Spin relaxation in a GaAs quantum dot embedded inside a suspended phonon cavity
Y. Y. Liao,1 Y. N. Chen,1D. S. Chuu,1,*and T. Brandes21Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan
2School of Physics and Astronomy, The University of Manchester, P.O. Box 88, Manchester M60 1QD, United Kingdom 共Received 23 September 2005; revised manuscript received 27 December 2005; published 10 February 2006兲
The phonon-induced spin relaxation in a two-dimensional quantum dot embedded inside a semiconductor slab is investigated theoretically. An enhanced relaxation rate is found due to the phonon van Hove singulari-ties. Oppositely, a vanishing deformation potential may also result in a suppression of the spin relaxation rate. For larger quantum dots, the interplay between the spin orbit interaction and Zeeman levels causes the sup-pression of the relaxation at several points. Furthermore, a crossover from confined to bulklike systems is obtained by varying the width of the slab.
DOI:10.1103/PhysRevB.73.085310 PACS number共s兲: 73.21.La, 71.70.Ej, 63.20.Dj, 72.25.Rb
Spin properties in nanostructures have become a field of intense research ranging from spin field-effect transistor,1 spin-polarized p-n junctions,2 up to quantum spin compu-ters.3The quantum dot共QD兲 may be a good choice for quan-tum electronics due to its zero dimensionality, quantized en-ergy levels, and long coherence times of spin states.6,7 For example, the spin of an electron confined to a QD can form a qubit.4,5However, some scattering processes will cause the change of the spin states. One important process is related to the phonon-induced spin-flip resulting from the spin-orbit interaction. This affects the time of spin purity in the QD. In order to keep the information unchanged, a long relaxation time is required.
In general, the spin-orbit共SO兲 coupling, which is one of the main causes of spin relaxation, is a relevant intrinsic interaction in nonmagnetic semiconductors. It is known that there are two different types of spin-orbit coupling as QDs are fabricated within semiconductors of a zinc-blende struc-ture. The first one is the Dresselhaus interaction, which is due to the bulk inversion asymmetry of the lattice.8–10 The second is the Rashba interaction caused by the structure in-version asymmetry.11 The spin-orbit couplings mix the spin states with different orientations in the Zeeman sublevels12–15 and therefore make spin relaxation possible in the presence of the electron-phonon interaction.
Relaxation times of electron spins in a QD have been measured by electrical pump-probe experiments.16The trip-let-to-singlet transition with emission of phonons was found with corresponding spin relaxation times of about 200s. Recently, the spin relaxation time in a one-electron GaAs QD was measured by a similar electrical pump-probe tech-nique.17,18As the magnetic field was applied parallel to the two-dimensional electron gas, the Zeeman splitting of QD was observed in dc transport spectroscopy. By monitoring the relaxation of the spin, the relaxation time was found to have a lower bound of 50s at an in-plane field of 7.5 T.17 On the theoretical side, spin relaxation between two spin-mixed states in semiconductor QDs has been studied re-cently. However, to the best of our knowledge, all previous studies of spin relaxation have concentrated on QDs embed-ded in the bulk material,19–24whereas studies of spin relax-ation induced by confined phonons are still lacking. In this work, we therefore consider a single QD embedded into a
free-standing structure 共semiconductor slab兲, where the rel-evant characteristic is the two-dimensional phonon wave vector for the acoustic-phonon spectrum as shown in Fig. 1.25–29Since the reduced dimension will enhance the defor-mation potential, we will mainly focus on the spin relaxation rate induced by the deformation potential.25–27In this paper we describe the model with spin-orbit coupling. Energy spectra of the QD can be solved by using an exact diagonal-ization method. We then apply the Fermi golden rule to cal-culate spin relaxation rates for typical parameters. We dis-cuss the dependence of the spin relaxation rates on the size of the QD, the phonon bath temperature, and the width of the slab.
We consider an isotropic QD with an in-plane parabolic lateral confinement potential. An external magnetic field B is applied perpendicularly to the surface of the QD as shown in Fig. 1共a兲. The electronic Hamiltonian of this system can be written as
He= H0+ HSO. 共1兲 The first term describes the electron Hamiltonian without the spin-orbit coupling
FIG. 1.共Color online兲 共a兲 Schematic view of single QD embed-ded in the semiconductor slab with a width of a.共b兲 The side view shows a QD is located at z = 0.
HR= R ប 共xPy−yPx兲, 共3兲 HD= D ប 共−xPx+yPy兲. 共4兲 The coupling constants R andD determine the spin-orbit strengths, which depend on the band-structure parameters of the material. In addition, the Rashba and Dresselhaus terms are also associated to the perpendicular confinement field and the confinement width in the z direction, respectively.
For the electron Hamiltonian H0, the well-known Fock-Darwin states⌿n,l,can be easily obtained. The correspond-ing electron energy levels are En,l,=ប⍀共2n+兩l兩+1兲 +បBl / 2 +EB, where n共=0,1,2, ...兲 and l共=0, ±1, ± 2 , . . .兲 are the quantum numbers. The renormalized fre-quency is ⍀=冑02+B2, with the cyclotron frequency B = eB / m* and the characteristic confinement frequency
0 limited by the effective QD lateral length l0=
冑
ប/m*0. Here, EB= gBB / 2 is the Zeeman splitting energy, and = ± 1 refers to the electron-spin polarization along the z axis. To solve the Schrödinger equation with共He= H0+ HSO兲, the 共spin mixing兲 wave function is expressed in terms of a series of eigenfunctions⌿l共r,兲=兺cn,l,⌿n,l,for each state l. After exactly diagonalizing the electron Hamiltonian, the corre-sponding eigenvalues Eland the coefficient cn,l,can be ob-tained numerically.Before calculating the spin relaxation rate, the confined phonon in the free-standing structure must be introduced here. Following Ref. 25, we consider an infinite film with width a共Fig. 1兲. For the effect of the contact with the semi-conductor substrate, we neglect the distortion of the acoustic vibrations. Under this consideration, one can ensure that the in-plane wavelength can be shorter than the characteristic in-plane size of the solid slab. For simplicity, the elastic properties of the slab are isotropic. Small elastic vibrations of a solid slab can then be defined by a vector of relative displacement u共r,t兲. Under the isotropic elastic continuum approximation, the displacement field u obeys the equation
2u t2 = ct 2ⵜ2u +共c l 2− c t 2兲 共 · u兲, 共5兲 where cland ct are the velocities of longitudinal and trans-verse bulk acoustic waves. To define a system of confined modes, Eq. 共5兲 should be complemented by the boundary conditions at the slab surface z = ± a / 2. Because of the
con-tan共qt,na/2兲 tan共ql,na/2兲= −
4q储ql,nqt,n 共q储2− qt,n
2 兲2, 共6兲
with the dispersion relation
n,q储= cl 2
冑
q储2+ ql,n 2 = ct 2冑
q储2+ qt,n 2 , 共7兲wheren,q储is the frequency of the dilatational wave in mode
共n,q储兲. For the antisymmetric flexual waves, the solutions
ql,nand qt,n also can be determined by solving the equation tan共ql,na/2兲
tan共qt,na/2兲= −
4q储ql,nqt,n 共q储2− qt,n2 兲2
, 共8兲
together with the dispersion relation共7兲.
The electron-phonon interaction through the deformation is given by Hep= Eadiv u, where Ea is the deformation-potential coupling constant. The Hamiltonian can be written as Hep=
兺
q储,n =d,f M共q储,n,z兲共aq储 + + aq储兲exp共iq储· r储兲, 共9兲where r储 is the coordinate vector in the x-y plane and the
functions Md and Mf describe the intensity of the electron interactions with the dilatational and flexural waves, and are given by Md共q储,n,z兲 = Fd,n
冑
បEa 2 2An,q储冋
共qt,n2 − q储2兲共ql,n2 + q储2兲 ⫻ sin冉
aqt,n 2冊
cos共ql,nz兲册
, 共10兲 Mf共q储,n,z兲 = Ff,n冑
បEa2 2An,q储冋
共qt,n 2 − q储2兲共ql,n 2 + q储2兲 ⫻ cos冉
aqt,n 2冊
sin共ql,nz兲册
, 共11兲 where A is the area of the slab, is the mass density, and Fd,n共Ff,n兲 is the normalization constants of the nth eigen-mode for the dilatational共flexural兲 waves. Although the fluc-tuation of the dot 共due to strain, etc.兲 may affect the spin-orbit and electron-phonon coupling, we, for simplicity, neglect the effect on the scattering rate in this work.We calculate the spin relaxation rates between the two lowest共spin mixing兲 states from the Fermi golden rule30
⌫ =2 ប q
兺
储,n=d,f
兩M兩2兩具f兩eiq储·r储兩i典兩2共Nq储+ 1兲␦共⌬E − បn,q储兲,
共12兲 where the energy ⌬E共=Ei− Ef兲 is the energy difference be-tween the first excited兩i典 and ground 兩f典 states. Nq储represents the Bose distribution of the phonon at temperature T. For the sake of simplicity, we consider the QD to be located at z = 0 so that the function Mf for flexural waves plays no role. Let us first focus on the dependence of the relaxation rates on the magnetic field B for lateral length l0= 30 nm. Unlike the situation in bulk system, an enhanced spin relaxation rate occurs as shown in Fig. 2共a兲 共arrow 1 in the upper inset兲. This phenomenon originates from the van Hove singularity that corresponds to a minimum in the dispersion relation
n,q储for finite q储. We further plot the phonon group velocity
共n,q储/q储兲 as a function of q储 around the van Hove
singu-larity as shown in Fig. 2共b兲. There are three modes contrib-uting to the relaxation rate. In particular, a crossover from positive to negative group velocity is observed for one mode. Because of the zero phonon group velocity, the rate behaves sharply at that magnetic field. However in a real system the van Hove singularity would be cut off or broadened because of the finite phonon lifetime. Contrary to the enhanced rate, we find a suppression of the spin relaxation rate共arrow 2兲 at small magnetic field 共also seen in the lower inset兲. This comes from a vanishing divergence of the displacement field
u. As can be seen from Eq.共10兲 in detail, the deformation
potential disappears at the condition of q储= qt 关Fig. 2共c兲兴, which causes a zero spin relaxation rate. Note that our results
for the van Hove singularity and the disappearance of the deformation potential are consistent with what was found in Ref. 27. Although the phonon model in our work is the same, the dot part is different.
The relaxation rate for larger QDs exhibits a qualitatively different behavior. As shown in Fig. 3, two van Hove singu-larities appear when varying the magnetic field. In addition, one also finds two suppressions of the relaxation rate共arrow兲 near the singularities. We have analyzed the energy spacing between the two lowest states in the inset of Fig. 3. For small lateral size, the gap increases monotonically 共dashed line兲. On the contrary, energy spacing for larger QDs shows a quite different feature. The value initially increases as B increases. However, after it reaches a maximum point, the energy spac-ing decreases with the increasspac-ing of the magnetic field B: although the Zeeman splitting increases with increasing mag-netic field, the spin-orbit interaction, on the contrary, tends to reduce the energy spacing between the two lowest levels. When the magnetic field is large enough, the spin-orbit effect overwhelms the Zeeman term and results in a decreasing tendency. Therefore, if the magnetic field is increased high enough, the dashed line 共small QD兲 also shows similar be-havior. This agrees well with the findings in Ref. 14. From the inset, one recognizes that if the energy spacing exactly matches the specific phonon energy 共dotted line兲, the van Hove singularity will appear. For the case of a large lateral length, there are two van Hove singularities and two suppres-sions of the relaxation rate共dashed-dotted line兲.
Figure 4 shows the specific energy spacings where rates are enhanced and suppressed as a function of the width. For the case of small widths, the enhanced rates共black mark兲 and suppressed rates共red mark兲 can be clearly distinguished, and their corresponding energy spacings are relative large. With the increasing of the width, the energy spacing between the enhanced and suppressed rates decrease monotonically. One can expect that if the width increases further, the system will approach the bulk system. This means that the van Hove
FIG. 2. 共a兲 Spin relaxation rate as a function of magnetic field for the lateral length l0= 30 nm, the width a = 130 nm, and tempera-ture T = 100 mK. The SO couplingsR and Dare set equal to 5
⫻10−13and 16⫻10−12eV m, respectively. The insets further show the enlarged regions of arrow 1共upper inset兲 and arrow 2 共lower inset兲. 共b兲 Three phonon group velocities vs the magnetic field. 共c兲 The values q储and qtvs the magnetic field.
FIG. 3. Spin relaxation rate for the lateral length l0= 60 nm, width a = 130 nm, and temperature T = 100 mK. The SO couplings Rand Dare set equal to 5⫻10−13and 16⫻10−12eV m,
respec-tively. Two enhanced and suppressed rates共arrow兲 occur. The inset shows the energy spacing⌬E vs the magnetic field B for different lateral lengths: l0= 30 nm共dashed line兲 and l0= 60 nm共solid line兲. Two horizontal lines in the inset indicate the corresponding energies for the van Hove singularity共dotted line兲 and the suppression of the rate共dashed-dotted line兲.
singularity and the suppressed rate will be inhibited and eventually disappear.
If one varies the vertical position of the dot, the rate will change due to different contributions from the dilatational and flexural waves. Accordingly, the van Hove singularities resulting from flexural waves will also be altered. For ex-ample, the ratio of dilatational to flexural wave’s contribution is about 2.8:1 under the condition of B = 1 T and vertical position z = 25 nm. However, if⌬E also changes, the contri-butions from two waves will also change. This is because the parameters 共q储, ql,n, qt,n兲 of dilatational and flexural waves independently satisfy the dispersion relations. On the other
tum dot embedded in a semiconductor slab, where an en-hanced rate was found due to the phonon van Hove singu-larity. We found that at certain magnetic fields one enters a regime with quite the opposite characteristics, where a van-ishing divergence of the displacement causes a suppression of spin relaxation rates. For larger dots there are multiple singularities and suppressions in the electron-phonon rates due to the interplay between spin-orbit coupling and Zeeman interaction. We believe our results to be useful for the under-standing of spin relaxation in suspended quantum dot nano-structures. Our findings also point at novel effects to be ex-pected from future nanoscale systems where spin and mechanical degrees of freedom are combined.
This work was partially supported by the National Sci-ence Council, Taiwan under Grant Nos. NSC 009-019, NSC 94-2120-M-009-002, and NSC 94-2112-M-009-024.
*Electronic address: dschuu@mail.nctu.edu.tw
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30The parameters for the GaAs QD: m = 0.067m
0, g*= −0.44, Ea
= 6.7 eV,=5.3⫻103Kg/ m3, ct= 3.35⫻103m / s, cl= 5.7⫻103
