ED(E)Extended states
1.2.1 The 111 State
fermion Fermi sea at ν= 1/2.
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Here, zj are the coordinates for the N1 electrons in the first component (e.g., the top layer in a bilayer system) and wr are the coordinates for theN2 electrons in the second component (e.g., the bottom layer in a bilayer system). Note that once again we have dropped the exponential terms for the sake of simplicity. If either m1 or m2 are even, then a composite fermion Fermi sea term associated with the appropriate component must be added to equation (1.22) to preserve antisymmetry with respect to electron exchange. By considering the number of vortices bound to each electron, the Landau filling factors for the two components are given by
ν1 = m2−n
m1m2−n2 (1.23)
and
ν2 = m1−n
m1m2−n2. (1.24)
From now on, we specialize to bilayer systems, in whichν1 andν2 represent the filling factors for the two layers. We also assume that the spins of the electrons are frozen out by the large Zeeman field, even though this will ultimately prove to be an over- simplification. In the case of n = 0, equation (1.22) would be the product state of two uncorrelated quantum Hall systems (e.g., a bilayer system with d/` =∞), with ν1 = 1/m1 and ν2 = 1/m2. As the strength of interlayer repulsions grow (i.e., d/` is reduced from infinity), one expects that states with n6= 0 would become more favor- able energetically, and electrons in one layer will become anticorrelated with electrons in the other layer. As n grows in value while the individual Landau filling factors re- main constant,m1 andm2 will consequently decrease from their original values when the two layers were uncorrelated with each other. This represents electrons unbinding themselves from vortices associated with electrons in their own layer and becoming attached to the vortices of electrons in the other layer. The exponentsm1andm2 will switch back and forth between even and odd values, implying a series of transitions between compressible and incompressible bilayer states asd/` is tuned [134, 94].
If we consider the situation where m1 =m2 =n, then equations (1.23) and (1.24) seemingly imply that the filling factors for the individual layers are not well defined.
The focus of this thesis is the bilayer system occurring at νT = 1. We primarily restrict measurements to the case of equal densities in the two layers, N1 = N2. Following the previous discussion, in the limit of d/`=∞, the system will consist of two independent layers with m1 = m2 = 2 and n = 0. Both layers are compressible Fermi seas of composite fermions, with no Hall plateau. For d/` = 0, interlayer Coulomb energies are entirely equivalent to intralayer Coulomb energies. One would expect that the system should be described by the wave function in equation (1.22) with m1 =m2 =n= 1 (i.e., the “111 state”), such that each electron is bound to an equal number of upper and lower layer vortices.
Such a quantum Hall state at νT = 1 was first supported by numerical evidence from Chakraborty and Pietil¨ainen [10]. Experimentally, conventional transport mea- surements (i.e., driving a total currentIT that is equally split between the two layers) by Suen et al. [113] and Eisenstein et al. [25] found signs of an incompressible state in bilayer systems atνT = 1. However, we should note that the splitting ∆SAS of the symmetric and antisymmetric tunneling states can also generate an energy gap, even in the absence of Coulomb interactions. Such a splitting is analogous to the Zeeman splitting between spin-up and spin-down electrons. Murphy et al. [84] explored this possibility by examining a series of weakly tunneling bilayer systems with variabled/`
and ∆SAS. Their studies of samples with the smallest tunneling energies revealed that asd/` is reduced below a characteristic value ofd/` ≈2, an incompressible quantum Hall state develops at νT = 1. Samples with larger tunneling energies tended to have larger critical values of d/`, but they found evidence that the νT = 1 quantum Hall state remains even in the limit of ∆SAS = 0, leaving Coulomb interactions as the origin of the νT = 1 quantum Hall state. In figure 1.7 we show a summary of their results. Note that the critical d/` is finite even at ∆SAS = 0. An example of the evolution of the minimum inRxx withd/` is shown in figure 1.8. Here, the interlayer separation d is kept fixed, but the total density is tuned so as to alter the magnetic length ` atνT = 1. Thus, one may alter d/` within a single samplein situ.
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l
D
el
Figure 1.7: Phase diagram of νT = 1 QHE with respect to effective interlayer separa- tion d/` and single-particle tunneling energy ∆SAS, obtained by Murphy et al. [84].
Below the black curve, an incompressible QH state is observed at νT = 1. Beneath it, the bilayer is compressible. Note that the samples studied in this thesis are very weakly tunneling and would lie along the left boundary of this phase diagram.
W
l
n
n
Figure 1.8: Rxx in parallel flow versus magnetic field at various values ofd/`, which is tuned by changing the electron density and thus modifying the value of` atνT = 1).
The black dots denote the condition νT = 1 for each trace. Data taken using sample 7-12-99.1R atT = 50 mK.
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Similar to other quantum Hall states, the νT = 1 system is characterized by a minimum in Rxx and a quantized Hall resistance of Rxy ≡ Vxy/IT = eh2. In this respect, the νT = 1 state greatly resembles a single layer of electrons at filling factor 1, with charged excitations confined to the edge. However, the bilayer system has an additional degree of freedom in the form of whether a given electron occupies the upper or lower layer. Wen and Zee [125] argue that because the 111 state does not have a well defined ∆N ≡N1−N2, states with different ∆N have the same energy in the absence of capacitive coupling or interlayer tunneling. Charged excitations (as- sociated with changes in the total number of electrons, NT ≡N1+N2) have a finite energy cost and are said to be gapped out. This is connected with the appearance of an incompressible state in conventional transport measurements. But excitations that change ∆N apparently cost little or no energy and thus represent gapless ex- citations. Wen and Zee note that at finite d/` the gapless mode is associated with a spontaneously broken U(1) symmetry and should be accompanied by a superfluid mode. Of course, real samples have a finite tunneling energy that will explicitly break the U(1) symmetry by selecting the symmetric distribution of electrons between the two layers as the ground state. Nonetheless, it is assumed that the essential physics will remain so long as the tunneling energy is much smaller than any other relevant energy such as the Coulomb energy EC or thermal energy kBT.