Spin-flip phonon-mediated charge relaxation in double quantum dots
SSpin-flip phonon-mediated charge relaxation in double quantum dots
J. Danon
Dahlem Center for Complex Quantum Systems and Fachbereich Physik,Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany andNiels Bohr International Academy, Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark (Dated: December 7, 2018)We theoretically study the (1 ,
1) triplet to (0 ,
2) singlet relaxation rate in a lateral gate-defineddouble quantum dot tuned to the regime of Pauli spin blockade. We present a detailed derivationof the effective phonon density of states for this specific charge transition, keeping track of the con-tribution from piezoelectric as well as deformation potential electron-phonon coupling. We furtherinvestigate two different spin-mixing mechanisms which can couple the triplet and singlet states: amagnetic field gradient over the double dot (relevant at low external magnetic field) and spin-orbitinteraction (relevant at high field), and we also indicate how the two processes could interfere atintermediate magnetic field. Finally, we show how to combine all results and evaluate the relaxationrate for realistic system parameters.
PACS numbers: 03.67.Lx, 73.21.La
I. INTRODUCTION
The last decade has seen a great interest in spinqubits hosted in semiconductor quantum dots, moti-vated by the prospects of easy scalability, weak cou-pling to external perturbations, and flexible tunability.
Experimental advance has been substantial in the pastyears, and essential operations including qubit initializa-tion, manipulation, and readout have been convincinglydemonstrated.
This progress is not only exciting in thecontext of quantum computation and information, but italso provides a unique platform for studying fundamentalquantum properties of nanoscale systems. Ongoing effortis therefore directed at improving the quality of the spinqubits, mainly by trying to reduce qubit dephasing andincrease the measurement fidelity. A common method to read out a quantum dot spinqubit relies on the so-called Pauli spin blockade: A dou-ble quantum dot is tuned to a (1 ,
1) charge state, meaningthat each dot contains exactly one excess electron, andtwo of the four resulting (1 ,
1) spin states are used as aqubit basis. After qubit manipulation, the double dot po-tential is tilted such that a (0 ,
2) charge state becomes thetwo-electron ground state. For not too strong tilting, theonly accessible (0 ,
2) state is a spin singlet, which makesthe (1 , → (0 ,
2) charge relaxation spin-selective. Ifthe two qubit basis states contain a different spin-singletcomponent, then one can use charge detection to mea-sure the qubit’s final state before tilting, either by do-ing transport measurements coupling the doubly occu-pied dot strongly to an outgoing lead, or by detectingthe charge state with a nearby charge sensor.
The accuracy of such a readout depends crucially onthe effectiveness of the spin blockade: Any leakage outof the blocked triplet states reduces the readout visibil-ity and thereby distorts the measurement. A detailedunderstanding of the spin-flip relaxation responsible forsuch triplet leakage is thus essential in the context of spin qubit measurement. Most existing theoretical work alongthese lines was done for single-dot spin relaxation orconsists of numerical studies of the relaxation rates.
A thorough analytical study of interdot spin-flip chargerelaxation in double quantum dots is still missing.Here, we study in detail the (1 ,
1) triplet to (0 ,
2) sin-glet decay rate for a lateral double quantum dot. Weinvestigate two spin-mixing mechanisms which provide acoupling between the otherwise orthogonal states: (i) Forsmall externally applied magnetic fields the coupling isbelieved to be dominated by the effective magnetic fieldgradient over the two dots caused by the hyperfine cou-pling of the electron spins to the randomly fluctuatingnuclear spins in the host material. (ii) At larger fields,this coupling is suppressed for the two polarized tripletstates for which spin-orbit interaction takes over as dom-inating spin-mixing mechanism. The second ingredientnecessary for a finite leakage rate is the dissipation of theenergy difference ∆ between the initial (1 ,
1) triplet andfinal (0 ,
2) singlet state, which we assume to be providedby the coupling to acoustical phonons in the host mate-rial. We derive the function P (∆) for this specific chargetransition, which gives the probability that the transitionis accompanied by the dissipation of energy ∆ by a singlephonon (alternatively one could call this function the ef-fective phonon density of states for the charge transition).We include the contribution from piezoelectric as well asdeformation potential electron-phonon coupling, and wefind that the piezoelectric contribution to P (∆) is linearat low energies and ∝ ∆ − at high energies, whereas thecoupling to the deformation potential leads to a contribu-tion ∝ ∆ for low energies and ∝ ∆ − for high energies.We finally evaluate explicit relaxation rates using param-eters of the experiment of Ref. 19. We find a decay rate ∼ MHz, which agrees with experimental observations. Our results are not only relevant for spin qubit readout.In Ref. 19 it was suggested that the spin-orbit coupling ofthe (1 ,
1) triplet and (0 ,
2) singlet states could also be uti- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b lized to drive off-resonant microwave-stimulated Ramantransitions within the (1 ,
1) space. In that case transi-tions between the (1 ,
1) and (0 ,
2) states would contributeto qubit dephasing, and a detailed understanding of themechanisms responsible for these transitions would beessential in this context as well.The rest of this paper is structured as follows. In Sec. IIwe introduce our model of the double quantum dot andpresent an effective Hamiltonian defining the basis wewill work in. In Sec. III we then investigate the two spin-mixing mechanisms (spin-orbit interaction and a mag-netic field gradient over the double dot) and we derivethe matrix elements needed to calculate the leakage rate.In Sec. IV we study the coupling to the phonon bathin detail. We start from the standard Hamiltonian de-scribing the electron-phonon coupling and derive from it P (∆) for the (1 ,
1) to (0 ,
2) spin-flip charge transition.Finally, in Sec. V, we evaluate the leakage rate explicitlywith realistic experimental parameters.
II. MODEL
We consider a lateral double quantum dot, gate-definedin a two-dimensional electron gas (2DEG) formed at theinterface of an AlGaAs/GaAs heterostructure which hasbeen grown in the crystallographic (001) direction. Thesetup we have in mind is sketched in Fig. 1a: The twodots (gray circular areas) are separated by a distance d .Each dot is approximated to be a two-dimensional har- FIG. 1: (a) A lateral double quantum dot tunnel coupled to aleft and right lead. Two nearby gate electrodes with appliedvoltages V L,R can change the potential offset of the dots. Anin-plane external magnetic field B ext is applied in a directionparallel to the double-dot axis. (b) Charge stability diagramof a few-electron double dot. We investigate the gray region,where both (1 ,
1) and (0 ,
2) can be stable charge states. (c)The spectrum of the electronic Hamiltonian (2) as a functionof the detuning ε , where we chose t = 0 . B ext ). The detuning axis ε is also indicated in (b). We willfocus on the regime indicated with the red rectangle, wherethe two lowest-energy states are | S (cid:105) and | T + (cid:105) . monic potential well with a typical width a , correspond-ing to a single-dot level spacing ¯ hω = ¯ h / ma where m is the effective electron mass ( m ≈ . m e in GaAs).Nearby gate electrodes with applied voltages V L and V R couple capacitively to the dots and can change the po-tential offset of the two dots. We choose the z -axis to beparallel to the interdot axis and assume an external mag-netic field B ext to be applied in the same z -direction (aswas the case in the experiment of Ref. 19). Two nearbyleads, labeled L and R , are tunnel coupled to the twodots and act thereby as electron reservoirs.Part of the few-electron charge stability diagram ofsuch a double dot is shown in Fig. 1b. In the plane( V L , V R ) regions of stable charge configuration ( n L , n R )have a hexagonal shape, where n L ( R ) denotes the numberof excess electrons on the left(right) dot. The regimewe want to investigate in this work is the shaded trian-gle close to the (1 , ,
2) boundary. Here the groundstate of the system is a (0 ,
2) charge state, but excited(1 ,
1) states are also stable in the sense that they can-not decay through sequential tunneling processes like(1 , → (0 , → (0 , ,
1) or (0 ,
2) charge configuration.The single-dot orbital level spacing ¯ hω ∼ meV ∼
10 Kis typically comparable to the charging energy of thedot, i.e., the Coulomb energy it costs to add an ex-tra electron to the dot. This is such a large energyscale that we can disregard all electronic states involv-ing higher orbital states and focus on electrons in theorbital ground state. Explicitly, the two-dimensionalground state wave functions in the left and right potentialwells read ψ L ( r ) = (1 / √ πa ) exp {− [ x + ( z + d ) ] / a } and ψ R ( r ) = (1 / √ πa ) exp {− [ x + z ] / a } . Using aHund-Mulliken approach we orthonormalize these twowave functions, resulting in the basis states Ψ L ( R ) ( r ) = ψ L ( R ) ( r ) − gψ R ( L ) ( r ) (cid:112) − gs + g , (1)with the factor g = (1 − √ − s ) /s and the overlap in-tegral s = (cid:82) d r ψ ∗ L ( r ) ψ R ( r ) = exp {− d / a } .Including spin into the picture, we can construct fromthese basis states a (1 ,
1) spin-singlet state | S (cid:105) , three(1 ,
1) spin-triplet states | T (cid:105) and | T ± (cid:105) , and a (0 ,
2) spin-singlet state | S (cid:105) . We assume that in the basis spannedby these five states the Hamiltonian describing the ki-netic and potential energy of the two electrons as well astheir Coulomb interaction can be written as ˆ H = ˆ H + ˆ H t , (2)ˆ H = B ext {| T − (cid:105) (cid:104) T − | − | T + (cid:105) (cid:104) T + |} − ε | S (cid:105) (cid:104) S | , (3)ˆ H t = t {| S (cid:105) (cid:104) S | + | S (cid:105) (cid:104) S |} , (4)where we included a Zeeman term coupling to the spin ofthe two electrons. The field B ext is written in units of en-ergy (the sign chosen reflects the fact that the g -factor inGaAs is negative, so that a positive B ext corresponds toa positive field along the z -axis) and ε describes the de-tuning between | S (cid:105) and the unpolarized (1 ,
1) states. Increasing ε can be effected by changing V L and V R asindicated with the arrow in Fig. 1b.In Fig. 1c we plot the spectrum of ˆ H as a function ofthe detuning ε , where we chose t = 0 . B ext ). Our regime of interest (the gray area inFig. 1b) is where | S (cid:105) is the ground state, indicated withthe red dotted rectangle in Fig. 1c. We assume that inthis regime ε (cid:29) t , so that the tunneling Hamiltonian ˆ H t can be treated as a perturbation. An excited (1 ,
1) statecan only decay to the (0 ,
2) ground state if its wave func-tion is a spin-singlet (or contains a singlet component). Ifthe system is in a pure (1 ,
1) spin-triplet, it cannot decayand stays blocked in the excited state.We now focus on such a spin-blocked situation and as-sume that the system is initially in | T + (cid:105) . The purposeis to calculate the relaxation rate from | T + (cid:105) to | S (cid:105) (seethe wiggly line in Fig. 1c). For this we need two in-gredients, which we will investigate in detail in the nexttwo Sections: (i) We need a perturbation ˆ H sm with a fi-nite matrix element between the states | T + (cid:105) and | S (cid:105) .(ii) The energy difference between initial and final state E T + − E S ≡ ∆ has to be dissipated by the environmentof the double dot, for which we assume the coupling toacoustic phonons to be responsible. III. SPIN-MIXING PERTURBATIONS
As noted before, | T + (cid:105) and | S (cid:105) are orthogonal in spinspace, and the perturbation ˆ H sm thus has to be of aspin-mixing nature. We will investigate two such per-turbations: (i) spin-orbit interaction, which mixes thespin and orbital parts of the electrons’ wave functions,and (ii) a magnetic field gradient over the double dot,i.e., a difference between the effective magnetic fields atthe positions of the left and right dot. Below we will in-troduce the two perturbations and calculate the resultingmatrix elements coupling | T + (cid:105) and | S (cid:105) . A. Spin-orbit interaction
Spin-orbit interaction perturbs single-particle states inthe two dots resulting in mixed spin-orbital eigenstatesinstead of pure spin states. Thereby it can give rise to“spin-flip” tunnel coupling of states with apparent oppo-site spin.
For each electron the spin-orbit Hamil-tonian readsˆ˜ H so = α ( − ˆ p ˜ y ˆ σ ˜ x + ˆ p ˜ x ˆ σ ˜ y ) + β ( − ˆ p ˜ x ˆ σ ˜ x + ˆ p ˜ y ˆ σ ˜ y ) , (5)where ˆ p is the momentum of the electron and ˆ σ x,y,z arethe three Pauli matrices. The first term in (5) is the FIG. 2: The orientation of our xz -plane with respect to the(100) and (010) crystallographic directions. so-called Rashba term and the second the Dresselhausterm. For typical 2DEG’s in GaAs the correspondingspin-orbit length l so , i.e. the distance an electron has totravel to have its spin rotated by ∼
1, is of the order l so ∼ µ m, usually much larger than the size of thedots. The ratio of the two parameters, α/β , dependson the detailed confining potential of the 2DEG and canin practice be smaller as well as larger than 1.The spin-orbit Hamiltonian (5) is written such thatthe ˜ x -, ˜ y -, and ˜ z -axes point respectively along the (100),(010), and (001) crystallographic axes, and it is assumedthat the 2DEG lies in the ˜ x ˜ y -plane. Transforming thisHamiltonian to the coordinate system of Fig. 1 we findˆ H so = α [ − ˆ p x ˆ σ z + ˆ p z ˆ σ x ]+ β [ˆ p z (ˆ σ x sin 2 χ − ˆ σ z cos 2 χ )+ ˆ p x (ˆ σ x cos 2 χ + ˆ σ z sin 2 χ )] , (6)where χ is the angle between the double-dot axis and the(100) crystallographic direction, see Fig. 2.In the two-electron position representation, the basisstates | T + (cid:105) and | S (cid:105) read explicitly (cid:104) r , | T + (cid:105) = √ (cid:8) Ψ L ↑ ( r )Ψ R ↑ ( r ) − Ψ R ↑ ( r )Ψ L ↑ ( r ) (cid:9) , (cid:104) r , | S (cid:105) = √ (cid:8) Ψ R ↑ ( r )Ψ R ↓ ( r ) − Ψ R ↓ ( r )Ψ R ↑ ( r ) (cid:9) , where the Ψ( r ) are now in fact two-component spinors.Using that ˆ p = − i ¯ h∂ r we find straightforwardly thatonly terms with ˆ p z have a non-vanishing matrix elementbetween | T + (cid:105) and | S (cid:105) , the total matrix element reading T so = (cid:104) S | ˆ H so | T + (cid:105) = − i ¯ h d a s √ − s ( α + β sin 2 χ ) . (7)We see that T so depends non-trivially on the angle χ . Fora double dot with its interdot axis pointing in the crystal(110) direction the two spin-orbit terms add construc-tively and T so ∝ ( α + β ), whereas dots with the interdotaxis along the (¯110) direction have T so ∝ ( α − β ).We also note here that the direction of the externalfield B ext can have a great influence on the effectivenessof the coupling. For instance, for a double dot with theinterdot axis constructed along the (110) direction (i.e., χ = π/ H so = α ( − ˆ p x ˆ σ z + ˆ p z ˆ σ x ) + β (ˆ p z ˆ σ x + ˆ p x ˆ σ z ) . (8)If the external field now points in the x -direction, thenall terms with ˆ p z come with ˆ σ x , which cannot provide aspin-flip. In this case we have thus T so = 0.There is however another, second-order, spin-orbit me-diated process which can take place: ˆ H so couples theground state | T + (cid:105) to a (1 ,
1) spin-singlet which involvesan excited orbital state. This excited state can be as-sumed to be coupled to | S (cid:105) with a coupling energy of ∼ t . The spin-flip now takes place inside one of the dots,and does not depend on the orientation of the interdotaxis. An estimate of the magnitude of the resulting ef-fective matrix element coupling | T + (cid:105) to | S (cid:105) gives T so , ∼ ( α + β ) ¯ ha t ¯ hω , (9)If we assume that t ∼ µ eV and ¯ hω ∼ meV, then a com-parison with the T so found above yields that the “direct”coupling dominates as long as d/a < ∼ Ofcourse, if T so = 0 (due to a special orientation of the in-terdot axis and B ext ) then T so , still provides a spin-orbitassisted spin-mixing coupling between | T + (cid:105) and | S (cid:105) . B. Magnetic field gradient
A magnetic field gradient over the two dots generallymixes all four (1,1) states. All triplet states acquire aspin singlet component, and thus are coupled to | S (cid:105) bythe tunneling Hamiltonian ˆ H t . Such gradients could bedue to a deliberately fabricated on-chip micromagnet or to the effective magnetic fields set up by the randomlyfluctuating nuclear spins of the host material. The Hamiltonian describing the coupling of two dif-ferent effective magnetic fields δ B L and δ B R to the twoelectrons readsˆ H gr = − δ B L · ˆ S L − δ B R · ˆ S R , (10)where we again have set gµ B = − S L ( R ) is the (dimensionless) spin operator for the electronin the left(right) dot. In the basis of the (1 ,
1) singlet andtriplet states this Hamiltonian reads ˆ H gr = (cid:88) ± (cid:26) − δB ± s √ | T (cid:105) (cid:104) T ± | ± δB ± a √ | S (cid:105) (cid:104) T ± | + H . c . (cid:27) − δB zs (cid:8) | T + (cid:105) (cid:104) T + | − | T − (cid:105) (cid:104) T − | (cid:9) − δB za (cid:8) | S (cid:105) (cid:104) T | + | T (cid:105) (cid:104) S | (cid:9) , (11)where we used the notation δB ± s,a = δB xs,a ± iδB ys,a . Thesymmetric and antisymmetric fields we used are definedas δ B s = ( δ B L + δ B R ) and δ B a = ( δ B L − δ B R ).We assume that the fields δB L and δB R are muchsmaller than the externally applied field. Then we canuse first-order perturbation theory to find the singletadmixture in | T + (cid:105) caused by ˆ H gr , which yields | T + (cid:105) ≈| T + (cid:105) − ( δB + a / √ B ext ) | S (cid:105) . Therefore, the spin-flip ma-trix element due to the perturbation ˆ H t + ˆ H gr reads T gr = (cid:104) S | ˆ H t + ˆ H gr | T + (cid:105) = − t δB + a √ B ext . (12) If the gradients are caused by randomly fluctuating nu-clear spins, the typical magnitude of the effective fields isapproximately δB L,R ∼ A/ √ N , where A is the material-specific hyperfine coupling energy and N is the numberof nuclei in each dot. For GaAs dots A ∼ µ eV andtypically N ∼ , implying that δB L,R is in the regimeof 1–5 mT, which has been confirmed experimentally. IV. ELECTRON-PHONON COUPLING:SINGLE-PHONON PROBABILITY
Both perturbations outlined above provide effectivelya coupling between | T + (cid:105) and | S (cid:105) and can thus causerelaxation. The only ingredient still missing to fully de-scribe transitions between the two levels, is a mechanismdissipating the energy difference ∆ between initial and fi-nal state, typically 50–500 µ eV. We assume that this en-ergy is absorbed by the (acoustical) phonon bath in thehost material. Often the contribution to the electron-phonon coupling from the deformation potential is ne-glected, which is generally justified for phonons with en-ergies below ∼
10 K. However, phonons with a wavevector larger than the inverse in-plane dot size 1 /a areemitted almost exclusively in the y -direction (perpendic-ular to the plane of the 2DEG), and emission of piezoelec-tric phonons in this direction is strongly suppressed bythe crystalline anisotropy. Since for a typical GaAs-hosted double quantum dot system a phonon wave vectorof 1 /a corresponds to an energy of ∼ µ eV, we willkeep in our calculation both the coupling to piezoelectricphonons and deformation phonons.The Hamiltonian describing the coupling between theelectrons (density operator ˆ ρ ) and phonons (creation andannihilation operators ˆ a ( † ) ) readsˆ H e-ph = (cid:88) q ,p λ q ,p ˆ ρ q [ˆ a q ,p + ˆ a †− q ,p ] , (13)where λ q ,p are the coupling matrix elements and ˆ ρ q = (cid:82) d r e − i q · r ˆ ρ ( r ) is the Fourier transform of the electronicdensity operator. The sum runs over all allowed phononwave vectors q and includes three polarizations (one lon-gitudinal and two transversal), labeled by p = l, t , t .We will neglect the mismatch of phonon velocities at theGaAs-AlGaAs interface, and treat the phonon bath asthat of bulk GaAs.The matrix elements for electron-phonon coupling read λ q ,p = M ( p )ph (cid:115) ¯ h ρ V ω q ,p , (14)where ρ is the mass density ( ρ = 5 . × kg/m forGaAs), V the normalization volume, and we will assumethat the phonons have an isotropic linear dispersion re-lation at all energies of interest, i.e., ω q ,p = v p q , with v p the polarization-dependent sound velocity.The constant M ( p )ph = M ( p )pe + M def contains a contri-bution from both types of electron-phonon coupling, M ( p )pe = ieh A q ,p , (15) M def = Ξ q δ p,l , (16)where the coupling to the deformation potential only in-volves longitudinal phonons, expressed by the δ -functionin (16). We used here the piezo-electric constant, h =1 . × V/m in GaAs, and the deformation poten-tial, which is Ξ = 13 . The coupling topiezoelectric phonons involves the anisotropy factors A q ,p = 2 q (cid:104) q ˜ x q ˜ y e ( p )˜ z + q ˜ z q ˜ x e ( p )˜ y + q ˜ y q ˜ z e ( p )˜ x (cid:105) , (17)where e ( p ) is the unit polarization vector for the polariza-tion p . The factors as written in (17) are in the coordinatesystem of the crystal structure, i.e. the ˜ x -direction along(100), ˜ y along (010), and ˜ z along (001). We would liketo relate the factors to the coordinate system of Fig. 1a.We thus write in terms of the spherical coordinates of q A q ,l = 9 cos ( θ ) sin ( θ ) sin (2 φ + 2 χ ) , (18) A q ,t = [1 + 3 cos(2 θ )] sin ( θ ) sin (2 φ + 2 χ ) , (19) A q ,t = sin (2 θ ) cos (2 φ + 2 χ ) , (20)where θ = 0 corresponds to q parallel to our y -axis, and φ gives the azimuthal angle of q in our xz -plane. Theangle χ is the angle between the double dot axis and thecrystallographic (100) direction: A wave vector q withgiven φ thus has an azimuthal angle φ + χ in the crystal’scoordinate system (see Fig. 2).We see that we can write | λ q ,p | = ¯ h π v p q V (cid:16) g ( p )pe A q ,p + g def q δ p,l (cid:17) , (21)with the two dimensionless coupling constants, g ( p )pe ≡ ( eh ) π ¯ hρv p and g def ≡ Ξ π ¯ hρv l a , (22)the latter being dependent on the dot size a .The relaxation rate Γ of the excited (1 ,
1) triplet stateto the (0 ,
2) ground state will be calculated using a secondorder Fermi’s golden rule,Γ = (cid:88) f π ¯ h (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) v (cid:104) f | ˆ H (cid:48) | v (cid:105)(cid:104) v | ˆ H (cid:48) | i (cid:105) E i − E v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( E f − E i ) , (23)where ˆ H (cid:48) = ˆ H e-ph + ˆ H sm , with ˆ H sm being one (or both) ofthe spin-mixing Hamiltonians presented in Sec. III. Theenergy difference ∆ between initial state | T + (cid:105) and finalstate | S (cid:105) (which is equal to the energy of the emittedphonon) is assumed much larger than the temperatureand we therefore take as initial state a direct productof | T + (cid:105) and the phonon vacuum | vac (cid:105) , and as final state | S (cid:105) ⊗ | q ,p (cid:105) , where one phonon with wave vector q andpolarization p has been created.From the explicit wave functions of | T + (cid:105) and | S (cid:105) we calculate the diagonal matrix elements (cid:104) T + | ˆ H e-ph | T + (cid:105) and (cid:104) S | ˆ H e-ph | S (cid:105) , and find Γ = (cid:88) q ,p π ¯ h | T λ q ,p | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F q ∆ + F q + e i q · d F ∗ q − ¯ hω q ,p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ (¯ hω q ,p − ∆) , (24)where the spin-mixing matrix element T = (cid:104) S | ˆ H sm | T + (cid:105) and we used the Fourier transform of the squared elec-tronic wave function F q = (cid:82) d r e − i q · r | Ψ R ( r ) | . We canevaluate this Fourier transform explicitly and, anticipat-ing that the δ -function enforces ¯ hω q ,p = ∆, we writeΓ = 2 π ¯ h − s (cid:88) q ,p | T | ∆ | λ q ,p | e − a ( q x + q z ) × sin (cid:0) q z d (cid:1) δ (¯ hω q ,p − ∆) . (25)The exponential function exp {− a ( q x + q z ) } suppressesthe contribution from phonons having a wave vector within-plane components larger than the inverse system size1 /a . Indeed, the electronic density profile F q is expo-nentially small for these wave vectors. The sine functionsin( q z d ) describes the interference between the cou-pling to an electron in the left and right dot: A phononwave with given wave vector q has a phase difference δφ = q z d between the two dot positions. We convert the sum over q into an integral and thenfinally find that we can write for the relaxation rateΓ = 2 π ¯ h | T | P (∆) , (26)where the function P (∆) gives the total probability thatthe energy ∆ is absorbed by a single phonon, either bypiezoelectric coupling or by coupling to the deformationpotential. Alternatively, one could call this function theeffective phonon density of states for the (1 ,
1) triplet to(0 ,
2) singlet charge transition.The total single-phonon probability is the sum of thecontributions from the different types of phonons, P (∆) = P , def (∆) + (cid:88) p P ( p )1 , pe (∆) . (27)For the piezoelectric phonons we find P ( p )1 , pe (∆) = g ( p )pe ∆ (cid:90) π/ dθ sin θ − s f p (cid:18) ∆ d sin θ ¯ hv p (cid:19) × exp (cid:40) − ∆ a ¯ h v p sin θ (cid:41) , (28)where the dimensionless functions f p ( x ) are f l ( x ) = 92 cos ( θ ) sin ( θ ) g − ( x ) , (29) f t ( x ) = 18 [1 + 3 cos(2 θ )] sin ( θ ) g − ( x ) , (30) f t ( x ) = 12 sin (2 θ ) g + ( x ) , (31)in terms of the function g ± ( x ) = 1 − J ( x ) ± (cid:18) x − (cid:19) cos(4 χ ) J ( x ) ± (cid:18) x − x (cid:19) cos(4 χ ) J ( x ) , (32)with J n ( x ) the n -th order Bessel function of the first kind.The contribution from the coupling to the deformationpotential reads similarly P , def (∆) = g def ∆ (cid:90) π/ dθ sin θ − s (cid:20) − J (cid:18) ∆ d sin θ ¯ hv l (cid:19)(cid:21) × ∆ a ¯ h v l exp (cid:40) − ∆ a ¯ h v l sin θ (cid:41) . (33)The remaining integral over the polar angle θ has tobe evaluated numerically. We can however arrive at an-alytical results in the limits of small and large phononenergies. For small energies, meaning ¯ hv/ ∆ (cid:29) a, d , thesingle-phonon probabilities P can be expanded in ∆.Setting v t = v t ≡ v t we then find to leading order P ( l )1 , pe (∆) = 6105(1 − s ) g ( l )pe ∆ (cid:18) d ∆¯ hv l (cid:19) , (34) P ( t )1 , pe (∆) = 8105(1 − s ) g ( t )pe ∆ (cid:18) d ∆¯ hv t (cid:19) , (35) P , def (∆) = 16(1 − s ) g def ∆ (cid:18) a ∆¯ hv l (cid:19) (cid:18) d ∆¯ hv l (cid:19) , (36)where P ( t )1 , pe = P ( t , pe + P ( t , pe . We find for small energiesa linear piezoelectric P , pe (∆) and a cubic deformation P , def (∆), meaning that the phonon bath is superohmicin this setup. The result for the piezoelectric phononsagrees up to a prefactor with previous calculations of thephonon density of states for the (1 ,
0) to (0 ,
1) chargetransition where all anisotropy factors were set to one. In the opposite limit of large energies, ¯ hv/ ∆ (cid:28) a , wefind qualitatively different results compared to Ref. 29.We see from the exponential factors in (28) and (33)that in this regime only very small angles θ are rele-vant. Indeed, in this case only the confinement in the y -direction is strong enough to create an electronic den-sity profile with non-vanishing Fourier components of theorder ∼ ∆ / ¯ hv , and phonon emission takes place almostexclusively in the y -direction. For the piezoelectric cou-pling the anisotropy factors A q ,p now bring in small fac-tors of θ which cannot be ignored (see also Ref. 14). Since in this limit A l ∝ θ and A t , A t ∝ θ , we expect thedominating piezoelectric contribution for large energiesto come from transversal phonons.To evaluate the single-boson probabilities in this limit,we expand sin θ ≈ θ and extend the range of integrationover θ from 0 to ∞ . Then we find that to leading order P ( t )1 , pe (∆) = (cid:20) d a s (1 − s ) (cid:21) g ( t )pe ∆ (cid:18) ¯ hv t a ∆ (cid:19) , (37) P , def (∆) = g def . (38)The contribution from longitudinal piezoelectric phononsis P ,l ∼ ( g ( l )pe / ∆)(¯ hv l /a ∆) , which is much smaller than P ( t )1 , pe and therefore ignored. The large-energy result (37)for the piezoelectric phonons is qualitatively differentfrom the results presented in Ref. 29, which predictedthat P (∆) ∼ ( g pe / ∆)(¯ hv/a ∆) , the difference arisingfrom the inclusion of the anisotropy factors.We see from (37) and (38) that the contribution fromdeformation phonons becomes important when ∆ √ a ∼ FIG. 3: The total single-phonon probability P = P , pe + P , def as a function of ∆. The phonon energy ∆ is plotted inunits of ∆ a ≡ ¯ hv t /a and P in units of P a ≡ a ( eh ) / ¯ h v t ρ .For all plots we have set v l /v t = 1 .
73 and Ξ /aeh = 0 . P (∆) at χ = 0 for three different size ratios d/a . (inset) The dependence of the maximum P for d/a = 5on the angle χ . (b) The total P (∆), as well as the separatecontributions from piezoelectric and deformation phonons, for χ = 0 and d/a = 5 on logarithmic scales. The expected powerlaws are included as guides to the eye. ¯ hv t (cid:112) v l eh /v t Ξ, which is approximately 1 . · nm / using realistic parameters for GaAs. For a dot size of a = 20 nm (an orbital level spacing of ¯ hω ≈ . ∼ µ eV,which indeed lies inside our regime of interest.In Fig. 3 we plot the single-phonon probability P (∆)for typical double-dot parameters. In all plots ∆ is renor-malized to units of ∆ a ≡ ¯ hv t /a and P is plotted in unitsof P a ≡ a ( eh ) / ¯ h v t ρ . For v t = 3 . × m/s and a = 20 nm we find ∆ a = 99 µ eV. The parameter v l /v t was set to 1 .
73 and the ratio Ξ /aeh to 0 .
50. In Fig. 3awe show the total P at χ = 0 for three different size ra-tios d/a . The maximum of P always occurs on the scale∆ ∼ ∆ a , where the wave length of the emitted phonon iscomparable to the system’s in-plane dimensions. At lowenergies P is approximately linear and at high energiesit is suppressed, ultimately being dominated by the de-formation contribution making P ∝ ∆ − . In Fig. 3b weplot the total P as well as the two separate contribu-tions on logarithmic scales (for χ = 0 and d/a = 5), andwe added guides to the eye corresponding to the powerlaws expected in the different limits. The blue dottedline shows the piezoelectric contribution. Up to a few∆ a this contribution indeed dominates, being linear atvery small energies. For ∆ > ∼ ∆ a it becomes suppressedas ∝ ∆ − and at higher energies the dominating contri-bution comes from the coupling to the deformation po-tential, the green dotted line (see also Eqs. 37 and 38).The inset of Fig. 3a shows the dependence of the max-imum of P on the angle χ (for d/a = 5). We see thatthe density of states depends on the relative orientationof the double dot axis and the (100) crystal direction, itsvariation being however only ∼ V. SPIN-FLIP CHARGE RELAXATION RATE
Now we can use Eqs. (26)–(32) to evaluate explicit re-laxation rates for the T + → S transition. For a specificexperimental setup one can estimate the relative magni-tude of the matrix elements given in Eqs. (7) and (12),and decide which process dominates. Here, we will focuson the case of a large external magnetic field, such aswas the case in the experiments of Ref. 19. We assumethat B ext is large enough so that | T so | (cid:29) | T gr | . In thatcase the relaxation from | T + (cid:105) to | S (cid:105) is mainly causedby spin-orbit interaction.We thus use T so in (26) and take typical materialparameters for GaAs. For the dot size we take again a = 20 nm and the interdot distance is set five times aslarge, d = 100 nm. In the experiment of Ref. 19 the inter-dot axis was fabricated along the crystal (110) direction,so we set χ = π/
4. For this angle the two spin-orbit termsadd constructively, and we choose α = β = 100 m/s,such that l so ≡ ¯ h/m ( α + β ) ≈ . µ m. In Fig. 4a we plotthe resulting relaxation time T rel = Γ − , which is foundto be typically T rel ∼ µ s, the order of magnitude ofwhich agrees with experimental observations. The in-
FIG. 4: Relaxation time T rel from | T + (cid:105) to | S (cid:105) when relax-ation is mediated by spin-orbit interaction. To make thisplot we used v l = 5 . × m/s, v t = 3 . × m/s, h = 1 . × V/m, ρ = 5 . × kg/m , χ = π/
4, and α = β = 100 m/s. (a) T rel for a = 20 nm and d = 100 nm.(inset) The relaxation rate Γ for the same parameters calcu-lated at ∆ = 50 µ eV as a function of χ . (b) T rel as a functionof d/a at two different energies for fixed a = 20 nm (solidlines) and d = 150 nm (dashed lines). set to Fig. 4a shows the dependence of the relaxation rateΓ on the angle χ at ∆ = 50 µ eV, close to the minimumrelaxation time. We see that the rate indeed vanishesfor the angles χ = 3 π/ , π/
4, where the Rashba andDresselhaus terms add destructively.In Fig. 4b we show how the relaxation time depends onthe size ratio d/a , for two different energies ∆ = 50 µ eV(red lines) and ∆ = 250 µ eV (blue lines). The dashedlines have a fixed interdot distance d = 150 nm andthe solid lines have a fixed dot radius a = 20 nm. Allother parameters are the same as in Fig. 4a. We seethat a large size ratio d/a suppresses the relaxation effi-ciently: For widely separated dots the overlap of the twosingle-dot wave functions becomes exponentially small,and this suppresses the matrix element T so . In the limitof strongly overlapping wave functions, i.e. d/a going to-wards 1, we see that relaxation is much more efficientfor the smaller system with d = a = 20 nm. Indeed, for d = a = 150 nm we find ∆ a ≈ µ eV, so in this case bothenergies are larger than ∆ a where the electron-phononcoupling matrix elements are suppressed.For smaller external magnetic fields, or other angles χ ,one could be in the situation where the spin-orbit andfield-gradient give rise to matrix elements of the sameorder of magnitude, | T so | ∼ | T gr | . In this case one hasto use in (26) the total matrix element T tot = T so + T gr ,which possibly includes interference terms between thetwo mechanisms, T tot = − i (cid:26) t − iδB xa + δB ya √ B ext + ¯ hds ( α + β sin 2 χ )4 a √ − s (cid:27) . (39)We see that the spin-orbit mechanism interferes with the y -component of the difference field δ B a . By tuning δB ya or B ext one could thus enhance or counteract the spin-fliptunneling enabled by spin-orbit interaction. One word ofcaution is however required here: If the field gradientsare caused by the effective hyperfine fields, then the fi-nal state | f (cid:105) in (23) is different for a spin-orbit and ahyperfine mediated transition. Indeed, in the course ofhyperfine induced spin-flip tunneling the spin of one ofthe nuclei is raised by ¯ h , which does not happen duringa spin-orbit mediated transition. In that case one has tocalculate separately the two contributions to the relax-ation rate (26) or, equivalently, use | T | = | T so | + | T gr | . VI. CONCLUSION
We have studied the (1 ,
1) triplet to (0 ,
2) singlet re-laxation rate in a lateral gate-defined double quantum dot tuned to the Pauli spin blockade regime. We first de-rived an effective phonon density of states P (∆) for thischarge transition, and found that at small energies P islinear in energy, ∝ ∆, and dominated by the piezoelec-tric electron-phonon coupling, whereas at large energiesthe P is dominated by the coupling to the deformationpotential and is ∝ ∆ − . Then, we investigated two differ-ent spin-mixing mechanisms coupling the spin triplet andsinglet states: a magnetic field gradient over the doubledot (relevant at low external magnetic field) and spin-orbit interaction (relevant at high field). We showed howthe spin-orbit-mediated coupling depends on the devicegeometry as well as on the in-plane direction of the ap-plied magnetic field. We finally combined all results andtook realistic system parameters to evaluate the explicitdetuning-dependent relaxation rate, which we found tobe of the order of ∼ MHz.
Acknowledgments
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1) triplet states can be ne-glected, which is usually realistic when d > ∼ a . J. Danon and Y. V. Nazarov, Phys. Rev. B , 041301(2009). S. Nadj-Perge, S. M. Frolov, J. W. W. van Tilburg,J. Danon, Y. V. Nazarov, R. Algra, E. P. A. M. Bakkers,and L. P. Kouwenhoven, Phys. Rev. B , 201305 (2010). K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, andL. M. K. Vandersypen, Science , 1430 (2007). Of course t is also an (exponential) function of d/a anda more detailed comparison of t/ ¯ hω and s/ √ − s re- veals that the direct matrix element T so probably always dominates. In the text however we decided for clarity tofix t/ ¯ hω to a realistic value and then give the range of d/a where the coupling is mostly direct. O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. ,176804 (2006). S. Vorojtsov, E. R. Mucciolo, and H. U. Baranger, Phys.Rev. B , 205322 (2005). H. Bruus, K. Flensberg, and H. Smith, Phys. Rev. B ,11144 (1993). The same result follows for the T − → S and the T → S transitions. G. Granger, D. Taubert, C. E. Young, L. Gaudreau,A. Kam, S. A. Studenikin, P. Zawadzki, D. Harbusch,D. Schuh, W. Wegscheider, Z. R. Wasilewski, A. A. Clerk,S. Ludwig, and A. S. Sachrajda, Nature Physics8