Deep optical cooling of coupled nuclear spin-spin and quadrupole reservoirs in a GaAs/(Al,Ga)As quantum well
M. Kotur, D. O. Tolmachev, V. M. Litvyak, K. V. Kavokin, D. Suter, D. R. Yakovlev, M. Bayer
DDeep optical cooling of coupled nuclear spin-spin and quadrupole reservoirs in aGaAs/(Al,Ga)As quantum well
M. Kotur, D. O. Tolmachev, V. M. Litvyak, K. V. Kavokin, D. Suter, D. R. Yakovlev,
1, 4 and M. Bayer
1, 4 Experimentelle Physik 2, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany Spin Optics Laboratory, St. Petersburg State University, 198504 St. Petersburg, Russia Experimentelle Physik 3, Technische Universit¨at Dortmund, D-44221 Dortmund, Germany Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
The selective cooling of As spins by optical pumping followed by adiabatic demagnetization inthe rotating frame is realized in a nominally undoped GaAs/(Al,Ga)As quantum well. The rotationof 6 kG strong Overhauser field at the As Larmor frequency of 5.5 MHz is evidenced by thedynamic Hanle effect. Despite the presence of the quadrupole induced nuclear spin splitting, itis shown that the rotating As magnetization is uniquely determined by the spin temperature ofcoupled spin-spin and quadrupole reservoirs. The dependence of heat capacity of these reservoirs onthe external magnetic field direction with respect to crystal and structure axes is investigated. Thelowest nuclear spin temperature achieved is 0.54 µ K, which is the record low value for semiconductorsand semiconductor nanostructures.
I. INTRODUCTION
The hyperfine interaction is one of the main sourcesof spin relaxation and decoherence of charge carriers insemiconductor structures, which hinder their applicationin spintronics. This problem is especially strong in themost technologically versatile family of heterostructuresbased on compounds of the III and V groups of the pe-riodic table, where all the nuclear species have nonzerospins. One of the possible approaches to solve this prob-lem is to develop efficient methods of cooling the nuclearspin system down to the temperature of phase transi-tion into the antiferromagnetically ordered state [1, 2].The general principle of cooling the nuclear spin systemis to adiabatically demagnetize initially polarized nuclearspins by lowering the external magnetic field. The higherthe initial polarization, the lower spin temperature of nu-clei can be reached [3].The possibility to dynamically polarize nuclear spinsin a semiconductor via their hyperfine coupling withoptically oriented electron spins opens ways to realiza-tion of deep nuclear spin cooling by minimal technicalmeans, i.e. to avoid using dilution refrigerators and/orhigh magnetic fields. The key point here is the choiceof the optimal structure. The minimal list of necessaryprerequisites includes the possibility to optically polar-ize nuclear spins to a high degree and long nuclear spin-lattice relaxation. These conditions are realized in nomi-nally undoped GaAs/(Al,Ga)As quantum wells [4], mak-ing these structures prospective for further investigationof the properties of their nuclear spin system under deepcooling.The possibility to move down along the spin temper-ature scale depends on the interactions in which nuclearspins are involved. These include, in addition to theZeeman interaction with the external field, the dipole-dipole interaction between magnetic moments of nuclei,and their indirect coupling via electron states [5]. All theinteractions except the Zeeman one are usually lumped together under the name of spin-spin interactions, whichform the spin-spin energy reservoir. In addition, if nucleihave spins larger than , as it is the case in III-V semicon-ductors, they experience quadrupole coupling with elec-tric field gradients induced by strain [5–7]. In case ofstrong quadrupole splitting (e.g. in self-assembled quan-tum dots) it may prevent establishing of the thermody-namic equilibrium in the nuclear spin system [8]. How-ever, if the quadrupole and spin-spin interaction ener-gies per nucleus are comparable, a quadrupole and spin-spin energy reservoirs are effectively coupled, and the nu-clear spin system can be characterized by a unified spintemperature [9]. The nuclear magnetic ordering is ex-pected to develop when the coupled energy reservoirs arecooled down below a certain critical spin temperature.For this reason, understanding the properties of the spin-spin and quadrupole (SS&Q) reservoir under cooling iscrucial for realization of nuclear magnetic ordering in aspecific structure.The SS&Q reservoir can be cooled either together withthe Zeeman reservoir, or separately. The latter optionis realized by adiabatic demagnetization in the rotat-ing frame (ADRF) [10–12]. Within this approach, thestatic external magnetic field is kept unchanged, whilethe nuclear spins are manipulated by variation of ampli-tude and/or frequency of the applied radiofrequency field.The ADRF method allows one to considerably reduce thespin-lattice relaxation rate and to address independentlythe spins of specific isotopes in a multi-isotope crystallike GaAs.In this work, we use ADRF with optical pumping ofnuclear spins and optical detection of the free inductiondecay (FID) signals to selectively cool the As spinsin a GaAs/(Al,Ga)As quantum well. We demonstratethat the As spin polarization in the rotating frame iswell described by the spin temperature theory, and esti-mate the contributions of dipole-dipole and quadrupoleinteractions into the heat capacity of the SS&Q reservoir.Cooling of the As SS&Q reservoir down to ≈ . µ Kis realized for both positive and negative spin tempera- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b tures. II. THEORETICAL BACKGROUND
The adiabatic demagnetization in a rotating frame canbe realized in several ways, see Ref. [3]. Here we usethe spin lock technique, known to be one of the meth-ods ensuring lowest entropy gain. At the first stage ofthe experiment, the nuclear spins are polarized along theexternal magnetic field. The higher the initial polariza-tion, the lower spin temperature can be reached. In semi-conductors, high nuclear polarization can be reached byoptical pumping mediated by hyperfine interaction withphotoexcited electrons [13]. The high spin polarizationparallel to the external field can be interpreted as a re-sult of cooling of the nuclear spin system to a temeper-ature which is lower than that of the lattice by absolutevalue, and can be either positive or negative, dependingon whether nuclear spins are polarized along the externalfield or opposite to it. At the second stage, a π radio-frequency pulse is applied at the Larmor frequency of theselected isotope (in our case As), which tips the meanspin vector of the isotope so that it becomes perpendicu-lar to the external field. Once the mean spin is tipped at90 ◦ to the external field, the populations for all the Zee-man sublevels become equal. Correspondingly, the tem-perature of the Zeeman reservoir of As becomes infinite.However, the entropy of the spin system remains low asthe coherence of the tipped spins is maintained. Withinthe spin lock protocol, this is done by switching on theradiofrequency (RF) field at the Larmor frequency, calledlocking field, which differs in phase by π from that of thetipping pulse. In the coordinate frame rotating togetherwith the tipped nuclear spin around the static externalfield, the locking field has a static component parallel (orantiparallel, depending on the sign of the phase shift ofthe locking field) to the mean nuclear spin. The decay ofthe rotating nuclear magnetic moment must, therefore,be accompanied by relaxation of its Zeeman energy inthe locking field, which profoundly slows it down. In-stead of the spin-spin relaxation time T of about 0.1millisecond, the decay of the rotating spin polarizationin the locking field occurs on the scale of the spin-latticerelaxation time in the rotating frame T ρ [14], which canamount to many seconds. This is a manifestation of thequasi-equilibrium nature of this long-living spin polariza-tion, which is characterized by the spin temperature ofthe SS&Q reservoir Θ N . If the amplitude of the lockingfield, b , is changed slowly (that is, dbdt < B L T ), startingfrom its initial value b , Θ N should follow the adiabaticcurve given by the well-known equation [3]:Θ N ( b ) = Θ N ( b ) (cid:115) b + B L b + B L , (1)where B L is the local magnetic field due to spin-spinand quadrupole interactions. Although the magnetic field rate constrain is only relevant when b is compara-ble to B L , during our experiment the magnetic field waschanged so that the given condition was always fulfilledeven though the initial value of b exceeded B L severaltimes. The mean spin of the tipped isotope is determinedby Θ N and b : (cid:104) I (cid:105) = I ( I + 1) (cid:126) γ N b k B Θ N , (2)where I and γ N are spin and gyromagnetic ratio of thetipped isotope and k B is the Boltzmann constant.The Overhauser field acting upon the electron spins isproportional to the mean nuclear spin. In particular, theamplitude of the rotating field of the tipped isotope is: B ⊥ ( b ) = A N (cid:104) I (cid:105) (cid:126) γ e = A N I ( I + 1) γ N b k B Θ N ( b ) γ e == A N I ( I + 1) γ N b k B Θ N ( b ) γ e (cid:115) b + B L b + B L = B ⊥ ( b ) bb (cid:115) b + B L b + B L , (3)where A N is the hyperfine coupling constant for thetipped isotope and γ e is the electron gyromagnetic ra-tio. The only parameter that determines the shape ofthe function B ⊥ ( b ) is the local field B L , which is in factthe measure of heat capacity of the SS&Q reservoir. Itis defined by the relation: B L = Tr[ ˆ H ss + ˆ H Q ] (cid:126) γ N Tr[ ˆ I B ] , (4)where ˆ H ss and ˆ H Q are the secular, i.e. commutingwith the Zeeman Hamiltonian, parts of the spin-spin andquadrupole Hamiltonian correspondingly, and ˆ I B is theoperator of the projection of nuclear spin on the externalfield [3].As distinct from the case of adiabatic demagnetizationin the laboratory frame, B L depends on the orientationof the static external field with respect to the crystal andstructure axes. The quadrupole splitting of the nuclearspin energy levels results from the electric field gradient(EFG), which interacts with the quadrupole moments ofthe nuclei. The EFG is zero in unperturbed cubic lat-tices. In GaAs, it arises due to strain and electric fields.In thin planar structures like ours, one can safely assumethat shear strain in the XZ and YZ planes (where Z isthe structure growth axis) is zero. The remaining com-ponents of the strain tensor ε (that is, uniaxial along Zand biaxial in the XY plane), together with the electricfield E that could arise along Z due to spatial or sur-face charge, result in the following general form of thequadrupole Hamiltonian:ˆ H Q = E Qz (cid:20) ˆ I z − I ( I + 1)3 (cid:21) ++ E QR √ I x − ˆ I y ) + E QI √ I x ˆ I y + ˆ I y ˆ I x ) , (5)where the energies E Qz , E QR and E QI are related to thestrain tensor components and the electric field by thematerial tensors S and R : E Qz = 3 eQS I ( I + 1) (cid:18) ε zz − ε xx + ε yy + ε zz (cid:19) E QR = 3 √ eQS I ( I + 1) ( ε xx − ε yy ) E QI = 3 √ eQS I ( I + 1) ε xy + 3 √ eQR I ( I + 1) E, (6)where e is the electron charge and Q is the quadrupolemoment of the nucleus. The angle between the principalaxis of the EFG tensor in the plane and the [100] crystalaxis, ζ , is defined by the relations:cos 2 ζ = E QR (cid:113) E QR + E QI sin 2 ζ = E QI (cid:113) E QR + E QI . (7)In a strong static magnetic field, B (cid:29) ( (cid:126) γ N ) − (cid:113) E Qz + E Q ⊥ , with the direction definedby the polar angle θ and the azimuthal angle α , thesecular part of the quadrupole Hamiltonian in Eq. (5)can be written as: ˆ H Q = 14 (cid:104) E Qz (3 cos θ −
1) + E Q ⊥ √ θ cos 2( ζ − α ) (cid:105) ×× (cid:20) ˆ I B − I ( I + 1)3 (cid:21) , (8)where θ and α are the angles between the external mag-netic field and structure and crystal axes of the sample,respectively and E Q ⊥ = (cid:113) E QR + E QI . The quadrupolecontribution to the local field then reads: B LQ = (cid:34) Tr[ ˆ H Q ] (cid:126) γ N Tr[ˆ I B ] (cid:35) / == 14 √ (cid:126) γ N | E Qz (3 cos θ −
1) + √ E Q ⊥ sin θ cos 2( ζ − α ) | . (9)The secular part of the spin-spin Hamiltonian reads[15]: ˆ H ss = 12 (cid:126) γ As (cid:88) j>k r − jk (cid:0) − θ jk (cid:1) (cid:16) I jz ˆ I kz − (cid:126) ˆ I j (cid:126) ˆ I k (cid:17) ++ 12 (cid:126) γ As (cid:88) j>k (cid:48) γ k (cid:48) Ga (cid:104) ˆ A jk (cid:48) + ( ˆ B jk (cid:48) + r − jk (cid:48) )(1 − θ jk (cid:48) ) (cid:105) ˆ I jz ˆ I k (cid:48) z . (10)Here indices j and k numerate the As atoms, and k (cid:48) numerates the Ga atoms. Scalar and pseudodipole inter-actions are short-range, therefore, the constants ˆ A jk andˆ B jk are not zero for the four Ga nuclei nearest to the j -thAs nucleus only. The spin-spin contributions to the lo-cal field for a As nucleus in the GaAs layer, neglecting the effect of interfaces, have cubic symmetry. As all theinteractions are bi-linear in the spin operators, B Lss , cal-culated along Eq. (4) as a function of polar angle θ andazimuthal angle α , comprises cubic invariants of zerothand fourth order: B Lss ( θ, α ) = C + D (cid:2) sin θ (cos α + sin α ) + cos θ (cid:3) , (11)where the directions of the cubic axis correspond to θ = 0or θ = π and θ = π/ α = nπ/
2. Numerical summationover the zinc-blend lattice of GaAs yields the constants C and D expressed via the parameters of the spin-spinHamiltonian given by Eq. (10). Details of the calculationof the spin-spin and quadrupole contribution to the localfield are given in the Appendix B. III. SAMPLE AND EXPERIMENTALTECHNIQUE
The studied sample was grown by molecular beamepitaxy on a Te-doped GaAs substrate and consists of13 nominally undoped GaAs/Al Ga As QWs withthicknesses varying from 2.8 nm to 39.3 nm separated by30.9 nm thick barriers. The sample was placed in a He-flow cryostat and cooled down to T = 5 . σ + /σ − ) from a tunable diode laser was used withexcitation energy E exc set to 1.5498 eV. The PL was col-lected in the reflection geometry, passed through a spec-trometer and detected using an avalanche photodiode(APD). The PL intensity spectrum of the 19.7 nm QWand its circular polarization degree ρ = I + − I − I + + I − , where I + ( I − ) is the intensity of the right (left)-hand circularlypolarized PL emission, is shown in Fig. 1. The maximumof the PL intensity detected at 1.5267 eV is attributedto the neutral exciton (X ) emission [4]. It was chosenas a PL detection energy in all subsequently performedmeasurements. Accordingly, the excitation energy of thediode laser was tuned closer to resonance, E exc = 1 . ◦ deviation from the structure axis) magnetic field B ext is shown in Fig. 2. When the sample was excited bythe modulated ( σ + /σ − at 50 kHz) circularly polarizedlight, the transfer of spin polarization via the hyperfineinteraction from optically oriented electrons to nuclei ishindered and depolarization of the electron spins in theexternal magnetic field (Hanle effect) is well representedby a Lorentzian function [13, 16]. Otherwise, for exci-tation with fixed in sign circularly polarized light, thenuclei become polarized through hyperfine-induced flip-flop transitions of the electron and nuclear spins givingrise to the nuclear Overhauser field B N . The Hanle curveis affected by the presence of the Overhauser field, man-ifested, depending on the helicity of the laser beam, bythe appearance of an additional maximum where the twofields are anti-parallel ( B ext − B N ) or faster depolariza- N o r m a li z e d P L i n t e n s i t y , I / I N o r m a li z e d P L p o l a r i z a t i o n , ρ / ρ Energy, E (eV)E det
FIG. 1. Photoluminescence spectra of 19.7 nm QW measuredat T = 5 . E det = 1 . N o r m a li z e d P L p o l a r i z a t i o n , ρ / ρ Lorentz fi t σ + / σ - σ + Time (s) ρ / ρ ext =7.5kG π /2+spin-lock2 4 6 80 Magnetic fi eld, B ext (kG) FIG. 2. Depolarization of luminescence by oblique (66 ◦ fromFaraday geometry) magnetic field with σ + /σ − modulatedat 50 kHz (blue circles) and constant σ + excitation (redcrosses). The excitation and detection energies were equalto E exc = 1 . E det = 1 . ρ ≈
12 %. Inset: Time evolution of the nuclear spin polar-ization measured through the PL circular polarization degreeat B ext = 7 . tion of the PL when the two fields are parallel ( B ext + B N )[13, 16]. In order to observe the build-up of the nuclearspin polarization at a relatively high external magneticfield, B ext = 7 . As), spin polarization of the twoGa isotopes could be wiped out by sweeping the radiofre-quency field across their resonance frequencies. This way,at the external field of 7.5 kG the polarization of the Asspins theoretically could reach 28%. The time duration ofthe first stage was determined by the dynamics of nuclearspin polarization at a given angle θ of the external field( B ext = 7 . θ = 55 ◦ and θ = 66 ◦ ,the sample was pumped for 300 s, sufficient for the Over-hauser field to reach the saturation value of the externalfield. However, the pump period increased to 1000 s ifthe polarization of the Ga and Ga was suppressed bysending RF pulses at their resonant frequencies of 7.64and 9.70 MHz, respectively. When θ = 80 ◦ , saturation ofthe Overhauser field was reached after 1000 s, if the spinpolarization of the Ga isotopes was not erased. The stateof the Overhauser field was monitored through the PLcomponent detected by the APD after passing throughmonochromator.A second stage begins once the nuclear field reachessaturation, i.e. B N = − B ext , when the trigger signalis sent to the AWG and a sequence of radiofrequencypulses is applied (inset in Fig. 2). The π pulse withthe RF amplitude of 21 G turned the As mean spinperpendicular to the static 7.5 kG strong field. Then thelocking RF field: B ( t ) = 2 b cos( γ As B ext t + ϕ ) (12)was switched on, with the phase ϕ shifted by 90 ◦ or 270 ◦ with respect to that of the tipping pulse. In the coordi-nate frame rotating with the Larmor frequency of As( ω AsL = γ As B ext ), one of the circular components of thelocking field was static and directed parallel (positive spintemperature) or anti-parallel (negative spin temperature)to the tipped As mean spin (cid:104) I As (cid:105) . The initial ampli-tude of the locking field was 21 G, i.e. much strongerthan the local nuclear field of 1.5-5 G (see below). Wechecked that there was no noticeable decay of the rotat-ing As spin polarization during 3 seconds if the lockingfield was kept on, while in the absence of the locking field B e x t = . k G M Ga M As B B e x t = . k G M Ga M As B B e x t = . k G π /2 pulsespin-lock pulse φ =90 o φ =270 o Frequency (MHz)
10 150 5 FF T a m p li t u d e ( a r b . u . ) ν =5.48 MHz2 ν =10.96 MHz Glan λ /4Monochromator GlanAPD λ /4APD B ext B θ in outref Lock-in AWGAmpli fi erChopper OscilloscopeDiode laser (799-812nm) Lens LensLens B (a) (e) (d)(c)(b) z=z' z=z'z=z'x' x'x'y' y'y'M Ga M As FIG. 3. (a) Sketch of the experimental setup used for ADRF measurements. The experimental protocol presented on the rightpanel consisted of three stages: (b) During the first stage the sample was pumped with circularly polarized light in an externalmagnetic field, B ext = 7 . π/ As by 90 ◦ in relation to theexternal magnetic field. (d) Immediately after the π/ ◦ (270 ◦ ) whichmakes the RF field parallel (antiparallel) to the tipped nuclear spin magnetization of As in the rotating frame. (e) In thethird phase, after the end of the spin-lock pulse, the FID signal is measured and Fourier transformed from time to frequencydomain. the polarization decayed within ≈ µ s. This fact in-dicated that the As spin subsystem reached a thermo-dynamic equilibrium in the rotating frame, characterizedby a spin temperature of a few µ K. Then the lockingfield amplitude could be gradually changed to a variedfinal amplitude with the speed of 1 × G/s, providingadiabatic de- or remagnetization in the rotating frame.In the third stage, the locking field was switched offand the free induction decay (FID) signal was recordedby measuring the PL circular polarization as a functionof time with an assembly of a quaterwave plate and aGlan prism followed by a fast photodiode. The signalwas Fourier-transformed and the amplitudes of 1 st and2 nd harmonics of the As Larmor frequency were de-termined. The oscillating PL polarization used for FIDdetection results from the dynamic Hanle effect inducedby the superposition of the static effective field B Σ (ex-ternal field + Overhauser field from Ga isotopes, in casethey were not wiped out during the first stage) and therotating Overhauser field produced by the tipped meanspin of As (see Fig. 4).As shown in Appendix A, the amplitudes of the 1 st and 2 nd harmonics of the oscillating PL polarization areequal to: A = 2 T ∗ B (cid:107) B ⊥ (0) B sin θ cos θ (13)and A = T ∗ B ⊥ (0)4 B sin θ, (14)respectively. Here B = B (cid:107) + B ⊥ (0) is the rotating Over-hauser field just after switching off the locking field, T ∗ is z θ S B ext B Σ SB ⟂ B Ga I As FIG. 4. Graphical interpretation of the Hanle effect in a ro-tating nuclear field. As the Larmor period of the electronspin in the total fields B Σ is typically shorter than the elec-tron spin lifetime, the electron mean spin S is nearly parallelto B Σ . S Z oscillates while the field B ⊥ , created by the tippedmean spin of As, rotates with the As Larmor frequency. the decay time of the rotating nuclear polarization, and θ is the angle of the external field to the structure axis.The ratio of amplitudes of the second and first harmonicsequals to: A A = B ⊥ (0)8 B (cid:107) tan θ. (15)We used Eq. (15) to determine the initial value of therotating Overhauser field of As in a strong, when com-pared to B L , locking field, in order to obtain the abso-lute value of the spin temperature after ADRF. Since the2 nd harmonic frequency falls slightly outside the 10 MHzbandwidth of our photodiode, it could not be reliablymeasured in case of weak B ⊥ . For this reason, in themajority of experiments on ADRF only relative values of B ⊥ (0) were determined using Eq. (13). IV. RESULTS
The ADRF curves measured in the external magneticfield B ext = 7 . θ = 66 ◦ from Faraday geometry with spin-lock pulses shifted by90 ◦ or 270 ◦ from the π pulse are presented in Fig. 5.When the locking field differs in phase by 90 ◦ (270 ◦ )from the π pulse, the adiabatic demagnetization processstarts from a positive (negative) initial spin temperature.In this measurement, only As spins were pumped andthe polarization of the two Ga isotopes was erased withRF pulses at their resonance frequencies. Fitting thesecurves with Eq. (3) enables one to determine the valuesof the local fields for two different phases. Furthermore,knowing the value of the local field B L it is possible, usingEq. (1), to obtain the value of nuclear spin temperatureafter adiabatic demagnetization to zero field ( b = 0):Θ AsN ( b = 0) = Θ AsN ( b ) B L (cid:112) b + B L , (16)where Θ AsN ( b ) = (cid:126) γ As b I ( I + 1) / (cid:104) I (cid:105) . (cid:104) I (cid:105) is determinedfrom the nuclear field, created by rotating the As spins (cid:104) I (cid:105) = (cid:126) γ e B ⊥ (0) /A As , where γ e = 3 . × radG -1 s -1 isthe electron gyromagnetic ratio and A As = 43 . µ eV isthe hyperfine coupling constant for As [18]. The calcu-lated values of the nuclear spin temperature Θ
AsN ( b = 0)are given in Table I.Similar measurements were performed for θ = 80 ◦ and θ = 66 ◦ without RF erase pulses for the Ga isotopes. Inthis context, all three isotopes contribute to the overallOverhauser field and the calculated values for the nu-clear spin temperature after adiabatic demagnetizationare made under the assumption that 52% of the nuclearfield value stems from spin polarized As nuclei and theother 48% from the two gallium isotopes. This ratio istaken from the share of the three isotopes in the maxi-mum possible nuclear field of 53 kG for GaAs under 100%polarization [13]. For comparison, the values of nuclearspin temperature obtained this way are also added toTable I.The lowest spin temperatures of 2 µ K [9] and 5 µ K [19]reported to date for semiconductors and semiconductorstructures were measured in bulk GaAs by adiabatic de-magnetization in the laboratory frame or at the set valueof magnetic field B ⊥ = 0 . TABLE I. Experimentally determined values of the nuclearspin temperature after adiabatic demagnetization to zerofield. θ ( ◦ ) ϕ ( ◦ ) Θ AsN ( b = 0) ( µ K)80 90 1.266 90 1.6270 -2.190 0.54 a
270 -0.57 aa Spin polarization of the two Ga isotopes was erased. N u c l e a r fi e l d , B N ( k G ) Locking fi eld amplitude, b (G) -10 105-5 0 N u c l e a r s p i n t e m p e r a t u r e , Θ N ( μ K ) Nuc. spin temp. Θ N Phase 90 o Phase 270 o Nuc. fi eld B N Phase 90 o Phase 270 o B L = 4.6 GB L = 4.5 G A s As FIG. 5. ADRF curves measured for 90 ◦ (blue circles) and270 ◦ (red diamonds) phase shifts of the spin-lock from the π pulse when the spin polarization of the two Ga isotopes waserased. The oblique external magnetic field B ext = 7 . θ = 66 ◦ . The solid lines are fitsto Eq. (15) that determine the values of B L and Θ AsN . perature of As. Due to the absence of the strain in-duced quadrupole splitting of the nuclear spin states inbulk GaAs, the local field had the characteristic value of B L = B ss = 1 . B L = B ss + B Q , and consequently to a highervalue of nuclear spin temperature, if the energy values ofthe quadrupole and spin-spin interactions are compara-ble [9], or to a breakdown of the nuclear spin temperatureconcept for strong quadrupole-induced local fields [8]. Inour QW sample, the spin-spin and quadrupole contribu-tions to the local field are similar to each other so that thetwo reservoirs are effectively coupled and the nuclear spinsystem is characterized by a unique spin temperature. Inspite of considerable quadrupole effects, this coupling hasallowed us to enter the sub-microKelvin spin temperaturerange in a QW structure.The dependence of the local magnetic field on the ex-ternal magnetic field direction in and out of the sampleplane was studied by measuring ADRF for 66 ◦ and 55 ◦ B L Q ( G ) B L Q ( G ) B L Q ( G ) θ =55 o θ =55 o θ =55 o θ =66 o θ =66 o θ =66 o B L =2.7G − − A ( a r b . u . ) A ( a r b . u . ) A ( a r b . u . )
000 00 B L =5.0GB L =2.1G B L =2.7GB L =2.5G B L =3.2GPolar angle, θ ( o ) α =0 o α =90 o α =45 o − − − − − − − − − − Locking fi eld amplitude, b (G)Locking fi eld amplitude, b (G)Locking fi eld amplitude, b (G) FIG. 6. Dependence of the quadrupole part of the local field B LQ on the angle between the external magnetic field andthe sample surface θ for three azimuths α (45 ◦ , 90 ◦ and 0 ◦ ).The solid red lines are fits to Eq. (9). Values for the B LQ were obtained by fitting the ADRF curves, measured for dif-ferent polar and azimuthal angles and shown in the left andright insets, with Eq. (14) (solid blue and green lines). Sincefor θ = 80 ◦ we only performed one measurement when theazimuthal angle α was equal to 45 ◦ , the obtained value for B LQ was added to the graph (pink circle) without showingthe corresponding ADRF curve. polar and 0 ◦ , 45 ◦ and 90 ◦ azimuthal angles. Consider-ing the fact that the amplitude of the second harmonic A for θ = 55 ◦ was too small to be determined fromthe measured FFT spectra, the dependence of the am-plitude of the first harmonic A on the magnitude of theRF field are presented in Fig. 6. Using Eq. (13) with B = B (cid:107) + B ⊥ , where B ⊥ is represented by Eq. (3),to fit the measured ADRF curves we get the values ofthe local field B L . The measured local fields consist oftwo parts associated with the spin-spin and quadrupoleinteractions, B L = (cid:113) B Lss + B LQ . Using Eq. (11) tocalculate the values for the spin-spin contribution to thelocal field B Lss it is possible to extract the values for thequadrupolar part B LQ for various polar and azimuthalangles which are presented in Fig. 6. In order to compare our experimental data in theframework of the theoretical model, the obtained valuesfor B LQ were fitted with Eq. (9). For the two unknownparameters E Qz and E Q ⊥ we used E Qz = 2 × − eVand E Q ⊥ = 3 × − eV, while the best agreement withthe theory was achieved when the angle between the axisof the EFG tensor in plane and the [100] crystal axis wasequal to ζ = 134 ◦ . It can be clearly seen that overall, theexperimental results follow the general trend predictedby the theory presented in Section II, i.e. B LQ has thehighest values for α = 45 ◦ , in other words, when theexternal magnetic field is directed along the [110] axis.When changing the azimuthal angle by 45 ◦ clockwise( α = 0 ◦ ) or counterclockwise ( α = 90 ◦ ) the value of thequadrupole local field decreases and reaches a minimumat θ ≈ ◦ . V. CONCLUSIONS
We have studied the process of deep cooling of the nu-clear spins by adiabatic de(re)-magnetization in the ro-tating frame in a GaAs/(Al,Ga)As quantum well. Withinthis approach the nuclear spins were polarized in anoblique external magnetic field and the adiabatic trans-formation was achieved by sending an RF sequence, atfrequency set for the As isotope, consisting of π and”spin-lock” pulses. The RF field amplitude in the ”spin-lock” pulse was gradually decreased down to zero. Fol-lowing the change of the As spin polarization with thisslowly varying RF field we were able to confirm that thenuclear spin temperature concept is still valid for oursample regardless of the presence of strain-induced nu-clear quadrupole splitting manifested in an increase ofthe local magnetic field.We have also experimentally demonstrated that, fora semiconductor structure, in the case of adiabatic de-magnetization in the rotating frame, the local magneticfield, characterizing the heat capacity of the nuclear spinsystem, is dependent on the orientation of the externalmagnetic field with respect to the crystal and structureaxes. The local magnetic field at different polar and az-imuth angles was measured and compared with the the-ory taking into account both dipole-dipole interactionson the zinc-blend crystal lattice and quadrupole splittingdue to strain and electric field along the structure axis.Although certain deviation from the predicted angulardependence of the local field was found, the overall agree-ment between the experiment and theory is satisfactory.Finally, it turned out possible to cool the coupled nu-clear spin-spin and quadrupole reservoirs, characterizedby a unified spin temperature, by adiabatic demagnetiza-tion in the rotating frame down to the sub-microKelvinrange. The lowest thus far, in semiconductors, spin tem-peratures of +0.54 and -0.57 µ K are reached. We con-sider this an important step towards realization of nu-clear magnetic ordering, expected to appear at nuclearspin temperatures below 0.1 µ K.In our experiments, the nuclear spin temperature 10million times lower in absolute value than that of thecrystal lattice was reached. It is worth to compare thisreduction factor with previous works on nuclear spin cool-ing in the solid state. The lowest temperatures reachedso far, in the nanoKelvin range, were obtained in metalsby a “brute force” method with the reduction factor of ahundred thousand, which required two-stage pre-coolingof conduction electrons to sub-milliKelvin temperatures[2]. Cooling of the nuclear spins in dielectrics to frac-tions of microKelvin [21], in spite using initial microwavehyperpolarization of the nuclear spins via the solid stateeffect, still demonstrated a reduction coefficient not ex-ceeding one million. Distinct from those previous works,requiring unique purpose-built setups, we have realizedspin cooling to sub-microKelvin temperatures in a sam-ple that was held in a standard helium flow cryostat attemperature of 5.5 K. This advantage of our approach,which is a result of high efficiency of optical spin orien-tation in semiconductors, makes cooling nuclear spins toultra-low temperatures much more feasible than before,opening ways to their applications in e.g. quantum sim-ulators [22].
ACKNOWLEDGEMENTS
This work was supported by the Deutsche Forschungs-gemeinschaft within the International Collaborative Re-search Center TRR 160 (project A6) and Russian Foun-dation for Basic Research (project 19-52-12043). K.V.K.and V.M.L. acknowledge support from the St. Peters-burg State University (research grant No. 73031758).
APPENDIX A: HANLE EFFECT IN ROTATINGNUCLEAR FIELD
The characteristic relaxation times of the electron spinare much shorter than those of the nuclei. Therefore, one can consider the Hanle effect in a static oblique effectivefield [13]: S z S = T s (cid:107) ( B Σ ) τ cos θ + T s ⊥ τ
11 + B /B H sin θ, (17)where S z is the electron mean spin projection on the di-rection of light propagation in the sample, τ is the elec-tron lifetime and θ is the angle between B Σ and z -axis.The lifetimes of spin components along and perpendicu-lar to the effective field B Σ are given by the expressions: T s (cid:107) ( B Σ ) = T s ( ∞ ) − T s ( ∞ ) − T s (0)1 + B /B P RC T s ⊥ = T s (0) , (18)where B P RC is the half width at half maximum (HWHM)of the polarization recovery curve (PRC) [23], and B H is the HWHM of the Hanle curve. Introducing B z , thecomponent of B Σ along z , one can rewrite Eq. (17) as: S z S = B z B T s (cid:107) ( B Σ ) τ + B − B z B T s ⊥ τ
11 + B /B H == T s ⊥ τ
11 + B /B H + B z B (cid:34) T s (cid:107) ( B Σ ) τ − T s ⊥ τ
11 + B /B H (cid:35) . (19)The effective field B Σ has two components: the static B (cid:107) ( B (cid:107) (cid:107) B ext ) and the rotating B ⊥ ( B ⊥ ⊥ B ext ). B z canbe expressed via these components as follows: B z ( t ) = B (cid:107) cos θ + B ⊥ ( t ) sin θ cos ω N t, (20)where ω N is the Larmor frequency of the tipped isotopein the external field, θ is the angle between the externalfield and the z -axis and: B ⊥ ( t ) = B ⊥ (0) e − t/T ∗ . (21)From Eq. (19) we now obtain: S z S = T s ⊥ τ
11 + B /B H + B (cid:107) cos θB (cid:20) T s (cid:107) ( B Σ ) τ − T s ⊥ τ
11 + B /B H (cid:21) ++ (cid:20) T s (cid:107) ( B Σ ) τ − T s ⊥ τ
11 + B /B H (cid:21) (cid:20) B ⊥ ( t )2 B sin θ + B (cid:107) B ⊥ ( t ) B sin(2 θ ) cos( ω N t ) + B ⊥ ( t )2 B sin θ cos(2 ω N t ) (cid:21) . (22)Eq. (22) can be further rewritten as: S z S = Υ + Ξ( ω N t ) , (23) where: Υ = T s ⊥ τ
11 + B /B H ++ 2 B (cid:107) cos θ + B ⊥ ( t ) sin θ B (cid:34) T s (cid:107) ( B Σ ) τ − T s ⊥ τ
11 + B /B H (cid:35) (24)and: Ξ( ω N t ) = (cid:34) T s (cid:107) ( B Σ ) τ − T s ⊥ τ
11 + B /B H (cid:35) ×× (cid:34) B (cid:107) B ⊥ ( t ) B sin(2 θ ) cos( ω N t ) + B ⊥ ( t )2 B sin θ cos(2 ω N t ) (cid:35) (25)are the non-oscillating and oscillating parts of the z -projection of the electron spin polarization. Before the π pulse and after the decay of free induction, B ⊥ = 0 and: S z S = T s (cid:107) ( B Σ ) τ cos θ + T s ⊥ τ
11 + B /B H sin θ. (26)Before the π pulse B Σ corresponds to polarized nuclei(either all isotopes or only As). After the decay of freeinduction, B Σ is the same minus the contribution of As.During FID, the oscillating part Ξ( ω N t ) has contribu-tions at single and double Larmor frequency of As. Tak-ing the Fourier transform of Eq. (25), we find that theiramplitudes are equal to: A = 2 T ∗ B (cid:107) B ⊥ (0) B sin θ cos θ (27)and: A = T ∗ B ⊥ (0)4 B sin θ. (28)The ratio of amplitudes of the second and first har-monics equals: A A = B ⊥ (0)8 B (cid:107) tan θ. (29)Here B ⊥ (0) is the initial value of the rotating As field,and B (cid:107) is the static effective field (the same as remainsafter FID). Therefore, the amplitude of the rotating nu-clear field can be determined from experiment as: B ⊥ (0) = 8 B (cid:107) A A cot θ = 8 B (cid:107) A A tan (cid:16) π − θ (cid:17) . (30) APPENDIX B: CALCULATION OF THESPIN-SPIN LOCAL FIELD
The squared spin-spin local field: B L = Tr[ ˆ H ss ] (cid:126) γ As Tr[ ˆ I B ] (31)comprises three contributions: from the As nuclei, B As ;from the four Ga nuclei next to the As nucleus, B NGa ;and from the rest of Ga nuclei, B RGa . Since scalarand pseudo-dipole interactions are short-range, B As and FIG. 7. Fragment of the crystal lattice of GaAs and the coor-dinate frame used to calculate the spin-spin interactions forthe selected As nucleus outlined by the red circle. Left: viewalong [001], right: view along [110]. Bright and pale red ballsdenote As nuclei with even and odd k , while bright and paleblue balls denote Ga nuclei with even and odd k (cid:48) , correspond-ingly (see Eqs. (32) and (33)). The cubic cell of the fcc Assublattice with the lattice constant a is shown by blue lines.The four nearest Ga nuclei are outlined by blue circles. B RGa are of purely dipole-dipole origin. In order to cal-culate these contributions, we introduce the Cartesiancoordinate system with the axes x (cid:107) [110], y (cid:107) [1¯10], and z (cid:107) [001] (see Fig. 7). The coordinates of the As and Ganuclei in this frame are encoded by the indices n , l , k forAs: x Asnk = a (cid:34) √ n + √ (cid:18) − ( − k (cid:19)(cid:35) y Aslk = a (cid:34) √ l + √ (cid:18) − ( − k (cid:19)(cid:35) z Ask = a k (32)and n (cid:48) , l (cid:48) , k (cid:48) for Ga: x Gan (cid:48) k (cid:48) = a (cid:34) √ n (cid:48) + √ (cid:32) − ( − k (cid:48) (cid:33)(cid:35) y Gal (cid:48) k (cid:48) = a (cid:34) √ l (cid:48) + √ (cid:32) − ( − k (cid:48) (cid:33)(cid:35) z Gak (cid:48) = a (cid:18) k (cid:48) − (cid:19) . (33)We start from calculation of the purely dipolar contri-butions, B As and B RGa ˆ H ss = 12 (cid:126) γ As (cid:88) j>k r − jk (1 − θ jk )(3ˆ I jz ˆ I kz − (cid:126) ˆ I j (cid:126) ˆ I k )++ 12 (cid:126) γ As (cid:88) j>k γ kGa (cid:104) ˆ A jk + ( ˆ B jk + r − jk )(1 − θ jk ) (cid:105) ˆ I jz ˆ I kz . (34)0Using Eqs. (4) and (10) and applying the relationTr (cid:104) ˆ I jα ˆ I kβ (cid:105) = I ( I +1)(2 I +1)3 δ jk δ αβ we find: B As = 1 (cid:126) γ As Tr (cid:26)(cid:104)(cid:80) k ˆ I kB (cid:105) (cid:27) ×× Tr (cid:126) γ As (cid:88) j>k r − jk (cid:0) − θ jk (cid:1) (cid:16) I jz ˆ I kz − (cid:126) ˆ I j (cid:126) ˆ I k (cid:17) == 152 (cid:18) (cid:126) γ As a (cid:19) F As ( θ, α ) ≈ . × − [G ] F As ( θ, α ) (35)and: B RGa = 1 (cid:126) γ As Tr (cid:26)(cid:104)(cid:80) k ˆ I kB (cid:105) (cid:27) ×× Tr (cid:126) γ As (cid:88) j>k γ kGa r − jk (cid:0) − θ jk (cid:1) ˆ I jz ˆ I kz == 54 (cid:34) X (cid:18) (cid:126) γ Ga a (cid:19) + X (cid:18) (cid:126) γ Ga a (cid:19) (cid:35) F Ga ( θ, α ) ≈≈ . × − [G ] F Ga ( θ, α ) , (36)where X and X are the abundances of the correspond-ing Ga isotopes. The dimensionless functions F As ( θ, α ) and F Ga ( θ, α ) are defined as: F As ( θ, α ) = 12 (cid:88) nlk (cid:0) − φ Asnlk (cid:1) ( r nlk /a ) F Ga ( θ, α ) = (cid:88) n (cid:48) l (cid:48) k (cid:48) / ∈ NN (cid:0) − φ Gan (cid:48) l (cid:48) k (cid:48) (cid:1) ( r n (cid:48) l (cid:48) k (cid:48) /a ) , (37)where the indices n, l, k and n (cid:48) , l (cid:48) , k (cid:48) are defined inEqs. (32) and (33), r nlk or r n (cid:48) l (cid:48) k (cid:48) stand for the distance tothe corresponding nucleus from the selected As nucleusand φ Asnlk or φ Gan (cid:48) l (cid:48) k (cid:48) denote the angle between the externalfield and the direction to this nucleus (see Fig. 7). Ex-pressing the cosines of the angles φ nlk and φ n (cid:48) l (cid:48) k (cid:48) via thecoordinates of the corresponding nuclei and the directioncosines of the magnetic field: cos φ Asnlk = x Asnk sin θ cos α (cid:48) + y Aslk sin θ sin α (cid:48) + z Ask cos θ (cid:113)(cid:0) x Asnk (cid:1) + (cid:0) y Aslk (cid:1) + (cid:0) z Ask (cid:1) cos φ Gan (cid:48) l (cid:48) k (cid:48) = x Gan (cid:48) k (cid:48) sin θ cos α (cid:48) + y Gal (cid:48) k (cid:48) sin θ sin α (cid:48) + z Gak (cid:48) cos θ (cid:113)(cid:0) x Asnk (cid:1) + (cid:0) y Aslk (cid:1) + (cid:0) z Ask (cid:1) , (38)where φ (cid:48) = φ + π , we obtain: F As ( θ, α ) = 12 (cid:88) nlk (cid:2) x nk /a + y lk /a + z k /a − θ cos α (cid:48) · x nk /a + sin θ sin α (cid:48) · y lk /a + cos θ · z k /a ) (cid:3) ( x nk /a + y lk /a + z k /a ) (39)and F Ga ( θ, α ) = (cid:88) n (cid:48) l (cid:48) k (cid:48) (cid:2) x n (cid:48) k (cid:48) /a + y l (cid:48) k (cid:48) /a + z k (cid:48) /a − θ cos α (cid:48) · x n (cid:48) k (cid:48) /a + sin θ sin α (cid:48) · y l (cid:48) k (cid:48) /a + cos θ · z k (cid:48) /a ) (cid:3) ( x n (cid:48) k (cid:48) /a + y l (cid:48) k (cid:48) /a + z k (cid:48) /a ) . (40)The functions F As ( θ, α ) and F Ga ( θ, α ) can be expressedthrough cubic invariants of zeroth and fourth order, F ( θ, α ) = C + D (cid:2) sin θ (cos α + sin α ) + cos θ (cid:3) , wherethe coefficients C and D can be determined from the nu-merical values of F As ( θ, α ) and F Ga ( θ, α ) for two pairs ofangles, e.g.: C = 2 F ( π/ , π/ − F (0 , D = 2 [ F (0 , − F ( π/ , π/ . (41) This way, we find: C As = 127 . C Ga = 32 . D As = − . D Ga = 18 . . (42)Finally, the contribution of the nearest four Ga nucleican be calculated analytically: B NNGa = 1 (cid:126) γ As Tr (cid:126) γ As (cid:88) k (cid:48) ∈ NN γ kGa (cid:104) ˜ A + ( ˜ B + r − k (cid:48) )(1 − φ k (cid:48) ) (cid:105) ˆ I k (cid:48) z == 54 (cid:34) X (cid:18) (cid:126) γ Ga a (cid:19) + X (cid:18) (cid:126) γ Ga a (cid:19) (cid:35) (cid:88) k (cid:48) ∈ NN (cid:2) b sc + ( b pd + b dd )(1 − φ k (cid:48) ) (cid:3) == 54 (cid:34) X (cid:18) (cid:126) γ Ga a (cid:19) + X (cid:18) (cid:126) γ Ga a (cid:19) (cid:35) × b sc + 8( b pd + b dd ) − b pd + b dd ) (cid:2) sin θ (sin ϕ + cos ϕ ) + cos θ (cid:3) ≈≈ . × − [G ]4 b sc + 8( b pd + b dd ) − b pd + b dd ) (cid:2) sin θ (sin ϕ + cos ϕ ) + cos θ (cid:3) , (43)1where b sc = ˜ Aa , b pd = ˜ Ba and b dd = (4 / √ . Accord- ing to Ref. [24], b sc ≈ . b dd and b pd ≈ − . b dd . [1] M. Goldman, M. Chapellier, V. H. Chau, andA. Abragam, Principles of nuclear magnetic ordering,Phys. Rev. B , 226 (1974).[2] A. S. Oja and O. V. Lounasmaa, Nuclear magnetic order-ing in simple metals at positive and negative nanokelvintemperatures, Rev. Mod. Phys. , 1 (1997).[3] M. Goldman, Spin Temperature and Nuclear MagneticResonance in Solids (Clarendon Press, Oxford, 1970).[4] R. W. Mocek, V. L. Korenev, M. Bayer, M. Kotur, R. I.Dzhioev, D. O. Tolmachev, G. Cascio, K. V. Kavokin,and D. Suter, High-efficiency optical pumping of nuclearpolarization in a GaAs quantum well, Phys. Rev. B ,201303 (2017).[5] A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961).[6] M. Eickhoff, B. Lenzmann, D. Suter, S. E. Hayes,and A. D. Wieck, Mapping of strain and electric fieldsin GaAs/Al x Ga As quantum-well samples by laser-assisted NMR, Phys. Rev. B , 085308 (2003).[7] K. Flisinski, I. Y. Gerlovin, I. V. Ignatiev, M. Y. Petrov,S. Y. Verbin, D. R. Yakovlev, D. Reuter, A. D. Wieck,and M. Bayer, Optically detected magnetic resonance atthe quadrupole-split nuclear states in (In,Ga)As/GaAsquantum dots, Phys. Rev. B , 081308 (2010).[8] P. Maletinsky, M. Kroner, and A. Imamoglu, Breakdownof the nuclear-spin-temperature approach in quantum-dot demagnetization experiments, Nat. Phys. , 407(2009).[9] M. Vladimirova, S. Cronenberger, D. Scalbert, I. I.Ryzhov, V. S. Zapasskii, G. G. Kozlov, A. Lemaˆıtre, andK. V. Kavokin, Spin temperature concept verified by op-tical magnetometry of nuclear spins, Phys. Rev. B ,041301 (2018).[10] C. P. Slichter and W. C. Holton, Adiabatic demagnetiza-tion in a rotating reference system, Phys. Rev. , 1701(1961).[11] A. G. Anderson and S. R. Hartmann, Nuclear magneticresonance in the demagnetized state, Phys. Rev. ,2023 (1962).[12] A. G. Redfield, Nuclear spin thermodynamics in the ro-tating frame, Science , 1015 (1969). [13] F. Meier and B. P. Zakharchenya, eds., Optical Orienta-tion (North-Holland, Amsterdam, 1984).[14] D. L. VanderHart and A. N. Garroway, C NMR ro-tating frame relaxation in a solid with strongly coupledprotons: Polyethylene, J. Chem. Phys. , 2773 (1979).[15] S. Clough and W. I. Goldburg, Nuclear magnetic reso-nance study of electron-coupled internuclear interactionsin thallium chloride, J. Chem. Phys. , 4080 (1966).[16] M. I. Dyakonov, ed., Spin Physics in Semiconductors (Springer International Publishing AG, Berlin, 2017).[17] M. Kotur, F. Saeed, R. W. Mocek, V. L. Korenev, I. A.Akimov, A. S. Bhatti, D. R. Yakovlev, D. Suter, andM. Bayer, Single-beam resonant spin amplification ofelectrons interacting with nuclei in a GaAs/(Al,Ga)Asquantum well, Phys. Rev. B , 205304 (2018).[18] E. A. Chekhovich, A. Ulhaq, E. Zallo, F. Ding, O. G.Schmidt, and M. S. Skolnick, Measurement of the spintemperature of optically cooled nuclei and GaAs hy-perfine constants in GaAs/AlGaAs quantum dots, Nat.Mater. , 982 (2017).[19] V. K. Kalevich, V. D. Kulkov, and V. G. Fleisher, Onsetof a nuclear polarization front due to optical spin orien-tation in a semiconductor, JETP Lett. , 2 (1982).[20] D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov, Lowfield electron-nuclear spin coupling in gallium arsenideunder optical pumping conditions, Phys. Rev. B , 5780(1977).[21] M. Chapellier, M. Goldman, V. H. Chau, andA. Abragam, Production and observation of a nuclearantiferromagnetic state, J. Appl. Phys. , 849 (1970).[22] I. M. Georgescu, S. Ashhab, and F. Nori, Quantum sim-ulation, Rev. Mod. Phys. , 153 (2014).[23] D. S. Smirnov, E. A. Zhukov, D. R. Yakovlev, E. Kirstein,M. Bayer, and A. Greilich, Spin polarization recoveryand Hanle effect for charge carriers interacting with nu-clear spins in semiconductors, Phys. Rev. B , 235413(2020).[24] M. K. Cueman and J. F. Soest, Pseudodipolar and ex-change broadening of NMR lines in GaP and GaAs, Phys.Rev. B14