3. Composite Fermions with Spin
3.4 Zero Temperature Smoothening (ZTS)
Fig. 3.10:The experimental spin-flip gap for fixedν = 2/3as a function ofB[63].
A reentrant behaviour is clearly visible. The typical energy scale involved in the smooth transition region is of the order of 0.3 K.
we will now concentrate on the mechanisms producing the ZTS, while the ad-ditional polarization shoulders will be addressed in Chapter 5.
1.6 1.8 2 2.2 2.4 2.6 2.8 3 2.75
3 3.25 3.5 3.75 4 4.25 4.5
B(T) E(K)
Fig. 3.11:The SCFLL crossing of interest for the spin polarization transition at ν = 2/3 as functions of the magnetic field. Here the two crossing lev-els|0,↓iand|1,↑iare depicted as thick lines, together with the Fermi en-ergy (dashed line) passing through the crossing point. The two thin lines represent the Fermi energy shifted upwards and downwards by 0.15 K, so to mimic a characteristic energy window of 0.3 K (in agreement with the experimental indications) close to the transition. This window is as-sociated to a magnetic field region, ”guarding” the critical field Bc, de-limited by the two vertical lines. The magnetic field size of this region is in agreement with the transition region observed in the zero temperature measurements (see Fig (3.2)).
random scalar potential is via the momentum integral of the imaginary part of the self-energyΣfor the single-particle Green’s function, as we saw in section 1.3.2.
In the RMF case, this approach is questionable because the single-particle Green’s function is not a gauge invariant quantity. Moreover, for vanishing average field, within the consistent Born approximation (SCBA), the calculation of the self-energy is plagued by infrared divergencies [75] which are due to the long range nature of the correlator of the vector potentials.
To circumvent these difficulties, E. Altshuler et al. [77] calculated the DOS of elec-trons in a RMF in the semiclassical approximation, assuming that the energy E of the particle is much larger than both the cyclotron energy and1/τ, where1/2τ is the width of LLs due to the RMF (~= 1). This would correspond to consider large LL number (p1for CF).
In the analogous semiclassical calculation for spinless CF, we would start with the Hamiltonian
HDis=X
i
1 2m∗
hpi+e
c(A(ri) +δA(ri))i2
(3.10) whereA(r)is the vector potential generating the average mean magnetic fieldB∗ andδA(r)generates the RMF. We also assume a Gaussian,δ-correlated RMF
hδB(r)δB(r0)i=B02δ(r−r0) (3.11)
whereh...iindicates the average over the disorder. The corresponding vector po-tential correlator (in the momentum space) is
hδAα(q)δAβ(q0)i= B02
q2 δαβδ(q+q0). (3.12) In the semiclassical approximation, the Density of States DOS(E)(proportional to the imaginary part of the one point Green’s function G(E)) can be shown to be given by the disorder averaged path integral over the closed classical trajectories,xE
DOS(E) =m∗ 2π
1 +Re
I
DxE exp (iS[xE])
. (3.13)
In the same spirit of the theory of the weak localization, the approximation is made to take the orbits as the free ones (i.e. closed cyclotronic circles in the effective field B∗) but with a phase influenced by the vector potential. In particular, using Stoke’s theorem we can transform the phase contribution H dl·(A+δA) into a surface integral and write the action on a generic path as
S= e c Z
S
dr(B∗+δB(r)) (3.14)
where S is the area enclosed by the trajectory.
Assuming that the RMF is a small perturbation to the mean fieldB∗(~ωc∗m∗v20, wherev02=e2B02/4m∗2c2), the classical trajectories are not affected by the RMF and are circles with radiusRc =v/ωc∗(vis the velocity of the particle); more precisely, every circle gives rise to infinitely many trajectories labelled by their winding num-ber.For the disorder average over a Gaussian RMF we have
* exp
ie c
Z
S
drδB(r) +
= exp
− e2 2c2
Z
S
drZ
S
dr’
δB(r)δB(r’)
. (3.15) Hence, the DOS (3.13) can be calculated, with the averaged action on the trajectories
Sav= e
cSorB∗+ e2
2c2B20Snor (3.16)
whereSorandSnorare the oriented and non-oriented areas enclosed by the trajec-tory, respectively.
Carrying out the sum over the circles with different winding number, a Gaussian DOS is obtained (`∗the magnetic length associated toB∗)
DOS(E) = 1 2π`∗2
∞
X
n=0
τ(E)√π exp −τ2(E)
E−
n+1 2
~ωc∗ 2!
(3.17) and the width of the levels is
W(E)≡ 1 2τ(E) =
rEm∗v20
π . (3.18)
In our system the mean magnetic field can be relatively strong (only few Landau levels are filled) and the semiclassical approximation is not fully justified but we believe that the expression (3.18) yields a reasonable semiquantitative estimate of the broadening in this regime as well.
In the case of CFLLs,m∗is the effective mass of the CF around the magnetic fields of interest for the spin polarization transition andB02 ≈ ν(Φ0/2πl)2[80, 45]. The width of then-th LL,Wnis then
Wn≡W((n+ 1/2)~ωc∗) =
r(n+ 1/2)~ω∗cm∗v02
π (3.19)
showing weak LL number and magnetic field dependences.
Let us focus on theν = 2/3state: assuming thatB ≈2T (around which the polar-ization transition occurs) with the parameters of the GaAs-AlGaAs heterostructures we get, for the 0-th LL,W0≈0.34K. As mentioned in the last section this is the typi-cal energy stypi-cale involved in the experimental ZTS. Moreover the ZTS regions seem not to dependent significantly on the different spin polarization transitions, sug-gesting a weak energy scaling of the disorder induced CFLL broadening.
Of course, we have made some simplifying assumptions to get formula (3.19): in the experiments of Kukushkin et al., theδ-doped monolayer is separated from the 2DEG by an undoped spacer of widthd≈30nm. Therefore, the range of the RMF generated by the density fluctuations (of the order ofd) is longer than the magnetic length forB∗ > 1T. In order to describe more closely the experiments, a deeper analysis of the strong residual field and of the longer RMF correlations is needed.
In this direction, we performed preliminary calculations of the CF Green’s function in the SCBA for the strong residual magnetic field case. If Landau level mixing is neglected, in analogy with the approximate treatment of random scalar poten-tials [81, 82], the fermionic self-energy is finite. Surprisingly enough, the LL widths obtained in this approximation have the same dependence onB∗,B02andnas for-mula (3.19). However singular terms appearing due to Landau level mixing lead to divergencies as in the zero mean field case. The treatment of these divergencies is presently a highly debated problem and further investigations of this delicate issue are under study.
3.4.2 Spin-orbit effects
The disorder-induced broadening of the CFLLs is not the only origin of the ZTS.
One can also obtain it by anticrossing of the CFLL near the critical fields. In analogy with the IQHE [83], anticrossing could be driven by spin-orbit coupling.
In order to obtain the effective spin-orbit Hamiltonian for the CFLL involved in the transition, let us start with the single particle 2D Bychkov-Rashba term
VRashba2D =~ehEzi
4m20c2 ˆz·s×Π. (3.20) wherehEziis the average electric field built into the heterojunction along the growth directionz.
We can write the kinetic momentumΠin terms of the interLL operatorsa,a†(with a†|n, ki=√
n+ 1|n+ 1, ki, where|n, kidescribes the state of then-th LL with in-ternal momentumk):Πx=i`∗/√
2~(a†−a)andΠy =`∗/√
2~(a†+a)and the Pauli matricessx,sy in terms of the rising and lowering spin operatorss± = sx±isy. Then we obtain the effective Hamiltonian for two close CFLL with opposite spins
HSO=En↑+1,p,↑(B)c†n↑+1,↑cn↑+1,↑+En↓,p,↓(B)c†n↓,↓cn↓,↓+
+ VSOc†n↑+1,↑cn↓,↓+h.c. (3.21) withVSO=√
2~2ehEzi/4m20c2`∗andcns,sthe annihilation operator for a particle in the state |ns, kiand spins. Notice that the spin-orbit coupling is diagonal in the internal momentum and the corresponding index has been omitted for simplicity.
By diagonalizing (3.21) we get the resulting single particle split eigenmodesΨ±as linear combinations of the CFLL eigenfunctionsψns,
Ψ±=N
ψn↓+
∆(B) 2VSO ±
s ∆(B)
2VSO 2
+ 1
ψn↑+1
(3.22)
where∆(B) =En↑+1,p,↑(B)−En↓,p,↓(B)andN is a normalization factor. The new eigenenergies are
±(B) =En↑+1,p,↑(B) +En↓,p,↓(B)
2 ±
s ∆(B)
2 2
+VSO2 . (3.23) It can be seen how the eigenmodes (3.22) have expectation values of the spin that change smoothly from, say, ↓ to ↑ when B moves from the left to the right of Bn↑+1,n↓. By evaluating with these statesγe(B)atT = 0we obtain the cross-over behaviour shown in Fig (3.12) forν = 2/5(dashed line).
Fig. 3.12:Spin polarization of the GS atT = 0 forν = 2/5 as a function ofB (in Tesla), forα= 0.2. Full line: with level crossing. Dashed line: with anti-crossing (VSO= 0.1K, see text).
The width of the crossover region inBis a function ofVSO, which also represents half of the smallest energy separation (the gap) between the eigenmodes. The typ-ical spin-orbit-induced splitting in GaAs heterostructures is of the order of 0.2-0.3
K. Again, we obtain the right energy scale needed to produce the observed ZTS.
Similar results can be obtained for the otherν’s considered in [61].
In a real experiment both the disorder induced broadening of the CFLL and the spin-orbit anticrossing contribute to the ZTS.
Up to now we never considered the role of the residual fermionic interaction close to the degeneracy point. This seems to be a crucial issue for the explanation of the observed shoulder in the polarization experiments.
The first attempt we could try in this direction is to treat the CF Coulomb interac-tion at Hartree-Fock level. This would keep memory of the CF charge neglecting the role played by the attached fluxes, and by the consequent Chern-Simons inter-action. Of course, these are drastic approximations and we will see that, indeed, we will have to go beyond this level to grasp the nature of the partly polarized state.
In 1985 Giuliani and Quinn considered the problem of two LL of electrons brought to coincidence (via a tilted magnetic field configuration) and coupled by the Coulomb interaction [84]. The purpose of their investigation was to determine whether a spin-density wave GS close to the crossing is possible, or if otherwise the only sta-ble configurations are obtain by pure occupations of one mode only.
By performing a direct (non self-consistent) Hartree-Fock calculation they deduced that afirst-orderphase transition should be expected in the polarization, and that no spin-density wave GS should form. In the following years a similar calculation has been performed in presence of inhomogeneities with the result that the first-order phase transition should still be the expected behaviour [85].
The existence of the shoulder in the polarization experiments shows that more com-plex structures come up near the degeneracy of two CFLLs, and that the Hartree-Fock analysis is not enough to describe them. The nature of these partly polarized states will be addressed in Chapter 5.