Intervalley scattering of electrons by short-wave phonons in (GaAs)8(AlAs)8(001) superlattice

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Highlights

  • Phonon-assisted intervalley electronic transitions in a superlattice are studied.

  • Quantum-size effects in the electron-phonon scattering are analyzed.

  • Distinction between scattering in the superlattices and crystals is interpreted.

Abstract

Intervalley transitions induced by short-wavelength phonons in the conduction band of a superlattice (GaAs)8(AlAs)8(001) are investigated on the basis of the pseudopotential method and in the phenomenological model of interatomic forces. The main attention in the study centers around the transitions associated with the vibrations confined inside the layers. It is shown that the deformation potentials for the majority of intervalley transitions in a superlattice exceed the potentials of corresponding transitions in the binary crystals because of the localization of atomic displacements and wavefunctions of electrons inside the same layer of the superlattice.

The bottom of the conduction band in (GaAs)8(AlAs)8(001) superlattice corresponds to the states Γ1(1), Z3, M1, M4 originating from sphalerite's X¯ valleys localized in AlAs layers. Transitions between them are the most intense ones and they are caused by optical vibrations of Al atoms. “Semi-interface” vibrations being mainly localized in the one side of the GaAs layer are involved in the Γ1(2)X1, Γ1(2)R1 and X1Z1 transitions which are analogs of Γ¯L¯ transitions in binaries. The transitions Γ1(2)M1 and Γ1(2)M4 are governed by smooth parts of wave-functions and pseudopotentials. As a consequence their intensities are comparable with those of Γ¯X¯ sphalerite transitions in spite that these states are localized in the different layers of the superlattice.

Introduction

Due to the multivalley nature of a conduction band, the mixed nature of oscillations of atoms and a structural perfection of (GaAs)m(AlAs)n(001) superlattices they are of great interest for the theoretical research and a practical application of effects caused by the scattering of electrons at the intervalley short-wavelength phonons.

It is known that this scattering leads to a negative differential conductivity, leakage of currents in the cascade lasers, a reduction of electron mobility in the transistor channels, peculiarities in the photoluminescence spectra of hot electrons, the intersubband absorption in lasers at coupled quantum wells, etc [1], [2]. Due to the effects of a size quantization the electron-phonon interaction in (GaAs)m(AlAs)n(001) superlattices has a multi-channel character, so it is described by a great number of deformation potentials (DP) which makes it difficult to identify them from the analysis of experimental data and requires the use of theoretical methods. For the study of physical phenomena caused by the scattering of electrons by phonons the continuum models and the method of envelope functions are widely used [3] successively describing the long-wave vibrations in the bulk materials. However, in a case of the inter-valley scattering of electrons in superlattices the applicability of these approximations is violated and one must use the fundamental methods that take into account the atomistic structure of the materials. That is why in our papers [4], [5], [6], [7], [8] the intervalley scattering in (GaAs)m(AlAs)n(001) superlattices was studied on the basis of the pseudopotential method and the phenomenological model of binding forces. It allowed to determine DPs for the relevant electron transitions and to explain their variations in the series of superlattices from the analysis of quantum-size effects. In the present paper we apply this method to the investigation of the intervalley scattering of electrons by short-wavelength phonons in (GaAs)8(AlAs)8(001) superlattice which has all kinds of vibrational states typical in GaAs/AlAs nanostructures (the interface and semi-interface vibrations and vibrations locked inside the layers). We analyzed the influence of a quantum localization on the intensity of intervalley transitions and established the differences of the deformation potentials in the superlattice and binary crystals for the related scattering channels.

Section snippets

Method of calculation

The scattering of electrons by phonons was studied in the rigid ions' approximation, i.e. pseudopotentials of atoms are considered to shift rigidly with the atomic displacements from their equilibrium positions. The intensity of inter-valley electron scattering from the initial state nk to the final state n'k' on the (α,q)th phonon is determined by the deformation potential:Dnkn'k'α=|i,s(Mms)12(eis(αq)dis(nk,n'k'))|

Here α means a label of the phonon branch; q is the phonon wave vector; n is a

Deformation potentials in binary crystals

The features of electron bands and phonon spectra and the nature of inter-valley scattering of electrons by phonons in binary crystals are well established. There are three lowest valleys Γ¯, X¯, L¯ in the conduction band of GaAs, AlAs. Transitions between them define the majority of the kinetic phenomena. The discontinuities of bands calculated with pseudopotentials of [9] are as follows: ΔEΓ¯=EΓ¯(AlAs)EΓ¯(GaAs)=1.004eV;ΔEL¯=EL¯(AlAs)EL¯(GaAs)=0.556eV;ΔEX¯=EX¯(AlAs)EX¯(GaAs)=0.273eV.

So GaAs

Band spectrum of the superlattice

The folding of a band spectra from binary crystals into the tetragonal Brillouin zone and a subsequent mixing of sphalerite states by the non-cubic potential component of (GaAs)8(AlAs)8(001) superlattice leads to a complex, multi-valley structure of its conduction band (Fig. 3a, Table 3). There are several states: Γ1(1), Γ1(2), Γ3, M1, M4, R1, R3, X1, X3, Z1 with the close energy values.

The states in a central valley Γ1(1), Γ1(3), Γ3 correspond to three levels of the size quantization in the

Phonon spectrum of the superlattice

Calculated phonon spectrum ναq of (GaAs)8(AlAs)8(001) superlattice is shown in Fig. 3b. The low-frequency branches are associated with the vibrations of heavy As atoms. Optical branches with high 10÷ 12THz frequencies are associated with vibrations of light Al atoms and have their origin in the sphalerite states on lines Δ¯ and Λ¯. The mass-defect approximation leads to the coincidence of several frequencies that originate from the same sphalerite states. Some of the polarization vector of

Deformation potentials of the superlattice

The scheme of electronic transitions between the lowest states of the conduction band in (GaAs) superlattice caused by short-wavelength lattice vibrations is shown in Fig. 8. Corresponding deformation potentials are given in Table 4.

The analysis of partial contributions from different phonons shows that transitions ΓM and XX˜ are caused mainly by vibrations of atoms which are confined inside the layers. The maximum intensity have Γ1(1)M1 and Γ1(1)M4 tansitions, which are analogues of

Conclusion

Our study identifies the deformation potentials for the actual electronic transitions in the conduction band of (GaAs)8(AlAs)8(001) superlattice. From the analysis of wave functions and pseudopotential gradients a ratio of the deformation potentials for different inter-valley scattering channels is explained. It is shown that the intensity of transitions in the superlattice grows as compared with the related transitions in the binary crystals when both electron and phonon states are localized

Acknowledgements

The work was supported by RFBR, research project No.15-02-00293. V.G.T. acknowledges the support of Ministry of Education and Science of Russian Federation base part 101. The calculations have been performed using the SKIF CYBERIA cluster of Tomsk State University.

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