Spin relaxation in semiconductor quantum dots

Quantum dots ~QD’s ! are small conductive regions in semiconductor structures that contain a tunable number of carriers. The shape and size of quantum dots can be controlled by the gate voltage. The localized electronic states in QD’s can be significantly modified by a magnetic field. All this provides a valuable opportunity to study the properties of the electron quantum states in detail and manipulate the electrons in these artificial atoms in a controllable way ~see Refs. 1 and 2 for review! The spin states in quantum dots are considered to be promising for physical realization of the quantum computation algorithm. 3 Quantum computation requires coherent coupling between the dots, the coherence to be preserved on sufficiently long time scales. That makes it relevant to provide a complete theoretical estimation of the typical spin dephasing time of the electron in the QD. Transport experiments with QD’s have revealed that the current through a quantum dot can be influenced by the spin effects. 4 It opens up a possibility to estimate spin relaxation rates by means of transport measurement. 5 The origin of this effect is that the spin-flip process can provide a bottleneck for the energy relaxation in the dot, i.e., for transitions between the excited and ground states. Indeed, in the absence of the spin flip the total spin of the dot is a good quantum number and no transition is allowed between the states of different total spins. To illustrate, let us consider a QD with two electrons which can be placed in two levels. The ground state corresponds to two electrons in the lowest level having opposite spins ( S 50). One of the excited states corresponds to two electrons in different levels having the same spin direction ( S51). Due to the Pauli principle, the electron in the upper level cannot get to the lower level without changing its spin. Therefore, the relaxation to the ground state of the dot has to be accompanied by a spin flip. In contrast to the situation in two-dimensional ~2D! electron gas the electrons confined in the dot experience no electron-electron scattering ~see Ref. 6! The only source of dissipation is the interaction with phonons. Moreover, although the electron-electron interaction is quite important in determining the energies of the states and the number of electrons in the dot, it is less important for the structure of the wave functions. To calculate the matrix elements, we approximate the many-electron wave functions by the Slater determinants. In this way, we can treat the spin-flip processes within the one-electron approach. The electron-electron interaction can change only the numerical factors in our results. Since most controllable QD’s are made on the basis of 2D electron gas GaAs heterostructures with the @100# growth direction, we concentrate on the spin-flip mechanisms that are relevant for GaAs, and for this confinement direction. Such mechanisms are very specific for AIIIBV compounds. The unit cell has no inversion symmetry, which gives rise to a strong spin-orbit splitting in the electron spectrum. The splitting is known 7,8 to be the main source of the spin-flip both in the 3D and 2D cases. Besides, the piezoelectric effect provides a strong coupling of electrons to the acoustic phonons. Such coupling may be important for the inelastic relaxation in the GaAs crystal both with and without a spinflip.