Bottom arm gate Top arm gate
4.3 Mixed-fluid Models of the Phase Boundary
4.3.1 First-Order Phase Transition
d/`-dependence. Regions of the phase boundary at lower d/`and thus deeper within the correlated phase enter the regime of full spin polarization before those that are closer to the uncorrelated phase at highd/`. This raises the possibility that somehow the presence of the excitonic phase makes it easier to spin polarize the competing compressible phase. Later in this chapter we will discuss the possible mechanisms by which this could occur, using two different theories of the phase boundary.
We close this section by commenting on the temperature dependence of ∆(d/`), as shown in figure 4.4b. Here, we plot the width of the phase transition versus temperature forθ = 0 andθ = 66◦. As first reported by Kellogget al. [65], the width of the transition atθ = 0 extrapolates to relatively small value in the limit ofT →0.
However, in the high Zeeman regime the width of the transition is clearly nonzero even in the zero temperature limit. This eliminates the possibility that enhanced thermal fluctuations alone cause the broader phase transition at high tilt angles.
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D l
h
D l
q q
Figure 4.4: (a) Width, ∆(d/`), of the longitudinal drag peak atνT = 1 andT = 50 mK versus normalized Zeeman energyηat the peak center. (b) Temperature dependence of ∆(d/`) at θ = 0 (squares) and θ= 66◦ (triangles).
set of Clausius-Clapeyron relations seem to accurately and consistently describe the behavior of the phase transition in response to changes in not only Zeeman energy, but also temperature and density imbalance between the two layers [139].
A truly discontinuous first-order phase transition, however, is unlikely in real samples because of disorder. For example, Stern and Halperin conjecture that during the phase transition density fluctuations break up the system into spatially distinct regions of correlated and uncorrelated fluids [108]. At high d/`, small puddles of the correlated fluid occupy a fraction fcorr of the system. As d/` is lowered, the number and size of the puddles presumably grow until they percolate at some critical fraction fcorr∗ . Stern and Halperin find that before percolation is achieved, the very different transport properties of the two fluid types lead to a large peak in Rxx,D over a relatively narrow range of fcorr. Meanwhile, Hall drag rises monotonically with fcorr, from Rxy,D = 0 at fcorr = 0 to Rxy,D = h/e2 at the percolation point of fcorr =fcorr∗ . In particular, they derive a semicircle law for the longitudinal and Hall drag resistivities,
(ρDxx)2+ (ρDxy +π¯h/e2)2 = (π¯h/e2)2. (4.1) This predicts a peak ρDxx of h/2e2 coincident with ρDxy = h/2e2, which qualitatively agrees with measurements of Coulomb drag across the phase boundary [65, 120, 38].
While this model was originally constructed assuming fully spin polarized elec- trons, it could be modified to permit unequal spin polarizations of the two phases.
This leads to the possibility that since the density of states of the composite fermions in the uncorrelated phase drops by a factor of 2 upon full spin polarization, the ability to screen the disorder potential would be changed. Consequently, the transition width should grow in the high Zeeman regime, just as we observe. We depict the phase sep- aration of the correlated and uncorrelated fluids for the two different Zeeman regimes in figure 4.5.
However, recall that the boundary between the two spin polarization regimes (the
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d / !
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E Z
QH State QH State
CF fluid:
!
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CF fluid:
!
""
(d / ! )
cFigure 4.5: Depiction of first-order phase transition in the low Zeeman (left of dashed line) and high Zeeman (right of dashed line) regimes. In the high Zeeman regime, the uncorrelated composite fermion phase is fully spin polarized and only one Fermi sea exists to screen density fluctuations. Consequently, one anticipates disorder to be more prevalent and the phase transition to be broader than at low Zeeman energy.
shrinks as d/` is decreased. However, the local density (and Fermi energy) of each CF region will remain fixed once normalized by the Coulomb energy. Thus, the CF regions should become fully spin polarized at a single value of normalized Zeeman energy that is independent ofd/`. This is in conflict with the boundary between the two spin polarization regimes, which we observe to have a finite slope rather than being a vertical line. We might explain this discrepancy by noting that exchange interaction effects might lower the critical Zeeman energy at which the CF regions become fully spin polarized. Such exchange interactions will become relatively more important at lower density. Since we achieve lower effective interlayer separation in our sample at fixed d by reducing the density of the 2DESs and tuning the magnetic field appropriately, we would then expect the critical Zeeman energy to decrease with d/`. This model does not require the presence of the correlated fluid and thus could be tested by determining if the critical Zeeman energy is still density dependent in bilayer samples at high d/`, when interlayer correlations are unimportant.
A second, more exotic possibility is that the spin polarized correlated fluid induces an effective Zeeman energy in the uncorrelated fluids through a proximity effect. As fcorr grows at lower d/`, the influence of this proximity effect should also increase and lower the critical Zeeman energy required to fully spin polarize the uncorrelated regions.