Abstract
The Rashba spinorbit splitting of a hydrogenic donor impurity in GaAs/GaAlAs quantum wells is investigated theoretically in the framework of effectivemass envelope function theory. The Rashba effect near the interface between GaAs and GaAlAs is assumed to be a linear relation with the distance from the quantum well side. We find that the splitting energy of the excited state is larger and less dependent on the position of the impurity than that of the ground state. Our results are useful for the application of Rashba spinorbit coupling to photoelectric devices.
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In the framework of effectivemass envelope–function theory, excluding the relativity effect, the electronic states have been studied for a hydrogenic donor impurity in quantum wells (QWs) [15] and its important application in the photoelectric devices. The relativity effect introduces evidence of Rashba effects in the semiconductor materials. In recent years, spindependent phenomena was also proposed using spin fieldeffect transistor based on the fact that spin precession can be controlled by an external field due to the spinorbit interaction [6]. Gvozdić et al. studied efficient switching of Rashba spin splitting in wide modulationdoped quantum wells [7]. They demonstrated that the size of the electricfield induced Rashba spin splitting in an 80nm wide modulationdoped InGaSb QW depends strongly on the spatial variation of the electric field.
The interplay between Rashba, Dresselhaus, and Zeeman interactions in a QW submitted to an external magnetic field was studied by means of an accurate analytical solution of the Hamiltonian [8]. Hashimzade et al. presented a theoretical study of the electronic structure of a CdMnTe quantum dot with Rashba spinorbit coupling in the presence of a magnetic field. The multiband theory was used to describe electrons in Rashba spinorbit coupling regimes and an external magnetic field[9]. However, most importantly, many researchers anticipate that Rashba effects can introduce spin splitting of electron energy levels in zero magnetic field.
The Rashba term is caused by structural asymmetry, which is a position dependent quantity in a QW, significant near the interface but quickly falling to zero away from the interface. The electron position in the well can be modulated by the impurity center and will sensitively change the Rashba spinorbit splitting energy.
In this letter, we will introduce a linear Rashba spinorbit coupling module dependent on the electron position in a QW and study the change in spinorbit splitting energy as the impurity position and the QW width change.
For a hydrogenic donor impurity located atr_{0}= (0, 0,z_{0}) in a GaAs/GaAlAs QW, the electron envelope function equation in the framework of the effectivemass approximation is
where and The third item in Eq. 1 is the contribution of the Rashba spinorbit effect to the single electron Hamiltonian. , , and , respectively, are the Rashba parameter, the Pauli matrices, and electron momentum operator, respectively. The subscriptn = 0, 1, 2, corresponds to the ground, first excited, second, excited states. The units for length and energy are in terms of the effective Bohr radius and the effective Rydberg constant , where and are the effective mass and dielectric constant of an electron.
We adopt the square potential energy model as
where W and V_{0} are the width of the QW and the band offset of the electron, respectively.
We introduce a linear Rashba spinorbit effect model
where α_{0} is the maximum value of the Rashba spinorbit effect at the side of the QW. The Rashba parameter is a function of z and is dependent on the size of the QW, which is demonstrated in Fig. 1.
Figure 1. The Rashba parameter as a function ofz. The horizontal and vertical dashed lines indicate the value of α_{0}and the borderline of the QW, respectively
In the following sections, using the normalized plane–wave expansion method [1012], we give numerical results for the Rashba spinorbit splitting energy of a hydrogenic donor impurity in a QW. We take the effective mass parameters of [13] and the Rashba parameter [14].
The spinorbit splitting energy Γ is defined by the difference between the two splitting energy levels. Figure 2shows the change in spinorbit splitting energy Γ as the GaAs QW width increases for a hydrogenic donor impurity at the QW center under the linear Rashba model along thezdirection. The Rashba spinorbit splitting energy is very small for the narrow QWs. As the well width increases from zero, the splitting energy of the ground state increases first, then reaches a maximum value before decreasing monotonously. This is because the wave function of the ground state is localized at the QW center for the impurity at the QW center and the Rashba effects is very small at the QW center for the wide QWs. However, for the excited states, the wave functions are spread in space and the Rashba effects can affect the excited states for the wide QWs. So the spinorbit splitting energies of the excited states decrease more slowly than that of the ground state.
Figure 2. The change in spinorbit splitting energy Γ as the GaAs QW width increases when the hydrogenic donor impurity is at the QW center under the linear Rashba model along thezdirection. The low to top lines correspond to the ground state (n = 0), the first excited state (n = 1), the second excited state (n = 2), and the third excited states (n = 3), respectively
The spinorbit splitting energy Γ of the ground state decreases when the impurity moves to the QW side and the impurity positions in the QW do not sensitively affect the spinorbit splitting energy Γ of the excited states. This is because the ground state is more localizing than the excited states in QWs. These changing trends are found in Fig. 3.
Figure 3. The change in spinorbit splitting energy Γ as the position of the impurity under the linear Rashba model along thezdirection changes, when the GaAs QW widthWequals 5 nm. The low to top lines correspond to the ground state (n = 0), the first excited state (n = 1), the second excited state (n = 2), and the third excited states (n = 3), respectively
In summary, we proposed a linear Rashba model along thezdirection and calculated the splitting energy of a hydrogenic donor impurity in a GaAs/GaAlAs QW. We found that the Rashba spinorbit splitting energy of the ground state is more sensitively dependent on the QW width and the center position of the hydrogen donor impurity than those of the excited states.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 60776061, and 60521001.
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