,Solid State C o m m u n i c a t i o n s , P r i n t e d in Great B r i t a i n
Vol.53,No.l, pp.2729,
1985.
00381098/85 $3.00+ .00 P e r g a m o n Press Ltd.
ON THE F R A C T I O N A L Q U A N T U M HALL EFFECT Francisco Claro Facultad de Ffsica, P o n t i f i c i a U n i v e r s i d a d Cat61ica de Chile, (Received 5 September
Casilla
ll4D, Santiago,
Chile
1984 by M. Cardona)
We show in the H a r t r e e  F o c k a p p r o x i m a t i o n that the formation of a two d i m e n s i o n a l e l e c t r o n lattice allows for a natural e x p l a n a t i o n of the anomalous fractional q u a n t u m Hall effect. L a n d a u levels are b r o a d e n e d and split in a number of bands in such a way that if the n u m b e r of electrons per unit cell is a half odd i n t e g e r the Fermi energy is in a gap for an odd filling fraction denominator, and at the center of a band if the d e n o m i n a t o r is even.
b]e of e x T ] a i n i n g in a very n a t u r a l way the zero r e s i s t a n c e s t a t e for f r a c t i o n a l f i l l i n g s w i t h an odd d e n o m i n a t o r N o n l y , and an a r b i t r a r y n u m e r a t o r M p r i m e to N. We start our a r g u m e n t by c o n s i d e r i n g a W C in the H a r t r e e  F o c k a p p r o x i mation. T h i s m a n y b o d y g r o u n d s t a t e m a y be c o n s t r u c t e d in the L a n d a u gauge u s i n g a s i n g l e p a r tJc]e b a s i s set Ink > , w h e r e n = 0, 1, 2,... lab e l s the L a n d a u l e v e l s and k is a w a v e n u m b e r in the y  d i r e c t i o n . At large m a g n e t i c fields the m i x i n g of L a n d a u l e v e l s m a y be n e g l e c t e d and one can treat for s i m p l i c i t y just the l o w e s t level n : 0 and e x p r e s s the H a r t r e e  F o c k o r b i t a l s in t e r m s of the set 10k > £ ]k > only. This app p r o a c h h a s b e e n f o l l o w e d by s e v e r a l a u t h o r s in the past a n d we s h a l l omit d e t a i l s . 9 , ] ] , ]4 For the W C s t a t e one o b t a i n s the f o l l o w i n g s y s t e m of coupled equations:
The q u a n l u m H a ] ] e f f e c t is a zero Tesistance sta~e that develops in a I~o dimensional strip of e ] e c c r o n s in the pre~enee of a large magnetic field f o r an o c c u p a t i o n of Landau ]eve]s of the Iorm~ v = M/N, w h e r e M and E ar~ i n t e g e r s prime to e a c h o t h e r , and N is odd. ]3 The case of integral f i l l i n g (N = 1) is current]}, u n d e r stood in t e r m s of the energy gap that c h a r a c t e r izes the s e p a r a t i o n b e t w e e n i m p u r i t y b r o a d e n e d adjacent L a n d a u levels and for w h i c h the simple one e l e c t r o n p i c t u r e p r o v i d e s the b a s i c insight. ~7 For f r a c t i o n a l f i l l i n g a m o r e s o p h i s t i c a t e d a p p r o a c h is n e e d e d since the one e l e c t r o n m o d e l does not p r o d u c e gaps at v a ] u e s of N d i f f e r e n t f r o m unity. Two m o d e l s for tbe ground s t a t e h a v e b e e n e x p l o r e d to e x p l a i n the a p p e a r a n c e of f r a c t i o n a l n u m b e r s . The first a s sumes that a c h a r g e d e n s i t y w a v e or W i g n e r c r y s tal (WC) is f o r m e d t h r o u g h the e l e c t r o n  e l e c t r o n
e2 1
I
Es  h~c]
<
kls
i~ 2 k +
• >

x
_ _ _
Is >
=
0
(la)
Y G
4
1!n[
G
"2J
~
sk
+~
> e
i~ 2 k +
Gx
(]h)
e
y
interaction. Past w o r k u s i n g this a p p r o a c h found no e v i d e n c e for a s e l e c t i o n rule that ex
H e r e Es i s t h e e n e r g y a s s o c i a t e d with orbit;l] s, u~c is the e y c ] o c r o n frequency, c an a v e r n g e d i e l e c t r i c c o n s t a n t , [ = (hc/eB) ]/2 the Larmor length,.f~, a Fourier c o m p o n e n t of the c h a r g e in a unlt cell, O = (Gx, Gy) a r e c i p r o c a l lattice v e c t o r , n the o c c u p a t i o n n u m b e r of state s, s N the number of cells in the WC, and
cludes f i l l i n g s of even d e n o m i n a t o r from exh i b i t i n g z e r o r e s i s t a n c e . 8]7 The s e c o n d m o d e l is an i n c o m p r e s s i b l e q u a n t u m f l u i d (IQF) obtained using a Jastrowlike variational wave function. 18 T h i s s t a t e o c c u r s for f i l l i n g s of the lowest L a n d a u level ~ = I/N, N odd only, thus c o n s t i t u t i n g a g o o d c a n d i d a t e to e x p l a i n the e x p e r i m e n t a l facts. E x t e n s i o n s of this m o d e l to n u m e r a t o r s M g r e a t e r than ] h a v e a l s o been p r o posed. ]9 In this p a p e r w e show that the f o r m a t i o n of a two d i m e n s i o n a l l a t t i c e of e l e c t r o n s is capa
[2G2 U(G)
e
(i  6G,0) G[
27

I° 
28
Vol.
ON THE F R A C T I O N A L Q U A N T U M HALL EFFECT
53,,No.
spectively. It is convenient to make the transformation
a r a p i d l y d e c a y i n g f u n c t i o n of IGI = G, w i t h lo(x ) the m o d i f i e d Bessel function. The p a r a m e t e r ~ = ~ B / ( h c / e ) is an i m p o r t a n t q u a n t i t y that m e a s u r e s the n u m b e r of flux q u a n t a t r a v e r s ing the unit cell of the WC. It is also the ratio of the W C cell area ~ to that of the m a g n e t i c lattice. 20 The s y s t e m of e q u a t i o n s (1) m u s t be s o l v e d self c o n s i s t e n t l y . Once the s o l u t i o n s are
f(~)
el~
=
~(~p)
2~k < a
Is >
w h e r e the index s is now i m p l i c i t in the n e w l y defined function f(K). E q u a t i o n (5) then takes the form
_2~1J¢ p7 + m)n  (m + P)4~ F
m~n k n o w n the e n e r g y per p a r t i c l e expression
1 [ 7IrisEs
E :
iTi,n
e
~
 f ( <
0
(6)
This e q u a t i o n is invariant u n d e r the translation K + K + ~. In p a r t i c u l a r , q c o n s e c u t i v e t r a n s l a t i o n s make < ~ < + p so for a given value of K Eq. (6) may be closed by use of Floquet t h e o r e m
is given by the
+ 1 ~h~°cj
+ m)
(2)
s
f(K + p)
=
e2Wiuf(K),
(7)
or
h~c
:
rr ~
~
no i 2
C~,
[2 ~ 2
j7
272
Z
~ ~e0
i~p ~2
w h e r e n O is the total n u m b e r of p a r t i c l e s , =
,
U~G)e
w h e r e 0 ~< p < i is the Floquet i n d e x and ~ n o w also has the range 0 <~ K < I. Using (7) the e q u a t i o n may be w r i t t e n in the form
and
(4)
Cv
is the n u m b e r of e l e c t r o n s p e r unit cell. E q u a t i o n (la) was s t u d i e d first in the past a s s u m i n g the charge d i s t r i b u t i o n r e s p o n s i b l e for the e l e c t r o n  l a t t i c e c o u p l i n g is given, and k e e p ing the v e c t o r s n e a r e s t to the o r i g i n o n l y in the r e c i p r o c a l l a t t i c e e x p a n s i o n . 2122 R e s u l t s are a v a i l a b l e for the s q u a r e and t r i a n g u l a r l a t t i c e s and for our p u r p o s e s the most i m p o r t a n t facts are that the single p a r t i c l e d e g e n e r a t e L a n d a u level is b r o a d e n e d by the e l e c t r o n  l a t t i c e i n t e r a c t i o n and a c q u i r e s a s t r u c t u r e r e s u l t i n g in a s p e c t r u m of p bands, w h e r e p is the n u m e r a t o r in the p a r a m e t e r ~ = p/q w h i c h is a s s u m e d to be a rational. The coarse s t r u c t u r e of bands and gaps is u n a f f e c t e d by small d e v i a t i o n s from this rational. Furthermore, the p b a n d s are c l o s e d and the n u m b e r of states in each is the same. 2223 These results are still rigorously true if the reciprocal lattice vectors next to nearest the origin are kept in the square lattice calculation. 2~ The 8eneralization to c o u p l i n g to all Fourier components p~ is simple for showing the formation of p separate bands if the lattice has inversion symmetry. For d e f i n i t e ness we take a t riangula i lattice with G = 4<(n  m/2, \~m/2)/w/la, with m, n integers, and rewrite Eq. (]a) Jn the form
X 6
~L m,l~
F
[II i n
~
+
(~ +m),
2
where K = ka/2~ with a the lattice constant, and the e i g e n v a l u e s X s and c o e f f i c i e n t s Im,n contain the energy and charge density, re
2 Irm
a
(3)
Xf(K+£)
pI  ,[ R £ ~ , ( ~ , ~ , ~ ) f ( K + £ + ~ =0
i )
= 0
(8)
w h e r e ~ : 0,1,..., p  l . For every < Eqs. (7) and (8) form a s y s t e m of p e q u a t i o n s and p unknowns. The m a t r i x R~£' turns out to be h e n n i t i a n for a lattice with inversion s y m m e t r y w h i c h we take to be the case. Then there are p real e i g e n v a l u e s X],..., Xp that form as m a n y separate bands, as ~ and K vary throughout their range. The spectrum has thus p bands. A self consistent solution of Eqs. (I) exhibits the same property. In what follows we shall a6sume then lhat the s p e c t r u m of single p a r t ~ c ] e Btates E~ h a t , p b a n d s e a c h w i t h t h e s a m e n u m b e r Of states. We n o w show that this mode] is c a p a b l e of e>:p]aJnin~ t h e a p p e a r a n c e of o d d d e n o m i n a t o r s only in the q u a n t u m Hall effect. We start out by w r i t i n g the n u m b e r of e l e c t r o n s per unit cell as a half i n t e g e r ~ = Q/2. From (4) the L a n d a u ]eve] f i l l i n g factor is then ~ = Qq/2p w h e r e ~ is t a k e n to he the rational p/q as h e l o r e . Since there are p b a n d s with equal n u m b e r of states the n u m b e r of filled bands is g i v e n by o = Qq/2. If this n u m b e r is an i n t e g e r the
I
:s>
0
(5)
F e r m i e n e r g y lies in a gap and no f r i c t i o n w i l l o p p o s e the flow of a current at low e n o u g h temperatures. If it is a fraction the s y s t e m
Vol. 53, No.
]
ON THE FRACTIONAL QUANTUM HALL EFFECT
is an ordinary metal. Assume first that Q is even. Then ~ is always an integer and therefore all rational filling fractions lead to a frictionless current. This is the type of state considered by Y o s h i o k a and Lee. ]4 If Q is odd, two cases may be distinguished. For q odd as well, ~ is a half integer and the system will exhibit metallic bebaviour. The denominator of v is even in this case. For an even q = 2£ then ~ is an integer and one expects a frietionless current. Since q even forces p to be odd then v = Q£/p has an odd denominator. Thus, provided the n u m e r a t o r of T is odd, a zero resistance s t a t e is formed for N odd and a metallic state for N even, in agreement with experiment. This completes our proof. An estimate of the energy gap for v = i/3 may be obtained using results available in the literature. 22 Taking 7 = ]/2, c = 13 and £ = 80A we find AE ~ h w 6 / 4 , which is consistent with experiment. 25 A/so, decreasing m the gap decreases as well, demaning lower temperatures for the effect to be observable, also in agreement with experiment. 26 The HartreeFock approximation gives for 7 = 1 a ground state energy for the WC up to 5% higher than the IQE state.16, ]8 Results for half odd integral 7 as required by our analysis are not available in the literature. Current
29
results therefore make the IQF model preferable to the WC. The HartreeFock and the Jastrowlike wave functions are only an approximate representation of the true ground state of the system, however. Exact calculations for the ground state done for up to six electrons in a cell that repeats periodically give lower energies than HartreeFock for the crystal state,and appear to faw~ur a non periodic distribution of charge within each cell .2"3 No clear clue for the appearance of tile fractional values of found in experiment is given by this system, probably due to its small size. Our work shows that the formation of a lattice by the electrons allows for a natural explanation of the experimental facts that is not available in previous treatments of the problem. We therefore propose that such a lattice contains the key physical idea to explain the fractional quantum Hall effect. Its main feature shown in the HartreeFock approximation that the bottom Landau level is broadened and split into an appropriate number of bands is likely to persist as the approximation is improved, provided the underlying crystalline arrangement of electrons is maintained. This work was supported by DIUCGrant and CONICYTGrant ]030/83
48/83
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