πne
(2.26) Therefore the phase boundary curve does not depend on the temperature .
The Wigner crystal can melts and turns into an uncorrelated state of degenerate Fermi gas. This region in the phase diagram is not able to be done with experiments on bulk helium. Even with a maximum achievable electron density of bulk helium 109cm−2(see section 2.1.6) and a temperature of 10−2K, the degenerate Fermi gas is not reached, and the crystal would be just cool [PP83]. In helium films, as it will be discussed in the next section, one could reach much higher electron densities with a good stability, so that one can move through all areas of the phase diagram.
2.2 Electrons on helium films
The maximum electron density on bulk helium is limited to nc = 2.4×109cm−2. At this density, the melting temperature is about 1K and the Fermi energy compared to approximately 50mK. The system of electrons on bulk helium therefore is limited in the phase diagram only on the classical regime. Ikezi and Platzman realized that with thin helium films d≈ 100˚A higher electron densities can be reached than in bulk helium, because the van der Waals forces between the helium atoms and the substrate for this small film thickness are very large and thus stabilize the film [IP81]. As shown by H.
Etz et al. a very high electron densities can be stable on helium films, which is limited only by tunneling of electrons in the substrate[EGIL84]. This is also shown theoretically by Hu and Dahm [HD90]. In order to study the electrons on the helium surface, one should consider the super-fluidity behavior of the helium at low temperatures below the lambda point. The film thickness on the substrated0, which is located at a vertical
distance of h from the helium bath level, results from the balance of the energies of the helium atoms and the atoms at the film surface given by:
d0 = α
ρlgh 1/3
(2.27) It should be noted that this equation is only valid for d ≤ 100nm. For d > 100nm d∝ h−1/4 [CC88]. For example for h = 1cm, using Eq. 2.27 one should have a film thickness of about 30nm(see Fig. 2.15).
Figure 2.15: Thickness of the uncharged helium film as a function of the vertical distance from the substrate to the helium bath level [Wr06]. The retardation of the van der Waals potentialαis not considered
2.2.1 Influence of substrate below the helium film
When the thickness of the film (d) is smaller compared to the distance between the electrons (r = 1000˚A) , the force between the electrons changes because of the presence of the image charge in the surface under the helium film [PP83]. For a film thickness of 65˚Athe energy gap between the first and the second state is E12= 1200K [AGP81].
F.M.Peeters in 1987 found that the highest average probability densityhzi shifts from hzi= 114˚A ford= 104˚Atohzi<15˚Afor d= 10˚A [Pee87].
The potential of electrons on a thin helium films changed to [PP83]:
Vd(r) =e2(1
r − ∆
p(r2+ 4d2)) (2.28)
here ∆ = r−1
r+1 and d = d+hzi. For bulk helium d → ∞ and so Vd(r) ∝ er2 as expected. The additional image charges in the substrate cause a screening effect so that with increasing dielectric constant (i.e. increasing of ∆), the substrate material is
important. For electrons on helium films for r ¿¿ d the interaction of electrons with each other still has a Coulomb character, and the potential due to the strong polarizability of the metal is assumed to have a dipole form [Sch07]:
Vd(r)∝ 1
r3 (2.29)
here r =∞ andr >> d.
Fig. 2.16 shows a schematic representation of electrons on thin helium films.
Figure 2.16: Schematic representation of electrons on thin helium films. Image charges are now produced in both, the helium film and the substrate.[Sch07]
2.2.2 Stability of electrons on helium films
As already discussed for bulk helium, there is also for helium films at a critical electron density nc where the helium surface becomes unstable. Using the ripplon dispersion relation, one can show that nc on helium films is higher than on bulk helium [IP81]:
ω2= ρs By comparing this equation with the dispersion relation for charged bulk helium (Eq.
2.20) one finds that there is an additional term which includes the van der Waals con-stant α. So the stability of a film is increased due to van der Waals forces. The last factor on the rhs of Eq. 2.30 arises from the finite depth d of the liquid film, the function F(k, ) take the effect of image charges in the solid substrate into account, and the factor ρρs indicates that in a thin film only the superfluid component can move.
With Eq. 2.30 one could find that charging a film reduces the ripplon frequency in a
similar way as for bulk helium, but as I said before, the critical charge density where the instability occur is higher.
Film thickness depression
Experimentally it was shown that it is possible to reach very high electron densities [EGIL84]. Saturated helium films on a hostaphan polymer foil e.g. could be charged with electrons up to 9.8×1010electron/cm2 before breaking through to the solid sub-strate. Also, this instability threshold appeared to be independent of the thickness d0 of the uncharged films in the investigated range 200˚A < d0 <400˚A[Lei92].
This apparent discrepancy can be resolved when the electronic pressurePel exerted on the film is taken into account. In the chemical potential,Pel adds to the gravitational and van der Waals terms. Therefore the equilibrium thickness of a charged film will be reduced to: the film thickness d0 before charging given in Eq. 2.27 (here retardation effects were neglected). The value of d from Eq. 2.31 is an average, around which the local film thickness is expected to vary because of the formation of a dimple underneath each electron [And84, PJ85]. As a result of the reduction of d the instability threshold, calculated from Eq. 2.30, is raised, and a higher charge density can be reached[Lei92].
The change in film thickness during charging the film was confirmed experimentally by an ellipsometric determination of d [EGIL84]. In Fig. 2.17, some data are plotted for two values of the intial thicknessd0. The two curves converge for high electron densities as expected from Eq. 2.31. As a result the stability limit for high electron densities on helium films is not affected by the initial film thickness, as observed in the experiment.
On the other hand, tunneling of the electrons through the helium into surface states of the substrate could happen when the film is very thin (d ≤ 50˚A)which limits the charge density on the helium film.
2.2.3 Phase diagram of the 2-DES on helium films
The phase diagram for electrons on helium films was investigated by Peeters and Platz-man [PP83]. To obtain the phase diagram of 2-DES on helium film of thickness d supported by a dielectric substrate of permittivity r, Eq. 2.24 has been solved with
Figure 2.17: Thickness d of charged saturated 4He films wetting a glass substrate (atT = 1.6K) as a function of electron density on the film [Lei92]. The thickness d0 of the uncharged films was 220 and 420˚A.
Eq. 2.28. In Fig. 2.18 the phase diagram for several thicknesses of a liquid helium film supported by a substrate with ∆ = 0.9 is shown.
For a metallic substrater=∞, ∆ = 1 by using Eq. 2.28 the phase diagram in this case shown in Fig. 2.19. The difference between these two diagrams is the appearance of a liquid dipole phase at T = 0. This is because on the metallic substrate the Coulomb term in the electron-electron energy vanishes and the interaction is diploar. By using Eq. 2.23 for the Fermi energy, one can see that for charge densitiesne≤
~2 md2e2
2
the quantum fluctuations will dominate the electron-electron interactions promoting the liquid phase.
For a metallic substrate r =∞, ∆ = 1 by using Eq. 2.28 the phase diagram in this case shown in Fig. 2.19. The difference between these two diagrams is the appearance of a liquid dipole phase atT = 0. This is because on the metallic substrate the Coulomb term in the electron-electron energy vanishes and the interaction is diploar. By using Eq. 2.23 for the Fermi energy, one can see that for charge densitiesne≤
~2 md2e2
2
the quantum fluctuations will dominate the electron-electron interactions promoting the liquid phase.
Figure 2.18: Phase Diagram of 2-DES on a helium film for different helium film thick-nesses for a dielectric substrate[Ska06].
Figure 2.19: Phase Diagram of 2-DES on a helium film for different helium film thick-nesses for a metalic substrate[Ska06]
2.2.4 Mobility of electrons on helium films
In helium films, the mobility of the SSE not only depends on the gas atom scattering, and the electron-ripplon coupling [JP81], but also on the roughness of the substrate.
As above-mentioned the mobility depends on the temperature (see section 2.1.5). For a helium film thickness of 350˚A and at 1.3K a mobility of 200cmV s2 has been obtained
[JSD88]. This value is 5 times less than the mobility on bulk helium at the same tem-perature (1000cmV s2). It is shown that for very thin helium films≈100˚Aone should take into account the deformation of a single electron due to the strong electrostatic attrac-tion of the helium surface [Shi70, Shi71]. L.M. Sander found that for a thin helium film
≈ 100˚A there is a localization or self-trapping of the motion of SSE and he observed a trapping energy of about 8K [San75]. This polaronic effect with a strong electron-ripplon coupling can be interpreted as an increase in the effective mass. The mobility of the electrons moving such as a dimple is≈100cmV s2 [TMP+96](also see[RD99]). A num-ber of review articles has been devoted to the electron mobility in helium films. They found that the mobility can change strongly due to substrate roughness. For example Kono measured the conductivity of SSE on a quench-condensed hydrogen film. He found that the surface has very high roughness just after the quench condensation, the SSE conductivity is unobservably small, and annealing of the hydrogen film improved the surface quality. For smoother substrates, however, the movement is based on ther-mally excited jumps and there are strong changes of the mobility in the growth of the helium film. This behavior was explained quantitatively by a re-trapping structural transition within the disordered system of localized charges [MAKL93]. The SSE bind to the roughness and are no longer free to move. For even thinner films, however, the electrons are in the valleys between the peaks from the lowest energy state. Between these states a transition must take place which can actually explain the behavior of the mobility [KAL91]. Also to study the mobility of SSE on helium films, Andrei used sap-phire substrate, which appears to have a smooth scale of 100˚A(see Fig. 2.20)[And84]
which showed a marked decline in mobility (dip problem).
Shikin suggested distinguishing between the SSE which can move freely on the helium film and the electrons at asperities (active tops) which are tightly bound, and so the total number of the electrons should be the summation of these two. This so-called two-fraction model, which can derive the position of the dips on the roughness of the substrate. This also depends on the equilibrium density of electrons. The model can be confirmed by additional experimental data from I.Doicescu [SKD+01, KGW+01].
Figure 2.20: The mobility of SSE as a function of helium film thickness d at two different temperatures. triangles denote increasing d; circles denote decreasing d.[And84]