The first experimental method employed for establishing and proving the solid TDES phase, involved the detection of ripplon resonances caused by oscilla-tions of the electron crystal in the verticalz-direction and it was implemented by Grimes et al.[39] (left plot of fig. 1.5. Subsequently, the existence and properties of the electron crystal were investigated by different indirect methods, like mobility measurements (Mehrotra et al.[79], right plot of fig. 1.5).
Grimes et al.[39] understood that when the electron crystal is formed, the application of a uniform AC-electric field in the vertical direction will force the crystal to oscillate up and down with respect to the surface of liquid helium.
However, for certain frequencies, the condition for the excitation of standing ripplon waves will be satisfied and a measurable resonance will develop. The condition for the excitation of standing ripplon waves can be formulated as λn=2L/n, whereλnis the wave length of then-th mode andLis the nearest neighbor distance in the electron lattice. Ripplons are observed below0.7K, meaning that the method ofGrimes et al.[39] cannot be used for detecting the electron crystal at higher temperatures.
The continuity of the transition was investigated byGlattli et al.[37] through thermodynamic measurements. Electrons on liquid helium are in very good thermal insulation from their environment and it is possible to heat them at a higher temperatureTefor a relatively long time by applying a short pulse of powerP(≈0.1nW) for a∆tperiod of time (≈100µs). The temperatureTeof the TDES can be determined from the phonon velocities by means of eq. (1.25) and the theoretical results ofMorf [83]. The TDES specific heat is by definition C=P∆t/∆Teand its behavior near the melting point should reveal whether the transition involves the emission of latent heat, determining thus whether the transition were continuous or not. No latent heat of melting was observed, meaning that the transition is a continuous phase transition as predicted by KT-theory.
The experimentally accessible electron densities on bulk liquid helium vary in the range from109m−2to1013m−2. One KT-type transition of a TDES from the liquid to the solid phase is then the only possible and the phase diagram is thus fully determined.
1.5. Electron mobility
The transport properties of electrons on liquid helium are determined by the scattering mechanisms and the dimensionality of the system. Electron mobility (µ) is a measure of transport properties, showing how easily an electron is moving and defined as
µ= eτ
m∗. (1.26)
It should be noticed that electron mobility is defined in terms of the effective massm∗, which might be greater or less than the usual bare electron mass.
However, for an electron on liquid helium it may be considered that the effective mass is with good accuracy equal to the bare electron mass.
The total single-electron mobility is fully determined, if one takes into account that the total inverse relaxation timeτ−1is, according to Matthiesen rule,τ−1=τ−1g +τ−1r , that is, the sum of the inverse relaxation time for each scattering mechanism, which for electrons on helium are scattering with ripplons, eq. (1.17), and helium gas atoms, eq. (1.14).
The collective,N-body nature of the system should provide an additional formula, relating the mobility of electrons on liquid helium with such collective properties as its densitynand conductivityσ. The theories devised for that purpose treat electrons on liquid helium as a highly correlated, nondegenerate quantum fluid. Perhaps surprisingly, it is found that although electron corre-lations are strong, electron mobility follows essentially a classical Drude-type behavior, making it possible to write
µ= σ
ne. (1.27)
The previous equation implies the main method for the experimental deter-mination of electron mobility: it suffices to measure the conductivity of a TDES under constant electron density. The resulting mobility values, in conjunction with eq. (1.26), give experimental information on the scattering mechanisms, enclosed in the total relaxation timeτ.
Sommer et al.[101] measured electron mobility at temperatures ranging from 0.9K to3.2K, for a liquid helium thickness of1mm–2mm. Below2K, electron mobility increases with decreasing temperature, following an activation-type law. This can be attributed to scattering of surface state electrons with helium gas atoms. Above2K, electron mobility drops abruptly, in agreement with the findings ofLevine et al.[73] who predicted and measured a transition to an electron complex, or a bubble state. Notably, the measured electron mobility is much lower forT > 2K than the electron mobility measured byLevine et al.[73], a fact whichSommer et al.[101] attribute to the lower dimensionality of TDES in comparison to free electrons inside helium gas.
Grimes et al.[38], using plasmon resonance, excited standing wave res-onances of surface state electrons and verified the two-dimensional plasmon dispersion relationωp∝√
k, whereωpis the plasmon cyclic frequency andk the wave vector. These resonances are observable as peaks of the experimentally measured quantity dA/dns, which is the derivative of the AC-electric field ab-sorbtionAwith respect to the densitynsof the TDES, when dA/dnsis plotted against the cyclic frequencyωpof the AC-electric field.
The resonance linewidthΓpp(peak-to-peak linewidth) gives the scattering timeτ, which is equal toτ= (2π√
3Γpp)−1, if the resonance peak is assumed to obey a Lorentz distribution. Electron mobility can then be calculated by means of the fundamental eq. (1.26). By means of this methodology,Grimes et al.[38]
1.5. ELECTRON MOBILIT Y 25
(A) Measured Electron Mobility (B) Measured Electron Mobility
Figure 1.6. Electron mobilityµversus temperatureTand helium-vapor densityN. In (A), reprinted fromGrimes et al.[38] (with data fromSommer and Tanner[101], and Brown and Grimes[13]),µfor different vertical holding fieldsE
⊥are plotted. Below 0.68K the mobility is limited by ripplon scattering while at larger temperatures vapor-atom scattering predominates. The solid line is a theoretical calculation for vapor-vapor-atom scattering only. In (B), reprinted fromIye[48] (with data fromGrimes and Adams[38], and Rybalko et al.[93]),µis measured for excitation frequencies100kHz (•) and30kHz (N) and liquid helium thicknessd=1mm, while measurement () has excitation frequency 100kHz andd=1.3mm. The dashed line, indicates mobility due to gas atom scattering alone. The solid curve is the theoretical prediction ofSaitoh[J. Phys. Soc. Japan42, 201 (1977)], forE
⊥=100V/cm. Scale of both axes is logarithmic.
measured electron mobility for temperatures as low as0.3K and verified the theoretical predictions with regard to gas atom scattering and electron-ripplon scattering. In particular, for electron-electron-ripplon scattering, it was found that:
• Electron mobility remains almost constant for T < 0.68K, where electron-ripplon scattering dominates, as predicted byCole[19].
• Electron mobility decreases when the magnitudeE⊥of the holding field increases, as predicted byShikin[Sov. Phys. JETP31, 936 (1970)].
• Electron mobility increases when guard voltage is set above a threshold value, asCrandall[24] predicted.
Inspection of eqs. (1.17) through (1.20) along with the associated fig. 1.3 suggests that—with the exception of guard voltage dependence, which is not
covered by these equations—the theoretical predictions are in good agreement with the experimental data shown in fig. 1.6.
Iye[48] measured electron mobility at low AC-excitation frequencies (in the kHz range, as compared to the MHz range ofGrimes et al.[38]) for different AC-excitation frequencies and bulk liquid helium thicknesses. The temperature range from0.5K to1.8K included both the ripplon scattering regime and the gas atom scattering regime. Despite the different experimental methods the various experimenters followed, all mobility measurements establish the strong increase of electron mobility at the gas-atom scattering regime for decreasing temperature, and an almost constant electron mobility at the ripplon scattering regime for further decreasing temperatures (see fig. 1.6).
CHAPTER 2