6.4 Measurement of DC conductivity
6.4.3 Control of electron density and holding electric field
A problem frequently encountered in experiments on SE is how to reliably determine the electron density ns. Determining ns via the CR line shape, as described on page 83, is rather cumbersome, moreover it requires comparison with a theoretical line shape, which is often an approximation. Another important motivation for the numerical analysis described in the previous section is that it provides a convenient method to control ns.
Generally, the method consists of varying the DC potentials applied to the cell walls at nonsaturated conditions with constant total surface charge Qs, which changes the charge pool radius r(p). This in turn changes the admittance, and by comparison with the result of the numerical analysis, ns can be determined.
It is here best to have a very low value ofω/σxx, so thatξk,l −→0 and the matrixH in Eq. (6.15) is real. The admittance is then purely capacitive,Iac/Vac = iωCx, with Cx
depending solely on geometry, i.e., on the4He liquid level h and the SE pool radius r(p). The usual configuration is to have the same negative potential −Vdc applied to the cell’s top-plate, side wall, and outermost bottom electrodeG, and to measureCx between center Corbino diskC and second Corbino ringA(seeFig. 6.10). At an increase ofVdc, the pool radius r(p) always decreases, but Cx depends nonmonotonically on Vdc, with a maximum that corresponds in this configuration roughly to r(p) being equal to the outer radius r(A) of electrodeA, see Fig. 6.14: At sufficiently highVdc, whenr(p)< r(A), decrease of Vdc, i.e., increase of r(p), leads to increase of Cx, because the capacitive coupling between SE layer and electrode A increases. But at r(p)> r(A), increase of r(p) rather increases the coupling between SE layer and outermost electrode Gand cell wall, which bypasses current to AC ground and leads to decrease ofCx. This effect is probably enhanced by the density perturbation being especially large at the pool edge, as discussed on page 89 and visible in Fig. 6.10(b). The decrease becomes very pronounced when r(p) approaches the cell wall radius a. At r(p)=a, the SE layer is at saturation, and the electron density n(sat)s at the pool center is forh≪r(A) in good approximation
en(sat)s ≃ǫHeǫ0
Vdc
h , (6.16)
derived assuming a uniform vertical electric field between SE layer and cell bottom. At further decrease ofVdc, charge must leave the bulk4He surface. In the experiment, charge loss even seems to occur somewhat earlier, at least at low electron density, as can be seen in Fig. 6.14, where the theoretical Cx(Vdc) is compared to measured data.
The maximum value of Cx depends only on the4He level h, and can in principle also be used for its determination. In our case, h is rather determined independently, which gives a slight offset between theory and experiment inCx (see Fig. 6.12) but an overall better description. In Fig. 6.14, this offset is corrected for better comparison, apart from this the only adjustment is the scaling of the Vdc-axis for the theoretical Cx to fit it to the measured one, which determines the total charge.
For our setup, Cx attains its maximum value when the density ns at the pool center is about 25% below the saturation value n(sat)s for the same Vdc, as can be seen from Fig. 6.14. For practical purposes, it is therefore sufficient to just ensure that one is well on the low-Vdc side of the maximum to knowns to be about 10% below saturation, with an acceptable error of less than±10%. This provides an easy way to control the density in the experiment without any calculation. It is also better not to approach saturation too closely, as otherwise spontaneous charge loss may occur during the measurement.
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Figure 6.14: Dependence on holding voltage−Vdcof: SE pool radiusr(p)(dashed), SE density at the pool center ns (dash-dotted), and capacitance Cx (solid), all calculated numerically for constant total surface charge Qs. Here, r(p) is normalized to the cell radius a, and ns to the saturation density corresponding to the momentary value of Vdc. The dotted vertical line indicatesr(p)being equal to the outer radiusr(A)of Corbino electrodeA; this roughly coincides with the maximum ofCx(see text). The circles show experimental data ofCx vs. Vdc, the sense of variation ofVdcis indicated by the arrows. Starting far below saturation (highVdc), firstVdcis decreased until near saturation charge begins to leave the surface. The theoreticalCx has been fitted to this branch by scaling the Vdc-axis, this determines the total chargeQs= 1.65×107e.
Saturation, corresponding to r(p) = a, is in theory reached at Vdc = 2.15 V ≡ V(sat), with n(sat)s = 2.67×107cm−2. After the charge loss, Vdc is again increased, the measured Cx then follows a curve corresponding to a lower total charge Q′s= 0.88Qs (thus V(sat)′= 0.88V(sat), n(sat)′s = 0.88n(sat)s ), determined by a new fit of the theoretical Cx (not shown).
At high electron density, surface deformation effects (see Sec. 1.2) may also be im-portant, but are difficult to account for in the numerical analysis beyond a uniform reduction ofh. The CR measurement analyzed on page83indicates that saturation can be approached more closely in this case.
Note that not too far below saturation, the holding field E⊥ is still almost equal to the saturation value E⊥(0)=ens/2ǫ0 [Eq. (1.7)]: Assuming uniform electric fields between SE layer and cell top and bottom, one estimates (E⊥/E⊥(0)−1)∼2 (h/l) (n(sat)s /ns−1). In our cell, typically h/l∼0.1, so that for instance if ns is 10% below n(sat)s , one estimates E⊥ to be only about 2 % above E⊥(0). Better accuracy requires a numerical treatment anyway, because the field between SE layer and cell top is not really uniform. Uniformity requires l≪a, which is not fulfilled at low MW resonance frequency of the cell.
The setup also allows to perform measurements at constant ns in dependence of the holding electric field in undersaturated conditions. In this case, the top-plate potential Vtopis varied independently of the potentialVguardof cell wall and bottom electrodeG(see Fig. 6.1). The desired value of the holding field is adjusted via Vtop, while Vguard, which mainly determines the radial electric field, is used to keep the SE pool radius constant, this is controlled by observing Cx. The electron density stays then also approximately constant, apart from a usually small, radial redistribution of charge within the pool. As noted above, for the geometry of our cell (l/a∼1), the exact value of the holding field must then also be calculated by the matrix method outlined in Sec. 6.4.2.
Chapter 7
Experimental results and discussion
In the following, first are presented the new experimental results in quantum cyclotron resonance (CR) in the vapor scattering regime (Sec. 7.1), and compared to previous theories and the new theory for Coulomb effects outlined in chapter 4. Also, they are compared to previous CR experiments (Sec.7.1.5). Complementary measurements of the DC magnetoconductivity in the vapor scattering regime are presented and discussed in Sec.7.2. Finally, Sec.7.3discusses experiments on the nonlinear conductivity at heating.
To experimentally investigate Coulomb effects in magnetotransport of SE, both in the AC case (CR) and DC case, the key experiment is to study the dependence on the electron density ns, as the latter only affects the strength of Coulomb interaction, i.e., the mean value of the fluctuational electric field (FEF) Ef(0)∝n3/4s Te1/2 [Eq. (4.2), Te is the electron temperature], which enters the parameter ΓC=√
2eEf(0)lB describing both the Doppler narrowing of Landau levels (Sec. 4.3) and, more importantly, the Doppler broadening of the dynamic structure factor (DSF) (Sec. 4.4). For convenience, these measurements (see Sec.7.1.1for the AC, and Sec. 7.2.1for the DC case) were performed at saturation, where changes of ns also change the holding electric field, this gives an additional contribution at highns(see Sec.7.1.2) but does not change the general picture.
Coulomb effects may also be studied by varying the magnetic field B, which changes the parameter ΓC, the Landau level width, and the level separation with different de-pendencies. Generally, Coulomb effects depend on the size of ΓC compared to Landau level width and level separation. In DC transport, it is rather easy to cover a sufficiently large range of B in a single measurement (see Sec. 7.2). In CR, one is usually limited to a single value of B, but the variable microwave (MW) frequency of the present setup allowed to study for the first time the B-dependence in a limited range here, too (see Sec. 7.1.4).
Also investigated was the influence of the strength of the interaction with the scat-terers. In the vapor scattering regime, this can be done by varying the temperature T of the measurement cell, which changes the density of scatterers and thus scattering rate and Landau level width, but essentially nothing else, see Sec. 7.1.3 for the AC, and Sec. 7.2.2 for the DC case. Another possibility is to vary the holding field (Sec. 7.1.2).
As the FEF not only depends on the electron density but also on the electron temper-ature Te, it is also of interest to study variations ofTe at fixed T. In the present setup, a simple way to only heat the SE is to increase the MW power under CR conditions, see Sec. 7.3. The resulting nonlinear effects were studied both in the AC (Sec. 7.3.1) and the DC conductivity (Sec. 7.3.2). Moreover, these experiments allow to study also the energy relaxation of SE, and the population of higher surface levels (Sec. 7.3.3).
95