filld. Th totl numbr of ltrons is N. Th nrg du to th prtill oupid n s stt is ( N ρ Bs) ω( s ). Thn th totl nrg is s U U ρ B( n ) ω ε F N ( N ρbs) ω( s ) n ω ( ) ( ρ B s ε F N N ρ Bs ω s ) (78) whr ρ Bs < N < ρb( s ) nd s ω < ε F ( s ) ω. Hr w introdu λ s λ N ρbs. Th prmtr λ stisfis th inqulit < λ < ρb ρs ρ( s ) for < <. Th prmtr λ dnots th numbr of ltrons prtill N B N oupid in th n s stt Th prmtr µ ρbs is th totl numbr of ltrons oupid in th n ρs ρ( s ) s- stts for < <. N B N (iii) Th n s stts r oupid. n s stt is mpt. Fig.9 In this s w hv ε ( s ). F ω 34

35 N ρ B( s ). U U ω ρb s n ρb( n ) ω ( s ) ε F N ω ρb[ s( s ). Totl nrg vs B W now disuss th totl nrg s funtion of B. N( s ) Th totl nrg hs lol minimum t B. ρs( s ) ((Proof)) Sin B ω ( ) B µ BB m m th totl nrg is prssd b U U ω ρb s ε F N ( N ρbs) ω( s ( ) s )] µ BρB s ε F N µ BBN(s ) µ BρB s(s ) ε F N µ BBN(s ) µ BρB s( s ) f ( B). f '( B) µ BN(s ) µ BρBs( s ). N( s ) Thn f(b) hs lol mimum t B ρs( s ) or ρ s( s ). B N s W lso show tht th totl nrg f (B) boms zro t sρ ( s )ρ nd. B N B N ((Proof)) W not tht U - U t ε F ω s nd N ρbs. s N Thn ε F ωs B sb µ B. m m ρ or N f ( B) µ B µ BBN(s ) µ BρB s( s ) ρ 35

36 µ B f ( B) [ ρ B s( s ) NρB(s ) N ] ρ or N N N N f ( B) µ Bρ[ B s( s ) B(s ) ] µ ρ( sb )[( s ) B ] B. ρ ρ ρ ρ Th solution of f(b) is sρ ( s )ρ nd. B N B N. Mgntiztion M vs B Th mgntiztion M is givn b U B U U N µ B F M (79) B ρ whr B ρ ρ F s( s ) (s ) N N F ρ ρ s( s ) (s ) 3 N N N µ B F N µ B ρ ρ M [ s( s ) (s )]. ρ ρ N N M t s( s ) ρ. s N ((Mthmti 5.-5)) Th Mthmti progrm is in th Appndi. In this numril lultion w us n µ B nd ρ. for simpliit. (*d Hs vn Alphn fft*) U s B s s B EF B N B N EF. N B BN s B N s s B B B qd[ub] N s B s B Solv[q Bs B N B] s s! "" s U#_ s_$ :% B' && U&. Simplif U[s] 36

37 N B Ns s s U s U N PowrEpnd N Simplif BN s s s N Solv[U s ]//Simplif s N N m!." # s$% s&' *!! N(s% U s ) Simplif N B 8s - 8s rul{n. /. B. } {N 3 B } U4 s6 7 U5 UnitStp89 s: N ;9UnitStp89 < >: s ;? BAB C ND Ns E F sgd shf E IJK L UnitStpME sf GD shf N E E NO N UnitStpM N U4U/.{. /} P BQR ST NU NsV W X W st N U sv XYZ[ QRS UnitStp\ W sx N ]WUnitStp\ W T U svx ] Y Z[ N N N (*Fr nrg s funtion of /B*) pplot[evlut[tbl[u/.rul{s}]]{.}plots tl. Hu[]Prolog. AbsolutThiknss[]Bkground. GrLv l[.5]plotpoints. 5AsLbl. {"/B""U-U"}] U_U `.5.5 Grphis` ^B pplot[evlut[tbl[u/.rul{s}]]{.4}plot Stl. Hu[]Prolog. AbsolutThiknss[]Bkground. GrLv l[.5]plotpoints. AsLbl. {"/B""U-U"}] 37

38 U_U ` Grphis` ^B Fig.3 Th plot of U-U vs /B (th dtil). pplot[evlut[tbl[u4/.rul{s}]]{.}plo tstl. Hu[]Prolog. AbsolutThiknss[]PlotPoints. 5Plo trng. {{}{7}}Bkground. GrLvl[.5]AsLbl. { "B""F"}] F ` Grphis` Fig.3 Plot of U-U vs B. D M U Simplif N Ns B s s DirDlt BN s s s UnitStp B N (*Mgntiztion s funtion of /B*) s s N DirDlt N UnitStp s N s N pplot[evlut[tbl[m/.rul{s}]]{.}plotst l. Hu[.4]Prolog. AbsolutThiknss[]Bkground. GrLv l[.5]plotpoints. AsLbl. {"/B""M"}] 38

39 M ^B - -4 ` Grphis` Fig.33 of M vs /B Show[ppPlotRng. {{}{-88}}] U_U ^B ` -8 Grphis` NNN-/ B s N-B s NN[_s_]NN/.B. ///Simplif Fig.33 Plot of U-U nd M s funtion of /B. (* Th prmtrs N-/ Bs nd / Bs*) N s NN3 s NN UnitStp s N s UnitStp s N UnitStp s N UnitStp s N N NN4 s UnitStp s N UnitStp s" s! N #UnitStp$% ' % UnitStp$% ()s*& ' s&n N bplot[evlut[tbl[nn3/.rul{s}]]{.}plot Stl. Hu[]Prolog. AbsolutThiknss[]Bkground. GrLv l[.5]aslbl. {"/B"" "}] 39

40 5 4 3 `.5.5 Grphis` ^B Fig.34 Plot of λ vs /B (rd). bplot[evlut[tbl[nn4/.rul{s}]]{.}plot Stl. Hu[.5]Prolog. AbsolutThiknss[]Bkground. GrL vl[.5]aslbl. {"/B"" "}] ` Grphis` Fig.35 Plot of µ vs /B (blu). Show[bb] ^B 6 4 `.5.5 Grphis` ^B Fig.36 Plot of λ vs /B (rd) nd µ vs /B (blu). 4

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