LaAlO$_3$ capping layer is thin, the energy gap of the QW is the energy difference between the Ti-$d$ states of the embedded TiO$_2$ layer and the O-$p$ states of the SrTiO$_3$ substrate, which remains at a constant energy. The more relevant quantity is the energy difference between the Ti-$d$ states of the embedded TiO$_2$ layer and the O-$p$ states on the surface. We refer to this quantity as the ‘Ti-O energy difference’ in the following. Increasing the thickness of the [LaAlO$_3$ ]{}capping layer reduces the Ti-O energy difference until it disappears and electron transfer occurs. The minimum thickness of the LaAlO$_3$ capping layer that enables this electron transfer is defined as the critical separation of the QW systems.

Table \[tab:gap\] shows the Ti-O energy difference and the critical separation of the QW systems. The critical separation of the QW coincides with that of the $n$-type interface, reflecting the fact that the band gap of SrTiO$_3$ is mainly determined by the Ti-$d$ states and the O-$p$ states. Therefore, the absence of Sr atoms does not change the critical separation. The local density of states at the Fermi level of the QW are shown in Fig. \[fig:ldosf\]b. Unlike the $n$-type interface, at 