Cite this Article
Yin Ji-Wen, Li Wei-Ping, Li Hong-Juan, Yu Yi-Fu. Direct spin–phonon coupling of spin-flip relaxation in quantum dots. Chinese Physics B, 2017, 26(1): 017201  
Permissions
Direct spin–phonon coupling of spin-flip relaxation in quantum dots
Yin Ji-Wen1, †, Li Wei-Ping2, Li Hong-Juan1, Yu Yi-Fu1
Department of Physics and Electronic Informational Engineering, Chifeng University, Chifeng 024000, China
Department of Basic Courses, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China

 

† Corresponding author. E-mail: ji-wenyin@163.com

Abstract

Within the frame of the Pavlov–Firsov spin–phonon coupling model, we study the spin-flip assisted by the acoustical phonon scattering between the first-excited state and the ground state in quantum dots. We analyze the behaviors of the spin relaxation rates as a function of an external magnetic field and lateral radius of quantum dot. The different trends of the relaxation rates depending on the magnetic field and lateral radius are obtained, which may serve as a channel to distinguish the relaxation processes and thus control the spin state effectively.

PACS: 73.21.La;72.10.Di;72.25.Rb
Keyword:spin relaxation;spin–phonon coupling;quantum dot
1. Introduction

The electron spin relaxation in quantum dots (QDs) is of central importance for the applications on the spin polarization and quantum computation devices.[1,2] In the semiconductor QD, extensive theoretical works have researched the main phonon mediated spin-flip relaxation mechanism, such as the admixture processes due to spin–orbit coupling,[2–5] phonon coupling due to interface motion (ripple mechanism),[6] the direct spin–phonon coupling mechanism,[10] the fluctuating magnetic field due to the fluctuating electron density or the modulation of the hyperfine coupling with nuclei by lattice vibration,[2] and so on. Among them, what remains in discussion is which mechanism is the dominant one and needs more endeavor in detail. The direct spin–phonon coupling mechanism is an intrinsic one that cannot be eliminated in principle, so it can provide the most fundamental upper bound on the lifetime of the electron spin state in QDs when the spin–orbit coupling does not produce a significant admixture between spin states. This situation can be reached in symmetrical samples in the absence of an electric field and in materials with negligible Dresselhaus contribution. In particular, recent experiments[7–9,11,12] suggest that the spin–phonon coupling dominates the spin relaxation in GaAs QDs in the strong magnetic field and some nonpolar or monolayer materials, such as silicene and graphene.[13,14]

In the present paper, we study the spin relaxation between the first-excited and ground states based on the spin–phonon interaction model proposed by Pavlov and Firsov,[15,16] which describes the transitions with spin reversal of the conduction band electrons due to the scattering by the longitudinal lattice vibrations, and has the advantage of being adaptable to the study of the other scattering mechanism by various phonon modes. Romano et al.[17] have used this Hamiltonian to investigate the spin-flip process of a single electron between the Zeeman sublevels in the same orbital state by considering the deformation potential mechanism as the dominant electron–phonon coupling. They give the regions in which the spin relaxation rates can be practically suppressed by choosing the suitable magnetic field and lateral QD size. We investigate the spin relaxation rates for the and processes as a function of the magnetic field and lateral radius. We find that the magnetic field dependence of the relaxation rates for the two cases are very different. This may serve as a channel to distinguish the relaxation processes in experiments, and thus control the spin state effectively. We hope that our results can stimulate and highlight related experimental work to identify this conclusion.

2. Theoretical model

We consider a single electron quantum dot with different parabolic confinement potentials in lateral and vertical directions , denotes the polar coordinate. The uniform magnetic field is applied in the vertical direction. The Hamiltonian is given by

(1)
in which with stands for the electron momentum operator, is the electron effective mass, and is the Zeeman energy with σ representing the Pauli matrices.

By solving

(2)
one can determine the eigenenergy and the eigenfunction of the Hamiltonian H 0 as follows:
(3)
(4)
where and are quantum numbers, and , is scaled radius with , is the Zeeman splitting energy, refers to the spin polarization along the vertical direction, and is the generalized Laguerre polynomial.

The spin–orbit interaction mediates a coupling between the spin and the electron orbital bath. Based on the effective mass theory, the spin–phonon interaction Hamitonian in the presence of an external magnetic field, which describes the transitions with spin reversal of the conduction band electrons due to scattering with lattice vibrations, can be expressed as[17,18]

(5)
where is the annihilation (creation) operator of the acoustic phonon with wave vector , where and are unitary vectors along the x and y axes; is the unit vector in the direction of the phonon polarization; ρ is the mass density; and V is the system volume. We assume the linear dispersion , ν being the average sound velocity. The form factor[19,20] equals unity for . The electron phonon scattering due to the deformation potential coupling described by the parameter is defined as
(6)
where denotes the deformation potential constant, is the semiconductor band gap, is the spin–orbit splitting, and .[17,18]

Based on the Fermi golden rule, the spin-flip relaxation rate assisted by one phonon scattering between an initial electron state and a final state is determined by

(7)
We mainly consider the spin-flip processes between the first-excited and ground states. After complexly calculations, the expression of the spin relaxation rate can be written as
(8)
The parameter and for the and processes, respectively. Here, ( ) is the zero (first) order Bessel function, is the lateral radius, denotes the height of QD in the vertical direction. We assume that the radius is much longer than the height, that is . The dynamics along the vertical direction is restricted to the lowest subband, corresponding to the ground-state wave function .[18–20] Throughout the paper, the ratio is fixed.

3. Numerical results

We only consider the contribution of the electron-longitudinal acoustic (LA) phonon coupling via the deformation potential mechanism to the relaxation rates, which plays the predominant role when the energy separation between the orbital states is in a several meV regime. The calculation of the relaxation rates is performed at temperature K. The temperature dependence of the relaxation rate for one-phonon emission is determined from , where n q is the Bose–Einstein distribution for the LA phonon and W 0 is the rate at T = 0 K. In the present paper, we consider the temperature regime K. Therefore, the approximation and is satisfied. In numerical calculation, we adopt the parameters in GaAs material:[21] , /m3, m/s, , , , and .

Figure 1 plots the magnetic field dependence of the spin relaxation rate between the ground state and first-excited state for different radii of QD. In the process of (Fig. 1(a)), we can see that the rates decrease until a minimum and then increase. Hence, the spin relaxations show the cusplike structures, which means that the maximum of the relaxation time can be obtained by modulating the magnetic field. This fact is very beneficial to controlling the coherence of the spin qubit in quantum information. Moreover, the minimums of the relaxation rates are moved to the small value of the magnetic field and the second turn points appear with the increase of the QD radius, which are caused by the interplay effects between the variation of the QD radius and magnetic field confinement. We emphasize that the cusplike structure of the relaxation rates in the present model are contrary to that in Refs. [19], [20], and [22] based on the spin admixture mechanism due to the Dresselhaus and Rashba spin–orbital coupling. In fact, the same structures are obtained and show the oscillatory behavior with magnetic field for the process between the Zeeman sublevels of the ground orbital state in InSb QD in Ref. [12]. In the present processes, we considered that the oscillatory behavior does not appear. However, the cusplike structures disappear in the process (Fig. 1(b)). In Fig. 1(b), one can see that not only is the variational magnitude of the relaxation rates smaller than that of the process shown in Fig. 1(a), but also the magnetic field and radius dependence of the rates differ from that in Fig. 1(a). From Eq. (9), we know that the factors X for two relaxation processes are different. This factor is related to the magnetic field and radius of QD directly, and thus results in the discrepancy between them. Several experiments have verified that the spin–phonon coupling mechanism dominates the spin relaxation in GaAs QDs under a strong magnetic field.[11,12] Therefore, this discrepancy may be useful to distinguishing the spin states between the Zeeman levels of the ground and first-excited states, and to control and readout the spin state in experiments effectively.

Fig. 1. (color online) The spin relaxation rate as a function of the magnetic field for different radii of QD, (a) for the process and (b) for the process , respectively.

The spin-flip relaxation rates as functions of the magnetic field and radius are contour plotted in Figs. 2(a) and 2(b) for the process and , respectively. From them, we can see that the relaxation rates can be varied in several orders of magnitude by modulating the radius of QD and strength of external magnetic field, so one can obtain the desirable relaxation time for the coherent states of spin qubit in quantum computation and information. From the comparison between two cases, we also clearly see that the variational magnitude of the relaxation rate for the process in Fig. 2(a) is larger than that in Fig. 2(b). In addition, the cusplike structures of the process (Fig. 2(a)) show the slight oscillatory behaviors arising from the interplays between the magnetic field and radius confinement, not for the process in Fig. 2(b).

Fig. 2. (color online) Contour plot of the spin relaxation rates as a function of magnetic field and radius for processes (a) and (b) , respectively.
4. Summary

In conclusion, we study the spin relaxation rates for the transitions between Zeeman sublevels of the first-excited and ground state in quantum dots based on the Pavlov–Firsov spin–phonon coupling mechanism. We find that the magnetic field and radius dependence of the relaxation rates for the two cases are very different, which may serve as an effective channel to distinguish the relaxation processes in experiment, and thus control the spin coherent state for the application of spin qubit.

Reference
1 Hanson R Kouwenhoven L P Petta J R Tarucha S Vandersypen L M K 2007 Rev. Mod. Phys. 79 1217
2 Khaetskii A V Nazarov Y V 2000 Phys. Rev. B 61 12639
3 Khaetskii A V Nazarov Y V 2001 Phys. Rev. B 64 125316
4 Wang Q W Hong L 2012 Acta Phys. Sin. 61 017107 (in Chinese)
5 Kang N L Choib S D 2014 Chin. Phys. B 23 087104
6 Woods L M Reinecke T L Lyanda-Geller Y 2002 Phys. Rev. B 66 161318
7 Hanson R Witkamp B Vandersypen L M K Willems L H Beveren V Elzerman J M Kouwenhoven L P 2003 Phys. Rev. Lett. 91 196802
8 Heiss D Schaeck S Huebl H Bichler M Abstreiter G Finley J J 2007 Phys. Rev. B 76 241306
9 Jan F Loss D 2009 Science 324 1277
10 Calero C Chudnovsky E M Garanin D A 2005 Phys. Rev. Lett. 95 166603
11 Elzerman J M Hanson R Willems L H Witkamp B Vandersypen L M K Kouwenhoven L P 2004 Nature 430 431
12 Amasha S MacLean K Iuliana P R Zumbuhl D M Kastner M A Hanson M P Gossard A C 2008 Phys. Rev. Lett. 100 046803
13 Struck P R Burkard G 2010 Phys. Rev. B 82 125401
14 Droth M Burkard G 2013 Phys. Rev. B 87 205432
15 Pavlov S T Firsov V A 1966 Sov. Phys. Solid State 7 2131
16 Pavlov S T Firsov V A 1967 Sov. Phys. Solid State 9 1394
17 Romano C L Marques G E Sanz L Alcalde A M 2008 Phys. Rev. B 77 033301
18 Wang Z W Li S S 2012 Solid. State. Commun. 152 1098
19 Bulaev D V Loss D 2005 Phys. Rev. Lett. 95 076805
20 Bulaev D V Loss D 2005 Phys. Rev. B 71 205324
21 Madelung O 1982 Semiconductors: Physics of Group IV Elements and III-V Compounds Berlin Springer 172 p
22 Prabhakar S Melnik R Bonilla L L 2013 Phys. Rev. Lett. 87 235202