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
