Dominant channels of exciton spin relaxation in photoexcited self-assembled (In,Ga)As
quantum dots
Yu-Huai Liao,1Juan I. Climente,2and Shun-Jen Cheng1,*
1Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China 2Departament de Qu´ımica F´ısica i Anal´ıtica, Universitat Jaume I, Castello E-12080, Spain
(Received 15 October 2010; revised manuscript received 2 March 2011; published 22 April 2011) We present a comprehensive theoretical investigation of spin relaxation processes of excitons in photoexcited self-assembled quantum dots. The exciton spin relaxations are considered between dark- and bright-exciton states via the channels created by various admixture mechanisms, including electron Rashba and Dresselhaus spin-orbital couplings (SOCs), hole linear and hole cubic SOCs, and electron hyperfine interactions, incorporated with single- and double-phonon processes. The hole-Dresselhaus SOC is identified as the dominant spin-admixture mechanism, leading to relaxation rates as fast as∼10−2 ns−1, consistent with recent observations. Moreover, due to significant electron-hole exchange interactions, single-phonon processes are unusually dominant over two-phonon ones in a photoexcited dot even at temperatures as high as 15 K.
DOI:10.1103/PhysRevB.83.165317 PACS number(s): 78.67.Hc, 71.35.Cc, 71.70.Ej, 78.47.−p
I. INTRODUCTION
Spin dynamics in semiconductor quantum dots (QDs) is a subject of interest in the current endeavor to develop spintronics and quantum information processing applications.1 It has been widely believed for a long time that the discrete nature of QDs can make the spin relaxation times of confined carriers long enough for further applications.2Indeed, the spin lifetimes of electrons confined in QDs have been experimen-tally confirmed to reach up to 1 s.3 Based on such long-lived electron spins, coherently controlled quantum gate devices made of electrode-defined QDs have been recently realized.4
In quantum photonic applications, InGaAs self-assembled QDs have been recently demonstrated as useful quantum light sources used in photonic quantum teleportation and cryptography.5The generation rate of single-photon emission from the dots is, however, severely limited by the undesired fast spin relaxation of excitons,6,7 reported to be as fast as ∼102ns in recent experiments.8Such transitions mainly occur between bright-exciton (BX) and dark-exciton (DX) states split by the e-h exchange interaction, which is of hundreds of μeV.9 What is more, the DX-to-BX transitions have been shown to be ultimately responsible for spin transitions within the BX doublet,10 so that they also limit the performance of entangled photon pair generators.
By contrast to existing extensive research for electrons or holes in QDs,1 the fundamental understandings of the spin relaxation processes of excitons in QDs are still incomplete. As a two-component quasiparticle, the spin dynamical processes of a quantum-confined exciton involve more complications, mixing various spin-flip mechanisms and phonon processes via the intrinsic e-h mutual interactions. To date, only the intrinsic mixing of heavy- and light-hole states11 and the linear-in-p spin-orbit coupling (SOC) of valence band holes12 have been theoretically studied as possible exciton spin-flip mechanisms, which, however, predict spin relaxation rates far below the observed values.8
In this work, we attempt to fill the gap between existing experiments and theoretical predictions. We present a compre-hensive investigation of spin relaxation of single excitons in InGaAs self-assembled QDs using both an analytical method
and a numerical exact diagonalization technique, with a full consideration of e-h exchange interactions, all possible electron and hole SOCs, hyperfine interactions, and particle-phonon couplings in single- and two-particle-phonon processes. We explain the fast exciton spin relaxation observed in QDs in terms of pronounced hole-Dresselhaus SOCs and e-h exchange interactions in predominant single-phonon processes.
II. THEORY A. Model and Hamiltonian
We start with an interacting Hamiltonian for a single neutral exciton confined in a phonon-free quantum dot:
HX0 = He+ Hh+ Veh+ Vehxc+ H e
SO+ H
h
SO. (1)
Here Hj denotes the noninteracting single-electron (j = e) or single-hole (j = h) Hamiltonian in a parabolic QD:
Hj = p2 j 2mj + 1 2mjω 2
jrj,2+ Vj(zj) , where pj are the oper-ators of linear momentum, mj are the effective masses of particles, ωj parametrizes the lateral confining poten-tial, Vj(zj) is the vertical square confining potential of thickness dz, and rj, = (xj,yj) is the in-plane coordi-nate. Within the model, the single-particle wave function can be written in a separable form: nj,nz(xj,yj,zj)=
ψn(rj,)gnz(zj). The in-plane part of Hj yields the
ex-plicit two-dimensional (2D) Fock-Darwin (FD) energy spectrum εjs = ¯hωj, ε
j
p± = 2¯hωj, . . ., and the single-particle wave functions ψs(rj,)= l 1 0√π exp(−r 2 j,/2l20), ψp±(rj,)= rj, l2 0 √π exp(−r2
j,/2l20)× e±iφ, . . .,13where the subscripts s, p+,
and p− indicate the atomiclike s- and p-shell orbitals of QD with orbital angular momentum projection Lz= 0, + 1, and−1, respectively, and l0≡
√
¯h/meωe= √
¯h/mhωhis the characteristic lateral extent of wave functions (as depicted by the lower-right schematics in Fig.1).
The terms Veh and Vehxc are the e-h Coulomb direct and exchange interactions, respectively. It is mainly the attractive direct interaction Vehmaking an e-h pair bind together to form an exciton. Veh does not, however, affect the spin structure of exciton states, and we shall treat it as a constant offset
3 4 5 6 7 8 9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 0 l e / h s ↑⇓ ↓⇑ ↓⇓ ↑⇑ xc 2 eh l0 − ∝ 1 tot − 1 tot − (ns -1 ) l0(nm) -1 tot(numerical) -1 tot(perturb.) -1 tot( xc eh=0.4meV) -1 h-D -1 e-D -1 h-lin -1 e-R -1 Hy
FIG. 1. (Color online) Exciton spin relaxation rates in the DX to BX transition and the rates contributed from each spin-flip mechanism, as functions of the QD size. Results with fixed xc
eh=
0.4 meV (dashed line) are also presented for comparison. Inset: Schematic of the DX to BX conversion channels.
of energy. The fourfold spin degenerate single-exciton states are split by e-h exchange interaction Vehxc into a lower-energy optically inactive doublet with Mz= sz+ jz= ±2 (the so-called DX states |sz= +12,jz= +32 = | ↑ ⇑ and |sz= −12,jz= −32 = | ↓ ⇓) and a higher-energy doublet with Mz= ±1 (the BX states | ↑ ⇓ and | ↓ ⇑) with an energy separation xc
eh∼ 102–103 μeV. V xc
eh thus acts as an effective field coupled to the exciton spin but, unlike an externally applied field, itself is associated with the dot structure. Treating the short-range part of the corresponding
e-h exchange interaction as the dominant component, the
BX-DX splitting xc
ehfor the low-lying exciton states can be modeled as14
xceh= xceh,bulk×π aB∗3 d3rs,e0(r)s,h0(r) 2
∝ l−2 0 , (2)
where xceh,bulk is the bulk e-h exchange energy and aB∗ the effective exciton Bohr radius for InGaAs. Note that the smaller the dot, the larger the xc
eh. The term He
SO (H
h
SO) denotes the electron (hole) SOC. For
electrons, the SOC Hamiltonian consists of the Rashba and Dresselhaus terms, HSOe = HRe+ H
e D,1being
HRe = (αe/ i¯h)(p+σ−− p−σ+), (3) HDe = −(βe/¯h)(p+σ++ p−σ−), (4)
where p±≡ px± ipy (σ±≡ σx± iσy) are orbital (spin) ladder operators for (s= 1/2) electron and αe (βe) is the e-Rashba (Dresselhaus) SOC constant.
For holes, we consider the h-SOC Hamiltonian as Hh
SO =
Hh
lin+ H h
D consisting of the relevant p-linear and -cubic terms:12,15
Hlinh = −(αh/¯h)(p+σ−+ p−σ+), (5) HDh = −(βh/¯h3)(p+p−p+σ−+ p−p+p−σ+), (6) where αh (βh) is the p-linear (-cubic h-Dresselhaus) SOC constant. Equations (5) and (6) are expressed in terms of Pauli operators for the pseudospin of heavy holes defined by the spin replacement j = 3/2 → sh= 1/2 and jz= ±3/2 → szh= ±1/2. For brevity, the superscripts h for the hole spin
operators are removed. The h-Rashba SOC is irrelevant in the problem here since Hh−R∝ p±3 involves only the remote h states above the d shell.15
The eigenenergies and eigenstates of a spin exciton in a phonon-free QD can be numerically obtained by solving the eigenequation HX0|X; i = EX;i|X; i using an exact
diagonal-ization method for the matrix of HX0 in the basis of exciton configurations built up from the FD orbitals (with a typical number of FD orbitals≈ 15 and that of exciton configurations ≈ 900). The parameters used in the calculations throughout this work are summarized in TableVin the Appendix.
B. Relaxation rates
Next, we consider the QD coupled to the acoustic phonon bath by introducing the particle-phonon couplings into the QD system, being
Hj−ph=
νq
Mνj(q)(bqeiq·rj + bq†e−iq·rj). (7) Here q is the phonon wave vector, ν= LD,T D,LP,T P denote the kinds of phonon modes (longitudinal or transversal modes of deformation phonons or piezoelectricity phonons),
bqand b†qare the phonon creation and annihilation operators, and Mνj are the phonon scattering matrix elements. TableI summarizes the expressions for Mνj(q) as functions of the phonon wave vector q= (qx,qy,qz) and relevant material parameters.
The exciton spin relaxation rate for the transition from DX states to BX ones involving single-phonon processes (as illustrated by the schematics in the inset of Fig.1) are evaluated using Fermi’s golden rule:
τtot−1=2π ¯h f j=e,h ν q f|Vνqj|i2δ(Ef i− ¯hωνq) Nqν. (8)
TABLE I. Expressions for phonon scattering matrix elements Mj
ν as functions of the phonon wave vector q= (qx,qy,qz) and phonon
parameters for InGaAs. ν(= LD,T D,LP,T P ) denotes the kinds of phonon modes (longitudinal or transversal modes of deformation phonons or piezoelectricity phonons). denotes the crystal volume. Other symbols for phonon parameters are summarized in TableVin the Appendix.
ν= LD ν= T D ν= LP ν= T P |Mj ν(q)| 2 ¯hDj,LD2 2dcLDq ¯hD2 j-T D 2dcT D q2 z(qx2+qy2) q3 32π2¯he2h2 14 2dcLP (3qxqyqz)2 q7 32π2¯he2h2 14 2dcT P | (qxqy)2+(qyqz)2+(qzqx)2 q5 −(3qxqyqz)2 q7 |
Here Vνqj = M j
ν(q)eiqrj, ¯hωνq is the phonon energy, and|i and|f are the initial (DX) and final (BX) states, respectively.
Ef i≡ Ef− Eiis the energy difference between|f and |i. The δ function in Eq. (8) ensures the resonance condition ¯hωνq = |Ef i| in the single-phonon processes, which indicates ¯hcνqν0= xceh∝ qν0 (cν is the speed of acoustic phonon). Nqν = 1/(e
¯hωνq/kBT − 1) is the phonon population, with T
being the temperature and kB the Boltzmann constant.
III. NUMERICAL RESULTS A. Single-phonon processes
Figure1presents the numerically calculated total rates τtot−1
of single excitons in QDs of fixed thickness dz= 3 nm but with varying lateral sizes. It can be seen that the total spin relaxation rate of the exciton is at the scale of 10−3–10−2ns−1 consistent with recent observations.8For further insight, one can extract the individual contributions τj−1−SO from each j (electron or hole) SOC term in the total rate τtot−1. Among all spin mechanisms, the p-cubic h-Dresselhaus SOC leads to the fastest spin relaxation rate, τh−1−D∼ 10−3–10−2 ns−1. This is 1 order of magnitude higher than that of e-Dresselhaus SOC (τe−1−D∼ 10−4 ns−1) for self-assembled dots with typical l0<6 nm and faster than those due to intrinsic heavy-hole
(hh)-light-hole(lh) mixing in excitons.8,11 The linear hole SO interaction, which was proposed as the exciton spin relaxation mechanism in large QDs,12is found to be a weak effect in self-assembled QDs, leading to a slow rate of τh−1−lin∼ 10−6ns−1.
For spin transitions mediated by the electron-nuclei hyper-fine (Hy) interaction, an exact diagonalization procedure like that used for the SOCs is quite challenging because the number of involved nuclei is over millions. We thus separately evaluate the spin relaxation rate via the Hy interactions by using the perturbation method presented in Ref. [16]. The calculated Hy-interaction-mediated spin relaxation rate of exciton in a QD is quite slow, τH y−1∼ 10−8ns−1, incomparable to the spin relaxation rate arising from other SOCs.
A remarkable feature of Fig.1is that the spin relaxation of excitons, as opposed to that of single electrons, is not really suppressed by the reduced dot sizes. Instead, the size effect of QDs makes the spin relaxation rate even faster. This is because the smaller the QD, the stronger the Coulomb interaction and hence xc
eh. Larger interlevel spacing translates into larger density of acoustic phonons, more efficient carrier-phonon coupling,17,18 and stronger spin-orbital admixture. Further understanding of this is provided by a perturbational analysis we carry out in Sec.IV.
B. Two-phonon processes
Next, we examine the influence of two-phonon processes on the spin relaxation of excitons. The examination of the two-phonon process effect is necessary here since there has been evidence that, in the absence of magnetic fields, the spin relaxation of holes is dominated by two-phonon processes starting from temperatures below 1 K.19We consider processes involving an initial absorption to a virtual state|n, followed by an emission to the final state, as illustrated in the inset of Fig. 2(a). The main spin-admixture mechanism is
hole-0.0 0.1 0.2 0.3 0.4 10-8 10-7 10-6 10-5 10-4 10-3 10-2 ωk ωq ↑⇓ ↑⇑ n (meV) xc eh
(ns
-1)
One phonon Two phonon 3 4 5 6 7 8 9 10-6 10-5 10-4 10-3 10-2 (b) (a) 0(nm) l(ns
-1)
One phonon Two phononFIG. 2. (Color online) Exciton spin relaxation rates due to one- and two-phonon processes as a function of the e-h exchange interaction (a) and the QD size (b) at T = 15 K. In panel (a), l0= 5
nm. In panel (b), size-dependent xc
ehis taken. The inset in panel (a)
is a schematic of the two-phonon process.
Dresselhaus SOC, and the rates are calculated numerically using a second-order Fermi golden rule:
τh−1−D(2)= 2 π ¯h f ν q,k n f |Vh νk|n n|V h νq|i (En− Ei− Eq) 2 × δ(Ef i+ ¯hωνk− ¯hωνq) Nqν Nkν+ 1 . (9)
Figure2(a)compares the relaxation rates obtained with one-and two-phonon processes as a function of xceh. One can see that single-phonon processes become rapidly inefficient for small xc
eh, which is due to the decreasing density of phonon states. By contrast, two-phonon rates remain roughly constant. This is because there is no resonance condition for the transition to the virtual state. As a result, even if xcehis small, two-phonon processes may rely on the absorption and emission of energetic phonons, as long as the sum of their energies matches Ef− Ei. The insensitivity of two-phonon processes to xceh is analogous to that of individual holes to external magnetic fields,19 except that here the role of the magnetic field is played by the inherent exciton e-h interaction.
Figure2(a)shows that, for excitons, two-phonon process dominate over single-phonon ones only if xc
ehis small. To test if this is actually the case in self-assembled QDs, in Fig.2(b) we compare one- and two-phonon rates as a function of the dot size, considering the size dependence of xc
eh. Clearly, one-phonon processes dominate up to very large dot sizes, where xc
eh becomes small enough. It is worth noting that one-phonon processes dominate despite the moderately large temperature, T = 15 K. This is because the e-h exchange acts as a fairly strong effective magnetic field. In what follows, we present an analysis for the exciton spin relaxation rates in main single-phonon processes to provide more understanding of the numerical data.
IV. ANALYSIS A. Exciton wave functions
For analysis, we begin with the energy spectrum of an exciton confined in a QD subject to relatively weak SOCs. Under the condition, the spin-mixed DX states as possible initial states can be expanded in the reduced basis formed by a few relevant exciton configurations and approximately
expressed as |X; i ≡ |DX × ∝ψse↑; ψsh⇑− CeDX-Rψpe+↓; ψsh⇑ + iCDX e-Dψ e p−↓; ψ h s ⇑ + iCDX h-linψ e s ↑; ψ h p+⇓ + iCDX h-Dψ e s ↑; ψ h p+⇓ , (10) |X; i ≡ |DX × ∝ψse↓; ψsh⇓+ CeDX-Rψpe−↑; ψsh⇓ + iCDX e-Dψ e p+↑; ψ h s ⇓ + iCDX h-linψ e s ↓; ψ h p−⇑ + iCDX h-Dψ e s ↓; ψ h p−⇑ , (11)
where |ψieσe; ψihσh denotes the single-exciton
configu-ration where an electron (a valence hole) occupies the
ieth (ihth) Fock-Darwin orbital with spin σe=↑ / ↓(σh=⇑ /⇓), and Cjξ-SO(∈ R) are the (real) coefficients of the
coupled configurations of dark exciton (ξ = DX,DX) aris-ing from the j -SO couplaris-ings (j -SO= e-R,e-D,h-lin,h-D). For brevity, the normalization constants are not shown in Eqs. (10) and (11).
Likewise, the spin-mixed bright-exciton (ξ = BX,BX) states, as possible final states in Eq. (8), are written as |X; f ≡ |BX × ∝ψse↑; ψsh⇓− CeBX-Rψpe+↓; ψsh⇓ + iCBX e-Dψ e p−↓; ψ h s ⇓ + iCBX h-linψ e s ↑; ψ h p−⇑ + iCBX h-Dψ e s ↑; ψ h p−⇑ , (12) |X; f ≡ |BX × ∝ψse↓; ψsh⇑+ CeBX-Rψpe−↑; ψsh⇑ + iCBX e-Dψ e p+↑; ψ h s ⇑ + iCBX h-linψ e s ↓; ψ h p+ ⇓ + iCBX h-Dψ e s ↓; ψ h p+⇓ . (13)
Figure 3 (4) depicts the schematics of the main low-lying exciton configurations for a dark-exciton (bright-exciton) state. By treating each SOC term separately and perturbatively
p p
s
p p h - lin. e h - D e p ph h Ds
h ; e h s s p p ; e h s p e - D e - R ; e h e h ; s p ep ; hsFIG. 3. (Color online) Schematics of the main low-lying exciton configurations coupled by various spin-orbital couplings for the dark-exciton state|DX defined by Eq. (10).
p p
s
p p h - lin. h D p p h - Ds
; e h s s p p ; e h s p e - D e - R ; e ; h e h s p ep ; shFIG. 4. (Color online) Schematics of the main low-lying exciton configurations coupled by various SOCs for the bright-exciton state |BX defined by Eq. (12).
in Eq. (10), one can derive the explicit expressions for the configuration coefficients Cjξ-SO in terms of SOC constants (αj,βj), characteristic length of wave function extent l0, energy
quantization of QD εjsp≡ εjp±− εjs, and BX-DX splitting xc
eh. We assume that, prior to the inclusion of SOC effects, s-shell DX and BX states are split by xc
eh. Then, the p-shell DX-BX splitting is xceh/2. The smaller splitting follows from the reduced orbital overlap in the short-range electron-hole exchange integral, d3r|e
p,0(r)s,h0(r)|2.14
Higher shells (used in numerical calculations) are consid-ered to have no sizable DX-BX splitting. The coefficients
Cjξ-SO derived under these conditions are summarized in TableII.
B. Characteristic rate of spin relaxation
Substituting Eqs. (10)–(13) into Eq. (8), the total spin relaxation rate in the transition from a DX state to BX ones can be decomposed into the individual rates arising from each SOC mechanism, τj−1−SO= 2π ¯h P 2 j-SO νq Mνj(q)2|p,0|eiq·rj|s,0|2 × δxceh− ¯hωνq Nqν. (14)
TABLE II. Explicit expressions for configuration coefficients of spin-mixed BX and DX states in terms of SOC constants (αj,βj),
characteristic length of wave function extent l0, energy quantization
of QD εj sp≡ ε j p±− ε j s, and BX-DX splitting xc eh. The formulations
are derived by treating separatively and perturbatively each SOC mechanism. The parameters used in the calculations are summarized in TableV. ξ Ceξ-R C ξ e-D C ξ h-lin C ξ h-D DX,DX αe/ l0 εe sp+3xceh/4 βe/ l0 εe sp+3xceh/4 αh/ l0 εhsp+3xceh/4 2βh/ l03 εsph+3xceh/4 BX,BX αe/ l0 εe sp−3xceh/4 βe/ l0 εe sp−3xceh/4 αh/ l0 εh sp−3xceh/4 2βh/ l03 εh sp−3xceh/4
TABLE III. Explicit expressions for the spin-mixture factors Pj-SOin Eq. (14) are in terms of SOC constants (αj,βj), characteristic
length of wave function extent l0, energy quantization of QD εsp, and
BX-DX spin splitting xc eh. Pj-SO Pe-R Pe-D Ph-lin Ph-D 3αe/ l0×xceh 2(εe sp)2 3βe/ l0×xceh 2(εe sp)2 3αh/ l0×xceh 2(εhsp)2 3βh/ l03×xceh (εhsp)2
The form factors Pj-SOare yielded by the slight spin admixture
between DX states and BX states and, under the condition of weak SOC, are derived as
Pe-R ≈CBX e-R − CeDX-R, (15) Pe-D ≈CBX e-D − CeDX-D, (16)
Ph-lin≈ChBX-lin− ChDX-lin, (17)
Ph-D≈ChBX-D− C DX
h-D. (18)
Note in Eqs. (15)–(18) that it is the asymmetry of the spin admixture of DX and BX states that makes transitions between them feasible. Substituting the formulations for Cξj-SO in
TableIIinto the above equations for Pj-SO, the spin-mixture
factors Pj-SOare expressed as
Pj-SO= 3j-SOxceh
2εspj 2, (19) in terms of the characteristic energies of SOCs defined as
e-R≡ αe/ l0, e-D≡ βe/ l0, h−lin≡ αh/ l0, and h-D≡
2βh/ l03, energy quantization of QD (ε j
sp), and the BX-DX spin splitting (xc
eh), as summarized in TableIII. The expressions account for the fact that the probability of the DX-to-BX spin-state transition directly depends on the relative strength of the involved SOCs, the BX-DX spin splitting, and the energy quantization of the QD.
The spin relaxation rate for j -SO coupling can be reformu-lated as τj−1-SO= 2π ¯h P 2 j-SO ν ¯Mνj(qν0)2ρν(qν0)Fν(l0,qν0)Nqν0, (20)
where qν0= xceh/(¯hcν) is the magnitude of the wave vectors, ρν= 2π2 q2 ν0 ¯hcν ∝ xc eh
2 is the density of states of
the resonant phonons involved in the relaxation process
( denotes the crystal volume), and M¯νj(q) is the mean q-anisotropic phonon scattering matrix element, separated from the angular part Iν(θq,φq) in the matrix element Mνj(q)≡ ¯Mνj(q)Iν(θq,φq). The function Fν(l0,q)≡
(1/4π )02π0πIν(θq,φq)
2
p,0|eiq·r|s,0 2
sin θqdθqdφq considers the anisotropy of phonon coupling and the localization of the particle wave function in the QD. TableIV summarizes the expressions for ¯Mνj, Iν, ρν, and Fνas functions of the wave vector of resonant phonons qν0= (qν0,θqν0,φqν0)
and/or the characteristic length of the wave function extent l0
defined by the 2D parabolic model.
Since | ¯MT P|2∝ 1/qT P0, while | ¯MLD(T D)|
2∝ q
LD(T D)0,
transversal piezoelectric phonon interaction is dominant in transitions between BX and DX states, which in self-assembled dots are split by only∼102μeV.17Thus, the total coupling strength of phonons involved in a spin relaxation can be estimated by | ¯MT P|2ρT P ∝ qT P0∝
xc
eh, a product of the dominant transversal piezoelectric phonon coupling and the phonon density of states.
The numerical results of Fig.1show that the Dresselhaus SOCs are generally dominant over other possible spin-admixture mechanisms in exciton spin relaxation. Among all p-linear terms, the e-Dresselhaus SOC leads to faster relaxation rates than other Rashba terms since βe αe,αh (see TableV). Yet, the cubic h-Dresselhaus SOC plays the main role because of the heavier mass of the hole and the strong confinement of the dot, as shown by the analysis below.
Taking the fact that l0qT P0 1 and Nqν0≈ kBT / xc eh (since kBT xceh), the characteristic spin relaxation rates of excitons in a QD via the main e- and h-Dresselhaus SOCs are derived as τe−1 ≈ Kβe2m4exceh4l08T , (21) τh−1≈ K(2βh)2m4h xceh4l04T , (22) respectively, where K≡ 48π m40e2h214kB 35¯h92dc2 T P
is a constant. The domi-nant role of h-D SOC in the exciton spin relaxation is identified by the high ratio of τe-Dto τh-D,
τe-D τh-D = 2βh/ l03 βe/ l0 2 ×mh me 1, (23)
TABLE IV. The expressions for the functions Iν, ¯Mνj, ρν, and F in terms of phonon parameters for InGaAs, the wave vector of phonon qν= (qν,θqν,φqν) represented in the spherical coordinate, and the characteristic length of wave function extent l0defined within the 2D parabolic
model. The symbols and values of the phonon parameters for InGaAs appearing in this table are summarized in TableV.
ν= LD ν= T D ν= LP ν= T P | ¯Mj ν(qν)|2 ¯hD2 j,LD 2dcLDqν ¯hD2 j,T D 15dcT Dqν 96π2¯he2h2 14 352dc LP 1 qν 128π2¯he2h2 14 352dc T P 1 qν |Iν(θqν,φqν)| 2 1 15 4 sin 22θ qν 105 4 sin2θ qνcos θqνsin 2φqν 2 35 16 sin4θ qνsin 22φ qν+ sin 22θ qν −315 16 sin2θ qνcos θqνsin 2φqν 2 ρν(qν) 8π3 4π q2 ν ¯hcν 8π3 4π q2 ν ¯hcν 8π3 4π q2 ν ¯hcν 8π3 4π q2 ν ¯hcν Fν(l0,qν) 16 2 5 + 1 l2 0q2ν −1 1 7 1 3+ 1 l2 0q2ν −1 1 6 4 11+ 1 l2 0qν2 −1 1 6 13 33+ 1 l2 0q2ν −1
which is explicitly shown to be much greater than 1 for InAs self-assembled QDs with typical l0<10 nm. The above
equations account for the faster exciton spin relaxation rate via
h-SOC than via e-SOC due to the heavier mass of the hole,
which results in the hole having a weaker quantization and a higher value of spin-mixture factor P , and a strong quantum confinement of the dot (small l0). For very large QDs, however,
the l0dependence indicates that e-SOC will eventually become
dominant.15
Figure5(solid lines) shows the dominant form factors P2
h-D,
| ¯MT P|2ρT P, FT P, and NqT P0as functions of l0. It is clearly seen
that, with reducing dot sizes, the three former functions make increasing contributions to the total spin relaxation rate. This is because the increased xcehand qν0 in small dots increase
the spin admixture between the DX states and the BX ones, the number of involved phonons, and the strength of effective phonon coupling.
C. Power-law dependencies
If xcehis treated as a constant, Eqs. (21) and (22) predict decreased relaxation rates by reducing size, which is indeed the behavior of independent electrons and holes.2 Taking the size dependence of spin splitting (xc
eh∝ l0−2) into
account, however, the power laws Eqs. (21) and (22) are reformulated as τe−1 ∝ βe2× m4e× T , (24) τh−1∝ βh2× m4h× l0−4× T . (25) 3 4 5 6 7 8 9 10-5 10-4 10-3 P 2 h-D l (nm) (a) (c) (d) (b) 0.0 0.1 0.2 0.3 F(l 0 ,qTP0 ) l 0(nm) 3 4 5 6 7 8 9 0 20 40 60 80 100 120 140 (/ 2π )| TP | 2 TP ( ns -1 ) l0(nm) 0 2 4 6 8 10 N qTP0 l 0(nm)
FIG. 5. (Color online) Calculated (a) spin-mixture function P , (b) product of mean phonon coupling and density of states of involved phonons| ¯M|2ρ, (c) correction factor F , and (d) phonon
population NqT P0 as functions of l0for the dominant h-D SOC and
transversal piezoelectric phonon (TP) couplings. The product of the four quantities determine the main exciton spin relaxation rates of QDs as shown by Eq. (20). The results calculated with fixed xc
eh= 0.4
meV are indicated by dashed lines for comparison.
These power laws account for the features of roughly constant
τe−1-R/Dand enhanced τh−1-Dby the reduced size of QDs observed
in Fig.1. To highlight the significance of the size dependence of the e-h exchange splitting, Figs.1and5also show the total spin relaxation rate calculated with a fixed xc
eh= 0.4 meV
TABLE V. Summary of the parameters used in the analysis and numerical calculations for InGaAs quantum dots throughout this work. In some cases (marked with a superscript∗) for which only the parameters for binary compounds (InAs or GaAs) are available, the parameters for InGaAs are determined by taking interpolated values.
Parameter Symbol Value Refs.
Electron effective mass me 0.05 m0 21*
Hole effective mass mh 0.2 m0 19*
Thickness of QD dz 3 nm
Bulk exciton Bohr radius aB∗ 25 nm 22*
Bulk e-h exchange energy xc
eh,bulk 4.3 μeV
9
e-Rashba coupling constant αe 0.1 eV· ˚A 23
e-Dresselhaus coupling constant βe 0.65 eV· ˚A
h-linear coupling constant αh 11 meV· ˚A 12*
h-Dresselhaus coupling constant βh 190 eV· ˚A3 15
Longitudinal sound velocity of acoustic phonon cLD/LP 4720 m/s 17
Transversal sound velocity of acoustic phonon cT D/T P 3340 m/s 17
Density of material d 5310 kg/m3 17
Hydrostatic deformation potential constant for electron (ac) De,LD −7.17 eV 21
Hydrostatic deformation potential constant for hole (av+ b/2) Dh,LD −2.16 eV 21
Uniaxial deformation potential constant for hole (−3b/2) Dh,T D 3 eV 21
Piezoelectric constant h14 1.41× 109V/m 17
Static dielectric constant 12.9 17
Dresselhaus constant γD 100 eV· ˚A3 15*
Split-off gap energy/(band gap energy+ split-off gap energy) η 0.35 15*
Heavy-hole and light-hole splitting lh
hh 0.15 eV
15*
Luttinger parameter γ2 4.2 21
Conduction band offset Vco 0.3 eV 21*
(dashed lines). The obtained behavior is drastically different from that with size-dependent xceh.
V. SUMMARY
In conclusion, we have calculated the relaxation rates between DX and BX states in InGaAs QDs for a wide number of spin-flip mechanisms and shown that hole-Dresselhaus SOC assisted by single-phonon processes is the dominant channel. The e-h exchange splitting acts as an internal magnetic field enhancing SOC mechanisms. Since the splitting grows with the confinement, the smaller the dot the faster the exciton spin relaxation. This is contrary to the well-known behavior of individual electrons or holes, for which relaxation is suppressed by the confinement.
ACKNOWLEDGMENTS
SJC is grateful to the National Science Council of Taiwan (Contract No. NSC-98-2112-M-009-011-MY2), the National Center of Theoretical Sciences and the National Center for
High-Performance Computing of Taiwan in Hsinchu for supporting. Support from MCINN Project CTQ2008-03344 and the Ramon y Cajal Program (JIC) is acknowledged.
APPENDIX: PARAMETERS
Table V summarizes the symbols and values of the parameters used in the analysis and the numerical calculations throughout this work. The determination of the Dresselhaus constant βe follows the formalism βe= γDpz2, where the bulk Dresselhaus SO constant γD= 100 eV· ˚A3 is taken for InGaAs and the term p2
z is evaluated by solving a one-dimesional Schr¨odinger equation for a square well in the
zdirection using a finite-difference method.20 The values of the conduction (valence) band offset Vco(Vvo) for the vertical square well is given in TableV. The parameter βhis determined by the evaluation of βh= 3γDγ2pz2/2m0ηlhhh, with γ2 as
the Luttinger parameter, m0 as the free electron mass, lhhh as the energy splitting between heavy holes and light holes, and the factor defined as η= so/(Eg+ so), where sois the split-off gap energy and Eg is the band gap energy for InGaAs (see TableV).15
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