Non-Abelian Fractional Quantum Hall Effect for Fault-Resistant

41 Pages·2013·0.89 MB·English

SANDIA REPORT
SAND2013-10574
Unlimited Release
October, 2013

Non-Abelian Fractional Quantum Hall
Effect for Fault-Resistant Topological
Quantum Computation

W. Pan, M. Thalakulam, X. Shi, M. Crawford, E. Nielsen, and J.G. Cederberg

Prepared by
Sandia National Laboratories
Albuquerque, New Mexico 87185 and Livermore, California 94550

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2
SAND2013-10574
Unlimited Release
October, 2013

Non-Abelian Fractional Quantum Hall
Effect for Fault-Resistant Topological
Quantum Computation

Wei Pan, Madhu Thalakulam, Xiaoyan Shi, Matthew Crawford, Erik Nielsen
Quantum Phenomena Department

Jeff G. Cederberg
Advanced Materials Sciences Department

Sandia National Laboratories
P.O. Box 5800
Albuquerque, New Mexico 87185-MS1086

Abstract

Topological quantum computation (TQC) has emerged as one of the most promising
approaches to quantum computation. Under this approach, the topological properties
of a non-Abelian quantum system, which are insensitive to local perturbations, are
utilized to process and transport quantum information. The encoded information can
be protected and rendered immune from nearly all environmental decoherence
processes without additional error-correction. It is believed that the low energy
excitations of the so-called =5/2 fractional quantum Hall (FQH) state may obey non-
Abelian statistics. Our goal is to explore this novel FQH state and to understand and
create a scientific foundation of this quantum matter state for the emerging TQC
technology.

We present in this report the results from a coherent study that focused on obtaining a
knowledge base of the physics that underpins TQC. We first present the results of
bulk transport properties, including the nature of disorder on the 5/2 state and spin
transitions in the second Landau level. We then describe the development and
application of edge tunneling techniques to quantify and understand the quasiparticle
physics of the 5/2 state.
.

3
ACKNOWLEDGMENTS

4
CONTENTS

1. Introduction ................................................................................................................................ 7
2. Nature of Disorder Matters on the 5/2 Fractional Quantum Hall Effect .................................... 9
3. Spin Polarization of the 12/5 Fractional Quantum Hall Effect ................................................. 15
4. Spin Transition in the 8/3 Fractional Quantum Hall Effect ...................................................... 23
5. Edge Channel Tunneling Spectroscopy of the 5/2 Fractional Quantum Hall Excitations
in Etch Defined Quantum Point Contacts ..................................................................................... 29
6. Summary and Outlook .............................................................................................................. 33
7. References ................................................................................................................................. 35
Distribution ................................................................................................................................... 39

5
FIGURES

Figure 1. Electron mobility versus density in HIGFET A. ........................................................... 10
Figure 2. FQHE states in diagonal magneto-resistance around =5/2 in HIGFET B. (a)
R versus n at various magnetic fields. (b) Temperature dependence at B = 4.68T. (c)
xx
Arrhenius plot for the R minimum at =5/2 at B = 4.68T. The line is a linear fit to the
xx
data. An energy gap of ~ 0.1K is obtained from the slope of this linear fitting. (d)
Density dependence of the 5/2 energy gap in HIGFET A. The bottom x-axis is electron
density, the top magnetic field. The red line is a fit according to a spin polarized model ........... 11
Figure 3. (a) normalized 5/2 energy gap,  /e2/l , as a function of ; and (b) as a
=5/2 B
function of 1/ for HIGFET A and modulation doped quantum well samples. The two
lines in (b) are guides to the eye. .................................................................................................. 12
Figure 4. R trace taken at = 0 under perpendicular magnetic fields.. .................................... 16
xx
Figure 5. The left column shows the R traces at three selected tilt angles, θ = 0o, 14o,
xx
27o. The arrows mark the positions of the 12/5 state. The middle column shows the
corresponding R traces at  = 0o, 14o, and 22o, respectively. The horizontal lines show
xy
the expected Hall value of the 12/5 state. The right column shows in semi-log plot the R
xx
versus 1/T at =12/5 at  = 0o, 14o, and 27o. Lines are linear fit. ................................................. 17
Figure 6. (a) R and R at =12/5 versus B , measured at T ~ 10 mK in the second
xx yy total
cool-down. (b) B versus B plot for the maximum at B = 4.8T in (a). The star is
total eff total
the experimental data point. The lines are for B /B = j × 2m /(g*m*) with j=1 and 2,
total eff e
respectively. (c)  as a function of B from two cool-downs. For the third cool-
12/5 total
down,  from the temperature dependence of both R and R is shown. In the gray
12/5 xx yy
region, magneto-resistance is not activated and  is not obtainable. ....................................... 18
12/5
Figure 7. (a) The energy gap of the 5/2 state as a function of B . (b) R and R at
total xx yy
=5/2 versus B , measured at T ~ 10 mK. ................................................................................. 20
total
Figure 8. Magneto-resistance R for sample C (Fig. 1a) and A (Fig. 1c). Arrows mark
xx
the positions of the FQHE states at =8/3, 5/2, and 7/3. Fig. 1b and Fig. 1d show the
temperature dependence of R at =8/3 (filled squares) and 7/3 (open squares) in these
xx
two samples, respectively. The lines are linear fit. ....................................................................... 25
Figure 9. Activation energy gap at =8/3 (a) and 7/3 (b) as a function of density. ...................... 26
Figure 10. (a) Optical micrograph of a representative QPC device with a schematic
diagram of the measurement setup superposed. (b) R as a function of gate voltage V
QPC g
showing well defined quantized conductance plateaus. ............................................................... 29
Figure 11.  =5/2 FQH state in QPC: (a) R vs B field showing the v =3, 5/2 and 2
D
plateaus. (b) Numerical derivative of the data shown in (a) vs B field. The dip at B = 2.73
T correspond to 5/2 FQH state. (c) g vs DC source drain bias current I for 15 mK
T DC
(green), 100 mK (magenta) and 150 mK (red) sample temperatures. Blue curves represent
the global fit to the data using the weak-tunneling formula. (d) R vs I for B = 3.25T
D DC
corresponding to the 2nd Landau level .......................................................................................... 30
Table 1. The quantum well width (W), 2DES density and mobility, as well as the
magnetic length (l ) at =8/3 and the ratio of W/l for the samples studied in this work. ........... 24
B B

6
1. INTRODUCTION

The most secure modern encryption method is based on the assumption that it is
impossible to prime-factorize a large digit number within a reasonable time frame.
Indeed, it is estimated that factorizing a 200-digit number would require 170 CPU years
using an Intel computer. This estimate, however, is drastically changed with the use of a
quantum computer. In a paper published in 1994 by Peter Shor, he showed that a
quantum computer (QC) could readily factorize a 300-digit number. This began a world-
wide interest in QC research. Yet, after 15 years of research, many fundamental issues
remain unresolved. For example, the strong coupling between electrons and their local
environments greatly reduces electron coherence time and requires complex error-
correction schemes to manipulate quantum information before it is lost. As such, there is
a pressing need to identify new paradigms which can potentially enable revolutionary
advances in the field of quantum computation.

Topological quantum computation (TQC) has emerged as one of the most promising
approaches. Under this approach, the topological properties of a non-Abelian quantum
system, which are insensitive to local perturbations, are utilized to process and transport
quantum information. The encoded information can be protected and rendered immune
from nearly all environmental decoherence processes without additional error-correction.
It is now generally believed that the low energy excitations of the so-called =5/2
fractional quantum Hall (FQH) state may obey non-Abelian statistics. Our goal is to
explore this novel FQH state and to understand and create a scientific foundation for
exploiting this quantum matter state in order to build a knowledge base for the emerging
TQC technology.

In this LDRD research, we propose to carry out a coherent study to obtain a scientific
foundation of TQC. Success of this project is expected to have a great impact on the
feasibility of eventually building a topological computer.

Our proposed work is based on our world-leading research in the fractional quantum Hall
(FQH) physics. We will study the physics of the 5/2 ground state; the elementary
excitations of the 5/2 state and their transport properties; and the quantum interference of
these elementary excitations that might be relevant to constructing a quantum bit (or, in
short, qubit).

7
8
2. NATURE OF DISORDER MATTERS IN THE 5/2 FRACTIONAL
QUANTUM HALL EFFECT

Among all observed fractional quantum Hall effect (FQHE) [1,2] states, the =5/2 FQHE
state [3,4] in the second Landau level remains the most exotic one. This state has been at
the center of current quantum Hall research due to the possibility of it being non-Abelian
[5,6] and, thus, having potential applications in fault-tolerant topological quantum
computation [7].Therefore, there is an urgent need to understand the impact of disorder
on this FQHE state, since a larger energy gap at =5/2 would exponentially reduce error
rates [7] and make the envisioned quantum computation more robust.

Today, our knowledge of the impact of various kinds of disorder on the 5/2 FQHE, or any
other of the FQHE states, remains limited. Indeed, it has long been observed [8,9] that in
the lowest Landau level, even after taking into account the finite thickness correction of
2DES and Landau level mixing effect, there is still a discrepancy of ~ 2-3K between the
experimentally measured energy gaps of the odd-denominator FQHE states and
theoretically calculated ones. This value of discrepancy has also been observed for the
5/2 state and the odd-denominator FQHE stats in the second Landau level [4]. In order to
reconcile the discrepancy, often, a very vague quantity termed “disorder broadening” is
employed [4], which does not seem to correlate with the 2DES mobility. Moreover, little
is known about the nature of disorder, e.g., the different influence of long range
Coulombic and short range neutral disorders, on the energy gap.

On the other hand, understanding the impact of disorder is essential for understanding the
nature of a quantum system. As shown in a series of papers [10], the long standing
controversies on the universality of quantum Hall plateau transition is directly linked to
the nature of disorder in the 2D channel. In 2DES samples with strong disorder from a
short-range neutral alloy potential a perfect power law scaling behavior of quantum Hall
plateau-to-plateau transition was observed over more than two decades of temperatures,
from 1.2K down to 10 mK. In contrast, in samples with weak disorder from a long-range
Coulombic potential a crossover behavior was observed from a non-universal scaling
region at high temperatures to a universal scaling region at low temperatures.

In this section, we present our results on how the nature of disorder affects the 5/2 energy
gap and, thus, the stability of this state. We compare the activation energy gap data
obtained in two types of samples: symmetrically doped modulation quantum well
samples and undoped heterojunction insulate-gated field-effect transistors (HIGFETs). In
modulation doped quantum well samples, where long-range Coulombic disorder
dominates, the energy gap drops quickly with decreasing mobility (or increasing
disorder). On the other hand, in HIGFET samples, where the short-range neutral disorder
dominates, the 5/2 energy gap shows only a weak mobility dependence. Our results
clearly demonstrate that the two types of disorder play very different roles in affecting the
stability of the 5/2 state. Possible physical mechanisms are discussed.

9
For this study we used two HIGFET (heterojunction insulate-gated field-effect transistor)
[11] devices, called A and B. Results from these two samples are consistent with each
other. Figure 1 shows the mobility () versus electron density (n) obtained in HIGFET A.
The mobility first increases with increasing n, reaches a peak of  ~ 14x106 cm2/Vs
around n ~ 1.8x1011 cm-2, and then decreases with increasing n. The decrease of mobility
with increasing density beyond n ~ 1.8x1011 cm-2 is due to surface roughness scattering
[12], which is of short-range. The weak kink at n ~ 3.5x1011 cm-2 is probably due to the
onset of occupation of the second electrical subband.

Figure 1: Electron mobility versus density in HIGFET A.

We studied the =5/2 state at various densities. In Figure 2a, we show the data obtained
in HIGFET B, taken at the high B/T facilities of National High Magnetic Field
Laboratory at the University of Florida. Different from the conventional magneto-
transport measurement, where the 2D electron density is fixed and the magnetic field is
swept, here, the magnetic field is fixed and the gate voltage and hence the carrier
densities is swept. This process minimizes heating from eddy currents. As shown in
Figure 2a, a well-developed 5/2 state is observed and its diagonal resistance (R )
xx
minimum reaches a very low value of ~ 10 ohms at a density of n=2.85x1011 cm-2. With
decreasing electron density, R at =5/2 increases, but the minimum remains visible
xx
down to n ~ 0.55x1011 cm-2.

The density dependence of the 7/3 state is similar to that of the 5/2 state. However, the
8/3 state shows a slightly different density dependence. Its strength first decreases with
decreasing density and then seems to saturate at lower densities.

Figure 2b shows the R traces at a few selected temperatures taken at a magnetic field B
xx
= 4.68T. Clearly, the R minimum shows an activated behavior, i.e., R increases with
xx xx
increasing temperature. This is different from our previous study [13] and thus allows us
to obtain an activation energy value for the 5/2 state, as shown in Fig.2c.

In Fig. 2d, we show the density dependence of the 5/2 energy gap in HIGFET A, in the
density range where surface roughness dominates. In this density range, the 5/2 energy
gap increases with increasing 2D electron densities. Two important features are apparent:
1) the increase of the 5/2 energy gap is not caused by an increase in electron mobility.

10