With the FSS known we can now start investigating various methods of modifying it and preferably tuning it to zero. A Hamiltonian containing only the heavy bright excitons has the matrix form
H = 1
in the basis{|↓⇑i,|↑⇓i}, whereEC is the exciton binding energy originating from the quantum dot confinement and Coulomb interaction. We would like to examine the possibility of tuning δ1 within the bright exciton subspace, and upon closer inspection of Eq. (2.50) we realize that the FSS depends on both the electron and hole envelope functions and their curvature. So in order to manipulate the exciton envelope function directly, we introduce an in-plane electric field. It should be mentioned that there are also many other methods of reducing the FSS, for instance by strain [48, 58], vertical electric fields [37, 58–61] as well as proposals to recover the entanglement in the presence of a finite FSS such as spectral filtering [45,62], time reordering [63], or embedding the quantum dot in an optical cavity [64].
When including the in-plane electric field, care must be taken when choos-ing the confinement potential for the quantum dot. A common model for a quantum dot is to assume a parabolic confinement potential so that the total mesoscopic potential in the effective Schr¨odinger equation (in 1D) has the form
where qn is the charge of the considered particle (−e for electrons and e for holes). The solutions to the effective Sch¨odinger equation are displaced harmonic oscillator wave functions
ψax,ay,az(r) = ψalxnx x−lEnx
ψalyny y−lEny
ψlaznz z−lEnz
(2.68) with characteristic lengths lnα =p
~/m∗nωnα defining the spread of the wave function andlEnα=eEα/m∗nωnα2 , the electric displacement,ax, ay, azare meso-scopic integer quantum numbers, and
ψal(x) = (2aa!)−1/2 s
√l
πe−x2/2l2Ha(x/l), (2.69) whereHa(x) are the Hermite polynomials [16]. We note that the electric field effectively separates the electron from the hole by a distance l = |lEe +lEh|. Except for very small electric fields, this is not very physical since the electron and the hole can be displaced well outside the region of the physical quantum dot. Instead we must consider some confinement having a more hard-wall character which prevents the electron and hole from escaping. The natural choice falls on the particle in a (finite) box model for which the confinement potential is given by where ∆En is the band offset and χA(r) is the characteristic function of the set A. The solutions to the effective Schr¨odinger equation are again product states c1, c2, a, b, andǫx generally have do be determined numerically from continu-ity conditions. Here we actually pretend that the electric field is only applied to the dot region to avoid having to deal with the infinite negative potentials appearing at infinity if the electric field were to extend over all space.
-100 -50 0 50 100 150 200 250 300 350 x [nm]
(b)
Fh(x)
Vh(x)
FhE(x) (a)
Fh(x)
Vh(x)
FhE(x)
lhx
lhx
lEhx
Ex= 0 kV/m Ex= 40 kV/m
Figure 2.5: Hole envelope functions with,FhE(r), and without,Fh(r), applied electric field Ex in x-direction, for (a) harmonic and (b) hard-wall confine-ment. For the case of harmonic confinement, the wave functions are trans-lated by lhE and are no longer inside the physical region of the dot when a field Ex is applied but retain their shape. For the hard-wall potential, the main effect of the electric field is to deform the wave functions; this is ac-companied by a relatively small shift that leaves the particles inside the dot.
The deformation affects the second derivative ofFhE(r) which determines the FSS, see Eq. (2.50).
The difference between the two confinement potentials is illustrated in Figure 2.5 which demonstrates the nonphysical, unrestricted displacement of the harmonic potential confinement and the more realistic hard wall confine-ment, for which the electron and holes remain within the dot.
In order to investigate the dependence of the FSS on the applied field, we consider a dot composed of In0.2Ga0.8As surrounded by GaAs. This is a het-erostructure of type I, meaning that the electron and hole both experience a potential minimum in the dot. We examine both the hard-wall confinement potential as well as the harmonic one and solve the effective Schr¨odinger equation numerically. We calculated the FSS numerically using Eq. (2.50)
0 1 2 3 4 5 6 7
0 5 10 15 20 25 30 35 40
δ [ µ eV ]
E [kV/m]
E k x ˆ E k x ˆ + ˆ y E k y ˆ
Figure 2.6: The FSS δ calculated for an harmonically confined InGaAs dot with characteristic lengths 30 ×20×7 nm3 and various directions of the electric field. Regardless of the direction of the field, the FSS is monotonically decreasing and vanishes only asymptotically.
and when we compare the FSS as a function for the two confinement poten-tials we find a significant difference. For the harmonic potential, shown in Fig. 2.6, the FSS decreases for any direction of the electric field. Further-more, the FSS vanishes asymptotically but is never completely eliminated.
This does not agree with experimental observations [65,66], in which both the complete elimination as well as non-monotonic behaviour was reported. If we instead consider the same situation for the hard wall confinement, shown in Fig. 2.7, we can see a clear dependence on the direction along which the electric field is applied. In addition, depending on the direction, the FSS may in fact increase. Finally we also note that this choice of confinement potential allows the complete elimination of the FSS, also observed in experiments.
-10 -5 0 5 10 15
0 20 40 60 80 100
δ [ µ eV ]
E [kV/m]
y ˆ
x ˆ E
E k x ˆ E k x ˆ + ˆ y E k y ˆ
Figure 2.7: The FSS δ calculated for In0.2Ga0.8As/GaAs dot of dimensions 55×50×7 nm3 with a hard-wall confinement in the presence of an electric field applied in various directions. The FSS is sensitive to the direction in which the field is applied: In the ˆx-direction the FSS decreases and changes sign at a critical field strength whereδ = 0 (here, atEx ≃35 kV/m). When the field is applied in the ˆy-direction, the FSS increases instead.