experiment of a two-dimensional electron system on liquid helium. From low-field ... interaction and the motion of the helium vapour atoms. ... Phase diagram:.
Causes of weak-localization of electrons on liquid helium
Universität Konstanz
Annie Wakata, Paul Leiderer, and Jürgen Klier Department of Physics, University of Konstanz, 78457 Konstanz, Germany
Abstract
Theoretical background
WL with fixed scattering sites:
Weak localization (WL) is a quantum effect caused by the coherence among multiple elastic-scattering paths of a conduction electron. This coherence leads to an enhanced backscattering probability and an increase in the resistivity over the classical Drude model. We have investigated the dephasing processes in a weak localization experiment of a two-dimensional electron system on liquid helium. From low-field magnetoconductivity measurements we can separate the damping of WL on the dephasing of electrons due to electron-electron interaction and the motion of the helium vapour atoms. We observe an intermediate regime where both damping mechanisms are of comparable importance and determine the transition from one dominant regime to the other. This is the first observation of a cross-over from the simple exponential decay to the cubic exponential damping in such a system.
For a non-degenerate 2D electron fluid the longitudinal conductivity in the low magnetic field including WL corrections is given by
NN = conductivity n = electron density m = electron mass, = electron mobility E? = localistion threshold B = perpendicular magnetic field, = quasi-elastic scattering time = depahsing due to e--e- interaction ! = depahsing due to motion of vapour atoms D = electron diffusion constant
Constructive interference increases the probability for the electron to backscatter: Weak Localization xx
B = 0:
n e WL m
A magnetic field destroys this constructive interference
total dephasing time: B = -1
-1
1
+
-1 3
Results
Electrons on liquid helium
obtained via measured inverse magnetoconductivity 1/sxx vs B . 2
56
(b) 11 -2 (a) n = 4.09 x 10 m at T = 2.15 K (b) n = 1.68 x 1011 m-2 at T = 2.12 K
1 / xx [M]
54 52
Solid lines show fits with parameters given in Table 1.
(a)
50
Here the dominating damping mechanism is electron-electron interaction: the values of t1and t3 correspond to a simple exponential damping regime.
26
25
Fig. 1
24 0,00
0,05
0,10
0,15
0,20
0,25
B2 [T2] 152 148
The electrons are trapped in a potential well and so form a nearly ideal 2D system with high mobility. The electron density can be varied from 109 to 1016 m-2, (classical to quantum system).
(b)
Solid lines show fits with parameters given in Table 1.
140
degenerate Fermigas
Electron density
(a) n = 2.08 x 10 m and 11 -2 (b) n = 1.72 x 10 m , both at T = 2.15 K. (c) n = 0.64 x 1011 m-2 at T = 2.12 K.
11
144
1 / xx [M]
Phase diagram:
(c)
67 66
Intermediate regime: in curve (a) t1 damping is more important, in curve (b) t3 damping is more important, and in curve (c) both damping mechanisms have comparable importance.
65 64 55
(a)
54
Wigner crystal
Fig. 2
Tmax
-2
53 0,00
0,05
0,10
0,15
0,20
0,25
B2 [T2] 11
290
classical Gas
Temperature 1 / xx [M]
280
2DES can exist in three phases: classical electron gas Wigner crystal degenerate Fermi gas Phase transitions depend on Coulomb screening.
Solid line shows fit with parameters given in Table 1. 270
260
Experimental set-up Fig. 3
Filament
-2
n = 0.34 x 10 m at T = 2.12 K. At this low electron density the extreme regime of damping dominated due to the motion of vapour atoms is reached. cubic exponential damping
250 0,00
0,05
0,10
0,15
0,20
0,25
Dashed line represents a fit assuming simple exponential damping. For that t is taken to be 4.90 ps, as measured for T = 2.12 K: By minimizing c we get t = 21.56 ps while t! . We also fitted the data by varying t and t simultaneously: As best fit (not shown) we get t = 4.60 ps and t = 19.32 ps while t! Obviously these calculated values for 1/sNN do not fit the data.
.
B2 [T2] Electrons
B1 Vexc
B3
B2
C
10 mV 40 kHz
C R
S C
R
R
dc
+UH
Liquid Helium
G LiA
dHe = 0.5 mm temp. range: 1.9 K to 2.2 K
C0
An ac current is capacitively coupled through the electrons to measure the resistance of the layer. 11
-2
11
-2
range of electron density: 0.2 x 10 m to 5 x 10 m uncertainity < 0.005 x 1011 m-2
Fig.
n [1011 m-2]
T [K]
1 [ps]
3 [ps]
0 [ps]
1a
4.09
2.15
19.78
890.1
4.30
2a
2.08
2.15
24.00
86.77
4.00
2b
1.72
2.15
82.43
23.92
4.12
1b
1.68
2.12
22.05
474.1
4.90
4
1.22
2.12
24.50
150.5
4.90
4
0.69
2.12
29.40
74.36
4.90
2c
0.64
2.12
44.10
40.35
4.90
3
0.34
2.12
2450
21.75
4.90
Table 1: Parameters used for the best fits of 1/sNN vs B (sorted by density): electron density n, dephasing time t (due to electron-electron interaction), dephasing time t! (due to the motion of vapour-atoms), and the quasielastic scattering time t . The uncertainty for t and t! for the data in Fig. 3 is about 1%, in all other cases it is much less than 1%.
We measure the conductivity, sxx, in a Corbino geometry
Fig. 4
Inverse dephasing times vs electron density at T = 2.12 K. One observes the cross-over from one regime to the other as the density varies. The lines between the data points are just a guide to the eye.
Corbino arrangement: G = guard ring, S = shaping electrode, B1 to B3 = measuring electrodes
B1 B2
The diameter of the electrodes B1 to B3 is about 11.0, 14.8, and 18.6 mm, the spacing between is 0.1 mm
S
B3
G
loading area for electrons
Conclusions We observed the presence of both predicted damping mechanisms: By changing only the electron density the two extreme regimes are presented:
damping of the electrons due to the electron-electron interaction. damping of the electrons due to motion of the helium vapour atoms. determine the cross-over from one dominant regime to the other. In the intermediate density regime both mechanisms are comparable.
One of us (A.W.) is grateful to the Alexander von Humboldt Foundation for a grant. This activity was supported by the DFG, Forschergruppe ‘Quantengase’, and the EU-RTN ‘Surface electrons on mesoscopic structures’.
