Spin Transition in the Half-Filled Landau Level
5.1 Composite Fermions and Spin
The notion of composite fermions was introduced in Chapter 3. However, the spin degree of freedom was neglected. Next, we introduce a simple picture that includes spin at ν = 1/2.
Even though ν = 1/2 occurs in the presence of a large perpendicular magnetic field, in a mean field approximation, at exactly ν = 1/2 the system can be treated as a Fermi sea of CF’s. The CF orbital degree of freedom behaves as if there were effectively zero magnetic field. However, the spin degree of freedom is still affected by the presence of the magnetic field. Adopting a simple picture, we assume that the effect of the magnetic field on the CF spin is to simply shift the energy of the up spins with respect to the down spins by the Zeeman energy, and that this shift is given by the electron spin g-factor g*
such that the Zeeman gap is EZ = μΒg*B. We set the CF g-factor equal to the electron g- factor [3]. We also assume that the CF’s have a parabolic dispersion relation E vs. k with an effective mass mCF such that, using the usual relation for the density of states for free fermions in 2D, the Fermi energy of a single spin branch of CF’s is EF = 2πħ2n/mCF. The relative magnitude of the Zeeman splitting and Fermi energy will determine the spin polarization of the system, as sketched in Fig. 5.1. For EZ < EF the system is partially spin polarized, and for EZ > EF, the system is completely spin polarized.
Fig. 5.1. Simple model of composite fermion spin polarization. The dashed line is the Fermi Energy. The two parabolas are the dispersion relations for up and down CF’s The left and right plot show the case of a partially and completely spin polarized electron gas, respectively.
The mass mCF is referred to as the “polarization mass” and is different from the effective mass commonly extracted from measurements of activation energies of FQH states via magnetotransport measurements [3]. A phenomenological, transport-derived effective mass mCF_transport for CF’s can be obtained by setting the activation energy Δ (measured via the temperature dependence of the resistivity) at a FQH state equal to the CF cyclotron energy. Then Δ = eBeff/mCF_transport, where Beff = B – B0 = φ0n
(
1/ν −2)
is the effective magnetic field experienced by a CF at filling factor ν, B0 =φ02n for a CF comprised of an electron bound with two flux quanta, φ0 = h/e is the quantum of magnetic flux, and n is the electron density. The activation gap at FQH states is determined by the Coulomb energy, so that Δ ~ n. Then, for fixed ν, mCF_transport ~ n.The polarization mass is not equal to the transport mass. The activation gap used to define mCF_transport contains contributions from both the bare CF cyclotron energy and the
E
k EF
EZ E
EF
EZ
k
self energies of an excited CF particle and CF hole. The polarization mass will not be determined by just the bare CF cyclotron energy alone, but the contribution due to interactions should be less than for the transport activation mass. However, the Fermi energy used to define the polarization mass is proportional to the Coulomb energy so that the polarization mass at fixed ν = 1/2 also scales like mCF ~ n~ B.
Within the model presented above, a spin transition between partial and complete spin polarization at ν = 1/2 should occur as a function of density and magnetic field (if n is held fixed, then n ~ B). The situation is sketched in Fig. 5.2. At fixed ν = 1/2, as a function of magnetic field, the Zeeman splitting rises more rapidly than the Fermi energy.
The Fermi energy is proportional to the Coulomb energy so that EF ~ B, while . Thus, EZ and EF will cross at some critical magnetic field BC. For fields below BC, EZ < EF so that the spin polarization is partial. As B is increased, the spin polarization will increase until
Z ~
E B
B B≥ C, at which point the spin polarization will be complete.
EZ ν= 1/2
EF
BC B
Fig. 5.2. Scaling of CF spin Zeeman and Fermi energies with magnetic field at ν = 1/2, showing the critical field BC at which the two energies are equal and the transition from partial to complete spin polarization should occur.
The value of the composite fermion effective mass mCF determines the critical magnetic field and density at which the 2DES becomes completely spin polarized. The transition occurs when the Zeeman splitting is equal to the Fermi energy:
(
/ 2 e)
B F2 / C ,g B E
g e m B n m F μ
π
=
=
= =
where me is the bare electron mass in vacuum and g is the g-factor for the composite fermions, which we assume is the same as that for electrons, g = -0.44. Now, at ν = 1/2, n = Be/2h, so at the critical magnetic field we find that
/ 1/
CF e
m m = g .
To start the chapter, we have presented a very simple picture of spin at ν = 1/2.
More sophisticated versions of this Pauli paramagetism picture that, for example, do not assume a parabolic dispersion relation for CF’s, can be found in Ref.’s [3, 4]. More speculatively, there is a possibility that the spin transition at ν = 1/2 is weakly first order.
Ferromagnetism, driven by residual interactions between CFs, has been theoretically predicted for CF’s at ν = 1/4 [5]. We will return to the topic of ferromagnetism for CF’s later in the Chapter. There is also theoretical evidence that the ν = 1/2 state is energetically near a state in which composite fermions form spin-polarized pairs [6].
Previous experimental evidence exists for a spin transition for CF’s at ν = 1/2. The first observation of this transition was by Kukushkin et al. using polarization-resolved photoluminescence; the electron spin polarization was observed to increase and then saturate when increasing the electron density and magnetic field while maintaining fixed filling factor ν = 1/2 [7]. Optically pumped NMR measurements of the Knight shift, using multiple quantum well samples (~ 100 closely spaced QW’s), also suggest that a spin transition occurs when the total magnetic field is increased by rotating the sample in the magnetic field while maintaining fixed perpendicular magnetic field to remain at ν = 1/2 [8, 9]. A combination of RDNMR and standard directly detected NMR has been used to measure the NMR Knight shift versus magnetic field in a variable density sample at ν
= 1/2; the Knight shift versus field data show a change in slope that is suggestive of a spin transition [10]. The electronic spin-flip excitations have been probed using inelastic light scattering; the spin-flip gap was shown to collapse as ν → 1/2 for a sample at relatively low magnetic field and remain finite for another sample at higher magnetic field [11].
The RDNMR measurements discussed in this chapter take a closer look at the spin transition at ν = 1/2, using higher quality samples and lower temperatures than previously achieved. Our measurements more thoroughly examine the nuclear spin-lattice
relaxation time T1 temperature and magnetic-field dependence for temperatures ranging from 35 to 200 mK over a wide magnetic field and density range. We also show the first measurements of how transport at ν = 1/2 depends on the electron spin polarization.