Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics

In two dimensions, the laws of physics permit the existence of anyons, particles with fractional statistics which are neither Fermi nor Bose. That is, upon exchange of two such particles, the quantum state of a system acquires a phase which is neither 0 nor $\pi$, but can be any value. The elementary excitations (Laughlin quasiparticles) of a fractional quantum Hall fluid have a fractional electric charge and are expected to obey fractional statistics. In this paper we report experimental realization of a Laughlin quasiparticle interferometer, where quasiparticles of the $1∕3$ fluid execute a closed path around an island of the $2∕5$ fluid and thus acquire statistical phase. Interference fringes are observed as conductance oscillations as a function of magnetic flux, similar to the Aharonov-Bohm effect. We observe the interference shift by one fringe upon introduction of five magnetic flux quanta $(5h∕e)$ into the island. The corresponding $2e$ charge period is confirmed directly in calibrated gate experiments. These results constitute direct observation of fractional statistics of Laughlin quasiparticles.

Figure 1

(Color online) Conceptual schematic of the Laughlin quasiparticle interferometer. (a) A quantum Hall sample with two fillings: an island of $2∕5$ FQH fluid is surrounded by the $1∕3$ fluid. The current-carrying chiral edge channels (shown by arrowed lines) follow equipotentials at the periphery of the confined 2D electrons; tunneling paths are shown by dots. The circles are the Ohmic contacts used to inject current $I$ and to measure the resulting voltage $V$. The central island is encircled by two counterpropagating edge channels. The current-carrying $e∕3$ quasiparticles can tunnel between the outer and the inner $1∕3$ edges, dotted lines. When there is no tunneling, $V=0$; tunneling produces $V>0$. The closed path of the inner $1∕3$ edge channel gives rise to Aharonov-Bohm-like oscillations in conductance as a function of the enclosed flux $\Phi$. No current flows through the $2∕5$ island, but any $e∕5$ quasiparticles affect the Berry phase of the encircling $e∕3$ quasiparticles through a statistical interaction, thus changing the interference pattern. (b) FQH liquids can be understood via composite fermion representation. A composite fermion energy profile of the interferometer shows the three lowest “Landau levels” separated by FQH energy gaps. Several $e∕3$ and $e∕5$ quasiparticles are shown as composite fermions in otherwise empty “Landau levels.”

Figure 2

(Color online) The Laughlin quasiparticle interferometer samples. (a) and (b) are atomic force and scanning electron micrographs of a typical device. Four $\mathrm{Au}∕\mathrm{Ti}$ front gates in shallow etch trenches define the central island of 2D electrons of lithographic radius $R\approx 1050\mathrm{nm}$. The 2D electrons are completely depleted under the etched trenches; the front gates are used for fine tuning the two wide constrictions. The chiral edge channels (blue) follow equipotentials at the periphery of the undepleted 2D electrons; tunneling paths are shown by dots. Four Ohmic contacts are shown schematically by the numbered circles, $R_{XX}\equiv V_{2–3}∕I_{1–4}$. The back gate (not shown) extends over the entire sample on the opposite side of the insulating GaAs substrate.

Figure 3

(Color online) The electron density $n$, normalized to the 2D density, as a function of radius $r$ in an electron island defined by an etched annulus of inner radius $R\approx 1050\mathrm{nm}$. The calculation follows model of Ref. 23. $W=250\mathrm{nm}$ is the depletion length parameter obtained in the same calculation. The blue circle gives the radius of the outer edge ring, $r_{\mathit{Out}}\approx 685\mathrm{nm}$, obtained from the Aharonov-Bohm period data for $f_B=1$ and 2, shown in Fig. 6. The magenta circle gives the radius of the inner edge ring, $r_{\mathit{In}}\approx 570\mathrm{nm}$ for $f_B=2∕5$ from the data of Figs. 9, 10, taking the same $r_{\mathit{Out}}$ for $f_C=1∕3$ and the ratio of densities $n\left(r_{\mathit{In}}\right)∕n\left(r_{\mathit{Out}}\right)=(2∕5)∕(1∕3)=1.20$.

Figure 4

(Color online) Magnetoresistance of the quasiparticle interferometer sample at temperature $10.2\mathrm{mK}$. The horizontal arrows show approximate ranges of filling factor ${\nu }_C=f_C$ plateaus in the constrictions. Note the quantized plateaus $R_{XX}\left(B\right)=h∕4e^2$ at $\sim 3.7\text{Tesla}$ ($f_C=1$, $f_B=4∕3$) and $R_{XX}\left(B\right)=h∕2e^2$ at $\sim 12.4\text{Tesla}$ ($f_C=1∕3$, $f_B=2∕5$). The overlap of the $f_C=1$ and $f_B=1$ plateaus for $4.2\mathrm{T}<B<4.8\mathrm{T}$ likewise results in $R_{XX}\left(B\right)=0$ in this range. The data were obtained with front gate voltage $V_{\mathit{FG}}=0$, and the front gates were not tuned for symmetry. The fine structure is due to quantum interference effects; some peaks can be identified as due to impurity-assisted tunneling.

Figure 5

(Color online) (a) Schematic of the sample when magnetic field is such that there is only one quantum Hall filling $f$ throughout the sample: $f_C$ in the constrictions is equal to $f_B$ in the 2D bulk and in the island. The numbered rectangles are Ohmic contacts, FG are the front gates. The chiral edge channels follow equipotentials at the periphery of the undepleted 2D electrons; tunneling paths are shown by dots. A closed edge channel path gives rise to Aharonov-Bohm oscillations in the conductance. (b) A sample with two quantum Hall fillings exhibits quantized diagonal resistance $R_{XX}=(h∕e^2)(1∕f_C-1∕f_B)$, where $R_{XX}\equiv V_{2-3}∕I_{1-4}$. Observation of a quantized plateau in $R_{XX}\left(B\right)$ provides definitive values for both $f_C$ and $f_B$. (c) Schematic of the sample when magnetic field is such that $f_C<f_B$. The sample exhibits a quantized $R_{XX}\left(B\right)$ plateau, and, upon fine tuning of front gates, exhibits Aharonov-Bohm oscillations in conductance as a function of the flux enclosed by the inner island edge ring.

Figure 6

(Color online) Interference of electrons in the outer ring of the device in the integer quantum Hall regime. Aharonov-Bohm-type oscillations in conductance are observed when one $\left(f=1\right)$ and two $\left(f=2\right)$ Landau levels are filled. The corresponding flux period $\Delta \Phi =h∕e$ gives the outer ring radius $r_{\mathit{Out}}\approx 685\mathrm{nm}$.

Figure 7

(Color online) Interference of electrons in the outer ring of the device in the integer quantum Hall regime. Application of a positive back gate voltage $V_{\mathit{BG}}$ attracts the 2D electrons one by one to the area of the outer ring, resulting in modulation of the interference amplitude. This calibrates the back gate voltage increment $\Delta V_{\mathit{BG}}$ necessary to increase the charge contained within the ring by $\Delta Q=e$. Note that $\Delta V_{\mathit{BG}}$ is independent of Landau-level filling.

Figure 8

(Color online) An enlargement of the data of Fig. 4. The horizontal arrows show approximate range of the $f_C=1∕3$ plateau in the constrictions. Note the quantized plateau $R_{XX}\left(B\right)=(h∕e^2)[{(1∕3)}^{-1}-{(2∕5)}^{-1}]=h∕2e^2$ at $\sim 12.35\text{Tesla}$ ($f_C=1∕3$, $f_B=2∕5$). The data were obtained with $V_{\mathit{FG}}=0$; the front gates were not tuned for symmetry. The fine structure is due to quantum interference effects; some peaks can be identified as due to impurity-assisted tunneling.

Figure 9

(Color online) Aharonov-Bohm effect of the inner ring $e∕3$ Laughlin quasiparticles circling an island of the $f=2∕5$ FQH fluid. Magnetic flux through the island period of $\Delta \Phi =5h∕e$ corresponds to creation of ten $e∕5$ quasiparticles in the island (one fundamental flux quantum $h∕e$ induces two quasiparticles in the $f=2∕5$ FQH fluid).

Figure 10

(Color online) Interference of the inner ring $e∕3$ Laughlin quasiparticles circling an island of the $f=2∕5$ FQH fluid. (a) Magnetic flux through the island period of $\Delta \Phi =5h∕e$ corresponds to creation of ten $e∕5$ quasiparticles in the island (one fundamental flux quantum $h∕e$ induces two quasiparticles in the $f=2∕5$ FQH fluid). (b) The back gate voltage period of $\Delta Q=10(e∕5)=2e$ directly confirms that the $e∕3$ quasiparticle consecutive orbits around the island are quantized by the condition requiring incremental addition of ten $e∕5$ quasiparticles of the $f=2∕5$ fluid. These observations imply relative fractional statistics, when a charge $e∕3$, statistics ${\Theta }_{1∕3}=2∕3$ quasiparticle encircles one $e∕5$, ${\Theta }_{2∕5}=2∕5$ quasiparticle of the $f=2∕5$ fluid.

Figure 11

(Color online) The ratio of the conductance oscillations periods $\Delta B∕\Delta V_{\mathit{BG}}\propto 1∕f$, which is independent of the edge ring area, from the data shown in Figs. 6, 7, 10. The fact that the ratios fall on a straight line forced through (0,0) and the $f=1$ point confirms the island filling $f=2∕5$. The magenta crosses give the neighboring $f=3∕7$ and $f=1∕3$, which clearly do not fit the data.