Relaxation of hole spins in quantum dots via two-phonon processes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Relaxation of hole spins in quantum dots via two-phonon processes
Mircea Trif , Pascal Simon , , and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland and Laboratoire de Physique des Solides, CNRS UMR-8502 University Paris Sud, 91405 Orsay Cedex, France (Dated: November 18, 2018)We investigate theoretically spin relaxation in heavy hole quantum dots in low external magneticfields. We demonstrate that two-phonon processes and spin-orbit interaction are experimentallyrelevant and provide an explanation for the recently observed saturation of the spin relaxation ratein heavy hole quantum dots with vanishing magnetic fields. We propose further experiments toidentify the relevant spin relaxation mechanisms in low magnetic fields.
PACS numbers: 72.25.Rb, 03.65.Yz, 71.70.Ej, 73.21.La
In the last decade, remarkable progress has been madein the manipulation and control of the spin of electronsconfined in semiconducting nanostructures such as quan-tum dots (QD) [1]. These achievements pave the waytoward quantum spin electronics and may lead to spin-based quantum computing [2]. In the past years, a newcandidate for a qubit state has been attracting growinginterest: the spin of a heavy hole (HH) confined in a flatQD. In a bulk semiconductor the HH ( J z = ± /
2) andlight hole (LH) ( J z = ± /
2) bands are degenerate givingrise to strong mixing and thus to strong HH-spin relax-ation. However, in a 2D system the HH and LH bandsare split due to the strong confinement along the growthdirection [3] implying a significant reduction of the HHspin relaxation via HH-LH mixing.The relaxation ( T ) and decoherence ( T ) times of aHH-spin localized in a QD are, like for electrons, deter-mined by the environment the hole spin interacts with:the nuclear spin bath in the QD and the lattice vibra-tions (phonons). The former interaction is weaker forHHs than for electrons (due to the p-symmetry of thehole) [4, 5]. More importantly, it is of Ising type, mak-ing it ineffective for HH-spins initialized along the growthdirection [4], as typically done in experiments [6]. Thisis in contrast to electrons, where the hyperfine interac-tion is isotropic and dominates the spin dynamics at lowB-fields [7, 8, 9, 10].The other relevant source of relaxation are phononswhich couple to the hole spin through the spin-orbit in-teraction (SOI) [11]. The predicted values [11] for theone-phonon induced relaxation time T agree quite wellwith data obtained in high B-fields [12]. However, for lowB-fields ( B ∼ . − T > T for very low or even vanishing B-field [6]. The relaxationtime was found to be unusually long, T ≈ . − B →
0. Theimportance of such two-phonon processes was noticed along time ago for electron spins in silicon-donors [13] andrare-earth ions [14], while for electrons in QDs, they havebeen recently analyzed and shown to be negligible com-pared to nuclear spin effects [15].To describe a HH confined to a QD and interactingwith the surrounding phonon bath, we start with thefollowing Hamiltonian H h = H + H Z + H SO + H h − ph + H ph , (1)where H = p / m ∗ + V ( r ) , is the dot Hamiltonian, V ( r ) ≡ m ∗ ω r / m ∗ being the HH mass.The second term in Eq. (1) is the Zeeman energy ofthe HH (pseudo-) spin H Z = gµ B B · σ /
2, with B be-ing the magnetic field and σ the Pauli matrices for theHH spin defined in the J z = ± / H SO = βp − p + p − σ + + h.c. (2)This Hamiltonian represents the effective DresselhausSOI (restricted to the HH subspace) due to bulk inver-sion asymmetry of the crystal [11], where p ± = p x ± ip y , p = − i ¯ h ∇ − e A ( r ), A ( r ) = ( − y, x, B/
2, and σ ± = σ x ± iσ y . We note that in Eq. (2) we have neglected theRashba SOI and other possibly linear-in-k but small SOIterms [11]. The fourth term in Eq. (1) represents theinteraction of the HH charge with the phonon field, i.e. H h − ph = P q j M q j X q j with M q j = F ( q z ) e i q · r p ρ c ω q j (cid:2) eβ q j − i (Ξ q · d q j − Ξ z q z d z q j ) (cid:3) , (3)and X q j = p ¯ h/ω q j ( a †− q j + a q j ), where q is the phononwave-vector, with j denoting the acoustic branch, ω q j = c j q the phonon energy, with c j the speed of sound in the j -th branch, d q j the polarization unit vector, ρ c the sam-ple density (per unit volume), and eβ q j the piezoelectricelectron-phonon coupling and Ξ ,z the deformation po-tential constants [11]. The form factor F ( q z ) in Eq. (3)equals unity for | q z | ≪ d − and zero for | q z | ≫ d − , with d being the dot size in the (transverse) z-direction. Thelast term in Eq. (1) describes the free phonon bath.In the following, we analyze the effect of the phononson the HH spin, both in the low and high temperatureregimes. The phonons do not couple directly to the spin,but the SOI plays the role of the mediator of an effectivespin-phonon interaction. Under the realistic assumptionthat the level splitting in the dot is much larger than theHH-phonon interaction, we can treat H h − ph in perturba-tion theory.Let us define the dot Hamiltonian H d ≡ H + H Z + H SO and denote by | nσ i , the eigenstates of H d (where n labelsthe orbital and σ the spin states). The product states | n i| σ i are then the eigenstates of H d in the absence of SOI( i.e. for H SO = 0). These states are formally connectedto | nσ i by an exact Schrieffer-Wolff (SW) transformation[16, 17], i.e., | nσ i = e S | n i| σ i , where S = − S † is the SWgenerator and can be found in perturbation theory inSOI. After this transformation, any operator A in theold basis transforms as A → e A = e S Ae − S in the newbasis (e.g., H d → e H d , H h − ph → e H h − ph , etc. ).Let us now derive the effective spin-phonon interactionunder the above assumptions. To do so, we perform an-other SW transformation of the total HH Hamiltonian H h to obtain an effective Hamiltonian H eff = e T H h e − T ,where T = − T † is chosen such that it diagonalizes theHH-phonon Hamiltonian e H h − ph in the eigenbasis of H d .In lowest order in H h − ph , we obtain T ≈ e L − d e H h − ph ,where the Liouvillean is defined as e L d A = [ e H d , A ], ∀ A ,and diagonal terms of H h − ph are to be excluded. In 2ndorder in H h − ph , we obtain then the effective spin-phononHamiltonian H s − ph = σ · X q j, q ′ j ′ (cid:2) δ q j, q ′ j ′ C (1) q j X q j + C (2) q j, q ′ j ′ X q j X q ′ j ′ + C (3) q j, q ′ j ′ (cid:0) P q j X q ′ j ′ − P q ′ j ′ X q j (cid:1)(cid:3) , (4)with σ · C (1) q j = h | f M q j | i , σ · C (2) q j, q ′ j ′ = h | [ e L − d f M q j , f M q ′ j ′ ] | i , σ · C (3) q j, q ′ j ′ = h | [ e L − d f M q j , e L − d f M q ′ j ′ ] | i , P q j = i p ¯ hω q j ( a †− q j − a q j ) is the phonon field momentumoperator, and | i is the orbital ground state. In Eq. (4)we have neglected spin-orbit corrections to the energylevels, being 2nd order in SOI. Note that for vanishingmagnetic field B → H s − ph is the last one since this is the only onewhich preserves time-reversal invariance and thus givesrise to zero field relaxation (ZFR) [13, 14, 15]. Quiteremarkably, this term is 1st order in SOI, whereas forelectrons it is only 2nd order [15]. This is one of the main reasons why eventually two-phonon processes aremuch more effective for HHs than for electrons.We now assume the orbital confinement energy ¯ hω much larger than the SOI, i.e. || H || ≫ || H SO || , so thatwe can treat the SOI in perturbation theory. We consideralso the B -field to be applied perpendicularly to the dotplane (as in Refs. [6, 12]). We limit our description tofirst order effects in SOI. The SW-generator S can bewritten as S = S + σ − − h.c. , and we then find S + = A p + p − p + + A [ p + p − P + − ( p + P − − P + p − ) p + ]+ A P + P − P + + A [( p + P − − P + p − ) P + + P + P − p + ] . (5)Here, A i ≡ A i ( ω Z , ω c ) with ω Z = gµ B B/ ¯ h and ω c = eB/ c . For ω Z , ω c ≪ ω , we obtain A ≈− (7 β/ h )(( ω Z + ω c ) /ω ), A ≈ − ( β/ h )( ω c /ω ), A ≈− (2 β/ h )( ω c ( ω c + ω Z ) /ω ), and A ≈ (2 β/ h )( ω c /ω ),while P ± = P x ± iP y with P x ( y ) = − i ¯ h ∇ x ( y ) ± ( m ∗ ω /ω c ) y ( x ). After somewhat tedious calculations, weobtain analytic expressions for C ( i ) = ( C ( i,x ) , C ( i,y ) , i = 3, the rest being too lengthy to bedisplayed here: C (3 ,x/y ) q j, q ′ j ′ = ± M q ′ j ′ q j m ∗ βe − b / λ d ¯ hω F ( b · b ′ ) × (cid:16) b y b ′ x − b ′ y b x ± ( b x − b ′ x )(2 b y b ′ y + 3 b x b ′ x ) (cid:17) , (6)where F ( b · b ′ ) = 1( b · b ′ ) (cid:0) e − b · b ′ / − b · b ′ / (cid:1) × (cid:0) γ + log( b · b ′ /
2) + Γ(0 , b · b ′ / (cid:1) . (7)Here, M q ′ j ′ q j = ( F ( q z ) F ( q ′ z )¯ h/ ρ c √ ω q j ω q ′ j ′ )(Ξ q · d q ,j − Ξ z q z d z q j )(Ξ q ′ · d q ′ j ′ − Ξ z q ′ z d z q ′ j ′ ) and b = q λ d , where λ d is the dot-diameter. We have also introduced γ ≈ . s, x ) the incomplete gammafunction. We note that C (1 , ∝ B , so that these twoterms vanish with vanishing B-field.Let us now analyze the relaxation of the spin inducedby all the phonon processes in the spin-phonon Hamilto-nian in Eq. (4). We first mention that all terms in Eq.(4) can be cast in a general spin-boson type of Hamil-tonian H ps − b = (1 / gµ B δ B p ( t ) · σ , p = 1 , ,
3, with thecorresponding identification of the fluctuating magneticfield terms δB j ( t ) from Eq. (4) (e.g. δ B ( t ) ∼ C (1) q j X q j ).Within the Bloch-Redfield approach, the relax-ation rate Γ ≡ /T can be expressed as Γ = P i = x,y [ J ii ( E Z / ¯ h ) + J ii ( − E Z / ¯ h )]. The correlation func-tions J ij are defined by J ij ( ω ) = ( gµ B / h ) R ∞ dte − iωt <δB i (0) δB j ( t ) > , where < · · · > denotes the average overthe phonon bath, assumed to be in thermal equilibriumat temperature T. The relaxation time associated withthe three types of spin-phonon processes in Eq. (4) isΓ = P i =1 , , Γ ( i ) withΓ (1) = 4 π ¯ h X q j | C (1) q j | (cid:18) n ( ω q j ) + 12 (cid:19) δ ( E Z − ¯ hω q j ) , Γ ( m ) ≃ π ¯ h X q j, q ′ j ′ | C ( m ) q j, q ′ j ′ | ( ω q j ω q ′ j ′ ) m − n ( ω q j ) × ( n ( ω q ′ j ′ ) + 1) δ (¯ hω q j − ¯ hω q ′ j ′ ) , (8)where n ( ω ) = 1 / (exp ( ω/k B T ) −
1) is the Bose factor and m = 2 , resp . We remark that in Eq. (8) wehave neglected some irrelevant processes in the limit oflow-B field [18]. Also, for B -fields perpendicular to thedot plane the decoherence time satisfies T = 2 T for one-and two-phonon processes since the spin-phonon fluctu-ations δ B j ⊥ B [11, 17].Note that for two-phonon processes the single phonon-energies do not need to match the Zeeman energy sep-arately (as opposed to one-phonon processes), so thatthere is only a weak dependence on the B-field left whichcomes from the effective spin-phonon coupling itself.In Figs. 1 and 2, we plot the phonon spin-relaxationrate Γ as a function of the B-field and of temperature,resp., for InAs and GaAs quantum dots. Fig. 1 showsa clear saturation of Γ at low magnetic fields which isdue to two-phonon processes, while Fig. 2 shows theknown saturation at low temperatures due to one-phononprocesses [11].For these plots, we used the following HH InAs QDs(labeled by A) [19, 20] and GaAs QDs (labeled by B)parameters [11]: Ξ = 1 . z = 2 . c At =2 . · m / s ( c Bt = 3 . · m / s), c Al = 3 . · m / s( c Bl = 4 . · m / s), ρ Ac = 5 . · kg / m ( ρ Bc =5 . · kg / m ), m ∗ A = 0 . m e ( m ∗ B = 0 . m e ), g A = 1 . g B = 2 . λ d = 3nm (¯ hω A = 35 meV,¯ hω B = 60 meV) and d = 3 nm (dot height). Also, β A ≈ . · m / s and β B ≈ . · m / s. From Fig.1 we can infer that the two-phonon processes becomedominant for magnetic fields B <
2T (
B < . T >
2K (
T > B -field [21].Next, we provide explicit expressions of the relaxationrates for low and high temperature limits. The rates Γ ( i ) can be written asΓ ( i ) = δ i r i X m =0 ω r i − mZ ω mc ω r i F ( m ) i ( t ) , (9)where δ ≈ π (¯ h eh β /κ m h λ d ρ c c l ), δ ≈ π ( m h β Ξ / ¯ h λ d ρ c c l ), δ ≈ π ( m h β Ξ / ¯ h c l λ d ρ c ), B[T] Γ [ s - ] T=0.85 KT=2.5 KT=5 KT=10 K
B[T] T=1 KT=2 KT=3 KT=8 K
FIG. 1: The heavy-hole spin relaxation rate Γ for InAs QDs(GaAs QDs in the inset) as a function of magnetic field B fordifferent temperatures T. The full curves represent the ratedue to one- and two-phonon processes, i.e. Γ = P i =1 Γ ( i ) as defined in Eq. (8) for different temperatures T, while thedotted lines present the one-phonon rate Γ (1) . r = 5, r = 2, r = 0, and t = k B T /E ph with E ph ≡ ¯ hc l /λ d . The functions F mi ( t ) depend on the F (0)1 F (1)1 F (0)2 ( t ) F (1)2 ( t ) F (2)2 ( t ) F ( t ) t ≪ .
004 0 .
015 10 t t · t t t ≫ . tω Z . tω Z t t t . t TABLE I: The asymptotic values for F ( m ) i ( t ). ratios t = k B T /E ph , d/λ , and c l /c t . In Table I we listthe asymptotic (scaling) expressions for F ( m ) i ( t ) in lowB-fields ω c,Z ≪ ω for low ( t ≪
1) and high ( t ≫ F (1)1 ( t ) ≈ F (2)1 ( t ) in bothregimes, and F (3 , , ≡ (2) = δ (cid:16) ω Z ω + ω Z ω c ω + 0 . ω c ω (cid:17) T , T ≪ E ph (cid:16) ω Z ω + ω Z ω c ω + 0 . ω c ω (cid:17) T , T ≫ E ph Γ (3) = δ (cid:26) T , T ≪ E ph . T , T ≫ E ph . (10)From Eqs. (10) we find that for T <
2K and for
B > .
5T the one-phonon processes dominate the relaxationrate Γ. On the other hand, for low B-fields (0 . < B < T >
T[K] Γ [ s - ] B=0.5 TB=1 TB=3 T
T[K] FIG. 2: The heavy-hole spin relaxation rate Γ in Eq. (8)for InAs QDs (GaAs QDs in the inset) as a function of tem-perature T for different B-field values. For finite B -field, Γsaturates at low temperatures due to one-phonon processes. experimentally by analyzing the temperature dependenceof Γ, scaling as Γ ∼ T for one-phonon processes and asΓ ∼ T for two-phonon processes. Also, the saturation ofΓ in vanishing B-field is a clear indication of two-phononprocesses. Note that the strong enhancement of the two-phonon HH spin relaxation arises because (i) the rate is2nd order in SOI (whereas for electrons it is 4th order)and (ii) the effective mass for HHs is much larger thanthat for electrons.In order to compute Γ (2 , , we took into account onlythe contribution from the deformation potential sincethis dominates the two-phonon relaxation for T /E ph > . ω Z , ω c ≪ ω . For the evaluation of Γ (1) instead,we considered both the piezoelectric and deformation po-tential contributions, both of them being important for B and T considered here. Surprisingly, we found thatthe ZFR rate Γ (3) increases when decreasing the dot sizeas Γ (3) ∼ λ − d , while the other two rates decrease withdecreasing the dot size as Γ (1) ∼ λ d and Γ (2) ∼ λ d . Thisbehavior strongly differs from the electronic case wherethe ZFR mechanism is efficient for rather large dots [15].Interestingly, the present results do not change muchif the B -field is tilted with respect to the QD plane. The g -factor for HHs is strongly anisotropic with g k ≪ g ⊥ sothat one can neglect the in-plane Zeeman splitting. Thisimplies performing the substitution ω c,Z → ω c,Z cos θ inabove results, with θ being the angle between the B -fieldand the z -direction. This will lead to a reduction of the B -dependent rates (Γ (1 , ), while the ZFR (Γ (3) ) beingindependent of B remains the same.In conclusion, we have shown that two-phonon pro-cesses give rise to a strong relaxation of the HH spin in aflat quantum dot. This time is predicted to be in the mil-lisecond range, comparable to the one measured in recentexperiments on optical pumping of a HH spin in QDs [6]. Though other sources of relaxation are not excluded, acareful scaling analysis of the measured relaxation timewith the magnetic field and/or the temperature should al-low one to identify the two-phonon process as the leadingrelaxation mechanism for the heavy-hole spin localized insmall QDs.We thank D. Bulaev, J. Fischer, V. Golovach, and D.Stepanenko for useful discussions. This work was sup-ported by the Swiss NSF, NCCR Nanoscience, and JSTICORP. [1] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,L. M. K. Vandersypen, Rev. Mod. Phys. , 1217 (2007).[2] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998).[3] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Heavy-Hole Systems , STMP,Springer-Verlag, Berlin (2003).[4] J. Fischer, W. A. Coish, D. V. Bulaev, and D. Loss, Phys.Rev. B , 155329 (2008).[5] B. Eble, C. Testelin, P. Desfonds, F. Bernardot, A. Baloc-chi, T. Amand, A. Miard, A. Lemaitre, X. Marie, M.Chamarro, arXiv:0807.0968.[6] B. Gerardot, D. Brunner, P. A. Dalgarno, P. Oehberg, S.Seidl, M. Kroner, K. Karrai, N. G. Stoltz, P. M. Petroff,and R. J. Warburton, Nature, , 441 (2008).[7] A. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. , 186802 (2002).[8] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson,L. H. Willems van Beveren, I. T. Vink, H. P. Tranitz, W.Wegscheider, L. P. Kouwenhoven, L. M. K. Vandersypen,Science 309, 1346 (2005).[9] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A.Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, A.C. Gossard, Science 309, 2180 (2005).[10] P. Maletinsky, A. Badolato, A. Imamoglu, Phys. Rev.Lett. , 056804 (2007).[11] D. Bulaev and D. Loss, Phys. Rev. Lett. , 076805(2005).[12] D. Heiss, S. Schaeck, H. Huebl, M. Bichler, G. Abstreiter,J. J. Finley, D. V. Bulaev, and D. Loss, Phys. Rev. B ,241306(R)(2007).[13] E. Abrahams, Phys. Rev. , 491 (1957).[14] H. Capellmann, S. Lipinski, and K. U. Neumann, Z.Phys. B , 323 (1989).[15] P. San-Jos´e, G. Zarand, A. Shnirman, and G. Sch¨on,Phys. Rev. Lett. , 076803 (2006); P. San-Jos´e, G.Sch¨on, A. Shnirman, and G. Zarand, Phys. Rev. B ,045305 (2008).[16] G. L. Bir and G. E. Pikus, Symmetry and strain-inducedeffects in Semiconductors , Wiley, New York (1974).[17] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.Lett. , 016601 (2004).[18] A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B ,125316 (2001).[19] M. Kroner, K. M. Weiss, B. Biedermann, S. Seidl, A.W. Holleitner, A. Badolato, P. M. Petroff, P. Ohberg,R. J. Warburton, and K. Karrai, Phys. Rev. Lett. ,1567803 (2008). [20] M. Kroner, K. M. Weiss, B. Biedermann, S. Seidl, S.Manus, A. W. Holleitner, A. Badolato, P. M. Petroff, B.D. Gerardot, R. J. Warburton, and K. Karrai, Phys. Rev.B, , 075429 (2008). [21] S. Amasha, K. MacLean, I. P. Radu, D. M. Zumbuhl,M. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys.Rev. Lett.100