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Quasiparticle charge of 13 FQH liquid from resistance fluctuation measurements
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Surface Science North-Holland
263
(1992)72-75
.:.
surface science
.; ,:...j ::::: .‘:i::.~i::i ..::. .:.:.:..A.. :...*> ,,:, :.+:. ~:.:,,:::.:., <::: ::.,: .,)... ““.:‘:‘-:l:i.::;i:.::~~::~,~:::~:~::..~.:::::: ,.:.: : .
Quasiparticle charge of l/3 fluctuation measurements S.W. Hwang,
J.A. Simmons
FQH liquid from resistance
*, D.C. Tsui and M. Shayegan
Department of Electrical Engineering, Princeton lJnk~ersi&, Princeton, NJ 08544, USA Received
30 June
1991; accepted
for publication
26 August
1YYl
We have studied resistance fluctuations as a function of magnetic field (B) and backgate bias tV,J at v = l/3, 1. and 2 in a high quality two-dimensional electron gas (2DEG). We find that, within an experimental uncertainty of - 25%, the relation among the fluctuation periods AB, at fixed n, is AB(v = l/3) = ~AB(Y = 1) = 3AB(v = 2), but at fixed B, the relation among the periods AVG is AV,(v = l/3) = AV,(v = 1) = (1/2)AV,(u = 2). These results provide experimental evidence that the charge of quasipartiof the semiclassical orbit is cle excitations from the l/3 FQH liquid is r * = e/3 and that the flux quantum relevant to quantization @* = 3h/e
The fractional quantum Hall effect (FQHEI is manifestation of a series of new ground states of interacting 2D electrons in a strong magnetic field [1,2]. The phenomenon is characterized by the vanishing of the diagonal resistance R,, of the 2D electrons at Landau level filling factors of odd denominator fractions, v =p/q, and, concomitantly, the appearance of Hall resistance plateaus quantized at (q/p)h/e’. The flow of dissipationless current, seen in vanishing R,, , results directly from the incompressible fluid nature of the ground states; the exact fractional quantization of the Hall resistance, on the other hand, is explained in terms of the theoretical construct that the quasiparticle (quasihole) excitations of the l/q FQHE ground state, which is separated by an energy gap from the ground state, carry charge e * = e/q( -e/q). Several recent experiments have attempted to determine the charge carried by the quasiparticle/quasihole excitations of the FQHE liquids. Clark et al. [3] have observed that, at v =p/q,
* Present querque,
address: Sandia NM 87185. USA.
0039-6028/92/$05.00
National
0 1992 - Elsevier
Laboratories,
Science
Albu-
Publishers
the T dependence of the conductance a,, in an Arrhenius plot extrapolates to (1/q2>e2/h as T + m. Chang and Cunningham [4] have observed the quantization of R,, for transmission of electrons from regions of v = 1 across barriers at v = 2/3, and from regions at v = 2/3 across barriers at v = l/3, and suggest an interpretation of their results in terms of e* = e/q. We have in an earlier paper by Simmons et al. [51 reported the observation of resistance fluctuations as a function of B, near the R,, minima of Y = 1, 2, 3, 4, and l/3, and the striking result that, while the quasiperiod of the fluctuations for integer v are for Y = l/3 is all - 0.016 T, the quasiperiod - 0.05 T, a factor of three larger. The fluctuations in the IQHE case is explained in terms of the Jain-Kivelson model [6] of resonant reflection of edge current channels by magnetically bound states encircling a potential hill of a diameter roughly the conducting width of the sample. In a similar semiclassical picture, the fluctuations near v = l/3 minimum is attributed to scattering by bound states of the fractionally charged quasiparticle excitations on the same potential hill and the difference in the quasiperiod is the difference in bound state energy due to the charge differ-
B.V. and Yamada
Science
Foundation.
All rights reserved
S. W. Hwang et al. / Quasiparticle charge of 1 / 3 FQH liquid from resistance fluctuation measurements
ence. The experiment, as already suggested by Kivelson and Pokrovsky [7], was interpreted as direct evidence that the charge of the quasiparticles of the l/3 FQH state is e* = e/3. Subsequently, Kivelson [8] has considered the theoretical problems in interpreting the experiment as measuring directly the Aharonov-Bohm CAB) interference effect. He found the dynamical conditions for the bound states, under which the experiment measures the quasiparticle charge directly and has suggested a possible experimental configuration, which will allow direct probing of the fractional statistics as well as the fractional charge of such semiclassically bound quasiparticle/quasihole excitations. Lee [93, on the other hand, has suggested that a Coulomb blockade may be responsible for the experimental observation. More recently, Thouless and Gefen [lo] have studied the AB oscillations in the l/q FQHE and discussed the possibility of observing crossover from a qh/e period to an h/e period in samples of finite size at finite T under the influence of disorder. We have since studied the dependence of the fluctuation amplitude on temperature and current [ll] and demonstrated that the Coulomb blockade is not relevant in our experiment. In addition, we have fabricated new samples, on which backgate biases Vo can be applied to vary the electron density nzD, and consequently been able to study the fluctuations as a function of V, as well as B. The data to be discussed below are taken from an - 2.5 pm wide Hall bar sample of GaAs/AlGaAs heterojunction, patterned by using photolithography and wet etching. The GaAs/AlGaAs heterojunction has a 2D electron density n 2D = 1.0 x 10” cm-’ and mobility p = 1.4 x lo6 cm*/V . s at 4.2 K. The measurements were done using the AC lock-in technique at 13.7 Hz, in a dilution refrigerator with a superconducting magnet. Fig. 1 is an example of the resistance fluctuations observed near the R,, minima of v = l/3, 1, and 2. In the B scan data (which are displayed on the left-hand-side panel), a dominant high frequency component is discernible in the fluctuations as R,, increases from its minimum value
oo13 13.1 13.2 13.3 13.4 13.5 0 1500~,
2
4
73
6
8
10
lsooy,
Fig. 1. Resistance fluctuations near the R,, minima of v = l/3, 1, and 2 as a function of B and Vo. Scans for Vo are taken when B is fixed at the values indicated by the arrows in the B scan data. Notice the scale change of Vo for the data taken at B = 2.27 T.
for all the filling factors. For the integer fillings, v = 1 and 2, there is no appreciable difference in periodicity and the period is - 100 G. But, for v = l/3, it is clearly longer and is N 300 G. These data are similar to those reported in ref. [5], though the periods in the earlier data are - 50% larger. Shown on the right-hand-side panel are the fluctuations in R,, as a function of Vo, measured at the fixed value of B, which are indicated by the arrows in the corresponding B scan data. Here, the perdiod A&, of the fluctuations near the v = l/3 and 1 R,, minima is - 1 V, but for v = 2, it is considerably larger and is - 2 v. In fig. 2, we summarize our data on AB and AVo by plotting them as a function of T. The periods are obtained by Fourier transforming the fluctuations and, due to the limited number of fluctuations and the changing background, the uncertainty is - 25%. The relation between the AB values for the three different v’s is AB(v = l/3) = 3AB(v = 1) = 3AB(v = 2), in agreement
0
40
‘0-O
80
120
160
T (mK)
Fig. 2. Fluctuation periods AB and AV, versus T. Typically, the ~uctuati~ns are not observed above 150 mK.
with that reported in ref. [5]. The relation of AV, among the three filling factors is: Al/,,(v = l/3) = AV@ = 1) = (1/2)Ai/,(v = 2). In the Jain-KiveIson model, the fluctuations in the IQHE result from resonant reflection of edge current channels by magnetically bound states encircling a potentail hill of a diameter roughly the conducting width of the sample. The bound states are the results of Bohr-SommerfeId quantization of the semiclassical orbits of the 2D electrons along the equipotentials around the potential hill. The quantization condition in general is given by e *@,/ fi = 2rrn,
(1)
where 4,, is the magnetic flux enclosed in the nth allowed orbit and e* is the charge of the semiclassical particle. For our 2D electrons e* = e and, in our experiment with a uniform 3, 4,, is simply related to the area A, of the orbit by 4, = BA,. A resistance peak occurs in this model whenever a bound state is aligned with the chemical potential, p, of the edge channel. The fluctuation period in the B scan data, given by the change in B needed to align the next bound state with p, is AB = A&[d( BA,)/dB]
-I.
(2)
In the IQHE case, A4,, = #,?+ I - 55, = &h/e. Consequently, AB is the same for all integer v (if dA,/dB is negligible), in agreement with experiment. In the V,; scan experiment, the period given by the change in p to align the next bound state is: AV, = A~~[d(B~~)/d~~]-‘. Since the fluctuations for the v = 1 minimum is observed at twice the B for the v = 2 fluctuations, the observed A&CV = 1) = (1/2)AVo(~ = 2) relation is also expected from the model. On the other hand, if the semiclassical orbits are assumed by the quasiparticles in the l/3 FQHE case, e* in the quantization condition is e/3 and A4, = h/e” = 3h/e. As a result, AB at v = l/3 should be three times that at v = 1 and AV,, because the fluctuations for v = l/3 is observed at a three times higher B, should be the same as at v = 1, all in agreement with experiment. In conclusion, we have presented new results from our study of the resistance fluctuations near the L’= l/3, 1, and 2 Rx,< minima. In particular, we have fabricated new samples, on which backgate biases can be applied to vary nzu to make it possible to study the fluctuations as a function of Vci as well as B. The fluctuation periods A B and AV, for the three different V’S are found to follow the relations: AB(v=
l/3)
=3AB(zr=
1) =3AB(v=2),
(3)
and AV,;( v = l/3)
= AV,,( v = 1) = (1/2)AV,;(
v = 2).
(4)
These relations are in agreement with the JainKivelson model, which is based on the existence of magnetically bound states resulting from Bohr-Sommerfeld quantization of the semiclassical 2D electron orbits in the IQHE, and its extension to the FQHE by assuming that the quasipartitles of the l/3 FQH liquid traversing the similar semiclassical orbits carry charge e” = c/3. Our observation of the latter relation that AVc,(v = l/3) = AV,(v = I) = (1/2)AV;;(v = 2) is incompatible with the Coulomb blockade model 191. We thank L.W. Engel, A.M. Chang and X.G. Wen for helpfu1 discussions. This work is sup-
S. W. Hwang et al. / ~~asi~arfic~
charge of l/3
ported by ONR through contract No. NOO014-89J-1567, and NSF through Grants No. DMR8719694 and No. DMR-8921073. One of us (S.W.H.) thanks the Korean Government Ministry of Education for its partial financial support.
References [l] D.C. Lett. [2] R.B. [3] R.G. CT.
Tsui, H.L. Stormer and AC. Gossard, Phys. Rev. 48 (1982) 1559. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. Clark, J.R. Mallett, S.R. Haynes, J.J. Harris and Foxon, Phys. Rev. Lett. 60 (1988) 1747.
FQH liquid from resistance ~~ctuutia~
measu~erne~t~
75
[4] A.M. Chang and J.E. Cunningham Solid State Commun. 72 (1989) 652. ]5] J.A. Simmons, H.P. Wei, L.W. Engel, DC. Tsui and M. Shayegan, Phys. Rev. Lett. 63 (1989) 1731. [6] J.K. Jain and .%A. Kivelson, Phys. Rev. Lett. 60 (1988) 1542. [7] S.A. Kivelson and V.L. Pokrovsky, Phys. Rev. B 40 (1989) 1373. [8] S. Kivelson, Phys. Rev. Lett. 65 (1990) 3369. [9] P.A. Lee, Phys. Rev. Lett. 65 (1990) 2206. [lo] D.J. Thouless and Y. Gefen, Phys. Rev. Lett. 66 (1991) 806. Ill] J.A. Simmons, SW. Hwang, DC. Tsui, H.P. Wei, L.W. Engel and M. Shayegan, Phys. Rev. B. 44 (1991).
Surface Science North-Holland
263
(1992)72-75
.:.
surface science
.; ,:...j ::::: .‘:i::.~i::i ..::. .:.:.:..A.. :...*> ,,:, :.+:. ~:.:,,:::.:., <::: ::.,: .,)... ““.:‘:‘-:l:i.::;i:.::~~::~,~:::~:~::..~.:::::: ,.:.: : .
Quasiparticle charge of l/3 fluctuation measurements S.W. Hwang,
J.A. Simmons
FQH liquid from resistance
*, D.C. Tsui and M. Shayegan
Department of Electrical Engineering, Princeton lJnk~ersi&, Princeton, NJ 08544, USA Received
30 June
1991; accepted
for publication
26 August
1YYl
We have studied resistance fluctuations as a function of magnetic field (B) and backgate bias tV,J at v = l/3, 1. and 2 in a high quality two-dimensional electron gas (2DEG). We find that, within an experimental uncertainty of - 25%, the relation among the fluctuation periods AB, at fixed n, is AB(v = l/3) = ~AB(Y = 1) = 3AB(v = 2), but at fixed B, the relation among the periods AVG is AV,(v = l/3) = AV,(v = 1) = (1/2)AV,(u = 2). These results provide experimental evidence that the charge of quasipartiof the semiclassical orbit is cle excitations from the l/3 FQH liquid is r * = e/3 and that the flux quantum relevant to quantization @* = 3h/e
The fractional quantum Hall effect (FQHEI is manifestation of a series of new ground states of interacting 2D electrons in a strong magnetic field [1,2]. The phenomenon is characterized by the vanishing of the diagonal resistance R,, of the 2D electrons at Landau level filling factors of odd denominator fractions, v =p/q, and, concomitantly, the appearance of Hall resistance plateaus quantized at (q/p)h/e’. The flow of dissipationless current, seen in vanishing R,, , results directly from the incompressible fluid nature of the ground states; the exact fractional quantization of the Hall resistance, on the other hand, is explained in terms of the theoretical construct that the quasiparticle (quasihole) excitations of the l/q FQHE ground state, which is separated by an energy gap from the ground state, carry charge e * = e/q( -e/q). Several recent experiments have attempted to determine the charge carried by the quasiparticle/quasihole excitations of the FQHE liquids. Clark et al. [3] have observed that, at v =p/q,
* Present querque,
address: Sandia NM 87185. USA.
0039-6028/92/$05.00
National
0 1992 - Elsevier
Laboratories,
Science
Albu-
Publishers
the T dependence of the conductance a,, in an Arrhenius plot extrapolates to (1/q2>e2/h as T + m. Chang and Cunningham [4] have observed the quantization of R,, for transmission of electrons from regions of v = 1 across barriers at v = 2/3, and from regions at v = 2/3 across barriers at v = l/3, and suggest an interpretation of their results in terms of e* = e/q. We have in an earlier paper by Simmons et al. [51 reported the observation of resistance fluctuations as a function of B, near the R,, minima of Y = 1, 2, 3, 4, and l/3, and the striking result that, while the quasiperiod of the fluctuations for integer v are for Y = l/3 is all - 0.016 T, the quasiperiod - 0.05 T, a factor of three larger. The fluctuations in the IQHE case is explained in terms of the Jain-Kivelson model [6] of resonant reflection of edge current channels by magnetically bound states encircling a potential hill of a diameter roughly the conducting width of the sample. In a similar semiclassical picture, the fluctuations near v = l/3 minimum is attributed to scattering by bound states of the fractionally charged quasiparticle excitations on the same potential hill and the difference in the quasiperiod is the difference in bound state energy due to the charge differ-
B.V. and Yamada
Science
Foundation.
All rights reserved
S. W. Hwang et al. / Quasiparticle charge of 1 / 3 FQH liquid from resistance fluctuation measurements
ence. The experiment, as already suggested by Kivelson and Pokrovsky [7], was interpreted as direct evidence that the charge of the quasiparticles of the l/3 FQH state is e* = e/3. Subsequently, Kivelson [8] has considered the theoretical problems in interpreting the experiment as measuring directly the Aharonov-Bohm CAB) interference effect. He found the dynamical conditions for the bound states, under which the experiment measures the quasiparticle charge directly and has suggested a possible experimental configuration, which will allow direct probing of the fractional statistics as well as the fractional charge of such semiclassically bound quasiparticle/quasihole excitations. Lee [93, on the other hand, has suggested that a Coulomb blockade may be responsible for the experimental observation. More recently, Thouless and Gefen [lo] have studied the AB oscillations in the l/q FQHE and discussed the possibility of observing crossover from a qh/e period to an h/e period in samples of finite size at finite T under the influence of disorder. We have since studied the dependence of the fluctuation amplitude on temperature and current [ll] and demonstrated that the Coulomb blockade is not relevant in our experiment. In addition, we have fabricated new samples, on which backgate biases Vo can be applied to vary the electron density nzD, and consequently been able to study the fluctuations as a function of V, as well as B. The data to be discussed below are taken from an - 2.5 pm wide Hall bar sample of GaAs/AlGaAs heterojunction, patterned by using photolithography and wet etching. The GaAs/AlGaAs heterojunction has a 2D electron density n 2D = 1.0 x 10” cm-’ and mobility p = 1.4 x lo6 cm*/V . s at 4.2 K. The measurements were done using the AC lock-in technique at 13.7 Hz, in a dilution refrigerator with a superconducting magnet. Fig. 1 is an example of the resistance fluctuations observed near the R,, minima of v = l/3, 1, and 2. In the B scan data (which are displayed on the left-hand-side panel), a dominant high frequency component is discernible in the fluctuations as R,, increases from its minimum value
oo13 13.1 13.2 13.3 13.4 13.5 0 1500~,
2
4
73
6
8
10
lsooy,
Fig. 1. Resistance fluctuations near the R,, minima of v = l/3, 1, and 2 as a function of B and Vo. Scans for Vo are taken when B is fixed at the values indicated by the arrows in the B scan data. Notice the scale change of Vo for the data taken at B = 2.27 T.
for all the filling factors. For the integer fillings, v = 1 and 2, there is no appreciable difference in periodicity and the period is - 100 G. But, for v = l/3, it is clearly longer and is N 300 G. These data are similar to those reported in ref. [5], though the periods in the earlier data are - 50% larger. Shown on the right-hand-side panel are the fluctuations in R,, as a function of Vo, measured at the fixed value of B, which are indicated by the arrows in the corresponding B scan data. Here, the perdiod A&, of the fluctuations near the v = l/3 and 1 R,, minima is - 1 V, but for v = 2, it is considerably larger and is - 2 v. In fig. 2, we summarize our data on AB and AVo by plotting them as a function of T. The periods are obtained by Fourier transforming the fluctuations and, due to the limited number of fluctuations and the changing background, the uncertainty is - 25%. The relation between the AB values for the three different v’s is AB(v = l/3) = 3AB(v = 1) = 3AB(v = 2), in agreement
0
40
‘0-O
80
120
160
T (mK)
Fig. 2. Fluctuation periods AB and AV, versus T. Typically, the ~uctuati~ns are not observed above 150 mK.
with that reported in ref. [5]. The relation of AV, among the three filling factors is: Al/,,(v = l/3) = AV@ = 1) = (1/2)Ai/,(v = 2). In the Jain-KiveIson model, the fluctuations in the IQHE result from resonant reflection of edge current channels by magnetically bound states encircling a potentail hill of a diameter roughly the conducting width of the sample. The bound states are the results of Bohr-SommerfeId quantization of the semiclassical orbits of the 2D electrons along the equipotentials around the potential hill. The quantization condition in general is given by e *@,/ fi = 2rrn,
(1)
where 4,, is the magnetic flux enclosed in the nth allowed orbit and e* is the charge of the semiclassical particle. For our 2D electrons e* = e and, in our experiment with a uniform 3, 4,, is simply related to the area A, of the orbit by 4, = BA,. A resistance peak occurs in this model whenever a bound state is aligned with the chemical potential, p, of the edge channel. The fluctuation period in the B scan data, given by the change in B needed to align the next bound state with p, is AB = A&[d( BA,)/dB]
-I.
(2)
In the IQHE case, A4,, = #,?+ I - 55, = &h/e. Consequently, AB is the same for all integer v (if dA,/dB is negligible), in agreement with experiment. In the V,; scan experiment, the period given by the change in p to align the next bound state is: AV, = A~~[d(B~~)/d~~]-‘. Since the fluctuations for the v = 1 minimum is observed at twice the B for the v = 2 fluctuations, the observed A&CV = 1) = (1/2)AVo(~ = 2) relation is also expected from the model. On the other hand, if the semiclassical orbits are assumed by the quasiparticles in the l/3 FQHE case, e* in the quantization condition is e/3 and A4, = h/e” = 3h/e. As a result, AB at v = l/3 should be three times that at v = 1 and AV,, because the fluctuations for v = l/3 is observed at a three times higher B, should be the same as at v = 1, all in agreement with experiment. In conclusion, we have presented new results from our study of the resistance fluctuations near the L’= l/3, 1, and 2 Rx,< minima. In particular, we have fabricated new samples, on which backgate biases can be applied to vary nzu to make it possible to study the fluctuations as a function of Vci as well as B. The fluctuation periods A B and AV, for the three different V’S are found to follow the relations: AB(v=
l/3)
=3AB(zr=
1) =3AB(v=2),
(3)
and AV,;( v = l/3)
= AV,,( v = 1) = (1/2)AV,;(
v = 2).
(4)
These relations are in agreement with the JainKivelson model, which is based on the existence of magnetically bound states resulting from Bohr-Sommerfeld quantization of the semiclassical 2D electron orbits in the IQHE, and its extension to the FQHE by assuming that the quasipartitles of the l/3 FQH liquid traversing the similar semiclassical orbits carry charge e” = c/3. Our observation of the latter relation that AVc,(v = l/3) = AV,(v = I) = (1/2)AV;;(v = 2) is incompatible with the Coulomb blockade model 191. We thank L.W. Engel, A.M. Chang and X.G. Wen for helpfu1 discussions. This work is sup-
S. W. Hwang et al. / ~~asi~arfic~
charge of l/3
ported by ONR through contract No. NOO014-89J-1567, and NSF through Grants No. DMR8719694 and No. DMR-8921073. One of us (S.W.H.) thanks the Korean Government Ministry of Education for its partial financial support.
References [l] D.C. Lett. [2] R.B. [3] R.G. CT.
Tsui, H.L. Stormer and AC. Gossard, Phys. Rev. 48 (1982) 1559. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. Clark, J.R. Mallett, S.R. Haynes, J.J. Harris and Foxon, Phys. Rev. Lett. 60 (1988) 1747.
FQH liquid from resistance ~~ctuutia~
measu~erne~t~
75
[4] A.M. Chang and J.E. Cunningham Solid State Commun. 72 (1989) 652. ]5] J.A. Simmons, H.P. Wei, L.W. Engel, DC. Tsui and M. Shayegan, Phys. Rev. Lett. 63 (1989) 1731. [6] J.K. Jain and .%A. Kivelson, Phys. Rev. Lett. 60 (1988) 1542. [7] S.A. Kivelson and V.L. Pokrovsky, Phys. Rev. B 40 (1989) 1373. [8] S. Kivelson, Phys. Rev. Lett. 65 (1990) 3369. [9] P.A. Lee, Phys. Rev. Lett. 65 (1990) 2206. [lo] D.J. Thouless and Y. Gefen, Phys. Rev. Lett. 66 (1991) 806. Ill] J.A. Simmons, SW. Hwang, DC. Tsui, H.P. Wei, L.W. Engel and M. Shayegan, Phys. Rev. B. 44 (1991).