Direct Measurement of the Coherence Length of Edge States in the Integer Quantum Hall Regime

We have determined the finite temperature coherence length of edge states in the integer quantum Hall effect regime. This was realized by measuring the visibility of electronic Mach-Zehnder interferometers of different sizes, at filling factor 2. The visibility shows an exponential decay with the temperature. The characteristic temperature scale is found inversely proportional to the length of the interferometer arm, allowing one to define a coherence length $l_{\phi }$. The variations of $l_{\phi }$ with magnetic field are the same for all samples, with a maximum located at the upper end of the quantum Hall plateau. Our results provide the first accurate determination of $l_{\phi }$ in the quantum Hall regime.

Figure 1

(a) Tilted scanning electron microscope (SEM) view of the “small” MZI. $G0$, $G1$, and $G2$ are QPCs whose split gates are connected with gold bridges over an isolator responsible for the black color of the SEM view. LG is a lateral gate. The white line on the SEM picture represents the outer edge state. The small Ohmic contact in between the two arms collects the backscattered current $I_B$ to the ground through a long gold bridge. (b) A 2D plot of $d{I}_T/d{I}_0$ as a function of the lateral gate voltage $V_{\mathrm{LG}}$ and the dc bias $V$, for the large sample at 20 mK. The visibility of interferences decreases with $V$ while the phase of interferences remains almost constant. (c) Phase of the large sample deduced from Fig. 1b. The dashed line is the energy dependence of the phase that would be necessary to explain our observed visibility decrease with thermal smearing.

Figure 2

Interferences revealed upon varying the magnetic flux through the surface defined by the two arms ($u$) and ($d$) of the interferometers. From the oscillation period $\delta B$ we deduce the surface $S=h/(e\delta B)$ of the 3 different studied MZI. (a) The small MZI ($S=8.7\pm 0.2\mu {\mathrm{m}}^2$). (b) The medium MZI ($S=15.5\pm 0.4\mu {\mathrm{m}}^2$). (c) The large MZI ($S=40.7\pm 0.8\mu {\mathrm{m}}^2$). All these surfaces are in good agreement with the lithographic ones (see text).

Figure 3

$\ln (\mathcal{V}/{\mathcal{V}}_B)$ versus temperature for the three samples, ${\mathcal{V}}_B$ is the visibility measured at $T_B=20\mathrm{mK}$. The measurement has been done at the magnetic field for which the visibility decay is the smallest. Inset: The slope $T_{\phi }^{-1}=\ln (\mathcal{V}/{\mathcal{V}}_B)/(T-T_B)$ is proportional to the arm length.

Figure 4

Upper panel: The dashed, solid, and dotted lines are the two point Hall resistance at $\nu =2$ measured for the small, the medium, and the large sample, respectively. $l_{\phi }$ has a general shape recovered by all three samples, with a maximum at the end of the Hall plateau. Lower panel: Coherence length at $T_B=20\mathrm{mK}$, $l_{\phi }=2L.T_{\phi }/T_B$ for the three samples studied ($L=5.6$, 8, and $11.3\mu \mathrm{m}$). The magnetic fields ($x$ axis) of the small and large sample have been shifted by $+0.25$ and $-0.1\mathrm{T}$, respectively, such that the plateau centers coincide.