We report on c-plane InGaN/GaN single quantum well (QW) light-emitting diodes (LEDs) of different well widths (3 or 9 nm) with and without doped barriers. QW barriers were doped with the aim of reducing the internal electric field (FQW) in the QW to increase the electron-hole overlap, therefore increasing the recombination rates and resulting in the reduction of the efficiency droop. We, indeed, observed, through biased photocurrent spectroscopy, a reduction in FQW with doped barriers, with FQW being in the same direction of the p-n junction field at zero bias as opposed to the junction field for LEDs without doped barriers. Even with the improvement in the ground state wavefunction overlap, the ground state transition rate remains low for thick QWs. Transitions through excited states were observed for both thick QW LEDs with and without doped barriers. The thick QW LED without doped barriers displayed low external quantum efficiency (EQE), likely as a result of the carrier overflow due to the poor confinement of carriers in the excited states. On the other hand, for LEDs with doped barriers, the flatter band in the QW resulting from the lower FQW reduces the energy separation between the eigenstates, leading to better confinement of carriers in the excited states. With doped barriers, we demonstrated a low efficiency droop 9-nm-thick single QW LED with a peak EQE of 42\% at 40 A/cm2 and an EQE of 36\% at 400 A/cm2.