Exclusion principle and screening of excitons in GaNÕAl x Ga 1 Àx N quantum wells Pierre Bigenwald, 1 Alexey Kavokin, 2 Bernard Gil, 3 and Pierre Lefebvre 3 1 De ´partement de Physique, UFR Sciences Universite ´ d’Avignon et des Pays de Vaucluse, 33 Rue Louis Pasteur, 84000 Avignon, France 2 LASMEA Universite ´ Blaise Pascal Clermont II, CNRS, 24 Avenue des Landais, 63177 Aubiere Cedex, France 3 CNRS GES, Universite ´ de Montpellier II, Place Eugene Bataillon CC 074 34095, Montpellier Cedex 5, France ~Received 19 May 2000; revised manuscript received 2 October 2000; published 2 January 2001! We model the variation of the exciton binding energy and of the oscillator strength versus temperature in strained GaN/Al x Ga 12x N quantum wells by using a self-consistent variational procedure. In addition, this method is extended to the case of high photoinjection conditions. We thus can properly account for the effect of a dense electron-hole plasma on the excitonic wave function, and we can quantitatively address the exciton bleaching phenomenon via quantum exclusion effects. A surprising behavior has been found: the robustness of the exciton to screening by the dense plasma increases with increasing temperature. In other words, the pumping intensity necessary to tear apart electrons and holes increases with increasing temperature. This is quantitatively interpreted in terms of the quantum exclusion effect as a straightforward result of the Pauli principle and of the fundamental prescriptions of quantum mechanics. The limitations imposed by this effect on the excitonic wave functions are relaxed with increasing temperature. DOI: 10.1103/PhysRevB.63.035315 PACS number~s!: 78.66.Fd, 71.35.Lk, 73.90.1f It is well known that excitons in quantum wells ~QW’s! containing a free-electron gas or an electron-hole plasma are subject to the quantum exclusion effect. 1 This effect arises from the Fermi principle and imposes strict conditions on the shape of the wave function of the relative motion of the electron and hole that form the exciton. Actually, the wave function of an electron bound to a hole can be represented as a linear combination of the eigenfunctions of all free- electron states of the system. This basis is substantially reduced if some of these states are already occupied by free electrons. This statement, of course, also holds for the holes. When some of the states of conduction and valence bands ~i.e., those having a wave vec- tor k close to zero! are occupied by free carriers, the eigen- functions of these states cannot contribute to the excitonic wave function. Then the construction of the exciton wave function is restricted to eigenstates having a substantial k, or, said differently, to states having a kinetic energy larger than the states at the zone center. This general argument is also valid for free-exciton states in QW’s or quantum wires, the k distributions of the eigenvalues and eigenstates just obey dif- ferent rules. From such an increase of the kinetic energy of the carriers, a decrease of the exciton binding energy results. 2,3 In GaN/Al x Ga 1 2x N quantum wells, internal electric fields have been predicted, 4 having both piezoelectric and sponta- neous polarization origins, and they have been shown by multiple experiments. 5 These fields are really huge, with in- tensities on the order of 1 MV/cm. They actually destroy excitons, which represents a major problem for the fabrica- tion of exciton-based optical devices emitting blue light. Photoinjected carriers can be used to compensate for the po- larization field and the quantum confined Stark effect ~QCSE!. 3,6 In such nitride systems, one can screen the polar- ization fields by injecting carriers, but one can also revive the excitons. This is not so simple since many phenomena are involved in the process: simultaneously with the reduction of the QCSE and the exciton revival, the intrinsic electron-hole plasma effect on the excitons ~screening and exclusion ef- fects! destabilizes them. The interplay of these two factors yields an interesting nonmonotonic dependence of the exci- ton binding energy on the pumping intensity that we inves- tigated in previous work, at T 50. 3 How can the suppression of excitons by the electron-hole plasma be reduced? This is a key question for the excitonic physics of nitride-based heterostructures. Here, we report a theoretical discovery that partly answers this question and has a quite general interest, in our opinion. We next address the theoretical ingredients that permit us to resolve this prob- lem. Then we discuss the results of the calculations. In this computation, we have combined a self-consistent procedure based on the adiabatic separation of the in-plane and normal-to-the-plane motions of carriers 7 with the formal- ism for variational solution of the excitonic problem in re- ciprocal space developed by Pikus. 2 Our method is described in detail in Ref. 3. We briefly list here the main equations used. For each carrier in the exciton, we first solve self- consistently Schro ¨ dinger and Poisson equation. z is the growth axis of the structure. We write, for each particle i ~electron, hole!, the Schro ¨ - dinger equation H 2 \ 2 2 ] ] z S 1 m i * ] ] z D 1V i ~ z ! 1q i @ F i ~ z ! 1F i ~ z !# z J x i ~ z ! 5E i x i ~ z ! , ~1! where x i ( z ) is the carrier eigenfunction associated with the eigenenergy E i , m i * is the effective mass, V i ( z ) is the po- tential, F i ( z ) and F i ( z ) are the electric fields, F i being static and F i being induced by the free carriers, and q i is the charge of the carriers. Taking into account the filling of con- duction and valence bands, we write F ( z ) in the following manner: 8 PHYSICAL REVIEW B, VOLUME 63, 035315 0163-1829/2001/63~3!/035315~4!/$15.00 ©2001 The American Physical Society 63 035315-1