Perfect Cooper pair splitting is proposed, based on crossed Andreev reflection (CAR) in a p-type semiconductor-superconductor-n-type semiconductor (pSn) junction. The ideal splitting is caused by the energy filtering that is enforced by the band structure of the electrodes. The pSn junction is modeled by the Bogoliubov-de Gennes equations and an extension of the Blonder-Tinkham-Klapwijk theory beyond the Andreev approximation. Despite a large momentum mismatch, the CAR current is predicted to be large. The proposed straightforward experimental design and the 100% degree of pureness of the nonlocal current open the way to pSn structures as high quality sources of entanglement. Spatially separated entangled electron pairs arise in hybrid normal-metal-superconductor structures. Andreev reflection (AR) is the conversion of an electron into a hole at the interface between a normal-metal or semiconductor and a superconductor [1]. This makes CAR a promising source of locally separated entangled electrons as building block for solid state Bell-inequality experiments, quantum computation and quantum teleportation [4,5].The Bell inequality can only be violated when the CAR fraction is larger than 1= ffiffiffi 2 p of the total current [6]. Despite the intensive investigation, the CAR fraction is usually small due to competing processes [7]. Aside from local AR, tunneling of a particle from one lead to another can occur. Elastic cotunneling (EC) is to lowest order in tunneling amplitude equal in magnitude and opposite in sign to CAR, resulting in a vanishing nonlocal conductance [8]. Including higher order terms, which become important in more transparent junctions, unfortunately provides EC to be the dominant process [9].Several proposals have been put forward to enhance the CAR current. Using ferromagnetic half-metals (F) as leads can result in dominant CAR in an antiparallel magnetization alignment [3], though spin entanglement is then questionable. Pure nonlocal Cooper pair splitting is predicted in a superconductor-topological insulator structure [10], but the fabrication will be challenging. Both the electromagnetic environment [11] and a change in the density of states (DOS) of a superconductor due to an ac bias [12] can result in dominant CAR, though the influences are expected to be small. A larger effect is predicted by Cayssol [13] in an n-type graphene superconductor-p-type graphene junction. However, full cancellation of EC and AR is only at precise biasing to the Dirac point and at a small range of the energy spectrum so that the current is not completely carried by CAR. Optimization of CAR may also be realized by using the Coulomb interaction [4]. Recent experiments indicated great potential of using the energy to discriminate CAR [15], although the splitting efficiency is yet small.Here, we propose a strategy for ideal 100% nonlocal Cooper pair splitting, with no contributions from AR and EC, in a relatively straightforward device. Making use of the energy difference between the incoming electron and the And...