Nuclear spin-lattice relaxation in p-type GaAs
M. Kotur, R. I. Dzhioev, M. Vladimirova, R. V. Cherbunin, P. S. Sokolov, D. R. Yakovlev, M. Bayer, D. Suter, K. V. Kavokin
NNuclear spin-lattice relaxation in p-type GaAs
M. Kotur, R. I. Dzhioev, M. Vladimirova, R. V. Cherbunin, P. S. Sokolov,
4, 3
D. R. Yakovlev,
4, 1
M. Bayer,
4, 1
D. Suter, and K. V. Kavokin
1, 3 Ioffe Institute, Russian Academy of Sciences, 194021 St-Petersburg, Russia Laboratoire Charles Coulomb, UMR 5221 CNRS/ Universit´e de Montpellier, F-34095, Montpellier, France Spin Optics Laboratory, St-Petersburg State University, St-Peterbsurg, 198504, Russia Experimentelle Physik 2, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany Experimentelle Physik 3, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany
Spin-lattice relaxation of the nuclear spin system in p-type GaAs is studied using a three-stageexperimental protocol including optical pumping and measuring the difference of the nuclear spinpolarization before and after a dark interval of variable length. This method allows us to measurethe spin-lattice relaxation time T of optically pumped nuclei ”in the dark”, that is, in the absenceof illumination. The measured T values fall into the sub-second time range, being three orders ofmagnitude shorter than in earlier studied n-type GaAs. The drastic difference is further emphasizedby magnetic-field and temperature dependences of T in p-GaAs, showing no similarity to thosein n-GaAs. This unexpected behavior is explained within a developed theoretical model involvingquadrupole relaxation of nuclear spins, which is induced by electric fields within closely spaceddonor-acceptor pairs. I. INTRODUCTION
Optical pumping of nuclear spins via their dynamicpolarization by photoexcited spin-polarized electrons is apowerful method for obtaining considerable nuclear po-larization in semiconductors even in weak magnetic fieldsof the order of a few Gauss . Creation and manipula-tion of the resulted Overhauser fields, acting upon thespins of charge carriers, presents multiple possibilitiesfor studying the dynamics of mesoscopic spin systems.It is considered as one of the possible ways towards re-alization of spin-based information processing. Galliumarsenide, a direct-bandgap semiconductor with the 100percent abundance of magnetic isotopes and strong hy-perfine coupling, has been used as a test bench of theelectron-nuclear spin dynamics since 1970s. It was knownto specialists in the field (though, to the best of ourknowledge, never explicitly mentioned in publications),that nuclear spin-lattice relaxation time T in p-GaAsremains short even at liquid-helium temperatures, whilen-GaAs demonstrates long T (hundreds of seconds oreven more) in this temperature range.The spin-lattice relaxation of nuclei in n-GaAs was in-vestigated in our recent works . It was found to bedominated by the diffusion-limited hyperfine relaxationand quadrupole warm-up in lightly doped dielectric crys-tals, and by hyperfine relaxation involving both itinerant(Korringa mechanism) and localized electrons in heavilydoped samples with metallic conductivity. In what con-cerns p-GaAs, even the time scale of the nuclear spin-lattice relaxation has not been exactly known.In this paper, we present measurements of nuclear spin-lattice relaxation time T as a function of magnetic fieldand temperature in two insulating p-GaAs layers withdifferent concentrations of acceptors. The measured nu-clear spin-lattice relaxation times are of the order of100 ms. They are independent of magnetic fields in therange 0 −
100 G, and demonstrate a slow increase with lowering the temperature in the range 10 −
30 K, whichsuddenly becomes sharp below 10 K. These findings aredrastically different from what is known about the nu-clear spin relaxation in n-GaAs. The fact that nuclearspin relaxation in the dark, i.e. in the absence of pho-toexcited conduction-band electrons, is three orders ofmagnitude shorter than in n-GaAs, where resident elec-trons are abundant, is counterintuitive, since hyperfinecoupling in the valence band is ten times weaker thanin the conduction band . We propose a theoreticalmodel that qualitatively explains the whole set of the ex-perimental data for p-GaAs, and allows us to quantita-tively reproduce the measured temperature dependenceof nuclear spin relaxation time T . II. SAMPLES AND EXPERIMENTAL SETUP
The studied samples are two germanium-doped GaAslayers grown by liquid phase epitaxy on [001] GaAs sub-strate. The corresponding acceptor concentrations are n A = 2 . × cm − (Sample A) and 6 . × cm − (Sample B). The samples are placed in a variable tem-perature cryostat (either helium flow or cold finger), sur-rounded by three pairs of Helmholtz coils. Such arrange-ment allows for the compensation of the geomagnetic fieldand application of an external mangnetic field in an ar-bitrary direction. Optical orientation of electron spinsis achieved by pumping with a circularly ( σ + ) polarizedlight from continuous wave (CW) titanium-sapphire laserat the wavelength λ = 800 nm. The light beam is di-rected along the sample axis. The spectra of photolu-minescence (PL) intensity and its circular polarizationdegree ρ = ( I + − I − ) / ( I + + I − ) for the two samplesare shown in Fig. 1 (a). Here I + ( I − ) is the intensityof PL emitted in σ + ( σ − ) polarization, respectively. Thetwo PL peaks can be identified as acceptor-bound exciton(ABX) emission and conduction band-to-acceptor (CBA) a r X i v : . [ c ond - m a t . o t h e r] F e b PL intensity (arb. units) ( c )( b )
S a m p l e BS a m p l e A W a v e l e n g t h ( n m )
S a m p l e A
S a m p l e B
P L
01 02 0 r (%) C B AA B X r
051 01 5 r = 1 3 . 5 % s + s + / s - L o r e n t z f i tB = 9 . 5 G r (%) B p u m p = 4 G ( a ) r = 1 0 . 8 %- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0051 01 5 s + s + / s - L o r e n t z f i tB = 2 0 . 5 G r (%) B ( G )
FIG. 1. (a) PL intensity (right scale) and polarization (leftscale) spectra for p-GaAs, Samples A and B at B = 0 and T = 5 K. The two PL peaks are identified as acceptor-boundexciton (ABX) emission and conduction band-to-acceptor(CBA) recombination. The black arrow indicates the cho-sen PL detection energy. (b, c) PL polarization as a functionof oblique magnetic field in Samples A (b) and B (c). Pumppolarization is either alternated by a photoelastic modulatorat the frequency of 50 kHz (no Overhauser field, red symbols),or fixed (black symbols). The onset of the Overhauser fieldresults in the asymmetry with respect to zero. Solid lines areLorentzian fits to the data, that allows for determination of B / and ρ in Eq. (1). recombination. Figs. 1 (b) and 1 (c) present the PL po-larization degree as a function of magnetic field B appliedat 80 degrees with respect to the structure axis, for Sam-ples A and B. When the pump polarization is alternatedbetween σ + and σ − by a photoelastic modulator operat-ing at the frequency of 50 kHz (red curve), nuclear spinsremain unpolarized, and ρ ( B ) obeys the Lorentzian law(the Hanle effect) . Under pumping by light with staticcircular polarization ( σ + ), nuclear spins get polarized,and the Hanle curve is affected by the Overhauser fieldwhich is either parallel or anti-parallel to the externalfield, depending on the sign of the latter (black curve). For studies of transient nuclear spin polarization P N we use the experimental setup shown in Fig. 2 (a). Toasses millisecond time scale, we use pump pulses cut outof the CW laser beam with an acousto-optical modu-lator (AOM), controlled by the pulse generator. Thesame pulse generator controls the power supply of the3D Helmholtz coils. The PL polarization is measuredin the reflection geometry at the spectral maximum ofthe PL polarization ( λ ≈
835 nm, see Fig. 1 (a)). Theemitted light passes through the spectrometer and is de-tected by the avalanche photodiode (APD), followed bythe multi-channel photon counting system (PCS). Thelatter is synchronized with the AOM via the pulse gen-erator.The three stage experimental protocol that we imple-ment here is very similar to that used in our previouswork on n-GaAs , but here it is adapted for measure-ment of subsecond nuclear spin relaxation times. The ex-periment timeline is shown in Fig. 2 (b). The first stageof the experiment is the optical pumping of nuclei bycircularly ( σ + ) polarized light from a titanium-sapphirelaser at λ = 800 nm during 500 ms. The excitation poweris P pump = 4 mW, focused on 90 µ m spot on the samplesurface. The magnetic field B pump = 4 G is applied at80 degrees with respect to the structure axis. At the sec-ond stage, the pump is switched off for an arbitrary time t dark (typically from 2 ms to 1 s), and the magnetic fieldis set to the value B dark at which we intend to measurethe nuclear spin dynamics. B dark is parallel to B pump and ranges from zero to 120 G. The switching time is ≈ B <
10 G and ≈
10 ms for
B >
10 G). At theend of the dark interval, B pump is restored and the pumpis switched on. At the same moment the photon count-ing system starts the PL detection in either right or leftcircular polarization. The PL signal is monitored during500 ms, which is sufficient to fully restore the nuclear spinpolarization corresponding to the chosen pumping condi-tions. At the end of this stage the cycle is repeated. Theresulting PL signal is averaged over 100 measurement cy-cles. The same procedure is performed for the oppositepolarization of PL. From each pair of measurements thedegree of circular polarization of PL is evaluated, andplotted as a function of the photon counting time t P CS .Two examples of ρ ( t P CS ) dependence for Sample Bare presented in Figs. 2 (c) and 2 (d). They correspondto dark intervals t dark = 400 ms and 2 ms, respectively.One can see that ρ decreases under pumping down to ρ pump ≈ B pump = 4 G, at which the Overhauser field adds up tothe external field and induces additional depolarizationof electrons (see also Fig. 1 (c), where the value of polar-ization under σ + pumping at B = B pump corresponds to ρ = ρ pump ). During dark intervals, nuclear polarizationdecreases, which results in a larger value of ρ measuredwhen the pump is back on. Fitting ρ ( t P CS ) by expo-nential decay gives the build-up time of the nuclear fieldunder optical pumping T NB , as well as the value of ρ dark .The knowledge of ρ dark allows for determination of the FIG. 2. (a) Experimental setup designed for three stage measurements of sub-second nuclear spin relaxation times. (b)Time-line of experiments. Two periods of pumping/magnetic field/photon counting sequence synchronously controlled by thepulse generator are shown (100 periods are used for each measurement at a given pump polarization). (c, d) Typical PLpolarization decays measured in Sample B (symbols) at T = 10 K, B = B dark = 4 G, t dark = 400 ms (c) and t dark = 2 ms (d).PL polarization ρ dark at the end of the dark interval ( t PCS = 0) is recovered from the exponential fit (solid line) of the data.It is used to calculate the Overhauser field B N from Eq. (1) at given duration of dark interval and magnetic field. The decaytime T NB of the exponential fit characterizes the nuclear spin relaxation time in the presence of optical pumping. It is alwaysshorter than T . nuclear field intensity . Indeed, the nuclear field can berecovered from the PL polarization degree using the fol-lowing formula, derived from the well-known expressionfor the Hanle effect: B N = B / (cid:114) ρ − ρ dark ρ dark − B pump . (1)Here ρ dark is the degree of the PL polarization at theend of the dark interval [ ρ dark ≡ ρ ( t P CS = 0)], ρ is thePL polarization in the absence of the external field, and B / is the half width of the Hanle curve, measured inde-pendently under conditions where nuclear spin polariza-tion is absent (pump polarization modulated at 50 kHz,see Figs.1 (b) and 1 (c)). Even after shortest dark in-tervals, B N is a bit lower than before switching off thepump, most likely because of nuclear spin warm-up bythe Knight field of photoexcited electrons, that rapidlychanges when the pump is switched off and on . By re-peating the protocol for different durations of t dark , weobtain B N relaxation curves for given values of tempera-ture and applied magnetic field B dark . Examples of suchdependences for two different temperatures are shown inFig. 3. One can see that B N decreases with increasingthe length of the dark interval t dark. . This is due to nu-clear spin-lattice relaxation ”in the dark”, that is in theabsence of perturbation by pumping. Exponential fittingof these curves yields the nuclear spin relaxation timein the dark T , that we aim to study as a function oftemperature and applied magnetic field B dark . FIG. 3. Overhauser field B N derived using Eq. (1) from PLpolarization measurements as shown in Figs. 2(c, d) for var-ious dark interval durations ( B = B dark = 4 G). The expo-nential decay fit (solid lines) yields T for given temperatures. III. EXPERIMENTAL RESULTS ANDDISCUSSION
Magnetic field dependence of nuclear spin relaxationtime T measured in Sample B at T = 10 K is shown inFig. 4. For comparison, the relaxation time in dielectricn-GaAs with donor concentration n D = 6 × cm − (from Refs. 4 and 5) is shown on the right scale. Onecan see that in p-GaAs nuclear spin relaxation is about Magnetic field (G) T ( m s ) T ( s ) T=10 K 0100200300 T (ms), p-type (Sample B)T (s), n-typeModel FIG. 4. Magnetic field dependence of nuclear spin relaxationtime. Sample B (circles, left scale) and n-GaAs with n D =6 × cm − (diamonds, right scale, data from Ref. 5). Solidline is a model prediction for p-GaAs, Eq. (19). Temperature (K) T ( m s ) T ( s ) Model Sample A, = ph =5 sModel Sample B, = ph =3 sSample ASample Bn-type, B=0 B=4G
FIG. 5. Temperature dependence of nuclear spin relaxationtime. In p-GaAs (left scale): Sample B (circles), Sample A(squares); in lightly doped n-GaAs n D = 6 × cm − (rightscale: diamonds, data from Ref. 5). Solid lines are modelpredictions for p-GaAs samples, calculated using Eq. (15). three orders of magnitude faster than in the n-GaAs.Moreover, in n-GaAs the relaxation time exhibits a pro-nounced field dependence, while in p-GaAs it does notdepend on the field.The temperature dependence of T is also surprising.Figure 5 shows spin relaxation time measured in Sam-ples A and B, as well as the comparison with n-GaAssample with n D = 6 × cm − from Refs. 4 and 5.It appears that in p-GaAs nuclear spin relaxation slowsdown significantly below T = 10 K: for example in Sam-ple B T = 45 ms at T = 30 K and 310 ms at 4 K. Thisbehaviour is not observed in n-GaAs. We discuss belowpossible mechanisms of nuclear spin lattice relaxation inp-GaAs that could account for these experimental results,taking advantage of the knowledge we already accumu-lated for n-GaAs.The main feature distinguishing the spin-lattice re- laxation of nuclei in p-GaAs from that in n-GaAsis its time scale, which is three orders of magnitudeshorter. This fact excludes the diffusion-limited hy-perfine relaxation from possible relaxation mecha-nisms. Indeed, with the nuclear spin diffusion constant D ≈ − cm /s the diffusion length during the time100 ms is just 1 nm, i.e. two lattice constants of GaAs .This means that the nuclear spin polarization does nothave time to reach any remote killer center (e.g. para-magnetic impurity or neutral acceptor site) by diffusion,and decays within the same area where it has been cre-ated. Thus, spin diffusion that controls nuclear spin re-laxation in n-GaAs can not account for fast nuclearspin relaxation in p-GaAs.This consideration leaves us two possible scenarios fornuclear spin relaxation in p-GaAs: either (i) nuclear spinsare polarized only in regions where some efficient relax-ation mechanism is at work, or (ii) a new, still unknownrelaxation mechanism is acting everywhere in the crystal.Let us start from the examination of the second sce-nario. So far, the only long-range relaxation mechanismknown for the dielectric GaAs at low temperatures, is thequadrupole relaxation due to fluctuating electric fields.Such fields result from hopping of localized charge car-riers in the impurity band . However, this mechanismis far too weak to explain the observed relaxation timescale. Indeed, the calculations reported in Ref. 4 showthat one cannot expect the relaxation times shorter than20 s induced by this mechanism. This conclusion is cor-roborated by the experiments on n-GaAs. Additionally,it was shown that the efficiency of quadrupole relaxationof bulk nuclei drops down in magnetic fields exceedingthe nuclear spin local field of the order of a few Gauss .By contrast, in the studied p-GaAs samples no magnetic-field dependence of T is observed at least up to 120 G(Fig. 4).Exploring the first scenario, we note that the efficiencyof dynamic polarization of nuclear spins by free photoex-cited electrons is very low . Nuclear spins are polarizedby electrons trapped to donor centers, which in p-typecrystals are empty in the absence of optical excitation.Under continuous optical pumping, the nuclear polariza-tion can, in principle, spread out from the vicinity ofdonors into the bulk of the crystal by spin diffusion. This,indeed, occurs in n-type crystals. This is illustrated inFig. 6 (b), where a sketch of spatial distribution of thenuclear spin polarization in n-GaAs is shown. After suf-ficiently long pumping time nuclear spins are not onlypolarised within the donor orbits, but also everywherein the bulk, due to spin diffusion. The relaxation ofthis polarization in the dark is then ensured by diffu-sion towards neutral donors D (which play here the roleof the killer centers), and, at sufficiently low magneticfields, directly in the bulk via interaction of the nuclearquadrupole moments with the fluctuating electric field ofhopping charges.In p-GaAs the situation is quite different. Let us firstdiscuss how nuclear spin polaization in p-GaAs is gener-ated under optical pumping. As shown by Paget, Amandand Korb , in p-doped III-V semiconductors under opti-cal pumping the nuclear polarization accumulates arounddonors only within the so-called ”quadrupole radius” δ ,see Fig. 6 (a). It is defined by competition betweenhyperfine polarization and quadrupole relaxation of nu-clei. The reason for this is as follows. Dynamic polar-ization of nuclei occurs due to their hyperfine couplingwith the spins of electrons captured by donors, at therate proportional to the electron spin density. The lat-ter falls down exponentially with increasing the distancefrom the donor center . Since photoexcited electronsspend at the donor only a limited time before recom-bination, the donor repeatedly changes its charge statefrom positively charged to neutral. This blinking chargecreates a time-dependent electric field which, obeying theCoulomb law, extends far beyond the Bohr radius of thedonor-bound electron a BD . In piezoelectric semiconduc-tors like GaAs, the electric field induces quadrupole split-ting of nuclear spin states ; fluctuating electric fieldsthus act similarly to fluctuating magnetic fields, causingnuclear spin relaxation. At some distance from the donorcenter, quadrupole relaxation overcomes dynamic polar-ization, because the electric field decreases with growingdistance slower than the electron density. According tothe calculations reported in Ref. 14, this happens at ap-proximately 0 . a BD . The numerically calculated depen-dence of the nuclear polarization on the distance fromthe donor is shown by red solid line in Fig. 7. Note, thatthe presence of an additional relaxation channel underpumping is corroborated by our experimental data: inboth p-GaAs samples that we studied, the build-up timeof the nuclear polarization is even shorter than its decaytime in the dark, about 50 ms, see Figs. 2 (c) and 2 (d).Thus, we conclude that the nuclear polarization in-duced by optical pumping in p-GaAs is confined neardonors, as sckethed in Fig. 6 (a). What relaxation mech-anism can be responsible for its rapid decay in the dark,when the donor is empty? We suggest that it is thequadrupole relaxation induced by the electric field of acharged acceptor located in the vicinity of the donor.The presence of charged acceptors is a result of recom-bination of one of acceptor-bound holes with the donorelectron. At zero temperature, the negative charge corre-sponding to the absence of a hole is located at the accep-tor nearest to the positively charged donor with 97 . . The distribution function of the distancesfrom the donor to the nearest acceptor is shown in Fig. 7by the blue dashed line (details of the corresponding cal-culation are given in Section IV). It has the maximumat R (1) DA = (2 πn A ) − / . For the studied range of n A ,this amounts to approximately 1 . a BD . At this distance,a charged acceptor produces electric field E of severalkV/cm in the vicinity of the donor. Since GaAs is a polarcrystal, this electric field induces an effective quadrupolefield: B Q = b Q E, (2) where b Q = eQβ Q γ N I (2 I − . (3) β Q is the experimentally determined and isotope-dependent constant, eQ is the nuclear quadrupole mo-ment, also isotope-dependent, e is the absolute valueof the electron charge, γ N is the nuclear gyromagneticratio, I is the nuclear spin. . For GaAs b Q ≈ . · cm/kV. Figure 8 shows the effective quadrupolefield as a function of the distance from the charged (redsolid line) and neutral (green dashed line) acceptor. Onecan see that at the distance R (1) DA from the A − acceptor B Q ≈ T (cid:54) = 0, a hole from a more remote, neutral, accep-tor can jump to this site, neutralizing it. Thus, fluctu-ations of the occupation number of the nearest acceptorproduce fluctuating quadrupole fields. In Section IV weshow that these fluctuations provide an efficient nuclearspin relaxation.Figure 6 summarises the above considerations via com-parison of nuclear polarization patterns that form as aresult of optical pumping in p-GaAs and n-GaAs. Inn-GaAs, most of the donors are neutral; nuclear polar-ization created under orbits of donor-bound electronsspreads into the inter-donor space. The number ofcharged donor-acceptor pairs is small and most of nu-clei are situated far from such pairs. To the opposite, inp-GaAs all the donors are charged; nuclear polarizationis concentrated near donors because of its quadrupole re-laxation during pumping . Almost each donor has anacceptor nearby, and the electric charge of this accep-tor fluctuates while it captures and releases a hole. Thisexplains why quadrupole nuclear-spin relaxation inducedby fluctuating charges is three orders of magnitude fasterin p-GaAs as compared to n-GaAs.This model explains also the T behavior as a functionof temperature and magnetic field in p-GaAs. Indeed,with lowering temperature the fraction of time, duringwhich the nearest acceptor site is charged, increases, sincethis state is energetically favourable. As a result, chargedistribution in the vicinity of the donor becomes frozen,and the electric field stops fluctuating. This obviouslyshould lead to an increase of T , and this is exactly whatis observed in experiments, see Fig. 5.The T independence of the applied magnetic field B means that ω B τ c <<
1, where ω B = γ N B is the nuclearLarmor frequency in the field B and τ c is the correla-tion time of the fluctuating field which causes the spinrelaxation . For nuclear species of GaAs, the average nu-clear gyromagnetic ratio (cid:104) γ N (cid:105) ≈ × rad/G · s. Usingthe well-known formula for spin relaxation under influ-ence of a fluctuating magnetic field :1 T = ω f τ c , (4)where ω f = γ N B f is the spin precession frequency in thefluctuating field B f , one can estimate what correlation A - A D + D D D d a BD a BA A A p-type n-type P N =0P N ≠ D + e A - a BD (a) (b) FIG. 6. Sketch of nuclear polarization patterns that form asa result of optical pumping in p-GaAs (a) and n-GaAs (b).The degree of nuclear spin polarization P N is schematicallyrepresented by the red color intensity. In n-GaAs most ofthe donors are neutral; P N created under orbits of donor-bound electrons spreads into the inter-donor space so than P N (cid:54) = 0 everywhere under the light spot. The relaxationof this polarization in the dark is provided by (i) diffusiontowards neutral donors D , and (ii) directly in the inter-donorspace via interaction of the nuclear quadrupole moments withthe fluctuating electric field of hopping electrons (between D and D + ). In p-GaAs all donors are charged; P N (cid:54) = 0only near donors because of its quadrupole relaxation duringpumping. Almost each donor has an acceptor nearby, and theelectric charge of this acceptor fluctuates while it captures andreleases a hole. This induces quadrupole relaxation which ismuch faster in p-GaAs than in n-GaAs. time would explain the observed T ≈
100 ms (see Fig. 4).Assuming the magnitude of the fluctuating quadrupolefield of 1 G as estimated above (see Fig. 8 and details ofthe calculations in Section IV) we get τ c ≈
100 ns. There-fore, throughout the range of magnetic fields applied inour experiment, the condition ω B τ c << T independence of the magneticfield, up to the maximum magnetic field B = 120 G ap-plied in our experiments. In Section IV we present thetheoretical model which quantifies the above considera-tions. IV. THEORETICAL MODEL
Let us assume that the fluctuations of the electric fieldexperienced by the optically polarized nuclei under thedonor orbit result from the charge fluctuations on thenearest acceptor, as shown in Fig. 6. When it is nega-tively charged, it creates an electric field E − in the vicin-ity of the donor (we neglect spatial variation of this fieldwithin the sphere with the radius δ , where nuclear spinsare polarized). When the nearest acceptor is neutral, weassume that the electric field takes certain value E ; indoing so, we neglect the variability of charge configura-tion at more remote impurities.We denote the average time during which the nearest Distance from donor position (a BD ) N u c l e a r p o l a r i z a t i o n , P N ( a r b . un i t s ) F D A P N 𝐹 𝐹 d 𝑅 𝑅 𝑛 $ = 6×10 &0 cm Sample B
FIG. 7. Left scale: nuclear polarization created by opticalpumping as a function of the distance from the donor (redsolid line, from Ref. 14). Right scale: Probability densityfor the first (blue dashed line) and second (green dotted line)neighbouring acceptors calculated from Eq. (16) for SampleB as a function of the distance from the donor. The distanceis expressed in the units of donor Bohr radius a BD = 10 nm. -5 -3 -1 Distance from acceptor position (a BD ) B Q ( G ) 𝑅 " 𝑛 = 6×10 %- cm Sample B 𝐴 𝐴 FIG. 8. Effective quadrupole field B Q in the vicinity of thecharged (red solid line) and neutral (green dashed line) ac-ceptor. It is plotted as a function of the distance from theacceptor position. The distance is expressed in the units ofdonor Bohr radius a BD = 10 nm. Blue arrow shows that atthe distance R (1) DA corresponding to the maximum probabilityto find the nearest donor ( cf Fig. 7), the charged acceptor A − creates B Q ≈ B Q in the vicinity of A is negligiblysmall. acceptor stays charged as τ − , and the average time duringwhich it is neutral as τ . The average electric field at thedonor is then equal to (cid:104) E (cid:105) = E − τ − + E τ τ − + τ . (5)The mean squared fluctuation of this field is, correspond-ingly, δE = (cid:104) ( E − (cid:104) E (cid:105) ) (cid:105) = ∆ E τ − τ ( τ − + τ ) , (6)where ∆ E ≡ ( E − − E ) . The autocorrelation functionof the fluctuating part of the electric field, E f = E −(cid:104) E (cid:105) ,which presents an example of an asymmetric randomtelegraph signal, is determined by the shortest of the twotimes, τ − and τ : (cid:104) E ( t ) · E (0) (cid:105) = δE exp (cid:16) − t ( τ − + τ τ − τ ) (cid:17) . (7)In other words, the correlation time of the electric fieldfluctuations is equal to τ c = τ − τ τ − + τ . (8)The Fourier transform of the autocorrelation function inEq. (7) allows calculating the spectral power density ofelectric field fluctuations at the donor: δE ω = δE τ c ω ( τ − τ ) / ( τ − + τ ) . (9)Therefore, the resulting spectral power density of thequadrupole-induced effective magnetic field is given by δB ω = b Q δB ω . (10)According to Abragam the spin relaxation rate ofthe nuclear spin system in presence of the fluctuatingmagnetic field reads: 1 T = γ N δB ω . (11)At Larmor frequency of nuclear spin in the external field B , ω = ω B . At low magnetic fields used in this study andsatisfying the condition ω B τ c <<
1, Eq. (11) reduces toEq. (4). Thus we arrive to the following expression for T − : 1 T ≈ γ N b Q ∆ E ( τ − τ ) ( τ − + τ ) . (12)The characteristic times τ − and τ are determined byprobabilities of phonon-assisted transitions between theconfigurations with charged and neutral nearest accep-tor. The most reasonable assumption, that can be doneto estimate these times, is to suppose that these transi-tions correspond to the hopping of a hole between twoacceptors closest to the donor, see Fig. 6 (a). Denotingthe hole energy at the nearest acceptor by (cid:15) − , and at thesecond nearest acceptor as (cid:15) we arrive to the followingexpressions for τ − and τ : τ − = τ ph n ph ; τ = τ ph n ph + 1 , (13) where τ ph is the characteristic time of the correspondingphonon-assisted transition, n ph is the number of phononsgiven by the Planck distribution: n ph = 1exp [∆ (cid:15)/ ( k B T )] + 1 , (14)and ∆ (cid:15) = (cid:15) − − (cid:15) . Finally we obtain:1 T ≈ γ N b Q ∆ E τ ph × [1 − exp ( − ∆ (cid:15)/k B T )] exp ( − ∆ (cid:15)/k B T )[1 + exp ( − ∆ (cid:15)/k B T )] . (15)Eq. (15) allows one to calculate the temperature de-pendence of T . In order to do that, we need to estimate∆ (cid:15) and ∆ E . They are given by the Coulomb energiesand electric fields of two charges located at distances r (1) DA and r (2) DA from the donor to two nearest acceptors. As anestimate for these distances we take the maxima of thefirst and second neighbouring acceptor distributions: F (1) DA = 4 πr DA n A exp ( − πr DA n A ); F (2) DA = 163 πr DA n A exp ( − π r DA n A ) . (16)These distributions are shown in Fig. 7 for Sample B.The maxima of these distributions are given by R (1) DA = (2 πn A ) − / ; R (2) DA = ( 4 π n A ) − / . (17)Thus, we obtain ∆ (cid:15) = − e π(cid:15)(cid:15) (cid:34) R (1) DA − R (2) DA (cid:35) ;∆ E = (cid:18) e π(cid:15)(cid:15) (cid:19) (cid:32) R (1) DA (cid:33) + (cid:32) R (2) DA (cid:33) (18)where we averaged the squared electric field over angulardistribution of the two acceptors.Eq. (15) together with Eqs. (3) and (18) leaves uswith the only fitting parameter, τ ph , to reproduce themeasured low-field temperature dependence of the nu-clear spin relaxation time shown in Fig. 5. The suppres-sion of the spin relaxation by application of the magneticfield can be calculated from this value of T using themotional narrowing formula : T ( B ) = T ω B τ c . (19)The results of the fitting procedure are shown in Figs.4 and 5 by solid lines. One can see that the agreementis quite reasonable: there is no suppression of the nu-clear spin relaxation up to 120 G (no magnetic field de-pendence) and the quenching of spin relaxation below T ≈
10 K is well reproduced assuming τ ph = 5 µ s inSample A and τ ph = 3 µ s in Sample B.We note that τ ph obtained by fitting experimentaldata yields a qualitatively correct trend as a functionof acceptor concentration (phonon-assisted hops becomemore frequent with decreasing the average distance be-tween nearest acceptors). However, since the overlapof wave functions of impurity-bound holes decreases ex-ponentially with growing distance, one would expect agreater difference in τ ph between the two studied p-GaAssamples. This might be an indication that our model,which takes into account only two nearest acceptors, istoo simplistic. In order to clarify this issue, an exten-sive experimental study of nuclear spin relaxation over abroad range of doping in p-GaAs is needed. Such studiescould be an interesting subject for the future work. V. CONCLUSIONS
In conclusion, we have investigated relaxation of nu-clear spin polarization created by optical pumping inbulk p-GaAs. The nuclear spin-lattice relaxation time T in the dark (that is in the absence of optical pump-ing) turns out to be longer than that under pumping, butstill remains in the sub-second range. This is three ordersof magnitude shorter than in n-GaAs. This fact seemscounterintuitive, since hyperfine coupling of holes is muchweaker than that of conduction-band electrons. This paradox can be solved by taking into account charge fluc-tuations at acceptors located in close vicinity of positivelycharged donor centers. Indeed, since optically inducednuclear spin polarization is created only in the vicinityof donors and cannot diffuse outwards, the nearby fluctu-ating charge effectively destroys the nuclear polarizationvia the quadrupole interaction. The proposed theoreti-cal model quantitatively describes the slowing down ofnuclear spin relaxation below T = 10 K (due to slowingdown of charge fluctuations), and magnetic-field indepen-dence (up to ≈
100 G) of T . Our results fill the gap inthe general picture of nuclear spin relaxation in dopedGaAs, where p-type doping has not been addressed sofar. They also suggest that in compounds with I = 1 / Acknowledgements . This work was supported by theRussian Foundation for Basic Research (RFBR, GrantsNo. 16-52-150008 and 15-52-12020), by the Ministry ofEducation and Science of the Russian Federation (con-tract No. 14.Z50.31.0021 with the Ioffe Institute, Rus-sian Academy of Sciences, and leading researcher M.Bayer), by Saint-Petersburg State University via a re-search grant 11.34.2.2012, by French National ResearchAgency (Grant OBELIX, No. ANR-15-CE30-0020-02)and National Center for Scientific Research (CNRS, PRCSPINCOOL No. 148362), and by Deutsche Forschungs-gemeinschaft in the frame of the ICRC TRR 160 (ProjectNo. A6). F. Meier and B. Zakharchenya, eds.,
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