Effective masses of compositive fermions

8urfaco sclonco ELSEVIER

Surface Science 361/362 (1996) 92-94

Effective masses of composite fermions N . d ' A m b r u m e n i l "'*, R. M o i l " b • Department of Physics, University of Warwick, Coventry CV4 7AL, UK b Paul Scherrer Institut, Badenerstrasse 569, CH 8048 Zflrich, Switzerland Received 16 June 1995; -__cor__ptedfor publication 31 August 1995

AIm~ct The close correspondence we find between the ground-state energy and that for free fermions as a function of system size in small systems allows us to make theoretical estimates of the CF effective mass m*, not based on the sealing of gap energies away from half-filling. The quaY-particle--quasi-hole excitations for systems with filled shells also yield an estimate of the effective mass of CFs. Both thcso ostimate~ agree. Keywords: Many body and quasi-particle theories; Quantum effects

Much is already understood about the properties of a CF liquid once it has formed and once the effective mass parameter m* is known [1]. Here we show how the results of finite-size studies can give a good estimate of the effective mass parameter. Rezayi and Read ['2-1 showed that ff a CF state is formed for a system of N particles on a sphere pierced by 2S = 2(N-- 1) flux units, then the angular momentum should be the same as that expected on the basis of Hund's second rule for electrons in zero magnetic field (we are considering spinpolarized systems, so Hund's first rule is trivially satisfied). It has also been shown ['4-1 that the ground-state energy per particle follows closely that of weakly interacting quasi-particles in zero flux. The idea is that the dominant contribution to the ground-state Coulomb energy comes from *Corresponding author. Far: +44 1203 692016; e-mail" phrjf~csv.warwick.ae.ul~

the CF "kinetic" energy characterized by the effective mass parameter m*, with the residual C F - C F interactions playing a subsidiary role. We now show how this correspondence in energies can be used to give estimates of the effective mass parameter at the filling fractions v = 1/2, v--1/4 and v = 9/4. The ground-state kinetic energy per particle of N non-interacting fermions on a sphere T(N), is given by N

T(N)=(2ti2/ma2N2) ~'. l,(l,+ 1).

(1)

t

Here a is the ion disc radius, where na 2 = 1/density, for the particles of mass m, and It are the angular momenta of the N lowest energy single-particle states. For filled shells, Tf,(N)--(~/ma2)(1 - 1/N). For the electrons in magnetic flux 2 S = 2 ( N - 1 ) , the ground-state energy in the filled-shell confignrations at N = 4 , 9 and t6 lie on a straight line

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N. d'AmbrwnenU,1L Morf/Surface Science361/362 (1996) 92-94

when plotted against 1IN (Fig. la). We can cheek the extrapolation to the thermodynamic limit using the L = 0 states in the 2 S = 2 N - 3 and 2 N - 1 families. All three families extrapolate to the same thermodynamic limit. We note in passing that the scaling with 1IN is a quite general phenomenon for electrons on a sphere, which is associated with curvature effects and the precise definition of the interparticle distance [3]. The linear extrapolation in 1IN for the family with 2 S = 2 ( N , 1) defines an equivalent "filled shell" energy for a system with N particles given by the solid line in Fig. la. We subtract this from the actual Coulomb energy to give a measure for the systematic variation of the energy with angular momentum, and compare the -0.93

-0.94

eL'°.\

Z

'"'O-.

,,~ -0.9e -0.97 ILl -0.98

(a)' t

"'-.

• - - 2S=2N-2 a ....... 2S=2N-1 0 ..... 2S=2N

"~

"'"'""-.. " ""'-... "'..

-0.99

~

0.16

t

I

• kinetic energy o Coulomb energy v0~1/2

t

I"

t

~

(b)

//o

'"<'0.12

~

0.04

W

0.0

0.05

0.1

0.15

0.2

0.25

system size 1IN

Fig. 1. (a) The ground-state energy per particle plotted as a function of 1IN at % = 1/2 for 2S = 2 N - - 2 and for the L = 0 states in the sequences 2 S = 2 N and 2 S = 2 N - - 1 . For the sequenoe 2S=2N, the N = 3 result is off-scale but the dotted line passes through it. Extrapolations linear in 1/N to the thermodyrmmic limit for all sequences yield --0.932. The solid line defines the "equivalent filled-shell energy" (see text) at each N for the sequence 2S = 2N--2. Energies are quoted in units ea/ea, where a is the ion disc radius. (b) The ground-state energy minus the respective =filled shell" equivalent energies for non-interacting free fermious and for interacting electrons at % = 1/2 and at vI = 1/4. The constant C in Eq. (2) is taken as 0.2 at % = 1 / 2 and 0.18 at v=9/4.

result

with

the

corresponding

93

difference

T(N)-- Tn(N) for non-interacting CFs in zero field. The results are shown in Fig. lb. Fig. lb is plotted with the effective mass parameter C=0.20 for v=l/2, where [1]

C e2 m*- (4rip) 1/2 e"

(2)

We also show the corresponding result for v = 9/4, where we estimate C=0.18 5:0.02. A similar analysis gives C=0.15:0.02 for v=l/4. There are expected to be logarithmic corrections to the effective mass as v-*l/2 [1,5]. These are associated with long wavelength fluctuations of the gauge field, which are cut off for v ~ 1/2 by the cyclotron orbit radius for CFs ~ (or alternatively at energy scales less that the CF cyclotron energy). In finite systems on a sphere, this cut-offis provided by the diameter of the sphere and the energy level spacing. Increasing the system size should be somewhat akin to increasing o/~ in an infinite system. We would therefore expect m* to increase (C to decrease) with increasing system size. If we average the values we obtain for C in the / = 2 ( N = 5 - 8 ) and l = 3 (N=10-15) shells, we find that there is indeed a slight decrease in C and a corresponding increase in effective mass of ~ 6%. A complementary route to an estimate of the effective mass parameter involves analysis of the largest angular momentum one-paxticie one-hole excitation for CFs above the filled shell configurations at N = 4 , 9 and 16 [6]. The excitation will involve one hole in the filled shell and a particle in the next angular momentum shell. By maximising the angular momentum, the two excitations will be as far apart as possible on the sphere. The expectation is that the dominant contribution to the particle-hole excitation energy will be from the kinetic energy, and that corrections resulting from any dipole moment and residual C F - C F interactions will be higher order in the inverse system size. The results of this analysis yields C=0.195 for systems at v = 1/2. To aid comparison with experiment we can translate the parameter C into a mass. For a sample with a number density of electrons 1015 m -2, and taking a relative permittivity c = 13

94

N. d'Ambrumentl, R. Morf/Surface Science361/362 (1996) 92-94

for G-aAs, we obtain m* =0.36m. at v= 1/2, m* = 0.7m. at v = l / 4 , and m*=0.13m, at v=9/4. Hero, me is the free electron mass. For samples with number densities n x 1015 m -2 these results are multiplied by n 1/2. The estimates are based on a bare Coulomb interaction between electrons and do not take account of the extent of the wavefunction in the perpendicular direction.

Acknowledgements We thank the ISI Foundation in Torino for their hospitality and the participants in the ISI work-

shop on the Quantum Hall Effect, June 1994, for many interesting seminars. We also thank B.I. Halperin and E. Rezayi for useful discussions.

References El] B.L Halperin, P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312. [2] E. Rezayi and N. Read, Phys. Rev. Lett. 72 (1994) 900. [3"1 R. Morf and B.L Halperin, Z. Phys. B 68 (1987) 391. [4] 1L Morf and N. d'Ambrumenik Phys. Rev. Lett. 74 (1995) 5115. [5"1 A. Stern and B.I. Halperin, SISSA cond/mat 9502032 (1995), preprint. [6"1 E. Rezayi, N. Read, R. Mo~ N. d'Ambrumenil and S.H. Simon, APS Meeting. March 1995, San Jose.