Discontinuities in the level density of small quantum dots under strong magnetic fields

Discon inui ies in he le el densi y o small quan um do s unde s ong magne ic ields Augus o Gonzalez* Ins i u o de Cibe ne ica, Ma ema ica y Fisica, Calle E 309, Vedado, Ciudad Habana, Cuba Robe o Capo e† Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea , Uni e sidad de Se illa, Facul ad de Fı ´sica, AP 1065, E-41080 Se illa, Spain 共Recei ed 30 May 2002; published 30 Sep embe 2002兲 Exac diagonaliza ion s udies o he le el densi y in a six-elec on quan um do unde magne ic ields a ound 7T共‘‘ illing ac o ’’a ound 1 2) a e epo ed. In any spin-pola iza ion channel, wo egimes a e isible in he do exci a ion spec um: one co esponding o in e ac ing quasipa icles 共i.e., composi e e mions兲 o exci a ion ene gies below 0.4 meV, and a second one o ene gies abo e 0.4 meV, in which he le el densi y 共exponen- ially兲inc eases a he same a e as in he nonin e ac ing composi e- e mion model. DOI: 10.1103/PhysRe B.66.113311 PACS numbe 共s兲: 73.21.La, 73.43.Lp, 73.63.K The lowes -ene gy s a es o ela i ely small quan um do s 共numbe o elec ons Ne⭓3) in s ong magne ic ields ha e been quali a i ely desc ibed whi hin he composi e- e mion pic u e.1–3 The posi ion o cusps in he g ound-s a e ene gy as a unc ion o he angula momen um, and he spin quan- um numbe s o he low-lying exci ed s a es a e nicely ep o- duced by his heo y. The esidual in e ac ion be ween com- posi e e mions 共CF’s兲is expec ed o be a weak, con ac in e ac ion. The p esen pape is aimed a gi ing a quan i a i e cha - ac e iza ion o he densi y o ene gy le els o small quan um do s in an ene gy in e al below 1–1.5 meV, whe e dozens o hund eds o s a es exis . Fully con e ged exac diagonaliza- ion esul s o a six-elec on GaAs do in magne ic ields co esponding o ‘‘ illing ac o s’’ ␯ ⬇1 2a e p esen ed below. The e o in compu ing ene gy eigen alues is es ima ed o be lowe han 0.02 meV o le els wi h exci a ion ene gies be- low 1 meV. The s udied exci a ion ene gy ange is small as compa ed o he cyclo onic (ប ␻ c⫽12 meV), Coulomb 共8.4 meV兲, o con inemen (ប ␻ 0⫽3 meV) ene gies o he model do . Th ee Landau le els 共LL’s兲a e included in he calcula- ions, and a 25-meV cu o in he ene gy o he nonin e ac - ing many-elec on s a es used as basis unc ions4allows us o deal wi h Hamil onian ma ices o dimension less han 850000, which a e diagonalized by means o a Lanczos al- go i hm. The main esul o he pape is he exis ence o wo egimes in he exci a ion spec a, co esponding o low (⌬E⬍0.4 meV) and in e media e (⌬E⬎0.4 meV) exci a- ion ene gies, in which he le el densi y inc eases a di e en a es. Thus, a an ene gy in e al ␦ Ea ound 0.4 meV he le el densi y expe iences a ‘‘discon inui y.’’ The model pa ame e s a e simila o hose used in Re . 5. The con inemen po en ial is pa abolic. The ba e Coulomb in e ac ion is weakened by a ac o 0.8 o app oxima ely ac- coun o quasibidimensionali y 共ins ead o exac bidimen- sionali y兲. The Zeeman ene gy is w i en in he o m 0.01432 BszmeV (Bin Tesla and sz⫽⫾ 1 2), co esponding o a 8-nm-wid h well in magne ic ields a ound 7 T.6 The lowes -ene gy s a es in each angula momen um and spin-pola iza ion owe 共 he y as spec um兲 o magne ic ields B⫽7 and 8 T a e shown in Fig. 1. Ene gy jumps be ween adjacen angula momen um s a es a e abou 0.6 meV in he spin-pola ized case, bu oughly h ee imes smalle in any o he spin-pola iza ion sec o . A his poin i is impo an o s ess he ole o he highe LL’s in he ene gy eigen alues. The absolu e con ibu ion o he second and hi d LL’s is a ound ⫺0.4 meV, a magni ude much g ea e han he Zeeman spli ing 共0.1 meV兲, and han he cha ac e - is ic ene gy spacing be ween s a es nea he absolu e mini- mum. Exci a ion ene gies a e pushed down 0.1–0.3 meV by he highe LL’s. We show in Fig. 2 he numbe o s a es as a unc ion o he exci a ion ene gy, ⌬E. In his igu e, he e e ence ene gy FIG. 1. The lowes -ene gy le els in each angula momen um and spin-pola iza ion sec o s. Lines a e guides o he eyes. PHYSICAL REVIEW B 66, 113311 共2002兲 0163-1829/2002/66共11兲/113311共3兲/$20.00 ©2002 The Ame ican Physical Socie y66 113311-1 is he minimal-ene gy s a e wi hin each pola iza ion sec o . A low and in e media e exci a ion ene gies he cu es a e well i ed by exponen ial unc ions,5 n共⌬E兲⫽n0exp⌬E ⌰,共1兲 i.e., desc ibed by he so-called ‘‘cons an - empe a u e app oxima ion.’’7In he pola ized case, he e a e only a ew s a es o low exci a ion ene gies, hus we do no i hem. The i ing pa ame e s n0and ⌰a e gi en in Table I. P imed quan i ies co espond o he low-ene gy sec o s. No ice ha he empe a u e pa ame e , ⌰, changes e y li le on going om Sz⫽2 oSz⫽0 o a gi en B. No ice also he appa en jump o he empe a u e pa ame e a ␦ E, which means a discon inui y in he le el densi y. The nonin e ac ing CF 共NICF兲unde s anding o he exci- a ion spec um s a s om a simpli ied i s LL pic u e in which he exci a ion ene gy is w i en in he o m2 ⌬E⫽ប ␻ CF⌬nLL⫹ប ␻ 0 2/ ␻ c⌬L.共2兲 ⌬nLL is he a ia ion, wi h espec o he quasipa icle g ound s a e, o he e ec i e LL occupa ion numbe s. By de ini ion, ⌬L⫽ 兩 L 兩 ⫺ 兩 Lgs 兩 , whe e Lgs is he g ound-s a e angula mo- men um. ប ␻ cis he elec on cyclo onic ene gy in GaAs, and he e ec i e cyclo onic ene gy o CF’s is ex ac ed om he pola ized y as spec um, Sz⫽3, as2 ប ␻ CF⫽E共L3⫹1兲⫹E共L3⫺1兲⫺2E共L3兲,共3兲 whe e L3is he angula momen um o he lowes pola ized s a e. I gi es ប ␻ CF⫽1.15 and 1.19 meV a B⫽7 and 8 T, espec i ely. The quasipa icles a e supposed o occupy s a es in e ec i e Landau le els, which a e sepa a ed by ប ␻ CF . The angula momen um o he quasipa icle sys em is LCF which, due o he special o m o he a ia ional CF wa e unc ion, is ela ed o Lby2 L⫽⫺Ne共Ne⫺1兲⫹LCF .共4兲 The NICF model gi es quali a i ely co ec answe s o ques ions such as he posi ion o cusps in he y as spec um, he na u e o he i s exci ed s a es, e c. One may ask o o he isible mani es a ions o he NICF in he spec um, and indeed, i gi es he co ec ⌰pa ame e a in e media e ex- ci a ion ene gies, ⌬E⬎ ␦ E. In o he wo ds, he NICF cu e g ow hs exponen ially wi h he same ⌰, al hough i is shi ed wi h espec o he ac ual spec um. This ac is illus a ed in Fig. 2 also, whe e he NICF cu e a a ce ain le el 共 he six h in he uppe igu e, o example兲is o ced o mee he ac ual cu e. F om his poin on he wo spec a ha e he same slope on a e age. The e is a na u al in e p e a ion o his beha io . In he g ound and i s exci ed s a es 共low ⌬E), he quasipa icles o m compac clus e s2and he in e ac ion be ween CF’s plays an impo an ole. The low-ene gy pa ame e ⌰⬘is a FIG. 2. The loga i hm o he numbe o le els as a unc ion o he exci a ion ene gy a B⫽7 T: exac esul s 共squa es兲, cons an empe a u e i s 共do ed lines兲, and he shi ed NICF cu es 共solid lines兲. TABLE I. The le el-densi y pa ame e s n0and ⌰. P imed quan- i ies co espond o exci a ion ene gies below ␦ E. SzB⫽7T B⫽8T ␦ E0.30 0.45 3n02.099 1.712 ⌰0.510 0.443 n0 ⬘1.398 1.614 ⌰⬘0.145 0.145 2 ␦ E0.35 0.30 n05.341 5.241 ⌰0.400 0.363 n0 ⬘1.982 2.069 ⌰⬘0.107 0.085 1 ␦ E0.30 0.20 n011.57 10.99 ⌰0.398 0.344 n0 ⬘1.506 1.635 ⌰⬘0.113 0.100 0 ␦ E0.40 0.25 n014.74 12.90 ⌰0.423 0.358 BRIEF REPORTS PHYSICAL REVIEW B 66, 113311 共2002兲 113311-2 consequence o his in e ac ion. On he o he hand, a in e - media e ene gies and assuming ha he in e ac ion is weak, one expec s he le el densi y o be domina ed by he ele an ene gy scales, i.e., ប ␻ CF and ប ␻ 0 2/ ␻ c. One may isualize his egime in e ms o Rydbe g-like exci a ions, in which a ew nonin e ac ing CF’s o bi a ound a co e o weakly in e - ac ing CF’s. Ene gy di e ences will, o cou se, ollow he NICF ules. The conclusion o be ex ac ed om his igu e is ha aces o he e ec i e LL s uc u e o CF’s may also be looked o a in e media e exci a ion ene gies, ⌬E ⬎0.4 meV. Fo ⌬E⬍0.4 meV he ansi ion o a egime o weakly in e ac ing quasipa icles akes place. Tempe a u e pa ame e s a e di e en om bo h sides o ␦ E, hus a dis- con inui y in he le el densi y is expec ed. The le el densi y may be di ec ly measu ed in Raman- sca e ing expe imen s unde ex eme esonance, whe e he Raman ampli ude depends on he densi y o ene gy le els in inal s a es.8Magne oconduc ance measu emen s unde equilib ium9o nonequilib ium condi ions10 could also gi e e idence abou he le el-densi y beha io a in e media e ex- ci a ion ene gies. Finally, we show in Fig. 3 he magni udes ប ␻ CF and ⌰Sz⫽3in a wide magne ic- ield in ensi y ange, 7⭐B ⭐12 T. Nea B⫽12 T, whe e he ‘‘ illing ac o ’’ is a ound 1 3, he e is a dec ease o ប ␻ CF which co esponds o an ab up inc ease o he quasipa icle mass.11 A simila beha - io is obse ed in ⌰, showing ha he le el-densi y pa am- e e ⌰depends on a nega i e powe o mCF . Pa o his wo k was ca ied ou a he Abdus Salam ICTP. A. G. acknowledges he ICTP Associa e and Fede a- ion Schemes o suppo . R. C. acknowledges suppo om he Minis e io de Educacio ´n, Depo es y Cul u a de Espan ˜ a, Sec e a ı ´a de Es ado de Educacio ´n y Uni e sidades. *Elec onic add ess: [email p o ec ed] †Pe manen add ess: Cen o de Es udios Aplicados al Desa ollo Nuclea , AP 100, Ciudad Habana, Cuba. Elec onic add ess: [email p o ec ed] 1C.W.J. Beenake and B. Rejaei, Physica B 189, 147 共1993兲. 2J.K. Jain and T. Kawamu a, Eu ophys. Le . 29, 321 共1995兲; R.K. Kamilla and J.K. Jain, Phys. Re . B 52, 2798 共1995兲. 3J.K. Jain and R.K. Kamilla, in Composi e Fe mions, edi ed by O. Heinonen 共Wo ld Scien i ic, Singapo e, 1988兲. 4The basis unc ions in a disk geome y a e desc ibed in T. Chak abo y and P. Pie ilainen, in The Quan um Hall E ec s 共Sp inge , New Yo k, 1996兲. 5A. Gonzalez and R. Capo e, Physica E 共Ams e dam兲10, 528 共2001兲. 6We use a pa ame iza ion by B. Rod iguez o he expe imen al g alues o quan um wells gi en in S.P. Najda e al., Phys. Re . B 40, 6189 共1989兲; M.J. Snelling e al.,ibid. 44,11345共1991兲; 45, 3922 共1992兲; N.J. T ayno e al.,ibid. 55, 15 701 共1997兲;M. Seck e al.,ibid. 56, 7422 共1997兲. 7T. E icson, Ad . Phys. 9, 425 共1960兲; A. Gilbe and A.G.W. Came on, Can. J. Phys. 43, 1446 共1965兲. 8D.J. Lockwood e al., Phys. Re . Le . 77, 354 共1996兲. 9T.H. Oos e kamp e al., Phys. Re . Le . 82, 2931 共1999兲. 10T. Fujisawa e al., Phys. Re . Le . 88, 236802 共2002兲. 11M. Onoda e al., Phys. Re . B 64, 235315 共2001兲. FIG. 3. The magni udes ប ␻ CF 共squa es兲and ⌰Sz⫽3共ci cles兲as a unc ion o magne ic- ield in ensi y B. BRIEF REPORTS PHYSICAL REVIEW B 66, 113311 共2002兲 113311-3