P H I L O S O P H I C A L M A G A Z I N E B , 2 0 0 1, V O L . 8 1 , N O . 5 , 4 6 1± 4 7 5
E n e rg y sp e ctra a n d p h o to lu m in esc en c e o f fra c tio n a l
q u a n tu m H a ll sy ste m s co n ta in in g a v a len c e-b a n d h o le
A rk a diusz Wo js y z } , K y u ng-S o o Yi y § a n d Joh n J. Q ui n n y
y U n iv er sit y o f T e n n es se e, K n o x v ille, T en n ess ee 37 9 9 6 , U S A
z W ro cla w U n ive rs ity o f T ech n o lo g y, W ro cla w 5 0 -3 7 0, P o la n d
§ P u s a n N a t ion a l U n ive rsity , P u sa n 60 9 -7 3 5 , K o re a
[ Received 13 November 2000 and accepted 2 January 2001 ]
A b str act
T h e en erg y sp ec tru m o f a tw o -d im en sio n al e lec tro n g a s (2 D E G ) in ter a ctin g
w ith a v a len ce- b an d h o le is stu d ied in th e h igh m ag n e tic ® e ld lim it a s a fu n c tio n o f
th e ® llin g f a c to r a n d t h e s ep ar at ion d b et w een th e e lect ro n a n d h o le lay er s. F o r
d sm a ller t h a n t h e m a g n e tic len g t h , t h e h o le b ind s o n e o r m o re e lect ro n s to fo rm
¡
n eu t ra l ( X ) o r ch a r ge d (X ) ex cito n s. T h e lo w -ly in g stat es ca n b e u n d e rst o o d in
ter m s o f L a u g h lin -lik e co r re lat io n s a m o n g t h e co n s tit u e n t c h a rg ed fe rm io n s
¡
(ele ct ro n s a n d X ) . F o r d c o m p ar ab le to , t h e ele ct ro n ± h o le in ter a ctio n is n o t
str o n g e n o u g h to b in d a f u ll e lec tro n , a n d f ra ct ion a lly c h a rg ed ex cito n s h Q E n
(b o u n d sta tes o f a h o le a n d o n e o r m o re L a u gh lin q u as i-e lect ro n s, Q E s) a re
fo rm ed . T h e e Œe ct o f t h e se ex cito n ic co m p le x e s o n th e p h o to lum in es cen ce
sp ect ru m is stu d ied n u m e ric ally fo r a w ide r an g e o f v alu es o f
and d .
} 1 . I nt roduct ion
T h ere h a v e b een m a n y exp e rim en ta l stu d ies o f p h o to lu m in e scen ce (P L ) in fra ctio n a l q ua n tu m H a ll sy stem s d u rin g th e p a st d ec ad e (H eim a n et al. 1 98 8 , B uh m an n
et al. 1 9 90 , 19 9 1 , 1 99 5 , G o ld be rg et al . 19 9 0 , T u rb er® eld et al. 1 9 9 0 , G o ld y s et al.
1 99 2 , K h en g et al. 1 9 9 3, K u k u sh k in et al. 1 99 4 , F in k elstein et al. 19 9 5 , 1 9 96 , S h ield s
et al. 1 9 95 , B ro w n et al. 1 9 96 , G rav ier et al . 1 99 8 , Jian g et al. 1 9 9 8, N ick el et al. 1 99 8 ,
T ak e ya m a et al. 19 9 8 , H a y n e et al. 19 9 9 , T isch ler et al. 1 9 99 , W o jto w icz et al . 1 99 9 ,
K im et al. 20 0 0 , M u n tea n u et al . 20 0 0 ), b u t th e d a ta ha v e b een ra th er d i cu lt to
in terp ret. In o rd er to o b ta in a m o re co m plete u n de rstan d in g o f the P L p ro cess, it is
essen tial to u n d ersta n d th e n atu re o f th e low -ene rg y states o f th e electro n ± h o le
sy stem , a nd to ev a lu a te th eir o scillato r stren g th fo r rad iative reco m b in a tio n. In
th is n o te w e in v estiga te th e elem en ta ry ex cita tio n s o f a sy stem co n sistin g o f N
elec tro n s (e) co n ® ne d to th e p lan e z ˆ 0 a n d in tera ctin g w ith a sin g le v a len ce -ba n d
h o le (h ) co n ® n ed to th e plan e z ˆ d . T h is m o de l is ap p ro p riate fo r sy stem s in w h ich
th e h o le co n cen tra tio n is v ery sm a ll co m p a red w ith th e e lectro n co n ce ntra tio n so
th at th e in ter action b etw een th e h o les is ne g lig ib le.
T h ere are thre e n ea rly d istin ct reg io n s fo r th e in ter ac tio n o f th e h o le an d th e
elec tro n sy stem , w h ich w e refer to a s w ea k , stro ng a n d in ter m ed iate co u p lin g . In th e
w ea k -co u p lin g re gio n ( d m u ch larg er th a n th e m ag n e tic len gth ), th e electro n ± h o le
} E m a il: a w o js@ u tk .e d u
Philosophical M agaz ine B I S S N 1 3 6 4 ± 2 8 1 2 p r in t / I S S N 1 4 6 3 ± 6 4 1 7 o n lin e # 2 0 0 1 T a y l o r & F r a n c i s L t d
h tt p : / / w w w . t an d f.c o . u k / j o u r n a ls
D O I : 1 0 . 10 8 0 / 1 3 6 4 2 8 1 0 1 1 0 0 3 6 7 6 1
462
A . W o js et al.
in tera ctio n is a w eak p e rturb a tio n o n the e igen sta tes o f th e in tera ctin g elec tro n s
(C h en an d Q u in n 1 9 9 3, 1 9 9 4a , 1 9 9 5 ). In th e stro n g -co u p lin g reg io n ( d < ), th e
¡
h o le b in d s o n e or tw o electron s to fo rm a n eu tral (X ) o r n e ga tively c ha rg ed (X )
ex cito n (K h en g et al . 1 9 93 , B u h m a n n et al. 19 9 5 , F in k elstein et al. 1 9 9 5, 1 9 96 , H ay n e
et al. 1 9 99 , S hield s et al. 1 9 9 5, W o js an d H aw rylak 19 9 5 , P a lacio s et al. 1 99 6 ,
W h itta k er a n d Sh ield s 19 9 7 , N ick el et al. 1 99 8 , W o js et al . 1 9 9 8, 1 9 99 a , b , 2 0 00 a ,
b , T isch ler et al. 1 99 9 , W o jto w icz et al. 1 99 9 , K im et al. 2 00 0 , M u n tea n u et al. 2 0 00 ).
W h en d is eq u a l to ze ro , th e n e utra l e x cito n is c o m p letely un c ou p led fro m th e
rem a in in g N ¡ 1 electro ns d u e to th e `h id d en sy m m etry ’ (L ern er an d L o zo v ik
1 9 81 , D zy u b e n ko a n d L o zo v ik 1 9 8 3, M acD o na ld an d R eza y i 1 9 90 ), a n d it is o n ly
¡
w ea k ly co u p led a t 0 < d < . T h e X is a n eg ativ ely ch a rge d ferm io n w ith a sim ilar
d eg en era te L a nd a u lev el (L L ) stru c ture to th a t o f a n electro n (W o js a n d H a w ry lak
1 9 95 , W o js et al. 1 9 9 8, 1 9 99 a ). It h a s L au g h lin -lik e c o rrelatio n s w ith the rem a in ing
N ¡ 2 electro n s tha t ca n b e d esc rib ed b y a g en era liz ed co m p o site ferm io n (C F )
m o d el (W o js et al. 19 9 9 b ). In th e inter m ed iate-co u p lin g regio n (
d
2 ), th e
h o le can n o lo n ge r bin d a full elec tro n to fo rm a n X ; h ow ev e r, it ca n b in d o n e o r
m o re L au g h lin q u a si-electro ns (Q E s) to fo rm fra ctio n a lly c h arg ed ex cito n s, so -ca lled
F C X s (W o js a n d Q u in n 2 0 01 a , b ). W e d en o te a co m p lex co n sistin g o f n Q E s bo u n d
to a h o le b y th e sy m b o l hQ E n . T o u n de rstan d th e sta b ility o f th e h Q E n sta te, it is
n ecessa ry to kn o w th e p seu d o p o ten tials d escrib in g th e in ter actio n s o f a Q E ± Q E p a ir
a n d o f a h ± Q E p a ir a s a fu n ctio n of th e pa ir a n g u lar m o m e nt u m (W o js a n d Q u in n
1 9 98 , 1 99 9 a , b , 20 0 0 a , Q u in n a n d W o js 20 0 0 ). In o rd er to d eterm in e th ese pse u do p o ten tials, a s w ell a s o th er p ro p erties o f b o u nd co m p lex es, w e h a v e p erfo rm ed e xa c t
(w ith in th e lo w est L L ) n u m erical d iag on a liza tio n s fo r a n in e-e lectro n ± o n e-h o le
sy stem as a fun c tio n o f th e layer se pa ra tio n d a n d the m ag n etic ® e ld . T h e ca lcu la tio n s a re p erfo rm ed in H a ld a n e’ s sp h erica l g eo m etry (H ald an e 1 9 83 , F an o et al.
2
2
2 1= 2
1 9 86 ) w ith th e elect ron ± h o le in terac tio n m o d elled b y V e h … r † ˆ e = … r ‡ d † , fo r
v a lu es o f d sa tisfy in g 0
d
5 .
} 2. M an y-el ect ron syst e m
T o in terp ret th e w ea k -co u p lin g reg im e w e m u st b egin w ith th e u nd e rstan d in g of
a n electro n system in th e a b se nc e of the h o le. In ® g u res 1 ( a )± ( d ) w e d isp lay th e lo w en erg y sp ectra o f n in e electro n s a t th e L L d eg en era cy g ˆ 2 S ‡ 1 c orre sp o n d in g to
th e m a gn e tic m o n o p o le stre ng th o f 2 S ˆ 2 4, 23 , 2 2 an d 2 1 , resp ective ly (on
H a ld a n e’ s sp h ere, the lo w est L L is rep res en ted b y a d eg en era te m u ltip let at a ng u lar
m o m en tu m l ˆ S ). T h e lo w est en erg y sta tes co n ta in ( a ) zero , ( b ) o n e, ( c ) tw o a n d ( d )
ˆ 13 state an d ca n b e sim p ly u n d ersto o d u sin g th e C F
th ree Q E s in th e L a u g hlin
p ictu re . T he eŒective m o n op o le stren g th (C he n a n d Q u in n 1 9 9 4c , W o js a n d Q u in n
1 9 98 , 19 9 9 a , b , 2 0 0 0a , Q u in n an d W o js 2 0 0 0) seen b y o n e C F is g iven b y
2 S ˆ 2 S ¡ 2 … N ¡ 1 † , a n d th e a n gu lar m o m en tu m o f the k th C F sh ell
( k ˆ 0 ; 1 ; . . . ) is lk ˆ S ‡ k . T h e Q E s a re the C F s in th e ® rst e xcited sh ell ( n ˆ 1 )
a n d th u s h av e lQ E ˆ l 1 . T h e q u a sih o les (Q H s) are th e em p ty sta tes in th e lo w est C F
sh ell ( n ˆ 0 ) a n d thu s h a ve l Q H ˆ l0 . F o r 2 S ˆ 2 4 th e n in e C F s ® ll co m p let ely th e
l0 ˆ 4 she ll g ivin g a tota l a n g u lar m o m en tu m L e ˆ 0 . T h is n o n -de gen era te g ro un d
sta te (G S ) is th e L a u g h lin in co m p ressib le ˆ 13 state. F o r 2 S ˆ 23 , eigh t o f th e C F s
® ll th e l 0 ˆ 72 sh ell, a n d th e n in th is a Q E w ith l Q E ˆ 92 . T h is g ives a to tal a ng u lar
m o m en tu m L e ˆ 92 fo r th e n in e-e lectro n G S . F o r 2 S ˆ 2 2 w e o b ta in tw o Q E s ea ch
w ith lQ E ˆ 4 ; a d d in g th e a n gu lar m o m en ta of th ese tw o ide n tica l ferm io n s g ives
L e ˆ 1 3 5 7 a s the lo w -en e rg y b a n d o f states. F o r 2 S ˆ 2 1 there are th ree
Energy spectra and photoluminescence of fractional quantum Hall systems 4 6 3
F ig u r e 1 . T h e en erg y sp ec tra (e n e rg y E v ers u s an g u lar m o m en tu m L ) o f th e n ine -e lect ro n
sy ste m o n H a ld a n e ’ s s p h er e w ith t h e m o n o p o le s tre n g t h b et w een 2 S ˆ 2 4 a n d 2 1. is
th e m a gn et ic len g th .
Q E s ea ch w ith lQ E ˆ 72 . A d d in g th eir thre e a n gu lar m o m en ta g ives th e lo w -lyin g n in e5
7
9
11
15
elec tro n states a t L e ˆ 32
.
2
2
2
2
2
In the ab se nc e o f th e Q E ± Q E in tera ctio n (d e® n ed b y a p seu d o p o ten tial
V Q E ¡ Q E … L † , i.e . p air in tera ctio n en erg y as a fu n ctio n o f p a ir a n g u lar m o m en tu m ),
a ll o f th e 2 Q E a n d 3Q E sta tes w o u ld b e d e ge ne rate (C h e n an d Q u in n 19 9 4 c, S itk o et
al. 1 99 6 ). H o w ev er, th e intera ctio n b e tw ee n (ch a rg ed ) Q E s exists an d rem o v es th is
d eg en era cy . In ® g u re 1 ( c ), th e 2 Q E states w ith L ˆ 3 a n d 7 are lo w ered relative to
th o se w ith L ˆ 1 a n d 5 . F o r th e 3Q E sta tes in ® g u re 1 ( d ), th e m u ltip let a t L ˆ 52 h as
th e lo w est en erg y a n d th ose at L ˆ 32 a n d L ˆ 92 h a v e the h ighe st en erg y. R em a rk a b ly,
th e 2 Q E a n d 3Q E `m o lec u le’ states w ith the m a xim u m a llo w e d a n gu lar m o m en tu m
( L ˆ 2 l Q E ¡ 1 ˆ 7 an d 3 lQ E ¡ 3 ˆ 125 , resp ec tive ly), tha t is w ith th e m in im u m a vera g e
Q E ± Q E sep a ra tio n , ha v e lo w en erg y (W o js a n d Q u in n 2 00 0 b ). W e ca ll th ese sta tes
Q E 2 and Q E 3.
A ll o f th e m a ny -electro n sta tes in th e lo w e st b an d ca n b e u n d e rstoo d o n th e b a sis
o f th e C F p ictu re; ex cellen t a g reem en t w ith th e n u m erica l resu lts is o b taine d w h en
in tera ctio n s b etw een q u a sip ar ticles (Q P ˆ Q E o r Q H ) a re inc lu d ed p h en o m en o lo g ica lly (S itk o et al. 19 9 6 ). A dd itio n a lly , in m a n y ca ses th e ® rst ex cited b a n d o f states
ca n b e id en ti® ed usin g th e C F p ictu re . In th is b a nd , o n e of the C F s is e xcited to th e
n ex t h igh er sh e ll. D ep en d in g o n w h e th er a C F fro m th e n ˆ 0 o r 1 sh ell is exc ited ,
th is co rresp o nd s to th e cre atio n o f an ad d itio n al Q E ± Q H p a ir o r to th e ex cita tio n of
A . W o js et al.
464
a Q E w ith lQ E ˆ l1 to th e Q E * state w ith l Q E ˆ l2 . L et u s id e ntify th e ® rst ex cited
b a n d s in ® g u re 1
F o r 2 S ˆ 2 4 , n o Q P s oc cu r in the low est b a n d , a n d th u s th e ® rst e xcited b a nd
co n ta in s a sin gle Q E ± Q H p air. B e ca u se lQ H ˆ 12 … N ¡ 1 † ˆ 4 a n d l Q E ˆ 12 … N ‡ 1 † ˆ 5 ,
th e m ultip lets fro m L ˆ lQ E ¡ lQ H ˆ 1 to lQ E ‡ lQ H ˆ N ˆ 9 a re ex p ected fo r su ch a
p a ir. It is k n o w n (S itk o et al. 1 9 9 6 ) tha t th e state a t L ˆ 1 (i.e. a t th e sm a llest a vera g e
Q E ± Q H sep a rat io n ) is p u sh ed to h ighe r en erg y (o r forb id d en ) b y th e stro n g h ard co re Q E ± Q H rep u lsio n . A s a resu lt, th e `m a g n eto ro to n ’ Q E ± Q H b an d e xten d s fro m
L ˆ 2 to N .
F o r 2 S ˆ 2 3 , tw o d iŒeren t Q P co n ® gu ratio n s o ccu r in th e ® rst ex cited b a nd . T h e
® rst co n sists of a sin g le Q E * w ith lQ E ˆ 121 givin g a lso the to ta l an g u lar m o m en tu m
o f L ˆ 121 . T h e seco n d co n sists of a sin g le Q H w ith lQ H ˆ 72 a n d a p air o f Q E s ea ch
w ith lQ E ˆ 92 . T o ta lly ign o ring in ter action s b e tw ee n th ese Q P s gives the fo llo w in g se t
o f d eg en era te a n g u lar m o m e n tu m m u ltiplets fo r th e 2 Q E ‡ Q H c o n® g u ra tio n :
2
3
4
4
4
3
3
2
2
21
23
… 52 †
… 72 †
… 92 †
… 121 †
… 123 †
… 125 †
… 127 †
… 129 †
L ˆ 12 … 32 †
.
The
2
2
Q P ± Q P in tera ctio n s (pa rticu larly, the Q E ± Q H h a rd -cor e rep u lsio n ) w ill rem o v e
th e d eg en era cy o f th is b a n d a n d p u sh so m e o f the m u ltip lets in to th e co n tin u u m
o f h igh er en erg y states. H o w e ve r, a lm o st a ll o f the pre d icted m u ltip lets ap p ear in th e
n u m erica l sp ectru m in ® g u re 1 ( b ).
F o r 2 S ˆ 2 2 th ere a re ag ain tw o p o ssib le co n ® g u ra tio n s fo r th e ® rst ex cited
b a n d . T h e ® rst co n tain s o n e Q E w ith lQ E ˆ 4 a n d o n e Q E * w ith lQ E ˆ 5 , a nd
g ives a b a n d of m u ltip lets e xten d in g fro m L ˆ 1 to 9. T h e seco n d co n ® gu ra tion
co n ta in s thre e Q E s ea ch w ith l Q E ˆ 4 a n d a Q H w ith l Q H ˆ 3 . N eg lec tin g Q P ± Q P
in tera ctio n s, th ese tw o Q P co n ® g u ratio n s w ou ld y ield a de g en era te b a n d o f m u lti2
3
5
6
7
6
7
5
4
3
2
p lets a t L ˆ 0
1
2
3
4
5
6
7
8
9
10
11
1 2. T h e
n u m erica l sp ectru m sh o w n in ® g u re 1 ( c ) co n tain s m a n y o f th ese sta tes in a ® rst
ex cited b an d w h ich is ra th er w e ll se pa ra ted fro m th e co n tin u u m o f h igh er states
fo r L < 3 a n d L > 6, bu t n o t w ell d e® n ed b etw een th ese regio n s.
F in a lly , fo r 2 S ˆ 2 1 th ere a re tw o c on ® g u ra tio n s for th e ® rst e xcited ba n d :
® rst w ith tw o Q E s ea ch w ith lQ E ˆ 72 a n d o n e Q E * w ith l Q E ˆ 92 , an d sec on d
w ith fo u r Q E s w ith lQ E ˆ 72 a n d o n e Q H w ith lQ H ˆ 52 . T h e fo rm er co n ® gu ratio n
2
3
3
4
3
3
… 32 †
… 52 †
… 72 †
… 92 †
… 121 †
… 123 †
y ield s th e set o f m ultip lets a t L ˆ 12
… 125 †
2
5
… 121 †
… 127 †
2
5
… 123 †
19
21
. The
2
2
3
2
15
…2†
… 127 †
latter
19
2
21
.
2
o ne
g ives
L ˆ … 12 †
2
… 32 †
4
… 52 †
6
… 72 †
6
… 92 †
6
A g a in , w e ex pe ct m a n y o f th ese m u ltip lets to b e
p u sh ed in to th e high e r e ne rgy co n tin u u m b y th e Q E ± Q H h ard -co re rep u lsio n . F ro m
th e n u m erica l resu lts in ® g u re 1 ( d ), it ca n b e see n tha t th e ® rst ex cited ba n d is rath er
3
13
w ell d e® n ed for L
a n d fo r L
, w h ere th e fo llo w in g m u ltip lets o ccu r:
2
2
2
2
1
3 3
13 6
15 4
17 3
…2†
…2†
… 129 †
… 221 † .
L ˆ 2 …2† and L ˆ … 2 †
} 3 . We ak-cou pli ng re gim e
In th e w ea k -co u p lin g lim it w e exp ect to ob ta in fairly w ell d e® n ed b a n d s fo r th e
electro n ± ho le sy stem b y trea tin g th e in tera ctio n o f the h o le w ith th e e lectro n s a s a
sm a ll p ertu rb a tio n (C h en a n d Q u inn 1 99 3 , 19 9 4 a , 1 9 9 5). T h e lo w -en erg y b a n d s a re
clear ly visib le in ® g u re 2 , a nd th ey a re ea sily u n de rsto o d o n the ba sis o f ® g u re 1 w ith
th e a d d itio n of the a n gu lar m o m en ta o f th e lo w -en erg y elect ron m u ltip lets to th at of
th e h o le. T h e a n g u lar m o m en tu m o f th e h o le, lh , is e qu a l to S , a n d so l h ˆ 1 2, 223 , 1 1
a n d 221 in ® g u res 1 ( a )± ( d ), resp ectively. In ® g u re 1 ( a ) the o n ly electro n m u ltip let in
th e lo w -en erg y secto r is th e L au g h lin G S at L e ˆ 0. T h erefo re, th e lo w -en e rg y b a nd
Energy spectra and photoluminescence of fractional quantum Hall systems 4 6 5
F ig u r e 2 . T h e en er g y s p e ct ra ( en erg y E v er su s a n gu la r m o m en tu m L ) o f t h e n in e- elec tro n ±
o n e- h o le s y s te m o n H a lda n e’ s sp h e re w ith th e m o n o p o le s tre n g th b et w e en 2 S ˆ 2 4
an d 21 . T h e se p a ra tio n o f elec tro n a n d h o le lay er s is d ˆ 4 (w ea k -c o u p ling r eg im e) .
is t h e m a g n et ic len g t h .
in ® g u re 2 ( a ) c on sists o f a sin gle m u ltip let a t L ˆ l h ˆ 1 2. In th e o ther fra m es of
® g u re 1 , th e a llo w ed lo w -en erg y electro n m u ltip lets are th o se o f ( b ) o ne , ( c ) tw o an d
( d ) thre e Q E s. W h en lh is a d d ed to ea ch o f th e v a lu es o f L e co rresp o n d in g to a lo w en erg y electro n state, a b a n d o f m u ltiplets is o b ta in ed w ith va lu es o f L satisfyin g
j lh ¡ L e j
L
lh ‡ L e . If the se pa ra tio n d b etw een electro n a n d h ole lay ers w ere
in ® n ite, so th a t V e h ˆ 0 , ev ery m u ltip let in a g iven b an d w o u ld b e d eg en era te an d
h a v e a n en erg y e q u al to th e en erg y o f th e ap p rop riate electro n sta te p lu s th e (cy clotro n ) en erg y o f the h o le, w h ich is a co n sta nt . T h e ® n ite electro n ± h o le in tera ctio n
ca u ses ® n ite a ttra ctio n b etw een th e h o le a n d (neg a tively ch a rg ed ) Q E s so th at th e
en erg ies w ith in ea ch b a n d in cre a se w ith in cre a sin g L (de crea sin g av era g e h ± Q E
sep a ra tio n ). F o r th e e lectro n ic m u ltip lets tha t a re clo se in e ne rgy , th e h ± Q E a ttra ctio n ca n a lso ca u se m ixin g o r ev en rev ers in g o f th e o rd er o f th e co rresp o n d in g
elec tro n ± h o le b an d s.
T h is v ery sim ply e xp lain s all o f the e lectro n ± h o le b a n d s a pp e arin g in th e lo w en erg y secto r o f ® g u re 2. F or ex am p le, in ® g u re 2 ( c ) the re a re fo u r h + 2 Q E ba n d s
star tin g at L ˆ 4 , 8 , 6 a n d 1 0 resu ltin g fro m th e lo w -en erg y e lectro n ic states in ® g u re
1 ( c ) a t L e ˆ 7 , 3 , 5 a n d 1 , resp ec tively. T h e ba n d s star tin g a t L ˆ 4 an d 8 h av e lo w er
en erg y tha n th o se sta rtin g at L ˆ 6 an d 1 0 b eca u se th e electro n ic m u ltip lets a t L e ˆ 7
a n d 3 h a ve lo w er en erg y th an th o se a t L e ˆ 5 a n d 1 . F o r th e ® rst ex cited secto r in
A . W o js et al.
466
® g u re 2 , w e d o ex actly th e sa m e th in g , ad d lh to the a llo w ed v a lu es o f L e in th e ® rst
ex cited secto r o f ® g ure 1 . F o r 2 S ˆ 24 this g ives b a n d s b egin n in g a t L ˆ 3 , 4 , 5 , . . . ,
1 0 , co rre spo n d in g to L e ˆ 9, 8 , 7 , . . . , 2 . A ll th ese b a n d s a re o b serv ed in ® gu re 2 ( a ).
A t 2 S ˆ 2 3, th ere a re a v ery large n u m b e r o f elect ron ic m u ltip lets in th e ® rst ex cited
secto r o f ® g u re 1 ( b ), bu t if w e co n cen trate o n th e lo w -en erg y m u ltip lets at
Le
lh ˆ S , w h ich w o u ld g ive lo w -en erg y e lectro n ± h o le b a nd s b eg in n in g a t lo w
v a lu es o f L ˆ j l h ¡ L e j , w e w o u ld ex p ect o nly a sin g le b a n d b e gin n in g a t L ˆ 0
a n d a sin g le b a n d b eg in n in g a t L ˆ 1, origin a tin g fro m the 2 Q E ‡ Q H m u ltip lets
a t L e ˆ 223 a n d 221 , resp e ctively. B eg in n in g a t L ˆ 2, w e w ou ld ex pe ct tw o ne w b a n d s,
o n e a t low er a n d o ne at h igh er en erg y , o rigin a tin g fro m tw o 2Q E ‡ Q H m u ltip lets a t
L e ˆ 129 . W e w o u ld a lso ex p ect o n e lo w - a n d on e h igh -en erg y b an d b eg in n in g at
L ˆ 3 a risin g fro m the tw o m u ltip lets a t L e ˆ 127 , etc. A ll th ese ba n d s c an b e id en ti® ed a t lo w L in ® g u re 2 ( b ). F o r 2 S ˆ 2 2 a n d 2 1, th e ® rst ex cite d b a n d s b eg in n in g a t
lo w va lu e s o f L ca n b e u n de rsto o d in ex ac tly th e sam e w a y , th a t is b y p ick in g o u t th e
lo w -en erg y electro nic sta tes a t v a lu es o f L e clo se to l h . T h u s, a ll th e lo w -en e rg y
electro n ± ho le b a n d s in ® gu re 2 c an b e ra the r w ell u n d e rstoo d in the w ea k -co u p lin g
lim it.
A ltho u gh th e a ttra ctio n b e tw ee n th e h o le a n d a given N -elec tro n eige n state
v a nish es in the d ! 1 lim it, th e m o st tigh tly b o u n d sta tes o f a h o le a n d on e , tw o
a n d th ree Q E s ca n be ide n ti® ed in ® g u re 2. In th ese sta tes, d en o ted a s h Q E , h Q E 2
a n d h Q E 3 , a Q E o r a n a p p ro p riate Q E m o lecu le m o ves a s clo se to th e h o le as
p o ssib le. T h ese sta tes h a v e th e sm a llest L w ith in th eir h + Q E , h + Q E 2 o r h + Q E 3
b a n d s a n d , to g eth er w ith th e `u n co u p led -h o le’ state h in ® g u re 2 ( a ), a re th e elem en ta ry ex cita tio ns o f th e w ea k -co u p lin g reg im e a t ® n ite d .
} 4 . St rong -coupli ng re gim e
In th e stro n g -co u p lin g reg im e th e h o le b in d s o n e o r tw o electro n s to form an X
¡
o r a n X . F o r d ˆ 0 , th e X is to ta lly u n co u p led fro m th e rem a in in g N ¡ 1 elec tro n s
(L ern er a nd L o z ov ik 1 9 8 1, D zyu b en k o a n d L o zo v ik 1 9 83 , M a cD o n a ld a n d R ez ay i
¡
1 9 90 ) a n d it is o n ly w e ak ly co u p led if d is sm all co m p are d to . In co n trast, th e X is
a n eg ative ly ch arg ed ferm io n w ith L L stru ctu re ju st lik e a n electron (W o js a nd
¡
H a w ry lak 1 9 95 , W o js et al . 1 9 9 8, 1 99 9 a ). B eca u se th e e± X p seu d o po ten tial rises
m o re q u ick ly w ith p a ir an g u lar m o m en tu m th a n th e h a rm o n ic p seu d o p o ten tial
(W o js a n d Q u in n 1 9 9 8, 1 9 9 9 a, b, 2 0 0 0a , Q u inn a n d W o js 20 0 0 ) th e lo w -en erg y states
¡
o f th e X a n d N e ˆ N ¡ 2 rem a in in g electro n s ca n b e d escrib ed b y th e g en era lized
C F p ictur e (W o js et al. 1 9 99 b ) w h ich acc o un ts fo r L a u g h lin -lik e c o rrelatio n s am o ng
th e tw o ty p es o f co n stitu en t ch a rg ed p a rticles.
¡
F o r th e sy stem co n ta in in g o n e X a n d N e re m a in in g elec tro n s, the eŒective
m o n o p o le stre ng th s seen b y a n electro n is g iven b y (W o js et al . 19 9 9 b )
2 Se ˆ 2 S ¡ … m e ¡ 1 † … N e ¡ 1 † ¡ m eX ¡ ;
w h ile tha t seen b y a n X
¡
… 1†
is
2 SX ¡ ˆ 2 S ¡ m eX ¡ N e :
… 2†
¡
H ere m e X ¡ is the ex p o n en t d escrib in g th e L au g h lin co rrelatio n s be tw ee n th e X a nd
ea ch electro n in th e tw o -co m p o n en t m an y -b o d y w av efu n ction . In the g en era liz ed C F
¡
¡
p ictu re , electro n s a n d X ’ s a re co n verte d in to tw o typ es o f C F s: C F -e a n d C F -X .
T h e an gu lar m o m en ta o f th eir lo w est sh ells a re l e ˆ S e a n d l X ¡ ˆ S X ¡ ¡ 1. T h e
fo llo w in g ty p es of L a u g h lin Q P s ca n b e d e ® n e d : th e p a rticles in the ® rst ex cited
Energy spectra and photoluminescence of fractional quantum Hall systems 4 6 7
C F -e sh ell a re `e-typ e’ q u asi-elec tro n s (Q E -e) w ith lQ E e ˆ le ‡ 1 , th e e m p ty sta tes in
th e lo w est C F -e sh ell a re `e-ty p e’ q u a sih o les (Q H -e) w ith lQ H e ˆ le , an d a sin g le
¡
¡
¡
p a rticle in th e lo w est C F -X sh ell is an `X -typ e’ q u a si-electro n (Q E -X ) w ith
¡
lQ E X ¡ ˆ lX ¡ . F o r sim p licity , in th e ® g u res w e d en o te Q H -e a n d Q E -X b y Q H an d
¡
X , resp ectively.
L et u s tu rn to th e n in e-electro n ± o n e-h o le sp ectra sh o w n in ® g u re 3 . F o r 2 S ˆ 2 1 ,
th e en erg ies o f th e m u ltip lica tive sta tes (co n ta in in g o n e u n c o up led X ) a re ex a ctly th e
sa m e a s th o se o f eigh t electro n s, w ith th e en erg y sh ifted b y th e X b in d in g e ne rgy . F o r
th e eigh t electro n s, th e lo w e st C F sh ell h a s l 0 ˆ 72 , a n d it is ® lled c o m p letely g ivin g a n
L ˆ 0 L au g h lin G S . A n ex cita tio n o f o n e C F to the n ex t sh ell g ives a `m ag n eto ro to n ’
Q E ± Q H b a n d ex ten d in g fro m L ˆ 2 to 8 , m ark ed w ith a d a sh ed lin e. T h e lo w est
¡
n o n -m u ltip licat ive state s c o nta in se ve n e lectro n s a n d an X . F o r m e ˆ 3 an d
m e X ¡ ˆ 2 w e o bta in 2 S e ˆ 2 S X ¡ ˆ 7 . T h e e igh t states in th e lo w e st C F -e sh e ll c o n¡
tain o n e Q H -e w ith lQ H e ˆ 72 , a n d th e sin g le Q E -X h a s lQ E X ¡ ˆ 52 . T h e a d d itio n of
lQ H e a n d lQ E X ¡ g ives th e b an d o f m u ltip lets ex ten d in g fro m L ˆ 1 to 6. T h ese states
¡
a re th e b o u n d states o f a Q H -e± Q E -X p a ir, co n n ected b y a solid lin e in ® g u re 3 ( d ).
T he ir en erg y in cre ase s w ith in crea sin g L , just a s th a t o f the n eu tra l ex cito n d o e s. O u r
in terp reta tio n o f th is lo w -lyin g b a n d o f n o n -m u ltip lica tive sta tes is tota lly d iŒeren t
fro m tha t o f a n eu tra l ex cito n w ith ® n ite m o m en tu m w h ich is d ressed b y m a gn et o-
F ig u r e 3 . T h e en er g y s p e ct ra ( en erg y E v er su s a n gu la r m o m en tu m L ) o f t h e n in e- elec tro n ±
o n e- h o le s y s te m o n H a lda n e’ s sp h e re w ith th e m o n o p o le s tre n g th b et w e en 2 S ˆ 2 4
an d 2 1. T h e sep a ra tio n o f ele ctro n an d h o le la y ers is d ˆ 0 (stro n g- co u p lin g r eg im e).
is t h e m a g n et ic len g t h .
468
A . W o js et al.
ro to n s o f th e L a u g h lin c o nd en sed sta te, su g g ested b y pre vio us a u tho rs (A p alk ov a nd
R a sh b a 19 9 2 , 1 9 9 3, M a cD o n a ld et al. 1 99 2 , W a n g et al. 1 9 9 2). T h e latter p ictu re
d o es n o t w ork be ca u se the co u p lin g o f an exc ito n w ith a ® n ite electric d ipo le m o m en t
to the electro n s is to o stro n g to b e trea ted p ertu rb a tive ly.
F o r 2 S ˆ 2 2 , there is a sin gle lo w -en erg y m u ltip let a t L ˆ 4 th a t is a m u ltip lic a tive state. It co rre sp o n ds to a sing le Q H in the l 0 sh ell. T h e n o n -m u ltip lica tive states
3
3
ex h ib it a lo w -en erg y b an d c on ta in in g th e m u ltip lets a t L ˆ 0
1
2
3
4
3
3
2
2
4
5
6
7
8
9
1 0. T h es e a rise fro m tw o Q H -e’ s ea ch w ith lQ H e ˆ 4
¡
¡
p lu s o ne Q E -X w ith lQ E X ¡ ˆ 3 . T h e G S ca lled X Q H 2 m ark ed w ith an o p en sq u a re
¡
is th e m o st tigh t ly b o u n d sta te o f Q E -X a n d a (Q H -e) 2 m o lecu le. Its a ng u lar
m o m en tu m L ˆ 4 re sults fro m a d d in g tw o lQ H e ’ s to o b tain l… Q H e † 2 ˆ 2 l Q H e ¡ 1 ˆ 7 ,
a n d the n a d d in g to it o ne lQ E X ¡ to o b tain l X ¡ Q H 2 ˆ j lQ E X ¡ ¡ l… Q H e† 2 j ˆ 4.
F o r 2 S ˆ 2 3 , th e lo w -en e rg y ba n d o f m u ltip lica tive sta tes co n ta in s tw o Q H s ea ch
w ith lQ H ˆ 92 , resu ltin g in th e m u ltip lets L ˆ 0
2
4
6
8 (co n n ected w ith a
d a sh ed lin e). T h e n o n -m ultip lic ative sta tes ha v e a lo w -en e rg y b a n d co n tain in g
¡
th ree Q H -e’ s eac h w ith lQ H e ˆ 92 a n d o n e Q E -X w ith lQ E X ¡ ˆ 72 . A dd itio n of th ese
4
6
7
a n gu lar m o m en t a gives a b a n d o f m u ltip lets at L ˆ 0
1
2
3
8
9
8
8
7
5
4
3
2
4
5
6
7
8
9
10
11
12
13
1 4. T h e a n g ular m o m en tu m
¡
o f th e m o st tigh tly b o u n d sta te o f a Q E -X a n d a (Q H -e) 3 m o lecu le is lX ¡ Q H 3 ˆ 7 .
T h is sta te is (m o st lik ely) th e o n e m a rk ed w ith a n o p en recta n g le.
F in a lly , fo r 2 S ˆ 24 the lo w -lyin g b a n d o f m u ltip lica tiv e sta te co n ta in s
2
2
Lˆ 0 2 3 4
5
6
7
8
9
10
1 2, arisin g fro m th ree Q H s ea ch
w ith lQ H ˆ 5 . T h e low est n o n -m u ltip lica tive ba n d co n tain s fo u r Q H -e’ s ea ch w ith
¡
lQ H e ˆ 5 a n d o n e Q E -X w ith lQ H X ¡ ˆ 4. It p ro d u ces 1 9 4 m u ltip lets startin g w ith
3
6
3
2
. . . a nd en d in g w ith 1 5
Lˆ 0
1
16
17
1 8 . O n ly th e lo w er-en erg y m u ltip lets o f th is set are sh o w n in ® g u re 3 ( d ) in w h ich w e h a v e restricted th e ra n ge of
en erg y a n d L .
It is re ally q uite rem a rk a b le th at th e m y riad o f m u ltip lets asso ciated w ith th e
lo w est b a n d o f b oth m u ltip lica tive a n d n o n -m u ltip lica tive state s ap p ea r in th e
n u m erica l sp ectra sh o w n in ® g u re 3 e xa ctly a s p red icted b y the g en era lize d C F
p ictu re . T h is sim p le m o d el ign o res th e in tera ctio n be tw ee n Q P s w h ich ca n lea d to
so m e o v erlap o f th e lo w est b a n d s w ith h igh e r b a nd s a t so m e va lu es o f L w h en th e
Q P ± Q P in tera ction en e rgy b eco m es large .
} 5. Int e rm e di at e-coupling r e gim e
W h en th e se pa ra tio n d in crea ses b ey o n d ro u g hly on e m ag n etic len g th , th e
a ttra ctiv e C o u lo m b p o ten tia l o f the h o le is n o t stro n g e n o u gh (a n d its reso lu tio n
is n o t h igh en o u g h ) to b in d a fu ll electro n . W e k n o w th is fro m n u m erica l ca lcu latio n s
fo r a sin gle electro n a n d h o le a s a fun ctio n o f d at d iŒeren t v a lu es o f 2 S . W h en m a ny
¡
electro n s a re p resen t, it is po ssib le th at the X a n d ev en th e X m ay p ersist to larg er
v a lu es o f d d u e to co rrelatio n s w ith in th e en tire electro n syste m . H o w ev er, fro m
k n o w in g th a t a t large v a lu es o f d , V e h ac ts a s a sm a ll p ertu rb a tio n o n th e N -electro n
eigen s tates, w e m igh t g u ess th a t L a u g h lin Q E s ra th er th a n `fu ll’ electro n s co u ld b in d
to th e h o le to fo rm F C X s. T h e p red ictio n th at Q E s can b in d to a h o le a t d >
at
w h ich `fu ll’ electro n s d o n o t is b a sed o n tw o sim p le facts: (i) n e g ative e lectric c h arg e
o f Q E s ( q Q E ˆ ¡ 13 e a t ˆ 13 ) ca u ses h ± Q E attra ctio n a n d (ii) b eca u se th e n u m be r of
Q E s is sm a ller tha n the n u m b e r o f e lectro n s, th eir ch a ra cteristic sep a ra tio n is larg er
a n d th eir in tera ctio n en erg y is sm a ller tha n th o se o f electro n s, an d thu s sm aller
Energy spectra and photoluminescence of fractional quantum Hall systems 4 6 9
stren g th a n d lo w er reso lu tio n o f th e ho le’ s attra ctive p o tential a re su  cien t to p ick
o n e o u t o f m a n y in tera ctin g Q E s to fo rm a b o un d sta te.
In ® gu res 4 a n d 5 w e p re sen t th e sp e ctra o b ta in ed fo r the n in e -electro n ± o n e-h o le
sy stem a t d ˆ
an d 2 . W e sha ll a tte m p t to in terp ret th e lo w -lyin g b a n d s in term s of
th e elem en ta ry e x cita tio ns a n d th eir in tera ctio n s. B eca u se d ˆ
a n d 2 a re o n th e
b o rd ers o f stro n g - a n d w e ak -c ou p lin g reg io n s , re spe ctively, th e in terp reta tio n is n ot
a lw a ys un iq ue . T h e sta ble elem en ta ry ex cita tio n s o f th e stro ng -co u p lin g reg im e a re
¡
¡
¡
X , X , X Q H o r X Q H 2 , an d tho se o f th e w ea k- co u p lin g re gim e a re h , h Q E an d
h Q E 2 . A t th e v alue s o f 2 S ˆ 2 1 an d 2 4, a t w h ich th e lo w e st-lyin g sta tes o r ba n d s at
sm a ll a n d larg e d o c cu r a t d iŒeren t L , the d -driv en p h a se tra n sitio n b etw e en th e tw o
reg im e s is of ® rst o rd er a n d the c ro ssin g of th e a p p ro p riate en erg y lev els is o b serv ed
¡
in th e sp ectru m . H o w ev er, this is n o t the ca se fo r 2 S ˆ 2 2 w he re X Q H 2 ha s th e
¡
1
sa m e L ˆ 2 … N ¡ 1 † a s h Q E 2 , o r at 2 S ˆ 2 3 w h ere X Q H 3 h a s the sa m e L ˆ N ¡ 2
a s h Q E . In th ese sp ectra , th e a n ti-cro ssin g o f en erg y lev els oc cu rs, a n d th e a n a lysis of
w av e fun ctio n s is n eed ed to d etect th e p h a se tra n sitio n (o f th e seco n d o rde r) b etw een
th e tw o re gion s .
F o r 2 S ˆ 2 1, it is clea r fro m ® gu res 4 ( d ) a n d 5 ( d ) th a t th e b a n d o f sta tes
ex ten d in g fro m L ˆ 0 to 6 a p p ea rin g in ® g u re 3 ( d ) is still p resen t b ey on d d ˆ .
F or L ˆ 5 a n d 6 it c ro sses in to th e co n tin uu m o f h igh er en erg y sta tes. F o r d ˆ 0 , th is
b a n d w a s id en ti® ed a s th e m u ltip lica tive state o f eigh t electro n s in a L a u g h lin
F ig u re 4 . T h e en erg y sp ect ra (e n e rg y E v er su s a n gu la r m o m e n t u m L ) o f th e n in e- elec tro n ±
o n e- h o le s ys te m o n H a ld a n e’ s s p h er e w ith th e m o n o p o le stren g th b e tw ee n 2 S ˆ 2 4
a n d 2 1 . T h e s ep a ra tio n o f e lect ro n a n d h o le lay e rs is d ˆ
(in te rm ed iat e- co u p lin g
re g im e) .
is t h e m ag n et ic len gt h .
470
A . W o js et al.
F ig u r e 5 . T h e en er g y s p e ct ra ( en e rg y E v er su s a n g u la r m o m en tu m L ) o f t h e n in e-e lec tro n ±
o n e- h o le sy stem o n H a lda n e’ s sp h e re w ith t h e m o n o p o le stre n g th b et w een 2 S ˆ 2 4
an d 2 1 . T h e s ep ar at io n o f elec tro n a n d h o le la ye rs is d ˆ 2 ( in ter m ed iat e- co u p lin g
reg im e) .
is th e m ag n etic len g t h .
¡
in co m p r essib le sta te p lu s a n X in its G S a t L ˆ 0 an d th e X Q H b a n d pr ed icted b y
¡
th e g en era lized C F pictu re a t L ˆ 1 , 2 , . . . , 6. H e nc e, w e co n clu d e th a t th e X a n d X
b o u n d sta tes pe rsist b e yo n d d ˆ
in this sy stem . T h e b a n d o f states ex ten d in g to
¡
L ˆ 8 w h ich cro sses th e X Q H b a n d a t L ˆ 5 ca n b e a sso ciated w ith a b o u n d state
o f th e h o le an d tw o L au g h lin Q E s, in tera ctin g w ith a th ird Q E . T h e hQ E 2 is th e m o st
stro n g ly b o u n d F C X , a s w e sh all se e later in th is se ctio n . Sin ce the L au g h lin Q E h as
lQ E ˆ l 1 ˆ S ¡ … N ¡ 1 † ‡ 1 ˆ 72 , th e a llow ed v alue s o f L 2 Q E a re 0 2 4 6 . T h e
2 Q E sta te w ith the sm a llest av era g e Q E ± Q E sep a ra tio n is the Q E 2 m o lecu le w ith
lQ E 2 ˆ 6 , a nd th e h Q E 2 state h a s lh Q E 2 ˆ j lh ¡ lQ E 2 j ˆ 92 . A d d in g to lh Q E 2 th e a ng u lar
m o m en tu m o f th e third Q E as if it w e re d istin g u ish a b le fro m th e tw o Q E s w ith in th e
h Q E 2 w o u ld g ive a b a n d o f states ex ten d ing fro m L ˆ j lh Q E 2 ¡ lQ E j ˆ 1 to
lh Q E 2 ‡ l Q E ˆ 8. H o w ev er, th e thre e Q E s in the h Q E 2 ‡ Q E b a n d a re id en tica l ferm io n s a n d m ust o b ey an d th e P a u li e xclu sio n p rin cip le. T h e ex clusio n p rin cip le
fo rb id s a n u m b er o f h Q E 2 ‡ Q E p air sta tes co rres p on d in g to th e sm a llest h Q E 2 ±
Q E sep a ratio n . T h e fo rb id d en sta tes ca n b e m o st ea sily id en ti® e d b y n o ticin g th a t th e
larg est an g u lar m o m en tu m o f thre e Q E s is lQ E 3 ˆ 3 l Q E ¡ 3 ˆ 125 w h ich , w h en ad d ed
to l h ˆ 221 , ca n n ot resu lt in a tota l h + 3Q E a n g u lar m o m en tu m sm aller th an 3 . T h u s,
th e L ˆ 1 a nd 2 sta tes o f th e h Q E 2 ± Q E p a ir a re fo rbid d en an d th e h Q E 2 ± Q E b a n d is
ex p ected to exten d fro m L ˆ 3 to 8 , e xa ctly a s o b s erv ed in ® g u re 5 ( d ). R em ark a b ly,
a lth o u g h th e h Q E 2 a n d Q E h a ve o p p o site ch a rg es ( q h Q E 2 ˆ ‡ 13 e a n d q Q E ˆ ¡ 13 e ),
Energy spectra and photoluminescence of fractional quantum Hall systems 4 7 1
th e in tera ctio n en erg y d o es n o t in crea se w ith in crea sin g L as it d o e s fo r th e elec tro n ±
h o le p a ir. T h is is d u e to th e co m p lex (no t p o in t p a rticle) stru ctu re o f th e co n stitu en ts,
w h ich m a k es th e h Q E 2 ± Q E inter a ctio n n o t gen e rally a ttra ctive (an d is resp o nsib le
fo r the in stab ility of th e h Q E 3 co m p lex ). In ® g ure 5 ( d ), the en tire h Q E 2 b a n d at
¡
L ˆ 3 to 8 is o b serv ed a t d ˆ 2 , as it is lo w er in en erg y th at th e re m n an ts o f th e X ±
Q H band.
¡
F o r 2 S ˆ 2 2, a se co n d -o rd er tra n sitio n b etw een the X Q H 2 G S o f th e stron g
co u p ling a t L ˆ 4 a n d th e h Q E 2 G S o f th e w e ak co u p lin g a t th e sa m e L is o b serv ed
in ® g u res 4 ( c ) a n d 5 ( c ). A lth o u g h it is n o t clea r fro m th e en erg y sp ectru m a lo n e, th e
tw o G Ss a n ti-cro ss a t d
. T h e lo w -lyin g ex cita tio n s o f th e h Q E 2 sta te a t d ˆ 2
a re co n n e cted w ith a solid lin e in ® g u re 5 ( c ). A t th is in term e d iate v a lu e of d , th ey a re
b est in terp reted as a disp ersio n o f a hQ E ± Q E p a ir. T h is ch a n g es a t large r d w he n th e
Q E ± Q E inter ac tio n b eco m es do m in a n t o ver the h Q E ± Q E in tera ctio n an d th e lo w lyin g ex citation s o f th e h Q E 2 sta te turn in to th e h ± Q E 2 p a ir exc itatio n s id en ti® ed in
® g u re 2 ( c ).
¡
F o r 2 S ˆ 23 , a sim ilar seco n d -o rd er tran sitio n b etw een the X Q H 3 an d h Q E
stat es o ccu rs a t L ˆ 7. A d d itio n a lly , a w ell-de ® n ed b a n d o f h Q E 2 ± Q H p a ir sta tes
o ccu rs at L ˆ 0 to 6 . T h is b a nd o ccu rs o n ly in the in term e diate-c o u plin g reg io n (at
d
) a n d c an n o t b e fou n d in ® gu re 2 o r 3 . T h e ra n g e o f d in w h ich it h a s lo w
en erg y is d eterm ine d b y th e co m p etitio n b e tw een th e h Q E ± Q E b in d in g en erg y g a in ed
th ro u g h th e fo rm a tio n o f a hQ E 2 sta te a n d th e L a u g h lin en erg y g a p to crea te a n
a d d itio n a l Q E ± Q H p a ir. T h e an g u lar m o m en ta o f th e h Q E 2 ± Q H states resu lt fro m
a d d in g lh Q E 2 ˆ 72 to lQ H ˆ 72 to o b ta in L ˆ 0 to 7 . T h e L ˆ 7 sta te co rres p o nd in g to
th e sm a llest a v era ge hQ E 2 ± Q H sep a ratio n is m o st lik ely p ush e d to a high en ergy
b eca u se o f th e Q E ± Q H h a rd co re.
F in a lly, at 2 S = 24 th e lo w erin g o f e ne rgy o f th e h sta te a t L ˆ 1 2 w ith in crea sing
d is ob se rved . In th is state th e h o le b eco m es co m p letely u n co u p led fro m the n in eelec tro n L au g h lin G S in th e d ! 1 lim it. O th er lo w -en erg y ba n d s th a t o cc ur in th e
in term e d iate-co u p lin g reg im e in vo lv e o n e o r tw o Q E ± Q H p airs sp o nta n eu sly
in d u ced in th e elec tro n sy stem to screen the ch a rg e o f the h o le. A t a sm a ller
d ˆ , th e co u p lin g o f th e h o le to th e L a u g h lin ex cita tio n s of th e e lectro n G S is
stro n g er a n d the h Q E 2 ‡ 2 Q H b a n d h a s the lo w est en erg y . A t a larg er d ˆ 2 , th e
co u p ling is w ea k er (a n d so is th e h ± Q E a ttra ctio n c om p ared to th e L a u g h lin g ap )
a n d the b an d o f h Q E ± Q H p a ir sta tes a t L ˆ 3 to 1 1 m o v es d o w n in en erg y relative to
th e h Q E 2 ‡ 2Q H b a n d. T h e b a n d sta rtin g a t L ˆ 4 an d go in g slig h tly a b o v e th e
h Q E ± Q H ba n d is b est in terp reted a s th e b a n d o f h Q E *± Q H sta tes in w hich h Q E *
is th e ® rst e xc ited sta te o f h Q E , w ith th e a n g ular m o m en tu m l h Q E ˆ lh Q E ‡ 1 . A s
sh o w n in ® g u re 2 ( a ), at ev en h igh er d , w h en th e d istan t h o le u n co u p les fro m th e
elec tro n syst em , a ll b a nd s in v o lvin g Q E ± Q H p a irs rec o n stru ct a n d th e h G S b e co m es
stab le. T h e a n g u lar m o m en ta o f th e h Q E ± Q H a n d h Q E * ± Q H p a irs ca n be ca lcu lated
b y a d d in g l h Q E ˆ 7 o r l h Q E ˆ 8 to lQ H ˆ 4 to o b ta in L ˆ 3 to 1 1 a n d L ˆ 4 to 1 2 ,
resp ective ly. F o r th e hQ E 2 ‡ 2 Q H b a n d , lh Q E 2 ˆ 3 m u st b e ad d ed to all p o ssib le
3
3
4
3
3
v alu es o f L 2 Q H ˆ 1
3
5
7 . T he resu lt is L ˆ 0
1
2
3
4
5
6
2
2
7
8
9
1 0 ; a ll th ese m u ltip lets o ccu r b elo w th e d o tted lin e in ® g u re 4 ( a ).
} 6. Photolum in e s ce nce
In the p rece din g sectio n s w e h a ve id en ti® ed th e lo w -lyin g elem en ta ry ex citatio n s
o f an id ea l sy stem co n ta in in g N electro n s co n ® n ed to a p lan e a n d o n e h o le co n ® n ed
to a n eigh b o u rin g p a ra llel plan e in the high m ag n etic ® eld lim it. In this lim it th e
472
A . W o js et al.
2
cy clo tro n en erg y ! c is so larg e co m p a red w ith the C o u lo m b en erg y e = tha t o n ly
th e lo w est L L n eed b e co n sid ered (W o js a n d Q u in n 2 0 0 1a , b ). In a ctua l ex p erim e nts
a t ® n ite m a gn etic ® elds, the m ixin g o f L L s b y th e C o u lo m b inter ac tio n oc cu rs. It is
¡
p a rticu larly im p or tan t in th e stro n g -co u plin g reg im e. W h ile on ly o n e b o u n d X state
ex ists in th e lo w est L L Рth e n o n -ra d iative trip let (W o js a n d H a w ry lak 19 9 5 , P a lacios
et al . 1 9 9 6) w ith p a ra lle l e lectro n sp in s id en ti® ed in ® g u re 3 Ð the L L m ixing lea d s to
¡
th e b in d in g o f o th er X states, p a rticu lar ly o f th e o p tica lly activ e sin g let (B u h m an n
et al . 1 99 5 , K h en g et al . 1 9 93 , W h itta k er a n d S h ield s 1 9 97 , W o js et al. 2 0 0 0a , b)
o b serv ed in P L . T he L L m ixin g is less im p o rtan t fo r th e in term ed iate an d w ea k
co u p lin g. R e al sy stem s are also c o m p lica ted b y ® n ite q u an tu m w ell w id th s, d iŒeren t
b a rrier h eigh ts fo r electro n s a n d h o les, n o n -p ara b o licity of th e en erg y b an d s, sp in±
o rb it co u p ling , e tc.
In this sectio n w e w ill discu ss P L (ra dia tive reco m b in ation o f an e lectro n ± h o le
p a ir) in th e `th eo retica l’ situ atio n in w h ich the inter ac tin g p a rticles a re co n ® n ed to
p lan es a n d o n ly the lo w est L L is tak en in to a cco u n t ( B ! 1 ). T he re a re tw o sym m etries th at lim it th e p o ssib le ra d iative d eca y p ro cesse s (W o js a n d Q u in n 2 0 00 a , b ,
2 0 01 a , b ). T h e m o st im p o rta n t on e is th e g eo m et rica l sy m m e try : tran slatio n al in v arian ce o n a p lan e (o r ro tatio n a l inv a rian ce o n H alda n e ’ s sp h ere). O n th e p lan e th ere
a re tw o co n serv ed o rb ita l qu a n tities M , the z co m p o n en t o f a n g u lar m o m en tu m ,
a n d K , an a d d itio n al q u an tu m n u m b er asso c iated w ith th e p a rtial d eco u p lin g o f th e
cen tre -of-m a ss m o tio n o f th e electro n ± h o le sy stem in th e m ag n etic ® e ld (A v ro n et al.
1 9 78 , D zy u b en k o 2 00 0 ). S tates in a g iven L L a ll h a v e th e sa m e va lu e o f L ˆ M ‡ K ,
a n d d iŒeren t sta tes in a L L are lab elled b y K ˆ 0 ; 1 ; 2 ; . . . . O n th e sph e rica l su rfa ce,
th e tota l a n gu lar m o m en tu m L a n d its z co m p o n en t L z a re co n serv ed . T he o th er
sy m m e try is th e `h id d en sym m etry ’ w hich is e xa ct in th e lo w est L L at d ˆ 0 a n d o n ly
w ea k ly b ro k en in th e en tire w ea k -co u p lin g reg im e. T h e `hid d en sy m m e try ’ , th a t is
th e p a rticle± h ole sy m m etry o f th e electro n ± va len ce-b a n d -h o le H am ilto n ian H ,
d ep en d s o n eq u a l m ag n itud e o f th e e lectro n ± elec tro n an d elec tro n ± h o le in tera ctio n s
in th e lo w est L L . T h is sym m etry m a ke s the co m m u ta to r o f th e P L o p era to r P w h ich
a n n ih ilates an o p tica lly a ctive electro n ± ho le p a ir w ith H p ro p o rtio n a l to P itself
(D zy u b en k o a n d L o zo v ik 19 8 3 , M a cD o n a ld an d R ez ay i 1 99 0 ). B eca u se o f th is,
o n ly th e m u ltiplica tive sta tes (co n ta in in g o n e u n co u p led ne u tra l ex cito n ) are ra diative a t d ˆ 0 . A t d > 0 , the states o rigin a tin g fro m o the r, n o n -m u ltip lica tive states
b eco m e rad iat ive, b u t the ir P L in ten sity rem a in s v ery lo w a t d < . T hu s the P L
sp ectru m in th e w e ak -c o up lin g reg im e g ives in form atio n a b o u t the b in d in g of th e X ,
b u t n o t a b o u t o rigin a l elec tro n ± electro n co rrelatio n s in the tw o -dim en sio n a l elec tro n
g a s (2 D E G ).
F o r w ea k a n d in term ed iate co u p ling th e syste m is b est d escrib ed in term s o f th e
1
h Q E n b o u n d states (F C X s). A t
, th e reco m b in atio n p ro ce ss c an b e th o u gh t o f as
3
(C h en an d Q u in n 1 9 9 4b )
h ‡ nQ E ! … 3 ¡ n † Q H ‡
:
… 3†
In o th er w o rds , th e h o le p lus n ˆ 0 , 1 , 2 o r 3 L a u g h lin Q E s co m b in e to give oŒ a
p h o to n
p lu s 3 ¡ n ˆ 3, 2, 1 o r 0 L a u gh lin Q E s. O the r p ro cesses, in vo lvin g a d d itio n a l Q E ± Q H pa irs , h av e m u ch sm a ller o scillator stren g th . In ® gu re 2 w e h av e
sh o w n th e lo w -lyin g b an d s fo r th e n in e-electro n ± o n e-h o le sy stem s th at co n tain ( a )
zero , ( b ) o n e, ( c ) tw o a n d ( d ) th ree Q E s, resp ective ly. A t zero tem p era tu re o n ly th e
lo w est sta te in eac h fra m e is o cc u p ied a n d ca n serv e a s an initial state. A t ® n ite
tem p e ratu re T , th e pro b a b ility o f the eigen sta te o f en erg y E b ein g o cc u pied is
Energy spectra and photoluminescence of fractional quantum Hall systems 4 7 3
p ro p o rtio n a l to e xp ‰ ¡ E = k B T Š . T h e P L sp ectr a a re o b ta in ed b y e va lu a tin g th e tran sitio n ra te w i! f b etw een lo w -lyin g in itial sta tes j i i o f th e N -electro n ± o n e-h o le sy stem
a n d ® n a l sta tes j f i o f th e … N ¡ 1 † -elec tro n sy stem ,
w i ! f ˆ co n st
2
j h f jP j i i j :
… 4†
W e h a v e u sed th e eigen fu n ctio n s o f th e n in e-electro n ± o n e-ho le sy stem a n d of th e
eigh t-elec tro n syst em o b ta in ed in n u m erical d iag o n aliza tio n to e va lu a te w i ! f . W e
® n d the follo w in g resu lts fo r w ea k a n d in term ed iate co up lin g (W o js an d Q u in n
2 00 1 a , b ).
(i) F o r w ea k co u plin g , th e P L in ten sity is w ea k . H o w ev er, th e P L sp ectra ca n
in vo lve o n e o r m o re p ea k s o f diŒeren t relative in ten s ities w h o se en erg ies
d ep e n d o n th e v a lu e o f n in eq u a tio n (3 ).
(ii) F o r in term ed iate c o u p lin g , th e stro n g est em issio n is th at o f th e stro n g ly
b o u n d a n d ra d iative h Q E 2 . T h e rec o m b in a tio n o f th e h Q E g ro u n d sta te is
fo rb idd e n b y th e co n serv a tio n o f L (o r K ), b u t th e ex cited sta te hQ E * is
ra d iative. T h e `u nc o up led -h o le’ sta te h is ra d iative a n d , ® n a lly , th e h Q E 3
`an y o n ex cito n’ p ro p o sed ea rlie r (R ash b a a n d P o rtn o i 1 9 9 3) is n eith er
b o u n d n o r rad iative.
L et u s illu stra te th e
L ˆ 0 o p tica l selectio n ru le o n th e ex a m p les o f th e h Q E 2
a n d h Q E reco m b in ation . A n isolated h Q E 2 sta te o ccu rs in th e n in e-electron ± o n eh o le syste m a t 2 S ˆ 2 2 ; see ® g u re 5 ( c ). Its a ng u lar m o m en tu m , lh Q E 2 ˆ 4, a ro se fro m
lQ E ˆ 4 an d l Q E 2 ˆ 7 co m b in ed w ith lh ˆ 1 1 . A fter th e reco m b ina tio n , w e a re left
w ith o n e Q H in th e eigh t-electron sy stem , w h ich h a s the sa m e a n g u lar m o m en tu m ,
lQ H ˆ S ˆ 4 . T h ere fore , j i i ˆ j h Q E 2 i a n d j f i ˆ j Q H i ea ch h a v e L ˆ 4 , a n d th e
o p tica l p ro cess is a llo w e d by th e
L ˆ 0 selec tio n ru le.
T h e iso lated h Q E an d h Q E * sta tes o ccu r a t 2 S ˆ 2 3; see ® g ure 5 ( b ). T h eir
a n g ular m o m en ta , lh Q E ˆ 7 a n d l h Q E ˆ 8 , resu lted fro m lQ E ˆ 92 co m b in ed w ith
lh ˆ 223 . In th e ® na l sta te, tw o Q H s o ccu r ea ch w ith lQ H ˆ 92 . T h e a llo w ed a ng u lar
m o m en ta fo r the 2Q H p a ir sta tes a re L 2 Q H ˆ 0
2
4
6
8 . C o m p a rin g a ng u lar
m o m en ta o f in itial a n d ® n a l states w e o b ta in th at th e j i i ˆ j h Q E i in itial g ro u n d sta te
ca n n o t reco m b in e to crea te a 2 Q H p a ir, j f i ˆ j 2 Q H i . H o w ev er, th e ex cited in itial
stat e j i i ˆ j h Q E i is o p tically a ctive, a n d the ® n a l sta te fo r its reco m b in a tio n is th e
j f i ˆ j Q H 2 i m o lecu le. B eca u se h Q E * is a n ex cited state, its P L lin e is ex p ected at
T > 0.
} 7 . Su m m a ry
W e h a v e ca lcu lated n u m erically th e ex a ct eigen sta tes of th e n in e-electron ± o n eh o le sy stem o n H a ld a n e’ s sp h ere in th e `id ea l’ th eo retical lim it o f ! c ! 1 a n d zero
w id th s o f electro n a n d h o le lay er s. W e h a ve sh ow n ho w th e lo w -lyin g b a n d s in
stro n g , w ea k a n d in term ed iate co u p ling ca n b e u n de rsto o d in term s o f ra ther sim p le
¡
elem en tary c o m p o site p a rticles (X , X , h Q E , h Q E 2 , etc.) a n d electro n s. W e h a v e
stu d ied th e o scillato r stre n g th fo r ra d iative reco m b in atio n a n d fo u n d tha t certa in
ra d iative d eca y p roc esses a re stro n g ly in h ib ited . In th e stro n g -co u p lin g reg io n , o n ly
th e m u ltip lica tive sta tes (o r, a t ® n ite b u t sm a ll v a lu es o f d , th e states a risin g fro m
th em ) h a ve a p p recia b le o scillator stren g th . F o r in term ed iate a n d stro n g co u p lin g th e
reco m b in ation o f the h Q E 2 bo u n d sta te h a s th e high e st in ten sity . T h e `u n co u p led
h o le’ h a n d th e ex cited sta te h Q E * a re a lso ra d iative, b u t th e reco m b in a tio n of th e
A . W o js et al.
474
latter sta te w ill o n ly b e o b serve d if th is stat e is o ccu p ied at a ® n ite tem p e ra tur e at
w h ich th e P L ex p erim en t is p e rform ed .
A CK N O W LED G EM EN TS
T h e au th o rs th a n k Iza b ela S zlu farsk a fo r h e lp o n th is p ro b lem , a n d a ck n o w led g e
th e su p p o rt o f th e M ater ials R esea rch P ro g ra m o f B a sic E n erg y S cien ces, U S
D ep a rtm e nt o f E n er gy . A . W . a c k no w led g es d iscu ssion s w ith P . H aw ry lak (IM S
N R C O tta w a) a nd M . P o tem s k i (H M F L G ren o b le), a n d p a rtial su p p o rt o f K B N
G ra n t 2 P 0 3 B 05 5 1 8. K .-S. Y . a ck n o w led g es su p p o rt fro m th e K o re a R esea rch
F o u nd a tion .
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