Relaxation of Shallow Donor Electron Spin Due to Interaction with Nuclear Spin Bath

Transcription

1 Relaxaton of Shallow Donor Electron Spn Due to Interacton wth Nuclear Spn Bath NANO LETTERS 2002 Vol. 2, No Semon Saykn,, Dma Mozyrsky, and Vladmr Prvman*, Center for Quantum DeVce Technology, Clarkson UnVersty, Potsdam, New York 13699, Department of Theoretcal Physcs, Kazan State UnVersty, Kazan , Russa, and T-13 and CNLS, Los Alamos Natonal Laboratory, Los Alamos, New Mexco Receved March 18, 2002; Revsed Manuscrpt Receved Aprl 29, 2002 ABSTRACT We study the low-temperature dynamcs of a shallow donor, e.g., 31 P, mpurty electron spn n slcon, nteractng wth the bath of nuclear spns of the 29 S sotope. For small appled magnetc felds, the electron spn relaxaton s controlled by the steady-state dstrbuton of the nuclear spns. We calculate the relaxaton tmes T 1 and T 2 as functons of the external magnetc feld and conclude that nuclear spns play an mportant role n the donor electron spn decoherence n S:P at low magnetc felds. Introducton. Recently, there has been much nterest n the study of decoherence of a sngle electron or nuclear spn for novel low-temperature semconductor applcatons consdered for realzaton of quantum nformaton processng. Several proposed desgns of quantum computers 1-7 utlze spn qubts. Interactons wth the envronment lead to devatons from controlled, coherent quantum-mechancal evoluton of the spn state. Tradtonally, the loss of the ntal spn polarzaton, has been characterzed by the longtudnal, T 1, and transverse, T 2, relaxaton tmes, relatve to the drecton of the external magnetc feld. 8 Interest n quantum computng has focused attenton on the tme scales of the processes nvolved and on the propertes of sngle spns. Much of the recent work has been devoted to the fndng 9-15 that ntal decoherence processes may be mportant for quantum computng. These processes occur on tme scales faster than energy exchange. In ths work, we focus on a system suggested by quantum computng desgns, but concentrate on the global tme dependence of decoherence and relaxaton. A spn qubt n a semconductor heterostructure at low temperatures can be nuclear spn 2,3 or spn of an electron bound to a donor mpurty 5 or trapped n a quantum dot. 1,4,6 Ths spn nteracts wth several types of envronments, such as phonons, conducton electrons, and other spns. It has been argued n the lterature that the spn envronment possesses fundamentally dfferent propertes from the bosonc one. 12,13,16,17 * Correspondng author. E-mal: prvman@clarkson.edu. Clarkson Unversty. Kazan State Unversty. Los Alamos Natonal Laboratory. In the present work, we study the effects of the nuclear spn bath on the mpurty-bound electron spn qubt. It s well establshed that the nuclear spn system can nfluence electron spn polarzaton. Ths has been demonstrated for GaAs, where polarzed nucle can create a strong (on the order of several Tesla) effectve nternal magnetc feld at an electron poston. 18 The nuclear spn system has long relaxaton tme as compared to the electron spn system, and for short tmes can be consdered as system of frozen magnetc moments, unless the nuclear spns are pumped externally by NMR radaton. Recently, electron spn dephasng by nuclear spns n GaAs quantum dots was studed n ref 19, where the case of fully polarzed nuclear spns was solved exactly and nonexponental decoherence processes were found. A dfferent mechansm for rreversble spnflp transtons wth energy dsspaton n quantum dots due to the hyperfne nteracton asssted by phonons was proposed n refs 20 and 21, but t cannot lead to sgnfcant decoherence and relaxaton. A more effectve phonon-medated decoherence mechansm 22 s due to spn-orbt nteractons, as referenced later. Nuclear spn drven dephasng of electron spns n GaAs quantum dots was consdered recently 23 wthn a model of electron spns movng n effectve magnetc felds created by contact hyperfne nteractons. In ths paper, we consder relaxaton of a shallow donor mpurty, 31 P(I P ) 1/2), electron spn n a defect-free S crystal at low external magnetc felds and low temperature. The nterest n ths system has been rekndled by the work of Kane, 3 n whch donor electron spns were proposed as medators of nteractons between nuclear spn qubts. We /nl CCC: $22.00 Publshed on Web 05/14/ Amercan Chemcal Socety

2 propose a novel mechansm for localzed electron spn relaxaton/decoherence n S:P, whch arses due to the precesson of the electronc spn n a randomly dstrbuted nuclear spn effectve magnetc feld. The nuclear spn bath couples to the electron spn va the hyperfne nteracton. A well-known dephasng mechansm was suggested n ref 24, wth electron spn dephasng by the nuclear spn polarzaton occurrng as a result of hoppng of the electron from one donor ste to another. In ths work, we establsh that nuclear spns contrbute to localzed electron spn dephasng even n crystals wth sparsely postoned donors, where such hoppng s not possble. Our results ndcate that nuclear spns are lkely to be an mportant source of the low-temperature donor electron spn relaxaton/dephasng n S:P at low magnetc felds. Donor Electron Spn Interactng wth a Nuclear Spn Bath. In the effectve mass approxmaton, donor electron n S s descrbed 25 by the lnear combnaton of modulated s-electron wave functons, 6 ψ(r) ) R F (r)u (r)e k r )1 where the envelope functon s F (r) ) (πb 3 ) -1/2 exp(-r/b). The effectve radus s b ) ɛr b m/m* Å, where ɛ ) 12 s the delectrc constant for slcon, R b s the Bohr radus, whle m and m* are the free and effectve electron masses, respectvely. In eq 1, coeffcents R are determned by the symmetry consderatons, and the summaton s carred over the sx mnma of the conducton band. 25 Natural slcon crystal contans about c 4.67% of 29 S atoms wth nuclear spn I ) 1/2. The lattce constant s a ) 5.43 Å, and thus the electron wave functon wth such effectve radus covers n I 80 nucle of 29 S. We wll assume that the electron spn nteracts wth the donor nucleus and nuclear spn bath of 29 S manly va the contact hyperfne nteracton. 8 At low temperatures, on the order of several 10 mk, the donor electrons are always bound and the role of phonons n relaxaton processes s dmnshed. We can then focus on the spn Hamltonan for a donor electron spn, S, nteractng wth a reservor of nuclear spns, I, n external magnetc feld, H. It can be approxmated by the Zeeman energy of the electron and nuclear spns, and by the hyperfne contact nteracton H ) gµ b SH + γ pi H + where summaton s carred over all the nuclear spns. The spn-orbt nteracton mxes ground and excted donor electron states descrbed by eq 1 and can gve rse to spnlattce relaxaton 26 and decoherence 22 by modfyng the electron g-factor. Ths s mportant for phonon-medated relaxaton mechansms. 22,26 In eq 2, the hyperfne couplng constant of the s nuclear spn to the electronc spn s A ) (8π/3p)g 0 µ b µ n ψ(r ) 2, where µ n s magnetc moment of the nucleus located at poston r, and g 0 s the free electron (1) A psi (2) Fgure 1. Energy level structure of a donor electron spn nteractng wth a system of 29 S nuclear spns va the contact hyperfne nteracton; see ref 27. g-factor. 18 Therefore, the devaton of the electronc g-factor from 2, and ts ansotropcty, can only somewhat rescale the value of the magnetc feld n eq 2 and wll be gnored n our calculatons. Our results, to be presented shortly, suggest that the man relaxaton effects due to the nuclear spn bath occur n the regme when the effectve magnetc feld owng to the hyperfne nteracton approxmately cancels the external appled magnetc feld. Therefore, we wll focus on the magnetc felds of less than order 100 G, so that the nuclear Zeeman part of the Hamltonan eq 2 s much smaller than the hyperfne part and can be neglected. Then the Hamltonan eq 2 can be transformed to the nteracton pcture, H nt (t) ) exp(h 0 t)h exp(- H 0 t), wth where we have ncluded the dagonal part of the hyperfne nteracton n the unperturbed Hamltonan H 0. The nteracton Hamltonan H nt (t) can be splt nto two parts: where H 0 ) gµ b S z H + A S z I z In the above equatons, H S and H P represent the off-dagonal part of the hyperfne nteracton wth the 29 S nucle and P nucleus, respectvely. It should be noted that the frequences are defned as operators n nuclear spn space, and ω Z ) (3) H nt (t) ) H S (t) + H P (t) (4) H S (t) ) 2 A {S + I - *0 e ωˆ S t + S - I + e ωˆ S t } (5) H P (t) ) 2A 0 {S + I - 0 e ωˆ Pt + S - I + 0 e ωˆ Pt } (6) ωˆ P ) ω Z + A I z ωˆ S ) ω Z + A 0 I 0 z + A I z (7) (8) 652 Nano Lett., Vol. 2, No. 6, 2002

3 gµ b H/p. The prme n the sum n eq 8 ndcates that the summaton s over *. The ) term can be ncluded wth neglgble error, assumng a large number of spns n the reservor. The energy level structure of the unperturbed Hamltonan H 0 s schematcally presented n Fgure 1. The hyperfne splttng produced by the donor nucleus s much larger than the splttng due to the nteracton wth the 29 S nucle, as determned n the ENDOR experments 27 for S: P, specfcally, H P 42 G and δ S 2.9 G, see Fgure 1. To evaluate dynamcs of the system governed by the Hamltonan eq 4, we use a Markovan approxmaton for the master equaton for the reduced densty matrx F(t) of the donor electron spn, 28 t F (t) )- 0 dτ{ [S+,S - F(t)]ξ 1 (τ) + [F(t)S -,S + ]ξ 2 (τ) + [F(t)S +,S - ]ξ 1 (-τ) + [S -,S + F(t)]ξ 2 (-τ)} (9) where ξ 1 (τ) ) (2A ) 2 I - *0 ξ 2 (τ) ) (2A ) 2 I + *0 I + e ωˆ Sτ + (2A 0 ) 2 I - 0 I + 0 e ωˆ Pτ (10) I - e ωˆ Sτ + (2A 0 ) 2 I + 0 I - 0 e ωˆ Pτ (11) In the above equaton, the angular brackets denote averages over the spn states, obtaned by tracng the approprate operators multpled by the densty matrx of the spn bath, θ(τ). We pont out that the approxmatons nvolved 28 n dervng eq 9 nclude the assumpton that the total densty matrx s factorzed at all tmes. Furthermore, the densty matrx of the bath s assumed to be tme ndependent. These assumptons of the Markovan approxmaton, are generally vald when the nuclear spn reservor s kept n ts reference state ether by external pumpng by NMR radaton, or by thermalzaton. These processes, as well as nteractons present n the system, wll defne the tme scales of the decay of ξ 1,2 (τ), whch should be smaller than the characterstc dynamcal tmes of the electron spn, T 1 and T 2. Thermalzaton processes alone mght not be suffcent to satsfy ths condton for expermentally relevant tmes. Ths lmtaton should be kept n mnd when usng the results of most recently publshed relaxaton calculatons medated by nuclear spns. 19,23 In our calculatons, we took the completely random θ ) 2 -n I, assumng that any expermentally relevant temperature s effectvely nfnte for nuclear spns, or that they are contnuously pumped. Averagng of exponental operators, exp(ωˆ S τ) ) exp(ω Z τ) exp(a 0 I 0 z τ) exp( A I z τ) (12) n eqs 10 and 11 can be done assumng that the number of nuclear spns s suffcently large, e ( A I z τ 1 2πσ 2 - dye -y 2 /(2σ 2 )(yτ ) e - τ2 σ 2 /2 (13) where the energy scale measures the root-mean-square hyperfne nteracton of the electron wth the nuclear bath. Thus, and eq 9 can be rewrtten n terms of dagonal, F VV, F vv, and offdagonal, F Vv, F vv, components of the spn densty matrx, and pσ ) p (A /2)2 (14) e ωˆ Sτ ) e -τ2 σ 2 /2+ω Z τ cos(a 0 τ/2) (15) e ωˆ Pτ ) e -τ2 σ 2 /2+ω Z τ (16) F vv (t) -F VV (t) )- dτ(fvv 0 (t) -F VV (t)) {σ 2 cos(ω Z + A 0 /2)τ + σ 2 cos(ω Z - A 0 /2)τ + (A 2 0 /2)cosω Z τ}e -τ2 σ 2 /2 (17) F vv (t) ) [F Vv (t)] * )- 0 dτ FvV (t){σ 2 cos(a 0 /2)τ + (A 0 /2) 2 }e ω Zτ-τ 2 σ 2 /2 (18) Solvng the above equatons, we fnd that the off-dagonal components of the densty matrx decay as exp(- t/t 2 ), where 1 ) π T 2 ( σ2 2σ 2 2 e-(ω Z+A 0 /2) 2 /2σ 2 + σ2 2 e-(ω Z+A 0 /2) 2 /2σ 2 + ( A 0 2 ) 2 e -(ω Z 2 /2σ )) 2 (19) The electron spn polarzaton, F vv -F VV, decays accordng to exp(- t/t 1 ), where from eq 17 one obtans that T 1 ) T 2 /2. The three terms n eq 19 correspond to the channels of dsspaton of the electron spn phase. The frst two terms descrbe donor electron spn-flps owng to ts nteracton wth the 29 S nuclear spns. The thrd term arses due to donor electron spn-flps at the P-donor nucleus. Dscusson and Summary. The transverse relaxaton rate, 1/T 2, eq 19, s shown n Fgure 2 as a functon of the external magnetc feld. It has a peak of Gaussan shape at H ) 0, and another Gaussan peak at H ) A 0 p/2gµ b. The ntensty and wdth of the peaks are determned by the electron-spn hyperfne nteracton constant, A 0, wth the donor nucleus, and by the energy-scale σ, see eqs 14 and 19. The latter parameter depends of the partcular lattce arrangement of 29 S nucle around the 31 P mpurty. Indeed, thus far we have consdered a sngle donor electron spn, and eq 19 descrbed a decoherence process. To obtan a specfc estmate used for Fgure 2, we have assumed that one can average eq 9 over a statstcal ensemble of spatally Nano Lett., Vol. 2, No. 6,

4 dstrbuted 29 S nucle surroundng the 31 P donors. The average value, σ*, then provdes a representatve measure of the ndvdual spn decoherence. The quantty defned va (σ*) 2 ) c l (A l /2)2 (20) where c s the concentraton of the 29 S nucle and the summaton s carred over all the lattce postons, can be obtaned from expermental data on the nhomogeneously broadened ESR lne of 31 P donors. 27 For a Gaussan lne, 29 σ* ) gµ b δ S 2p 2ln2 (21) where the wdth at the half ntensty of the ESR lne, δ S 2.9 G, s obtaned from experment. 27 The estmated value of σ* s thus s -1. The value of the hyperfne constant for the 31 P donor nucle s 25 A s -1. Thus, the averaged value of T 1,2 for the donor electron spn n the S:P system at low external magnetc feld and low temperatures can be as short as T 1,T s. Ths ndcates that the decoherence mechansm consdered n ths work s lkely to be the domnant one at low magnetc felds. We can also consder the case when the nuclear spn bath s polarzed, ether by thermalzaton or other means. For typcal temperatures approprate for quantum computng applcatons, as low as few mk, the polarzaton 0 e p e1 wll be descrbed by p ) (e -γph/kt - 1)/(e -γph/kt + 1), where for S, γ < 0. For magnetc feld values of nterest, H up to 100 G, we can assume that p, 1. Equaton 19 s then replaced by 1 ) π T 2 ( σ2 2σ 2 2 (1 + p)e-(ω Z + p A /2 + A 0 /2) 2 /[2σ 2 (1 - p 2 )] + σ 2 2 (1 - p) e-(ω Z + p A /2 + A 0 /2) 2 /[2σ 2 (1 - p 2 )] + ( A 0 2 ) 2 e -(ω Z + p A /2) 2 /[2σ 2 (1 - p )]) 2 (22) showng that the nuclear spn polarzaton partly cancels the effect of the external magnetc feld. In the quantum computer applcatons, the value of the qualty factor, Q ) T 2 /T clock s of nterest. Here T clock s the larger of the qubt-control and qubt-nteracton tme scales. For fault-tolerant error correcton, Q mn 10 4 to 10 6 s needed. 30 For electron-spn qubts, 4 T clock ps, and our results, whch provde the lower bound on the decoherence rate, suggest that large external magnetc felds should be used. Indeed, decoherence owng to nuclear spns alone wll volate the qualty factor condton for felds, H, below about 50 G. Isotopc purfcaton of the S crystal can mprove the qualty factor. In summary, we have consdered a model of a shallow donor electron spn nteractng wth a nuclear spn bath. We have shown that at low temperatures such system relaxes to Fgure 2. The transverse relaxaton rate 1/T 2, shown on a logarthmc scale, of a donor electron spn as a functon of the external magnetc feld. the state determned by the densty matrx of the nuclear spn bath, on tme scales of the order 10-7 to 10-9 s for low external magnetc felds. Wthn the approxmaton scheme used, the transton probabltes determnng T 1-1 and T 2-1 have Gaussan dependence of the appled magnetc feld. Acknowledgment. Ths research was supported by the Natonal Scence Foundaton, grants DMR and ECS , and by the Natonal Securty Agency and Advanced Research and Development Actvty under Army Research Offce contract DAAD References (1) Loss, D.; DVncenzo, D. P. Phys. ReV. A 1998, 57, 120. (2) Prvman, V.; Vagner, I. D.; Kventsel, G. Phys. Lett. A 1998, 239, 141. (3) Kane, B. E. Nature 1998, 393, 133. (4) Imamoglu, A.; Awschalom, D. D.; Burkard, G.; DVncenzo, D. P.; Loss, D.; Sherwn, M.; Small, A. Phys. ReV. Lett. 1999, 83, (5) Vren, R.; Yablonovtch, E.; Wang, K.; Jang, H. W.; Balandn, A.; Roychowdhury, V.; Mor, T.; DVncenzo, D. P. Phys. ReV. A 2000, 62, (6) Bandyopadhyay, S. Phys. ReV. B 2000, 61, (7) Mozyrsky, D.; Prvman, V.; Glasser, M. L. Phys. ReV. Lett. 2001, 86, (8) Slchter, C. P. Prncples of Magnetc Resonance; 3rd Cor. Ed., Sprnger-Verlag: New York, Berln, Hedelberg, (9) van Kampen, N. G. J. Stat. Phys. 1995, 78, 299. (10) Mozyrsky, D.; Prvman, V. J. Stat. Phys. 1998, 91, 787. (11) Palma, G. M.; Suomnen K. A.; and Ekert, A. K. Proc. Royal Soc. A 1996, 452, 567. (12) Shao, J.; Ge, M.-L.; and Cheng, H. Phys. ReV. E 1996, 53, (13) Tuptsyn, I. S.; Prokof ev, N. V.; Stamp, P. C. E. Int. J. Mod. Phys. B 1997, 11, (14) Manv, T.; Bychkov, Y. A.; Vagner I. D.; and Wyder, P. Phys. ReV. B 2001, 64, (15) Prvman, V. preprnt cond-mat 2002, (16) Prokof ev, N. V.; Stamp, P. C. E. Rep. Prog. Phys. 2000, 63, 669. (17) Caldera, A. O.; Castro Neto, A. H.; Olvera de Carvalho, T. Phys. ReV. B 1993, 48, (18) Paget, D.; Lampel, G.; Sapoval, B.; Safarov, V. I. Phys. ReV. B 1977, 15, Nano Lett., Vol. 2, No. 6, 2002

5 (19) Khaetsk, A. V.; Loss, D.; Glazman, L. preprnt cond-mat 2002, (20) Erlngsson, S. I.; Nazarov, Y. V.; Fal ko, V. I. preprnt cond-mat 2001, (21) Erlngsson, S. I.; Nazarov, Y. V. preprnt cond-mat 2002, (22) Mozyrsky, D.; Kogan, Sh.; Berman, G. P. preprnt cond-mat 2002, (23) Merkulov, I. A.; Efros, Al. L.; Rosen, M. preprnt cond-mat 2002, (24) D yakonov, M. I.; Perel, V. I.; Berkovts, V. L.; Safarov, V. I. JETP 1975, 40, 950. (25) Kohn, W. In Sold-State Physcs; Setz, F., Turnbull, D., Eds.; Academc Press: New York, 1957; Vol. 5, p 257. (26) Roth, L. M. Phys. ReV. 1960, 118, (27) Feher, G. Phys. ReV. 1959, 114, (28) Blum, K. Densty Matrx Theory and Applcatons; Plenum Press: New York, (29) Abragam, A. Prncples of Nuclear Magnetsm; Oxford Unversty Press: New York, (30) Aharonov, D.; Ben-Or, M. preprnt quant-ph 1999, NL Nano Lett., Vol. 2, No. 6,