Spin-orbit and electronic interactions in narrow-gap quantum dots

We present a detailed analysis of spin-orbit couplings in zinc-blende narrow-gap parabolic quantum dots built in the plane of a two-dimensional electron gas. Such couplings are related to both bulk (Dresselhaus) and surface (Rashba) inversion asymmetry terms in the Hamiltonian of the system. We start by focusing on how the pure Fock-Darwin spectrum of an InSb quantum dot is modified by the addition of separate terms of spin-orbit coupling; we then deal with the presence of all spin-orbit terms in the numerical diagonalization of the single-particle model. We also consider a two-electron quantum dot—by antisymmetrizing the one-electron basis—and study the competition between electron-electron and spin-orbit interactions. All these effects are analyzed in the presence of a magnetic field perpendicular to the quantum dot. Selection rules for spin-orbit-induced level anticrossings, as well as critical fields and energy minigaps related to them, zero-field energy splittings, and the role of the $g$-factor on the spectrum are also addressed.

Figure 1

$H_0$ FD spectrum of an InSb QD as defined by parameters in Table (no SO terms). Notice the energy shell structure at zero magnetic field, and the presence of several accidental degeneracies at finite fields. The states are pure, since the expectation values of spin and orbital angular momenta are only integer numbers at any field.

Figure 2

QD spectrum when only the diagonal SIA term $H_{\mathit{SIA}}^D$ is added to $H_0$. States continue to be pure at any field, and new accidental degeneracies are induced in the low-field spectrum as exemplified in inset of panel B; this introduces a competition between SO and magnetic field effects in QD electronic properties.

Figure 3

QD spectrum when only the Rashba SIA term $H_R$ is added to $H_0$. Strong level mixtures occur at $B_C\simeq 2.5\mathrm{T}$ for all states satisfying the selection rules, causing the observed ACs (panels C and D). Notice the inverted level crossing in inset of panel B when compared to Fig. 2.

Figure 4

QD spectrum when both SIA terms, $H_{\mathit{SIA}}^D$ and $H_R$, are added to $H_0$. The critical field range for ACs becomes wider (panels C and D), even though the first AC happens practically at the same field of the previous figure($B_C\simeq 2.55\mathrm{T}$ in insets of those panels). Anticrossings involving $l<0(l>0)$ orbitals are shifted to lower (higher) fields as seen on panel D.

Figure 5

QD spectrum when only the cubic BIA term $H_D^C$ is added to $H_0$. The influence of such term on the QD electronic properties is practically negligible for the parameters of Table ; it will be important for the full $H$ when we later consider the Rashba effect.

Figure 6

QD spectrum when only the linear BIA term $H_D^L$ is added to $H_0$. $H_D^L$ is not able to induce low energy ACs. However, it drastically alters the spectrum by introducing enormous zero-field splittings and large shifts of several crossings at finite fields (panels A and B); for example, the first crossing has moved to about $3.3\mathrm{T}$. Even the lowest energy states cease to be pure at low fields (insets of panels C and D). Notice only one crossing of the second shell levels in inset of panel B, instead of two as in previous figures.

Figure 7

QD spectrum when both BIA terms, $H_D^C$ and $H_D^L$, are added to $H_0$. Under the parameters of Table , the linear Dresselhaus contribution dominates over the cubic one. The strong mixings at low fields are due to $H_D^L$, while the low energy ACs are due to $H_D^C$.

Figure 8

One-particle full $H$ spectrum of an InSb QD as defined by parameters in Table (all SO terms simultaneously added to $H_0$). Basically, $H_R$ induces the main low energy ACs, while those due to $H_D^C$ are also noticed. Both sets are shifted by $H_D^L$ to higher fields. The $H_{\mathit{SIA}}^D$ term is able to shift higher energy ACs to slightly smaller fields. The remarkable feature of the simultaneous inclusion of all SO terms is that ACs related to distinct $n$ values are grouped at different values of critical fields.

Figure 9

Full $H$ spectrum of an InSb QD with doubled $z_0$ size (other parameters as in Table ). Because of the reduction of $H_D^L$ influence, the spectrum shows very similar features to the ones seen in Fig. 4, even though a wider range of critical fields is present here.

Figure 10

Full $H$ spectrum of an InSb QD with four times stronger Rashba field $dV∕dz$ (other parameters as in Table ). Notice the cancellation not only of the zero-field energy splittings but also of the spin splittings at low fields (panels A and B). Under such stronger Rashba coupling, the set of ACs related to the $H_D^C$ selection rules becomes visible around $1.1\mathrm{T}$ (panel B and insets of panels C and D), while $B_C$ for the first AC is shifted back to about $2.7\mathrm{T}$, the same occurring for the ACs around $5\mathrm{T}$ due to $H_D^C$. Notice also the enormous state mixture in panel C. At zero field, most states have $\mid ⟨s_Z⟩\mid <0.5$, except for the ground state (inset), with $\mid ⟨s_Z⟩\mid \simeq 0.7$.

Figure 11

Zero-field energy splittings for second energy shell states (left panel), critical magnetic fields for the lowest level AC (middle panel), and energy gaps opened at that AC (right panel) as function of the QD lateral radius $l_0$. Squares, circles, and triangles indicate, respectively, a QD defined by parameters in Table , a QD having doubled $z_0$, and a four times stronger $dV∕dz$. Dotted line shows $B_C^0$ as per Eq. (20). Arrows at $l_0=190\mathrm{\AA }$ show QD radius where spectra from Figs. 1, 2, 3, 4, 5, 6, 7, 8 were calculated.

Figure 12

Zero-field energy splittings for second energy shell states (left panel), critical magnetic field for the lowest level AC (middle panel), and energy gaps opened at that AC (right panel) as function of the QD vertical width $z_0$. Squares and triangles indicate, respectively, a QD defined by parameters in Table , and a QD having a four times stronger Rashba field $dV∕dz$. The dotted line shows $B_C^0$ as per Eq. (20). Arrows at $z_0=40\mathrm{\AA }$ show QD width where spectra from Figs. 1, 2, 3, 4, 5, 6, 7, 8 were calculated.

Figure 13

Two interacting particle QD spectrum when no SO terms are taken into account in Eq. (19). The left panel shows every state obtained from the numerical diagonalization, while the right panel shows a zoom focusing on the singlet ground state and on the zero-field splittings as induced by the exchange energy for the lowest excited states.

Figure 14

Two interacting particle QD spectrum when all possible SO terms are taken into account in Eq. (19). Comparing the ground and first excited states in panel B with those in right panel of Fig. 13, one can see that SO energy acts against the direct Coulomb term and favors the exchange electron-electron interaction. The first excited triplets, at zero field, are split according to the $M_J$ values. ACs at around $3\mathrm{T}$ [panels $C$ $\left(⟨S_Z⟩\right)$ and $D$ $\left(⟨M_Z⟩\right)$] are due to the Rashba term; the lowest ACs are indicated by the rectangle in panel B, detailed on Fig. 15.

Figure 15

$B_0$ dependence of $⟨S_Z⟩$ (upper panels) and $⟨M_Z⟩$ (lower panels) values for the lowest ACs shown in the rectangle of panel B in Fig. 14. Panels A and B include all the ten states present in that rectangle. The right panels include only the five lowest states, labeled by their ordering in increasing energy: panels ${\mathrm{A}}_1$ and ${\mathrm{B}}_1$ are related to the first AC between states 1 and 2, while panels ${\mathrm{A}}_2$ and ${\mathrm{B}}_2$ are related to the next two ACs. Although it appears that only one AC between states 3 and 5 is effective, state 4 acts as an intermediate state for this three-state mixture.