Exchange interaction effects in the thermodynamic properties of quantum dots

We study electron-electron interaction effects in the thermodynamic properties of quantum-dot systems. We obtain the direct and exchange contributions to the specific heat $C_v$ in the self-consistent Hartree-Fock approximation at finite temperatures. An exchange-induced phase transition is observed and the transition temperature is shown to be inversely proportional to the size of the system. The exchange contribution to $C_v$ dominates over the direct and kinetic contributions in the intermediate regime of interaction strength $(r_s\sim 1)$. Furthermore, the electron-electron interaction modifies both the amplitude and the period of magnetic field-induced oscillations in $C_v$.

Figure 1

(Color online) (a) Ground-state energy $E_{\mathrm{HF}}$ as function of temperature for $N=10$ electrons and $r_s=0.89$. The kinetic (b), exchange (c), and direct (d) contributions to $E_{\mathrm{HF}}$ are also plotted.

Figure 2

(Color online) Specific heat as function of temperature for the noninteracting (solid line) and interacting cases. In the latter, the interaction strength values are $r_s=1.07$ (up triangles), 2.14 (circles), and 3.30 (diamonds). The insets show typical charge distributions in the dot before and after the transition.

Figure 3

Reduced transition temperature $T^{*}∕T_L$ as function of $r_s^{-2}$ (see text for details). Inset, $T^{*}∕T_L$ increases linearly with the interaction strength $L∕a_B^{*}$.

Figure 4

(Color online) Specific heat as function of $r_s$ (circles). Also plotted are the kinetic (squares), direct (up triangles), and exchange (down triangles) contributions. Inset, $C_v$ as function of magnetic flux $\Phi ∕{\Phi }_0$ for the noninteracting (solid line) and interacting cases with $r_s=0.53$ (triangles), 0.89 (half-filled circles), and 1.78 (filled circles). Curves are off-set for clarity.